International Symposium Elementary Maths Teaching
Equity and diversity in elementary mathematics education
Charles University Faculty of Education Prague, the Czech Republic August 20 – 25, 2017 International Symposium Elementary Maths Teaching
Prague, the Czech Republic Charles University, Faculty of Education
August 20 - 25, 2017
Proceedings
Equity and diversity in elementary mathematics education
Edited by Jarmila Novotná and Hana Moraová
Prague 2017
International Symposium Elementary Mathematics Teaching
Prague, the Czech Republic Charles University, Faculty of Education August 20 - August 25, 2017
International Programme Committee Olive Chapman (Canada), Chair, Marie-Pierre Chopin (France), Brian Doig (Australia), Rose Griffiths (United Kingdom), Alena Hospesova (Czech Republic), Antonin Jancarik (Czech Republic), Esther Levenson (Israel), Jarmila Novotna (Czech Republic), Marta Pytlak (Poland), Petra Scherer (Germany) Advisory Board: Tuba Gökcek (Turkey), David Pugalee (USA), Annie Savard (Canada), Ewa Swoboda (Poland), Dina Tirosh (Israel)
Organizing Committee Jarmila Novotná, Hana Moraová, Antonín Jančařík
The papers presented at SEMT ‘17 have all undergone a rigorous blind reviewing process by members of the International Programme Committee and Advisory Board. Posters represent work in progress and are not reviewed.
Since all papers and other presentations here are presented in English, which is not usually the first language of the presenters, the responsibility for spelling and grammar lies with the authors of the papers themselves.
© Charles University, Faculty of Education, Prague, 2017
ISBN 978-80-7290-955-1 TABLE OF CONTENTS
Plenary lectures………………………………………...……………………………... 9 H. P. Ginsburg & C. Uscianowski: Stories, stories and more math stories ………… 9 M. Graven: Blending elementary mathematics education research with development for equity – an ethical imperative enabling qualitatively richer work …………… 20 T. Janík: From content to meaning: Semantics of teaching in the tradition of Bildung- centred didactics ……………..…………………………….……………………. 31 E. S. Levenson: Promoting mathematical creativity in heterogeneous classes….…... 42
Research reports ...……………………………………………………………………. 53 L. Bacon, N. Bednarz, C. Lajoie, J.-F. Maheux & M. Saboya: Two perspectives on diversity based on the pedagogical consultant’s work on problem-solving in a teaching context ……………………………………………………………….… 53 Y. Biton, S. Hershkovitz, M. Hoch, B. Ben-David & O. Fellus: Assessment issues that trouble mathematics teachers………………...... ………………. 63 N. Blundell, K. Bentley, B. A. Temple & D. K. Pugalee: Kinesthetic & Creative Approaches to Pre-K Spatial Geometry Learning: A Qualitative Case Study.….. 72 G. Bolondi, C. Cascella & C. Giberti: Highlights on gender gap from Italian standardized assessment in mathematics ………………………………………… 81 M. Brožová: The use of textbooks in different approaches – Hejny method, Montessori and traditional approach in the Czech Republic………………..….... 91 M. Bruna & R. Havlíčková: Parameters influencing difficulty of word problems – A case of four word problems in grade 4 ..……………..…………………………... 100 I. Budínová: Progressive development of perception of the concept of a square by elementary school pupils ……………...…………...……………………………. 109 R. Cabassut: Diversity of teachers’ beliefs on modelling through a French-Spanish comparison …...... ……………………………………………………………….. 119 C. Chambris: Changes in the teaching of numbers and ratios in the primary curriculum...………………………………………………………………...….… 128 O. Chapman: Prospective elementary school teachers’ learning trajectory of the numeration system……………………………………………………………….. 138 V. Cifarelli, M. Stephan, D. K. Pugalee & C. Wang: Implementing an instructional sequence for solving net worth problems ….………………………………….… 148 A. Duatepe-Paksu, Ľ. Rybanský & K. Žilková: The content knowledge about rhombus of Turkish and Slovak pre-service elementary teachers ...………….……………. 158 F. Favilli: Mirror curves and basic arithmetics …………...………………………… 168 D. Goetze: Language- and mathematics-integrated intervention for understanding division and divisibility…………………………………….……………….…… 177 K. Hähn: Analyses of learning situations in inclusive settings: a coexisting learning situation in a geometrical learning environment...…...……….….……………… 187 H. Haydar: Elementary Mathematics Teachers Celebrating Student Voice: The Clinical Interview Again and Again……………………………………..………. 197 A. Hošpesová, I. Stuchlíková & I. Žlábková: Introduction of formative peer assessment in primary mathematics from the pupils’ perspective....…………..… 205
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J. Hunter & J. Miller: Using cultural tasks to develop growing pattern generalisations For young culturally diverse students ……...... 215 A. Jančařík: The potential of the chess environment in mathematics education – Pre- service teachers’ perspective ...………………………………………….……….. 225 D. Jirotková & J. Slezáková: Do teachers understand their pupils?..….……………... 235 D. Jirotková & J. Slezáková: Student teachers’ didactical competences in mathematics …...………………………………………………………………… 245 M. Kaslová: Diversity of results in research in the domain of pre-school mathematics at kindergarten ...………….……………………………………………………... 255 L. Kasmer, D. Harrison & E. Billings: Equitable learning opportunities: Textbook language accessibility in English medium classes….…………………………… 265 J. Kopáčová & K. Žilková: Pyramid or triangle - Isn’t it and the same? (A case study) 272 L. Korten: The investigation of co-operative-interactive learning situations in an inclusive arithmetic classroom………………..……………………….……...…. 282 R. Lambert, P. Baddouh, E. Merrill, A. Ferrara, C. Wang & C. Martin: The implementation of a statewide kindergarten entry assessment of object counting... 292 B. Lazić & J. Milinković: Using multiple representations of fractions to enhance problem solving ……………………………………………………………….…. 301 A. Lipovec & J. Ferme: The use of the reference point strategy for measurement estimation ….……………………………………………………………….……. 311 A. Lipovec & M. Podgorošek: Students’ visual representation of fractions and exponentiation …………………………………………..……………………….. 319 C. Martin, R. G. Lambert, C. Wang & D. Polly: Supporting mathematics learning through project based learning: A fifth grade case study ………...……………... 328 H. Moraová & J. Novotná: Higher order thinking skills in CLIL lesson plans of pre- service teachers …...……………………………..…………………...………….. 336 A. Sáenz-Ludlow: Children’s progressive construction of number and numerals by means of numerical diagrams……………………………………………….….... 345 L. Samková: Planning and conducting inquiry based mathematics course for future primary school teachers ………………………………………………………..... 354 P. Scherer: Preparing pre-service teachers for inclusive mathematics classrooms – concepts for primary education …………………………………………….…..... 364 C. Schöttler & U. Häsel-Weide: Students constructing meaning for the decimal system in dyadic discussions: epistemological and interactionist analyses of negotiation processes in an inclusive setting ……………………………….….... 373 J. Slezáková, D. Jirotková & J. Kloboučková: Student portfolio as a tool for development of pre-service primary teachers’ competences in teaching mathematics ………………..……………………………………………….….... 383 D. Stott: Interactive gesturing: deepening understanding of mathematical activity with young learners in a South African context…………………………….….... 393 E. Thanheiser & R. Philipp: Do prospective elementary teachers (in the United States) notice cultural aspects of mathematics in a teaching scenario?……..….... 403 V. Tůmová: How do pupils of the 5th and 6th grade structure space ….…………..... 411 N. Venuto & L. C. Hart: First graders write about mathematics: A teaching experiment …………………………….………………………………………..... 421 J. Višňovská & J. L. Cortina: Learning to support all students’ fraction understanding 430 N. Vondrová & J. Novotná: The influence of context and order of numerical data on the difficulty of word problems for grade 6 pupils……...……………….…...... 440
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D. Walter: On the representation of quantities with multi-touch at the ‘Math- Tablet’……………………………………………………………………………. 449 R. Zemanová, D. Jirotková & J. Slezáková: Practical component of mathematics education of pre-service primary school teachers: students’ perspective ……….. 459 K. Žilková, J. Kopáčová & Ľ. Rybanský: Similarities and differences in identification of rectangles of fourth-grade pupils………………………………………….….... 466
Workshops .…………………………………………………………………………… 477 D. Clarke, C. Mesiti, J. Novotná, H. Moraová & M. C. E. Chan: Speaking in and about the mathematics classroom……………………………………….………. 477 B. Doig & S. Groves: Mathematical tasks stimulating student problem solving strategies ……………………………………………………………………….... 479 L. Samková: A pathway to inquiry..………………………….....…………………... 480 A. Savard & E. Polotskaia: Shifting paradigm in problem solving: algebraic versus arithmetic thinking………………………………………………………....…….. 482 P. Scherer, K. Hähn, C. Rütten & S. Weskamp: Primary students explore mathematics at the university – Activities for all ……………………………….. 484 D. Tirosh, P. Tsamir, E. Levenson & R. Barkai: Using tablets in preschool: patterns and number concept ……………………………………...……………………… 486
Posters ...... …………………………………………………………………………...... 489 A. Duatepe-Paksu & B. Boz-Yaman: Evoking and examining geometric habits of mind while constructing tetrahedron by modular origami..……………………... 489 A. Hošpesová & M. Tichá: Problem posing in prospective primary school teachers’ education: Case of concept cartoons...…..………………………………………. 491 E. Nováková: Primary school Mathematical Kangaroo: Mistakes in selected word problems……………………………………………………………...………….. 493 P. Scherer, K. Hähn, C. Rütten & S. Weskamp: Learning environments for diverse learners – Substantial mathematics for all…………………………………...…... 495 B. A. Temple, K. Bentley, N. Blundell & D. K. Pugalee: A transdisciplinary approach to elementary math literacy learning through visual art…...…..……...... 497
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Dear colleagues: Welcome to SEMT 2017! I am delighted to welcome you to the 14th biennial International Symposium – Elementary Mathematics Teaching [SEMT] at Charles University, Prague, Czech Republic, on August 20-25, 2017. Organized by Charles University, under the leadership of professor Jarmila Novotná and Hana Moraová with the support of an International Program Committee and an Advisory Board, SEMT offers participants a unique opportunity to focus on research at the elementary school level through plenaries, research reports, posters and hands-on workshops. All papers are refereed to ensure a high quality for the scientific program. This year, the symposium is framed by the theme, equity and diversity in elementary mathematics education, which is timely and important given the growing diversity of elementary classrooms in today’s world and the need for teachers to engage all students meaningfully in the mathematics classroom and the learning of mathematics. The papers in SEMT’17 Proceedings offer the mathematics education community a variety of ways in which this theme has been and can be addressed. SEMT is an opportunity to share your ideas and experiences and to learn from and connect with researchers in elementary mathematics education from around the world in a supportive, stimulating academic environment. The social activities, which are also important aspects to the SEMT experience, include a conference dinner with live music for listening and dancing and an educational excursion. Also, being in historic, beautiful Prague is a definite bonus for those of us who do not live there! Whether it is your first visit or not, exploring and enjoying the city should be on your agenda.
I wish you all a memorable and inspiring SEMT 2017.
Olive Chapman Chair International Program Committee
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P LENARY L ECTURES
STORIES, STORIES, AND MORE MATH STORIES Herbert P. Ginsburg and Colleen Uscianowski
Abstract This paper describes our work on three types of math stories. Some can help young children (roughly age 3 to age 8) understand math; others can help both prospective and practicing teachers (we will refer to both groups simply as “teachers”) understand children’s learning of math; and still others can help teachers to understand math. These various stories have the potential to reveal and promote a joyful and playful approach to early math education. Keywords: Math storybooks, mathematical thinking, teacher education Math storybooks What are they? There are so many wonderful children’s books! There are storybooks, dictionaries, letter books, shape books, board books, books about farm animals, fish, and trucks, and many, many others. New books appear every year, and timeless classics like the Three Bears are read by generations of readers.
Columbia University, USA; e-mail: [email protected]
9 The topic that we address in this paper is the use and value of math storybooks of two types, explicit and implicit. Explicit math books are deliberately designed to teach basic math concepts and skills. They cover many topics, including number, operations on number, shape, space, pattern, and measurement. They use a strong narrative, a story, to meet the goal. They are not textbooks and they aim to be the antithesis of boring. For example, Quack and count (Baker, 1999) introduces addition by showing different ways to arrange seven ducks. Children search for them in beautifully illustrated pages, and learn that whether there are three on one page and four on the other, or five on one page and two on the other, the total is still seven. Numbers can be composed and decomposed in different ways. Another type of storybook, the implicit, does not aim to teach math explicitly but contains important math ideas deeply embedded within the narrative. For example, Goldilocks and the Three Bears (Marshall, 1988) is usually seen as a moral tale (treat others with consideration and respect their belongings). But consider the math in the story. Goldilocks sees that the Baby Bear’s bed is the smallest, and that Mama’s bed is bigger than Baby’s but smaller than Papa’s. Also, Baby Bear is smaller than Mama, who is in turn smaller than Papa. The beds are in increasing order of size, and so are the bears. The order is more complex than it initially appears: momma is both bigger than baby bear and smaller than daddy bear. Also, there is a correlation between the size of the bears and the size of the beds: the bigger the bear, the bigger the bed. So the story of Goldilocks and the Three Bears can be used to help the child understand some fundamentally important math ideas about relative size, order, and the relations between two sequences. Why should teachers read math storybooks? Emotional experience. Many parents read to their children, even in the early years of their lives. The experience usually involves a warm bonding, as when the mother or father reads to the child before bedtime. As a result, years later, on entrance to school, children are predisposed to enjoying books. Preschool teachers often read to the whole class at circle time or quiet time, a practice that continues throughout the early years of elementary school. Reading is an opportunity for teachers to nurture and build upon the children’s feelings about reading and their love of books. Child and Teacher Enjoyment. One major reason for reading math storybooks is enjoyment. The books can be humorous, suspenseful, and charming. Surely the educational system should help children appreciate the pleasures of reading. Adventure for child and teachers. Reading of any type can be an intellectual adventure. Books introduce children to new worlds, new characters, and new ideas. When the teacher opens a book, children often sit in rapt attention, anticipating something new, exciting, unexpected, and even humorous.
10 Planned and unplanned learning. Teachers can use math storybooks in a deliberate attempt to teach math. One strategy is to begin a unit (like addition or spatial relations) with a math storybook that introduces the topic in a challenging and enjoyable manner. Another is to read a math storybook in conjunction with an ongoing unit. But teachers can also use implicit storybooks in an opportunistic manner. In the course of reading the Three Bears, for example, the teacher can make explicit and expand upon the interesting math ideas at the core of the story. Reduction of teacher anxiety. An unfortunate fact is that many early childhood teachers are afraid of teaching math and may even transmit their anxiety to their children (Beilock et al., 2010). But almost all teachers are comfortable reading storybooks. Professional development can use math storybook reading to provide math-anxious teachers with a safe and unthreatening entry into math education. Children’s learning. Last (like Baby Bear), but not least, reading math storybooks can help children learn math. For example, Casey and her colleagues (Casey, Kersh and Young, 2004) have created and evaluated a series of six adventure stories designed to teach math at the preschool through grade 2 levels. The results show improved learning of geometric concepts. Possible pitfalls. As with any educational practice, math storybook reading is not a panacea, comes with no guarantees, and may be, indeed often is, flawed. Teachers may not read stories in an effective way; the storybook itself may be of poor quality; and children may not enjoy reading or being read to. All this is normal in the real world: a potentially powerful method may not work under some conditions. But there is one particular danger that we wish to emphasize: strong pressure to use storybooks to raise math test scores may sabotage reading, turning it into joyless instruction that is boring, pedestrian, and pedantic. We should not make reading as dull as some math instruction. What is math storybook quality? Although reading high quality, engaging math storybooks, both explicit and implicit, is the goal, teachers may not know how to select them from the myriad of storybooks available. To help teachers understand quality, we have created guidelines for evaluating math storybooks. The guidelines stress several features including literary merit, math content, coherence of text, illustrations and math, and promotion of thinking. Consider each, beginning with literary merit. Literary merit. It is essential for math storybooks, implicit or explicit, to be good stories. If they are not, they may bore children and sour their interest in both reading and math. Teachers should ask several questions as they evaluate the literary quality of math storybooks. ● Does the book have an interesting plot that will hold children’s attention from beginning to end?
11 ● Is there a central theme that is developmentally appropriate and connected to children’s lives? ● Are the characters memorable or relatable? ● Are the illustrations beautiful and engaging? ● Does the book promote positive values? Consider The Growing Story (Krauss, 2007). It has an inviting, well-written narrative about a boy who is worried he is not growing taller, while the corn and the chicks seem to be growing bigger every day. The narrative is likely to capture children’s attention from beginning to end, as they want to learn how the boy’s dilemma unfolds; after all, they are extremely interested in growing—becoming a big boy or girl. The illustrations are charming and the language is often pleasingly rhythmic: “Little pears grew on the orchard trees. Little ears grew on the corn. The grass grew still faster. The chicks grew still taller. The puppy grew still taller.” The Growing Story provides opportunities for children to relate to the main character. They can also consider how they have grown older and taller, how growth might not be immediately apparent, and how rates of growth may differ. We also attempt to focus attention on issues of values, negative and positive. We encourage teachers to ask this question: Does the book contain negative stereotypes or values? An example is provided by The Littlest Rabbit (Kraus, 1961), a charming story that includes important math ideas. Yet in one scene, the main character punches and kicks two other rabbits that had been bullying him. When he runs home to tell his father about the victory, his father is pleased and responds, “I’m very proud of my big fellow.” Virtually all teachers would not like to encourage behavior of this sort in their own classrooms! At the same time, they may still wish to read the story for its mathematical merit and use the punching and kicking scene as an opportunity to discuss proper classroom behavior. We also want to encourage teachers to ask whether the book promotes positive values? We believe that children should be exposed to stories in which girls and boys are equally likely to be smart problem-solvers and mathematicians, as in A thousand Theos (Houran, 2015). Stories should also acknowledge and celebrate people of various ethnicities, as in Feast for 10 (Falwell, 2008). The math and its depiction. If the book has literary merit, we can then evaluate the following aspects of its math content and depiction. ● What math topics does the book address? ● Do the text and illustrations complement each other in the presentation of clear, accurate, and rich math ideas?
12 ● Does the book promote mathematical thinking? The first question concerning math topics may seem simple but really is not. Teachers need to closely analyze the text and illustrations to decide if the story is primarily about numbers, size, shape, measurement, or another topic. Answering these descriptive questions can help teachers notice the math content that they may have previously overlooked. Indeed, in our workshops, teachers are often surprised to discover the amount and types of math in storybooks. The next question is whether the text and illustrations complement each other in the presentation of clear, accurate, and rich math ideas. Circus Shapes (Murphy, 1998) provides an example. The page presents an informal definition of the concept of a square. The text might better have said same “length,” but the matter is clear in context. Also, the squares are in different colors, sizes, and orientations, which suggests (not in so many words) that shapes are invariant over these transformations. If the illustration did not vary color, size, and orientation, then the child may get the impression that squares always have to have one side horizontal to the bottom of the page.
Figure 1: Illustration of the idea of a square
In selecting storybooks to use for instruction, teachers must be careful to find any inaccuracies, such as mislabeled shapes or discrepancies between the text and illustrations. For example, a page from Ten Black Dots (Crews, 1968) reads, “Four dots can make seeds from which flowers grow.” Although the illustration presents four dots, they only produce three flowers, which does not support the idea of one-to-one correspondence that the book is intended to teach.
13 We have prepared an illustrative guide to quality analysis as shown below. A productive exercise in an in-service workshop or higher education classroom would be for teachers to create guides like these.
Math Picture Book Analysis Guide Use this guide to help you choose math storybooks to read with your child. A good math storybook will have all four of the pos s.itive qualitie POSITIVE NEGATIVE The illustration s c orrectly show the math. The illustrations incorrectly show the math.
Since this page is about the number four and there are four seeds, there should also be four flowers.
by Donald Crews by Stuart J. Murphy Circus Shapes Ten Black Dots
The text correctly shows the math. The text incorrectly shows the math.
This story isn’t about counting the ani mals. by Maurice Sendak by Stuart J. Murphy Instead, it’s about which animals arrive first, second, third, etc. The cat is not three, the cat is third. Elevator Magic One Was Johnny Development and Research in Early Math Education | h ttps://dreme.stanford.edu
Figure 2: Math Picture Book Analysis Guide
A Pig is Big (Florian, 2000) is relevant for the third question, concerning mathematical thinking. The first page shows an illustration of a pig, and every time children turn the page, they see another, larger animal. The text invites children to guess what they might find on the next page: “What’s bigger than a pig?” and then, “What’s bigger than a cow?” Children can delight in making predictions about what animal they may find next.
How should teachers read math storybooks? As we have seen, many books, even those not purposefully written with the goal to teach math, implicitly contain many math ideas. Yet our work with teachers suggests that many fail to notice that the stories contain underlying math concepts. Also, we found that teachers may not make effective use of the books that contain explicit math.
14 We have therefore developed several methods for helping teachers make the most of storybook reading to promote children’s math learning.
Figure 3: Math Picture Book Reading Guide
Guides to specific books. One method, shown above, involves guides to individual books. They are designed to help teachers notice and understand the math content and language, ask questions that support their children’s understanding of the math in the story, and then conduct a relevant classroom math activity. General Guidelines. Another approach is to encourage teachers to engage in interactive reading with the child. “Dialogic reading” has been shown to be successful in stimulating children’s emergent language and literacy skills (Lonigan and Whitehurst, 1998). Teachers learn to use open-ended prompts (“Tell me what’s happening on these pages.”) to prod the child to describe interesting and important ideas in the story. Teachers are also encouraged to expand on the child’s responses (When the child says, “Bears,” you can respond, “Yes, there are three bears!”) and relate the story to the child’s life (“Can you think of something shaped like a square at home?”) As they become more familiar with the story after repeated readings, children can take on a greater role in leading
15 the conversation. We suggest that the teacher follow children’s interests by noticing when children are intrigued during storybook reading and encouraging them to examine and discuss that part of the story. Dialogic reading, like any other teaching method, should be employed in a sensitive and flexible manner, should not be strictly scripted, and should not interfere with the story’s momentum. The second approach involves questions, prompts, and remarks that vary in level of abstraction and are designed to encourage children to think more and more deeply about the story. Basic, less abstract teacher language addresses content that is perceptually present on the page and places few cognitive demands on the child; more abstract teacher language prompts the child to think and analyze. When reading a storybook about number, a less abstract question that a teacher can ask is, “Can you see the six sleeping turtles?” While this less abstract question may focus the child’s attention on the illustration, the teacher can follow up with more abstract questions: “Are there more turtle or dragonflies? How do you know?” In asking questions of these types, teachers should consider the level of the children and the number of times they have previously read the story. Stories about children Teaching any topic requires an understanding of the topic itself as well as of the children learning it. Stories can help teachers improve both types of understandings. We begin with stories about children. Over the past 35 years or more, there has been a great deal of research on the development of children’s mathematical thinking (Sarama and Clements, 2009). It is impossible to summarize this extensive body of work here. One major theme is that by the time children enter school, around the age of 4, they already possess informal, everyday math knowledge of some complexity. They can distinguish more from less, understand simple ideas and procedures of everyday (non-written) addition and subtraction, differentiate shapes, identify simple patterns, locate objects in space, and more. The research provides a good deal of specific information useful for math education. One concerns the cognitive processes involved in children’s everyday math and the other concerns the overall sequence. We learn, for example, that given a verbally stated problem like “How much is 3 plus 4?” children may simply count all members of the two sets, perhaps on their fingers, from 1 to 7. Soon after that children tend to count on from the first number stated, from 3 to 7. Later, children count on from the higher number, from 4 to 7. And later still, others use a strategy of building known addition facts, for example, reasoning that because 3 plus 3 is 6, the answer must be 7 because there is 1 more than the 3 + 3. The order in which children learn these approaches—often without direct instruction—is approximate. Children vary in the ages at which they acquire the
16 strategies, may use several at the same time, perhaps skip some altogether, and may vary in the accuracy with which they execute the strategies. Our “thinking stories,” case studies in the form of text with embedded videos, are designed to help teachers understand a single child’s mathematical thinking and how to assess it. Here is an example involving Ben, interviewed at 5 years. We first establish that there are 6 toy bears under a piece of paper. Then, as he watches, I put 2 more under the paper and ask him how many there are altogether. At first, he is quiet, apparently whispering some numbers to himself. Then he pops up and triumphantly says, “eight”! When I asked him how he knew that, Ben says that 4 and 4 make eight, and also that he counted on two more from 6. I suspect he did that first, and then remembered that 4 and 4 is eight. And then he volunteers that, “if you make 2 more it’s ten.” When I asked him a series of questions involving the addition of 2, he got up to 14 and then made a mistake. Despite this, it’s clear that at 5 years, he seems to understand the equivalence between 6 + 2 and 4 + 4, and even creates and solves a new addition problem that begins with his previous answer. An essential feature of our thinking stories is an embedded video, in this case one that vividly captures the exchange described above. Stories of this type (available on our DREME website https://dreme.stanford.edu) are enormously powerful. The embedded videos bring the text to life and enthrall teachers: they not only read about mathematical thinking but see it directly. Thinking stories can help college education students bridge the gap between practice and theory as they relate their class reading assignment to the child’s behavior shown on the video. Thinking stories also illustrate the interviewer’s questioning methods, which can serve as a model for teachers. Indeed, this kind of clinical interviewing can and should be used by teachers as they read with children and engage them in learning activities. Further, thinking stories can help anxious teachers to realize that math is in fact an interesting topic. Indeed, the teachers may learn some basic math in the process of reading the thinking story. Thus, some teachers may not have closely and explicitly examined the equivalence relations involved in the expressions 6 + 2 and 4 + 4. To understand Ben’s thinking, teachers need to understand those equivalence relations. Two basic questions about the thinking stories deserve discussion. One is the issue of generality. Given that our thinking stories focus on individual children, how can we guarantee the generality of the results for other children of roughly the same age? The answer is that we cannot, but that is not our intention. The stories show interesting examples of thinking that we know are common, but may or may not be typical of children at the same age. What is important is that the story illustrates important aspects of mathematical thinking, shows the excitement and interest that potentially can infuse children’s thinking, and shows how an interviewer can uncover thinking that often gets ignored.
17 Another question is the relevance of the thinking stories for instruction. How are individual interviews relevant for teaching? Our answer is in the form of another question: how can you teach if you don’t know what the child is thinking? For example, Ben seemed to understand the equivalence between 6 + 2 and 4 + 4. A teacher could use this knowledge to elaborate on issues of partitioning, the meaning of the = sign, how to use the hundreds number chart to explore addition, and the like. And more fundamentally, the teacher could learn that Ben’s peers might understand more than is usually expected. In brief, the thinking stories can help teachers to understand children’s mathematical thinking as illustrated by engaging and informative cases and to use the information to design instruction that builds upon what children do and do not know. Further, the stories can help teachers to reduce their math anxiety and learn about the subject themselves. Stories about the math To some extent, learning about children’s thinking will help teachers learn about the math that children are thinking about. Thus, you cannot understand the child’s failure to understand the commutative principle without understanding it yourself. But understanding children’s thinking is not enough; other methods are necessary to help teachers learn the necessary math. Traditional methods like taking a math course or reading short accounts of the math can be useful. But we have been developing another approach, a hopefully humorous exposition by the fictitious and renowned Professor Ginsboo. Here is a short excerpt in which Professor Ginsboo is teaching his student, Menette, about the meaning of enumeration. Listen up, Menette. Professor Ginsboo is going to tell you something shocking— like totally. No one ever counts correctly. Well, hardly anybody (but Professor Ginsboo) counts the right way. Let me show you. Here are some frogs. When most people count, they do this. “This frog is one and this frog is two. So there are two altogether.” The last part is right: there are two frogs, but the rest of it is wrong. Yes, this pink frog—Feeble Phoebe—is one. But this purple frog is not two. That is not his name. It is Fast Freddy and he knows that he is one frog, not two. Number words are not names. They tell you how many. You are lucky that the frogs have decided to show you how to count. First, Phoebe says, “I am one” because she is one frog. Second, Fast Freddy says that he is one also. But then they hold hands and say, “We are two.” “Two” is a cardinal number that refers to the collection, not the individual unit. That is the right way to count.
18 You can access the Professor Ginsboo stories our DREME website https://dreme.stanford.edu. We do not know whether this approach is effective, but we intend to find out. Conclusions Implicit and explicit math storybooks hold great promise for promoting children’s math learning in the context of reading and literature. Thinking stories can bring to life children’s mathematical thinking and inform teaching. Stories about math can help teachers learn it and then teach it to good effect. Stories, stories, and more stories: intellectual adventures for all. Acknowledgments We wish to acknowledge the generous support of the Heising-Simons Foundation for our work, as well as the contributions of Mia Almeda in preparation of guidelines discussed in the paper. References Anno, M. (1977). Anno’s Counting Book. NY: HarperCollins. Baker, K. (1999). Quack and count. San Diego: Hara Brace. Beilock, S. L., Gunderson, E. A., Ramirez, G., Levine, S. C., Smith, E. E. (2010). Female teachers' math anxiety affects girls' math achievement. Proceedings of the National Academy of Sciences of the United States of America, 107(5), 1860–1863. doi:10.1073/pnas.0910967107 Casey, B., Kersh, J. E., Mercer, J. (2004). Storytelling sagas: an effective medium for teaching early childhood mathematics. Early Childhood Research Quarterly 19, 167–172. Crews, D. (1968). Ten black dots. New York: Scribner. Falwell, C. (2008). Feast for 10. New York: Clarion Books. Florian, D. (2000). A pig is big. New York: Greenwillow Books. Houran, L. H. (2015). A thousand Theos. New York: Kane Press. Krauss, R. (2007). The growing story. New York: HarperCollins. Kraus, R. (1961). The littlest rabbit. New York: Harper. Lonigan, C. J., Whitehurst, G. J. (1998). Relative efficacy of parent and teacher involvement in a shared-reading intervention for preschool children from low-income backgrounds. Early Childhood Research Quarterly, 13(2), 263–290. Marshall, J. (1988). Goldilocks and the three bears. New York: Dial Books for Young Readers. Murphy, S. J. (1998). Circus shapes. New York: HarperCollins. Sarama, J., Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge. Tompert, A. (1993). Just a little bit. Boston: Houghton Mifflin.
19 BLENDING ELEMENTARY MATHEMATICS EDUCATION RESEARCH WITH DEVELOPMENT FOR EQUITY – AN ETHICAL IMPERATIVE ENABLING QUALITATIVELY RICHER WORK
Mellony Graven
Abstract South Africa, as a result of its apartheid history, is a nation of extreme socio-economic and educational inequality. Three aspects of this context are important in understanding why it is essential that educational research and development be intertwined. The first is South Africa’s post-apartheid (1994) education context in which performance and opportunity gaps still persist along racial lines. The second is the in-service teacher education context, which – while aiming to support teachers in implementing three post- 1994 cycles of curriculum revision – has attracted criticism for having failed to provide appropriate kinds of support, and thus, of largely alienating teachers. The third is the education research context, and particularly mathematics education research, which mostly tells deficit stories both of learner performance and of teacher practice. In relation to each of these contextual aspects, establishing non-exploitative and trusting partnerships with teachers and communities in which meaningful dialogue and joint investigation, informed by a range of stakeholder perspectives, is essential for navigating what might be possible within our current context of a stubbornly persistent education crisis. In this paper I briefly explain each of these aspects. I then share the design of the South African Numeracy Chair Project (SANCP). This project was set up to enable a powerful dialectical relationship between research and development through merging the two in a network that has created multiple opportunities for dialogue across stakeholders, dialogue that has focused on mutual learning towards addressing the challenges of elementary mathematics learning. While beyond the scope of this paper in the presentation I share the learning trajectories of three SANCP participants. I argue that establishing a network of development programs is not just an ethical ‘nice to do give-back’ to research participants. These partnerships enable access to data and stakeholder perspectives that would be inaccessible without the relationships developed in these spaces of collaboration, thereby strengthening the design of more appropriate research projects and leading to richer and more valid research findings. In these collaborative spaces dialogue and active participation among all participants is critical in the joint enterprise of finding sustainable ways forward to the educational challenges.
Keywords: Professional development, communities of practice, research for development, intervention projects
Introductory context Under apartheid there were four racially segregated and differentially funded departments of education. The Department of Education for white citizens was well funded. The other departments were deliberately poorly funded. Education
Rhodes University, Republic of South Africa; e-mail: [email protected]
20 was used as a vehicle for maintaining white privilege. With South Africa’s first democratic elections in 1994 there emerged a single education department and education became the vehicle for social transformation and redress of apartheid inequities. However, 23 years since the advent of our democracy educational inequality persists along both socio-economic and racial lines. Participation in international and regional comparative research studies (such as Trends in International Mathematics and Science Study (TIMSS) and Southern and East African Consortium for Monitoring Educational Quality (SACMEQ)) tell a story of extreme failure of the post-apartheid aim that education should be the vehicle for redress of inequality. Our performance gaps of students in Mathematics and Science have been the highest of all countries participating in TIMSS (Reddy et al., 2015), and within the ten participating SACMEQ countries we have the highest between school performance inequality in mathematics (Taylor, 2009). South Africa has a bimodal education system: one system functions well and student performance in this system compares well to international benchmarks, while the other system, serving the vast majority of students, is largely dysfunctional resulting in among the lowest performance in TIMSS (Reddy et al., 2015). While we may question the validity of international comparative studies we do not perform much better in regional studies (e.g. SACMEQ) where we are compared to countries with similar language challenges and economic disparities. Furthermore South Africa’s own Department of Basic Education (DBE, 2014) Annual National Assessments (ANA) (begun in 2012 in Mathematics and Literacy from Grade 1 to 6 and Grade 9) show that in 2014 an average of only 3% of Grade 9 learners achieved 50% or more for mathematics. Of course there are many contributory factors (see Fleisch, 2008; Graven, 2014) including, for example, language of instruction practices that have 79% of Grade 4 students being taught mathematics in English while only 6.7% of learners are first language English speakers (Robertson and Graven, 2015). A common response to reports on the data described above is teacher blaming (Shalem and Hoadley 2009; Graven 2012). The logic then is, if teachers are the cause of the crisis, then the system (Department of Basic Education) should provide in-service support in the form of increased specification for what teachers must teach (e.g. enforced weekly plans and learner completion of workbooks) and increase the monitoring of this. This not only deflects attention from broader systemic issues that need to be addressed but lowers teacher morale and shuts down the space for the kind of teacher development that could build on teacher experiences as a basis for finding ways forward. Much teacher support has been in the form of short term, departmentally run ‘workshops’ in which teachers are provided documentation of what must be implemented. This seemingly ignores the data gathered in TIMSS, SACMEQ and
21 ANA that point to the need to address extreme backlogs in the mathematical knowledge of most learners which, Spaull and Kotze (2015) argue, is already 2 grades behind by Grade 4. In subsequent meetings or face-to-face school visits, teacher preparation files are inspected for compliance and curriculum coverage (Graven, 2016). Much needed support for developing teacher agency, to respond to local challenges of learners, is thus replaced with bureaucratically driven ‘one size fits all’ grade specific schemes of work. Widespread research shows the dominance of concrete methods of calculation (i.e. tally lines) even for large number calculations well into the Intermediate Phase grades 4 to 6 (Schollar, 2008; Hoadley, 2012). Schollar (2008) found, for example, that just under 80% of Grade 5 children solve problems with unit counting and argued that “learners are routinely promoted from one grade to the next without having mastered the content and foundational competences of preceding grades, resulting in a large cognitive backlog that progressively inhibits the acquisition of more complex competencies” (p.1). Forcing teachers to teach learners long multiplication of three digit by two digit numbers, when the children have not moved beyond tally counting results in teachers teaching long multiplication algorithms imitatively, often resulting in learner application of taught procedures without consideration of the reasonableness of the strategy or answer or the underlying place value of the digits being manipulated. So, for example learners taught the vertical addition algorithm, have been seen to answer 910 for 98 + 2 (Graven et al., 2013). In this dance of ‘playing school’ by learners, teachers, and departmental advisors mathematical learning appears to be completely out of focus. In a recent FLM paper (Graven, 2016) I shared the experiences of two teachers involved in one of our in-service teacher intervention projects. I wanted to illuminate how our current curriculum and systemic “support” work against teacher agency to respond to their local conditions and challenges. Here is what I wrote in respect of one of these teachers:
Zandi was asked how she managed the tension of revisiting work from earlier grades and keeping up with the grade 4-7 departmental schemes of work. She responded as follows: We tell the subject advisor that I am actually at grade 2, CAPS [Curriculum and Assessment Policy Standards] says I must teach this [grade 4]. But my learners are not yet on that level. That means I have to go to grade 3 work. They [district advisors] said no it is wrong they know that some learners struggle or whatever but we are wrong to go back to grade 2, or grade 3. We always argue about that and then they will say it is from the top not from them and then what do you do? (pp.9-10).
Zandi’s comments here illuminate the tensions that exist for teachers wanting to respond to the challenges they see in their classroom by deviating from the Department’s “one size fits all” schemes of work. These schemes assume that the majority of learners entering each grade have mastered the curriculum of previous
22 grades, even while this flies in the face of the ANA results which are intended to inform teaching and learning. Teachers are thus placed in unenviable positions of being blamed for not fixing a crisis they are pushed to perpetuate. This highlights the need for the development of teacher agency in all project work through promoting dialogue among teachers, district and provincial departmental advisors and researchers and teacher educators (Long et al., 2017). It also highlights the need for researchers to interrogate their responsibility in relation to the crisis context they are researching in and to question the role their research process and the research findings may play in helping to bring about improvements. As noted above much research, both under apartheid and since then, paints teachers in a bad light. While ethical requirements for education research are being tightened, many schools and teachers feel exploited by research that has offered no direct benefit in relation to their participation and that constantly reports on their deficiencies. Thus, even while anonymity is maintained, participation involves colluding in the construction of broadly damaging narratives that damage the status of their profession. Setati (2005) addressed this in a paper titled Researching teaching and learning in school from "with" or "on" teachers to "with" and "on" teachers, which I give as an introductory reading for SANCP researchers. The article pushes researchers to confront various ethical and political issues of researching teaching and learning in schools. Unidirectional power relationships between researchers and teachers, Setati argues, should be replaced with reciprocal relationships in which both teacher/s and researcher/s negotiate mutual benefits. Indeed, when Professor Setati spearheaded the setting up of the private- government funded Mathematics/Numeracy Education Chair initiative in 2010 (of which I am the incumbent at South African Numeracy Chair at Rhodes University), she set them up as research and development Chairs, the first of their kind within the South African National Research Foundation’s Research Chairs Initiative (SARCHI). In our Chairs, we were tasked with making a difference in the schools we work with through teacher development programs and researching ways forward to the many challenges we face in numeracy education. In other words, while we are asked to investigate the challenges, we need simultaneously to focus on possibilities within this context of challenge. We must act for change and not just state what is, or what should, be done. External monitoring and evaluation of the overall effectiveness of various programs implemented by SANCP has focused mainly on ‘performance indicators’ of the ‘impact’ of our work across the fourteen partner schools. (A positive evaluation led to renewal of a second 5-year term from 2016-2020). On the other hand, full time and part time doctoral students (over 20 students since the start in 2011) and SANCP team members mostly focused on smaller scale and more qualitative research that focused on a single programme or intervention with a small number of schools/teachers/learners (some with schools outside of SANCP partner schools)
23 in order to better understand the nature of learning enabled by specific interventions. My view is that focusing on research that does not contribute to addressing the many challenges faced is a luxury we cannot afford in our context of extreme educational inequality, and is inconsiderate of the needs of teachers and learners who participate in our research. This view I found to be shared by the teachers in the Grahamstown area. Despite being a university town, with a long history of participation of local schools in university-based research studies, government school performance in the area continues to be low. Thus when I first approached primary schools and teachers to partner with SANCP in a professional development community which would meet regularly to jointly explore sustainable ways forward to the mathematics teaching and learning challenges they faced, teachers were understandably sceptical and reluctant. It was only after our first meeting where the few teachers who initially attended spread the word through their community networks that they came to believe that our project would be different from previous research or development projects that many teachers had joined. Soon our programme was over subscribed. This enthusiasm of teachers for finding ways to strengthen mathematics teaching and learning contrasts the oft heard deficit narrative of teachers reluctant to participate in professional development (PD). In my experience, teachers have shown overwhelming willingness and commitment to participation and engagement for improving teaching and learning when they experience meaningful opportunities to engage as respected professionals whose expertise is central to the engagement. Having painted a picture of the context in which the South African Numeracy Chair Project (SANCP) emerges I now turn to discuss how SANCP responds to the above challenges. The design of the South African Numeracy Chair Project (SANCP) SANCP was set up as a ‘hub of mathematical activity passion and innovation’ focused on developing rich communication networks and spaces for collaboration and dialogue across all communities involved in mathematics education (teachers, parents, school management teams, district advisors, provincial curriculum planners, learners, national DBE mathematics/assessment specialists, local and international mathematics education professional and research associations) through projects grounded in the local needs of teachers, students, and communities. The design of SANCP, and the long-term teacher development communities of practice (CoP) created within SANCP, was informed by Wenger’s (1998) theory of learning in communities of practice which outlines four interrelated learning components. These are: practice (learning as doing), meaning (learning as experience), identity (learning as becoming), and community (learning as belonging). His work builds on his earlier work with Jean Lave in which they had
24 argued that learning is located in the process of co-participation and increased access of learners to participation, through which members develop changing ways of being and becoming. Here, access to quality resources is prioritized, and so, “to become a full member of a community of practice, requires access to a wide range of ongoing activity, old-timers, and other members of the community; and to information, resources, and opportunities for participation” (Lave and Wenger, 1991, p. 101). In this respect SANCP was based on the premise that teacher, student and researcher learning would be enabled through maximised access to participation in, and resources of various overlapping communities of practice. Wenger (1998, p. 214) defined communities of practice as “a living context that can give newcomers access to competence and also invite a personal experience of engagement by which to incorporate that competence into an identity of participation…and a good context to explore radically new insights”. He argued furthermore, that:
A history of mutual engagement around a joint enterprise is an ideal context for this kind of leading-edge learning, which requires a strong bond of communal competence along with a deep respect for the particularity of experience. When these conditions are in place, communities of practice are a privileged locus for the creation of knowledge (1998, p. 214).
The design of SANCP programs focused on establishing a partnership with teachers, students, school management teams, parents, and district advisors through setting up various platforms for regular engagement around mathematics teaching and learning. Two teacher development programs (the Numeracy Inquiry Community of Leader Educators (NICLE from 2011-2015) and the Early Number Fun programme (ENF from 2016-2017)), brought together Grade 2-5 and Grade R teachers respectively, as well as SANCP researchers and project staff, principals and deputy principals, and local/provincial departmental teacher advisors. Both programs foregrounded ‘deep respect’ for the teachers’ particularity of experience, and drew on this as a critically important resource when engaging with ‘new insights’ on research-informed teaching methods and interventions. Together we would all search for sustainable solutions to mathematics education challenges. Similarly, SANCP’s after-school learner clubs were designed to break down traditional learner/ teacher relationships and learner dependence on imitation of taught procedures. Instead club activities foregrounded explorative talk and individual learner sense-making focused on discussion and development of a wide range of efficient strategies for calculating and problem solving and the development of number sense (see Graven, 2015). SANCP’s family math forums brought caregivers and learners together on Saturday mornings in schools and community centres to work together on mathematics problems in a fun way. Thereafter a range of take home mathematics game resources, mostly using the
25 dice and cards given to families and oral games, were shared with some guidance on how caregivers might encourage children to more efficient ways of working while playing games. (So, for example, in a simple dice game where players must calculate the total of two dice, caregivers were encouraged to help children, through the regular playing of games, to move from counting all dots by touching each dot on the two dice (e.g. 6 and 3); to subitising (I see 6) and counting on from the largest number (6,7,8,9); to knowing the number bonds off by heart (6 + 3 = 9).) Thus, the guiding principle for all SANCP work was that effective learning would require that all participants in SANCP projects actively share their experiences and knowledge with others in the spaces created for dialogue between members of different communities. These newly created spaces of regular engagement were especially powerful in that they generated opportunities for dialogue in non- traditional spaces where new forms of relationships could be built. In each of these spaces the projects endeavoured to build reciprocal learning relationships in which all are learners and participants. Of course, the perceived hierarchies of status identities attached to positions of participants in various institutions - such as professor / researcher / teacher educator in a university; curriculum planner in the provincial Department of Education; or teacher / head of department/ principal in a school - cannot be wished away. However, by recognizing and conferring great value to teachers’ experiences (in engagement between participants from different institutions – i.e. universities, district/provincial departments of education, and schools) because teachers are the ones who have the critical and grounded experience of working almost daily with learners in classrooms, teacher agency is enhanced in shaping the joint enterprises and practices of the PD CoP and, subsequently, the practices in their classrooms. The experiences of others in academia, research or district or provincial subject specialist positions provide a more global, but no more important, perspective in the joint enterprise of seeking solutions to the many challenges faced. As Wenger (1998, p. 149) notes, learning involves defining who we are “by negotiating local ways of belonging to broader constellations and of manifesting broader styles and discourses”. In this way, reciprocal learning is facilitated by working towards more equal power relationships that recognize that expertise lies in the experience each member brings in relation to their belonging to specific mathematics education communities, and that the learning for all in the PD CoP is maximized by dialogue and discussion between members belonging to overlapping communities. So, for example, in the NICLE PD programme teachers, researchers, teacher development academics, and government employed teacher advisors engaged around the common aim of finding ways to improve student mathematical learning. The more traditional relationships of teachers being at the receiving end of short-term information dissemination (‘how to teach’) workshops were
26 replaced with mutually respectful and beneficial reciprocal learning relationships. These relationships optimised participation and learning by members and became ‘a privileged locus for the creation of knowledge’ (Wenger, 1998, p. 214). The diagram in Figure 1 (below) summarises the interrelationships between the various communities which SANCP has brought together through its three key projects (the PD CoPs (i.e. NICLE and ENF); the after-school math club programs; and the family math events). In these programs traditional bi- directional relations (i.e. between teacher educators and teachers in workshops or a monitoring meetings; between teachers and learners in classrooms; between learners and parents at home; between researchers and teachers in a data gathering situation) are opened up to multidirectional dialogue in which SANCP research and development members work to cultivate new forms of relationships in each of these project spaces focused on maximizing dialogue, participation and learning of all members in each community within each project space.
Figure 1: SANCP communities and key project spaces While it is beyond the scope of this paper to describe each of these project spaces (but see Graven and Stott, 2015; Pausigere and Graven, 2014; Graven, 2015), suffice it to say here that these spaces continue to provide rich learning opportunities from which sustainable, locally trialled and tested interventions can be developed and researched, and then shared more broadly through our freely available website and through professional/research platforms such as conferences, stake-holder think tanks, or task teams and publication forums.
27 SANC Projects as research spaces Figure 2 (below) captures SANC project researchers’ participation in the three key research spaces.
Figure 2: SANCP researchers and research spaces By way of example, let me focus in on the mathematics club space and its triadic dialogue between learners-teachers-R&D project contributors (as shown in the top central triangle of Figure 1 and the two club triangles in Figure 2). The mathematics after-school clubs were introduced in collaboration with teachers in order to provide: i) an after-school space for SANCP members to have direct grounded experience with local learners in partner schools; ii) a space for trialling research informed ideas before sharing them with teachers in PD CoPs more broadly; iii) supplementary support for the PD through direct learner interventions redressing learning gaps outside of the official curriculum. They were also designed to provide teachers a safe space to trial new ideas and more explorative ways of teaching, free from curriculum coverage demands, with a smaller group rather than a whole class of learners. Feedback from teachers over time indicated that club learners often became catalysts in their mathematics classrooms for modelling and demonstrating productive mathematical learning dispositions along with ways of talking and explaining their mathematical thinking (Graven, 2015). These club learners sometimes became the teachers’ helpers in class. I argue that this rich network of development work not only serves as an ethical ‘give-back’ to those participating in the research through responding to local
28 needs and challenges, but also enables more grounded and richer research that would not be possible without access to these networks and the relationships established across the communities of stakeholders in the various project spaces. With the growth of our SANCP research team of masters and doctoral students the mini laboratory developmental aspect of clubs provided useful after-school spaces which both enabled researchers to meet their obligation to ‘give something back’ to participating research schools and provided an empirical field for conducting research interventions aimed at finding sustainable ways forward to key challenges identified. So for example three researchers investigated the implementation of aspects of Bob Wright and colleagues’ mathematics recovery (MR) programme in the after-school club context. In this way the club spaces enabled research to proceed without interfering with ‘the normal school day’ which the Eastern Cape Department of Education insists should not be disrupted in any way by research. These research projects and their findings then fed back into the NICLE PD through presentations made by the researchers, some of whom happened to also be NICLE teachers and departmental teacher advisors (in the presentation I will share exemplar stories to illustrate this). Dissemination of research findings across the various after-school mathematics club interventions led to many requests for broader expansion of the after-school club model from NGOs and departmental teacher advisors and teachers from other districts and provinces. This then led us to develop a consolidated ten-week teacher development programme for club facilitators (coordinated by our graduated doctoral student and now full time SANCP research and development officer coordinating all mathematics club programs Dr Debbie Stott). This programme is now running in several provinces but of note is that three of our Masters and Doctoral students, who are also departmentally employed district/ provincial teacher advisors, are running these programs with teachers in three districts outside of the broader Grahamstown area. The two masters’ students, Nolunthu Baart and Gasanakaletso Hebe, are focusing their research on student learning, while doctoral student, Zanele Mofu is focusing her research on teacher learning, each in the context of the ten clubs run by ten teachers with whom they have partnered in relation to their departmental supporting of teachers in their districts. One student is researching across family and club intervention spaces. Her research focuses on stakeholder experiences of the introduction of these and other educational interventions in four after care centres in the Grahamstown area. These centres cater for at risk children in various communities by providing them with meals and safe spaces to be cared for in the afternoons. For other researchers, participating alongside teachers in the PD programs enabled trusting relationships to develop that led to teachers’ willingly participating in student research studies and willingly allowing researchers to enter their personal classroom spaces for data gathering. For example: doctoral students, Peter Pausigere and Roxanne Long researched the nature of teacher learning within
29 NICLE and ENF respectively, while doctoral student Sally-Ann Robertson researched two NICLE teachers’ classroom talk practices. In the presentation I will share three exemplar research-development trajectories to illuminate the powerful dialectic argued in this paper is created by a rich network of interconnected communities. Concluding remarks In this paper I have argued that establishing non-exploitative and trusting partnerships with teachers and communities is essential for navigating the way forward in the current South African education crisis. I have explained the way in which the South African Numeracy Chair Project (SANCP) was set up to enable a powerful dialectical relationship between research and development in a network of projects that create multiple opportunities for dialogue across stakeholders. What I hope to illuminate in the presentation, through exemplar trajectories of three members is the way in which the rich network of opportunities enabled participation in various roles across the multiple overlapping communities and projects, and that this provides momentum for increasingly central participation within each of these communities and a deepening of learning, stimulated by both grounded forms of belonging to the local practices of teaching, and participation in research and development work informed by a more global perspective. References Department of Basic Education. (2014). Report of the Annual National Assessments of 2014: Grades 1 to 6 & 9. Pretoria: DBE. Fleisch, B. (2008). Primary education in crisis: Why South African schoolchildren underachieve in reading and mathematics. Jhb: Juta. Graven, M. (2012). Changing the Story: Teacher Education through Re-authoring their Narratives. In C. Day (Ed.), The Routledge International Handbook of Teacher and School Development (pp. 127–138). Abingdon: Routledge. Graven, M.H., (2014). Poverty, inequality and mathematics performance: the case of South Africa’s post-apartheid context. The international journal of mathematics education ZDM, 46, 1039–1049. Graven, M. (2015). Strengthening maths learning dispositions through math clubs. South African Journal of Childhood Education, 5(3), 1–7. Graven, M. (2016). When systemic interventions get in the way of localised reform. For the Learning of Mathematics, 36(1), 8–13. Graven, M., Venkat, H., Westaway, L., Tshesane, H. (2013). Place value without number sense: Exploring the need for mental mathematical skills assessment within the Annual National Assessments. South African Journal of Childhood Education, 3(2), 131–143. Graven, M., Stott, D. (2015). Families enjoying Maths together-organising a family Maths event. Learning and Teaching Mathematics, 19, 3–6.
30 Hoadley, U. (2012). What do we know about teaching and learning in South African primary schools? Education as Change, 16(2), 187–202. Lave, J., Wenger, E. (1991). Situated learning: Legitimate peripheral participation. New York: Cambridge University Press. Long, C., Graven, M., Sayed, Y., Lampen, E. (2017). Enabling and constraining conditions of professional teacher agency: The South African Context. Contemporary Education Dialogue, 14(1), 5–21. Pausigere, P., Graven, M. (2014). Learning metaphors and learning stories (stelos) of teachers participating in an in-service numeracy community of practice. Education as Change, 18(1), 33–46. Reddy, V., Zuze, T, Visser, M., Winnaar, L., Juan, A., Prinsloo, C.H., Arends, F., Rogers, S. (2015). Beyond Benchmarks: What twenty years of TIMSS data tell us about South African Education. HSRC press: Cape Town. Schollar, E. (2008). Final report of the primary mathematics research project (2004- 2007). Towards evidence-based educational development in South Africa. Johannesburg: Eric Schollar & Associates. Spaull, N., Kotze, J. (2015). Starting behind and staying behind in South Africa. International Journal of Educational Development, 41, 13–24. Robertson, S-A., Graven, M. (2015). Exploring South African mathematics teachers’ experiences of learner migration. Intercultural Education, 26(4), 278–296. Setati, M. (2005). Researching teaching and learning in school from “with” or “on” teachers to “with” and “on” teachers. Perspectives in Education, 23(March), 91–102. Shalem, Y., Hoadley, U. (2009). The dual economy of schooling and teacher morale in South Africa. International Studies in Sociology of Education, 19(2), 119–134. Taylor, N. (2009). Standards-based accountability in South Africa. School Effectiveness and School Improvement, 20(3), 341–356. Wenger, E. (1998). Communities of Practice: Learning, meaning and identity. New York: Cambridge University Press.
FROM CONTENT TO MEANING: SEMANTICS OF TEACHING IN THE TRADITION OF BILDUNG-CENTRED DIDACTICS Tomáš Janík
Abstract The paper highlights images of the “new” culture of learning (evolving in the age of accountability) and argue that the focus on content is nowadays rather neglected in the school practice and in research on teaching. It presents the process of emptying the content as the great challenge of the “new” culture of learning. In an attempt to respond, the paper will turn to the tradition of Bildung-centred didactics. It will highlight the semantics of teaching and outline the theoretical background for the content-focused approach to (research on) teaching and learning that we develop and employ at Masaryk
Masaryk University, Czech Republic; e-mail: [email protected]
31 University in Brno (Czech Republic). This approach is based on fine-grained analysis of real-life teaching and learning situations in the classroom (as captured on video). The so-called 3A procedure (annotating-analyzing-altering classroom situations) as the key instrument in our approach will be presented. It will be highlighted how didactic case studies can be developed from the 3A procedure and whether it is possible to generalize the findings across individual cases. Towards the end of the paper, the power of content- focused approach in teacher education for developing and sharing pedagogical content knowledge for the improvement of instruction will be discussed. Keywords: Bildung, content, didactics, learning, meaning, teaching
To begin with… The “new” culture of learning evolving in the “age of accountability” is characterized by delivering teaching and learning in standardized ways, trying to reach objectives at expected levels and governing schools on the basis of student achievements. This image may represent such situation in school systems, where outcome-based educational policy has been adopted and competency standards and (high-stake) testing have been implemented. Another image of the “new” culture of learning could be presented for school “systems”, where the accountability is not an issue at all – e.g. due to limited capacity of educational policy or authority to control schools. Released curriculum liability, relaxing edutainment activities and pervasive lightness of being in the school are signifiers for this approach. However in individual images different features of learning culture are highlighted, something seems to be hidden across both images. We argue, it is the focus on content in teaching and learning. Emptying the content worked on in the school could be the phrase depicting aptly the grand challenge of the “new” culture of learning.
Emptying the content in school education Various reasons could be found that are responsible for emptying the content of school education in decades after the PISA shock. In school systems where testing is the prevalent instrument for managing schools and teaching, prescribed, narrow and target-oriented curricula are often implemented. These “new generation curricula” rest typically on explicit descriptors that standardize the content to be taught as well as ways of how it is worked on. This trend towards instrumentalism is characterized by stronger orientation to measure learning success (and make teachers accountable for it), which stands “in contrast to the traditional curriculum approach based on content and its mastery” (c.f. Hillen, 2015, p. 74 with reference to Karseth, 2008). The respective culture of learning – known as teaching to test – is characterized by focusing on “learning” tasks identical or similar to those used in tests. This has consequences in terms of not only choosing what to teach reduced to easily
32 testable content, but also instructional practices. In this respect, the current practice in schools, which is based on filling in and ticking options, is criticized. In the accountability approach, the content tends to be understood as given (prescribed), to be testable more easily. It is supposed, that the meaning is to the content inherent and that one can anticipate learners’ behavior as well as its results. In the consequence also ways of how content “is to be worked on” are given and thus restrained. Hillen (2015, p. 78) points out that the accountability approach strives for standardizing the ways of teaching – when learning outcomes are predefined, also the teaching methods are to a high degree predefined. In the means-ends logic, those teaching methods are “selected” that best attain a given goal. Teachers’ autonomy regarding teaching methods (“methodische Freiheit”) is in the accountability approach actually restricted which results in a limited sense for context and situatedness of teaching and learning in the classroom. Hillen (2015, p. 84) refers to Østrem (2011) who notices that “to predetermine the explicit objectives prevents the teacher from taking advantage of what happens in actual learning situations where the teacher, student, and teaching materials meet; that is in the didactic triangle”. These are some of the aspects that problematize the culture of learning in the age of accountability. It is evident that “the new language of learning” pushes questions of content and purpose of education into the shadows. Biesta (2006) is right, when he argues that we have to reinvent a language of education (Bildung) that would serve as an alternative to the language of learning. Hopefully, reinventing the language of Bildung will also address the issues of the content and purpose of education.
Turning to and reinventing the tradition of Bildung It seems the content is still a “missing paradigm” in teaching (Menck, 2000) as well as in research on teaching (Shluman, 1986). Losing the focus on the content could be surprising – particularly in continental Europe, where the tradition of Bildung-centred didactics is rooted in the fundamental writings of Jan Amos Komenský (Comenius) and Wolfgang Ratke and developed further by Weniger, Klafki and others. Also in the Scandinavian educational landscape the content- focused Bildung-centred didactics is developed further (see e.g. Englund 1997; Gustavsson, 1998; Hillen, Sturm and Willbergh, 2012). In France, similar developments could be found under the framework of transposition didactique (Chevallard, 1991; Brousseau, 2002). In the English-speaking world the tradition of Bildung did not have the same impact, however – as Hillen, Sturm and Willbergh (2012, p. 10) point out – the “questions of teaching and learning have been theorised within the framework of curriculum theory. These two sets of educational concepts still live separate lives even though there has been a dialogue over the last 20 years”. In the framework of curriculum theory as well as in the
33 field of domain-specific research on teaching and learning we can identify the presence of the content-focused approach (e.g. Shulman, 1986). Why do we think the content-focused Bildung tradition could offer valuable perspective and/or alternative to what we (today) experience in schools that are shaped by accountability and instrumentalism? First of all it should be noted that Bildung is rooted in the Humanistic tradition – it is linked with ethical and democratic values. In this context Kristiansen (2015) refers to Biesta (2006, p. 55), who points out, that “the most important question for us today is no longer how we rationally master the natural and social world. The most important question is how we can respond responsibly to it, and how we can live peacefully with what and with whom is the other”. Bildung – in contrast to Ausbildung – is free and never ending (long-life) process, which focuses on the cultivation of human beings and minds. Moreover, as Kristiansen (2015, p. 121) argues, Bildung can be used as a type of resistance in current (neoliberal) debate about education; it can widen the narrow instrumental perspective and help us to rethink the educational relationship. No less important is that the Bildung tradition and related didactics is by nature content-focused. As Künzli (2000) points out, the central task of didactics is to seek the character-forming significance of the knowledge and skills that a culture has. As these knowledge and skills enter the process of Bildung in the form of content, this is a content-based process. “Didactics is in this way understood as a teacher’s professionalised art of argument and deliberation over how to construct interpretations of teaching content that is perceived as meaningful by the students” (Hillen, Sturm and Willbergh, 2011, p. 11). In the tradition of Bildung-centred didactics various methodological approaches and instruments for analysing content transformations (relations between content to be taught and learned and content already taught and learned) were developed. To sum up (without ambition of being exhaustive) we can list some of them: Theory of didactical situations (Brousseau, 2002) Model of Educational reconstruction (Komorek and Kattmann, 2008) Critical Didactic Incidents Method (Amade-Escot, 2005) Qualitative analysis of content development in instruction (Gruschka, 2013) Mimetic perspective in didactics (Willbergh, 2016) 3A content-focused approach (Slavík, Janík and Najvar, 2016). Regardless how these approaches differ, they build on a common core, which we can refer as the semantic perspective. Semantics of teaching as a common core of Bildung-centred didactics It is not possible to teach nothing. Teaching is always about something. When the teacher teaches something and students learn something, it is the content – teaching is content-based.
34 Following the Bildung tradition, the content (alongside teacher and students) is one of three elements they constitute an educational relationship and enable to form an educational situation. Teacher arranging fruitful meeting between students and content – the fundamental proposition formulated by Klafki (2000) depict the core of the Bildung-centred didactics. However, it is another challenge to arrange fruitful meeting between the students and the content. The content is a possibility – it means, that various meanings could be created when different students and teachers work on the “same” content. The German Bildung tradition points this notion out using the Inhalt-Gehalt (matter-meaning) distinction (c.f. Klafki, 2000 and others). As Hopmann (2007, p. 117) explains, “as the connection of matter and meaning is no ontological or ideological fact, but rather an emerging experience which is always situated in unique moments and interactions, there is no way to fix the outcome in advance”. It means that the meaning is not inherent to the content (matter). The meaning must be interpreted from the content and it can be interpreted in different ways according to given (e.g. disciplinary) framing, according to students’ previous knowledge etc. Another point is that the outcome of knowledge construction (movement form content to meaning) is never predictable because the students construct meaning autonomously in response to the teacher’s presentation of the content (c.f. Midtsundstad, 2015, p. 30) A fruitful perspective in this context is offered by the mimetic didactics. In this perspective, the key issue is the semantics of teaching, i.e., the meaning construction “by doing imaginative acts with perceptible teaching objects representing the subject matter …” (Willbergh, 2011, p. 62). As Willbergh (2011, p. 62) further explains, these representations are “perceived by the students in the context of the classroom and are given meaning by the interaction of teacher and students. Teaching is, however, an intentional activity from the teachers’ point of view that does not guarantee success … The students are seen as autonomous individuals and a contribution to the individual’s Bildung demands that the learning is desired and actively sought out by the person”. This is why the transition between the content and the meaning(s) and also between the student’s and the teacher’s perspectives and interpretations on/of the “same” content should attract the interest of didactic research in the original sense (e.g. Gruschka, 2013, pp. 31-32). And this is way in which we have to strive in teacher education for approaches, that enable us to find 150 ways (using Shulmans’ rhetorical figure to get the point) to (re)present the content to students with diverse learning dispositions. We continue by presenting the 3A content-focused approach, which we have developed at the Masaryk University in Brno to reach these ambitions (Slavík, Janík and Najvar, 2016).
35 Content-focused approach 3A Analyses of teaching and learning situations are popular with researchers because they reflect the way teachers think about the quality of their own lessons. Here we present in more detail the 3A procedure as an approach to studying and improving the quality of instruction (Slavík, Janík and Najvar, 2016; Janík, 2016). It allows us to investigate how well the aims, content and concrete realisations of students’ activities are integrated. This is achieved by identifying teaching and learning situations that serve as examples of how educational aims, content and students’ learning meet and intersect in instruction and by building case studies around such teaching and learning situations. The proposed approach draws on the idea of reflective practice in teacher education; the procedure has great potential for teachers’ professional development. The ultimate aim of the approach is to generalize findings from particular case studies to arrive at abstract and theoretical categories that will help explain patterns in the deep structure of instruction and bring better understanding of the general aspects of teaching and learning quality. To achieve these aims we make use of the 3A procedure (Slavík, Janík and Najvar, 2016; Janík 2016). “3A” stands for a three-step methodology consisting of annotating, analysing and altering a particular teaching and learning situations. In line with the content-based approach to studying and improving the teaching and learning processes, the starting point for the proposed research approach is the way content is dealt with in the classroom. Annotation is a brief summary of the situation and its context. Situations are analysed from various perspectives (for example, from the perspective of “learning to learn” the situations are analysed in respect to various aspects: metacognition; gaining, processing and assimilating new knowledge and skills; applying knowledge and skills in variety of contexts etc. Analysis refers to a reconstruction of the situation – it focuses on specific aspects of the situation in order to reveal the potential for qualitative change (improvement). Conceptual structure diagrams based are used as tools for capturing and visualising the way the content was worked on in the situation. We strongly feel that only such logical-semantic analysis may provide grounds for suggesting alterations within the teaching and learning situations. Alteration is an alternative course of action. First, the analysed teaching and learning situations are assessed with regard to four qualitatively different levels: (1) failing, (2), undeveloped, (3) enabling and (4) supportive. It is by principle that it is the failing and undeveloped situations that are in need of alterations. Alterations are then suggested, reconsidered, and discussed in the professional community. Suggesting alterations within the situations is a way of professional learning.
36 In the first step (Annotation), teaching and learning situations (lesson segments) are accumulated, that include illustrative examples of working with educational content. Teaching and learning situations captured on video are developed into video cases (annotated). Finding a promising teaching and learning situation is vital for the success of the case study. This however is true for all case-study based research. Specifically, the proposed approach has been inspired by the Critical Didactic Incidents – CDIs method (Amade-Escot, 2005). In the second step (Analysis), the identified situations are analysed from various perspectives. They are considered as particular cases in a multiple case study (Yin, 2011). Such analyses help reconstruct the conceptual structure of the situation, i.e. its educational construction. Within the conceptual structure diagram, we distinguish three levels: the thematic level (i.e. students’ experience with phenomena), the concept level (i.e. the field-specific knowledge and procedures) and the competence level (i.e. transdisciplinary aims). Distinguishing these three levels makes it possible to take into account the relationships between the students’ everyday experience, the terminological and methodological (substantive and syntactic) structure of the content in the field and the educational aims (or competences). The relationships between the levels of the conceptual structure diagram that appear to be relevant for the assessment of the integrity of instruction are represented by arrows. Integrity of instruction is a central qualitative category that is defining for the procedure within the 3A approach. Quality of instruction is seen as dependent on the quality of functional relationships between (1) educational content, (2) instructional aims and (3) activities of the teacher and students. The better these three basic determinants of instructional quality are integrated, the higher quality the teaching and learning situation shows. Interpretation of the individual components (levels and relationships) within the conceptual structure diagram may unveil potential deficiencies in the deep structure of the teaching and learning situation, which threaten its integrity. Such deficiencies are the starting point for the formulation of alterations and prepares ground for analytic generalization. The focus of the third step (Alteration) therefore lies in the assessing of the quality of the situations and suggesting alterations within them. We operationalize categories for the assessment of quality of teaching and learning situations; these categories serve as intermediate frameworks for developing theory-based indicators of instructional quality and its aspects such as: working on content, cognitive activation, constructive dealing with mistakes, supporting metacognition (learning to learn) etc. The quality of teaching and learning situations is assessed (as: failing / undeveloped / enabling / supportive) and alterations are suggested, reconsidered and discussed. The resultant didactic case study may be seen as a precedent. It is an original solution (of a certain quality) to a situations of the same type. The
37 precedent is then discussed within the professional community of teachers and replaced, when a better solution is found. Developing case studies and generalizing over cases Building on the above mentioned analyses, in the fourth step we aim to develop relevant theoretical generalizations and verify them with other cases. The number of cases depends on the theoretical saturation. Accumulated theoretical constructs are the results of analytical generalization from various cases and they serve well in developing the understanding of instructional quality through reflection of practice. The general context of our research is thus the methodology of case studies that we adopt for the purposes of research in didactics. We build on Yin’s conception of analytic generalizations (Yin, 2011, pp. 98-102). As Yin suggests, it is possible to formulate propositions – based on the qualitative analysis of individual cases – that contain a set of theoretical constructs or hypothetical statements about a case. In the second step, this theoretical basis is applied to similar cases and verified. „The goal is to pose the propositions and hypotheses at a conceptual level higher than that of the specific findings“ (Yin, 2011, p. 101). In this approach, local experience is used to construct generalized propositions that can be verified in practice, which leads to testable theoretical generalizations. We find it inspiring for research that aims to support reflective practice in teacher education. Identifying didactic formalisms In our previous research, generalization over cases enabled us to identify didactic formalisms as alienated cognition and concealed cognition (c.f. Slavík, Janík and Najvar, 2016). Alienated cognition is characterized by classroom situation where a teacher substitutes students’ cognitive activities with his/her own presentation of content and evaluation of learning. The cause of failure is usually (a) disproportion between the complexity of task and conditions for its solution (lack of time, non-adjustment to students’ capabilities etc.), or (b) insufficient analysis of content (teacher does not grasp relations among work with content and students previous experiences). In the classroom, this kind of situation is characterized by excessive verbal activity of a teacher while there is lack of meaningful communication among students and its contribution to getting to understand the content and its generalization is rather insufficient. Low level of students’ autonomy in assessment represents a typical side effect of alienated cognition – even evaluation is “taken away” or “alienated” from students and is only conducted by a teacher; students are not encouraged to realize the consequences of their own assessment for learning. The practices listed above are typical of the so-called traditional approach to teaching and learning.
38 Concealed cognition is characterized by teaching and learning situations where students’ cognitive processes in the course of their own activities are separated from the development of basic concept knowledge. As a result, students do not understand what they are doing and how it relates to learning; they can’t really appreciate the cognitive benefits of what is taught. These kinds of situations are typical of utilizing teaching methods, which might actually facilitate the interconnection between the development of competences and of content and aims. The reason for failure is predominantly teachers’ lack of analysis of content – as a consequence, students only handle the concepts as words to be memorized – “the content is emptying”. Lack of metacognition is typical feature of these situations – conscious relation of the content to one’s personal experience, cognitive process, critical arguing in subjectively relevant personal, social and cultural contexts is underestimated. Whereas students may enjoy such lessons, they are unlikely to actually learn something. Such practice is typical of the so-called modern, playful or entertaining approaches. Discussion and outlook In this paper, the problem of emptying the content of school education was addressed. Turn to the tradition of Bildung and to content-focused Bildung- centred didactic was proposed as an answer to this challenge. Methodological approaches developed on the basis of semantics of teaching were reviewed and one of them – 3A content-focued approach – was presented in detail. In the core of the 3A approach there lies an analysis of educational content in three specific levels (thematic level, concept level and competence level) which makes it possible to identify those aspects that endanger the quality of teaching and learning situations and that can be traced in many different school subjects. We believe that the presented approach contributes to the development of professional knowledge that is relevant to instructional practice. Ideally, professional knowledge would be shared and used effectively in general, i.e. without notable differences between different types of professionals (e.g. in medicine). This supports the call for teacher professionalization. Professional pedagogical knowledge should be “clinically grounded” in the facts from the educational reality (c.f. Shulman, 1996). That is, professional pedagogical knowledge should support teachers’ ability to see relevant aspects of teaching and learning and help them assess, explain and improve their own practices. Such “grounding” will make it possible to help develop the language of the teaching profession (Wipperfürth, 2015). Such specific language enables to reflect teachers’ and students’ activities and forms the basis for improving them. It is to be regularly developed and cultivated and should build on empirical evidence and facts in order to precisely explain and justify quality of instruction.
39 Acknowledgement: This paper is a result of the research funded by the Czech Science Foundation as the project GA15-05122S Between acceptance and resistance: Teachers’ perceptions of curricular changes 10 years into the reform implementation.
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41 PROMOTING MATHEMATICAL CREATIVITY IN HETEROGENEOUS CLASSES Esther S. Levenson
Abstract In this talk I focus on factors that may affect the promotion of mathematical creativity in heterogeneous classes. One factor is teachers’ beliefs regarding the relationship between mathematical creativity and excellence in mathematics. Another factor is the teacher’s actions during classroom activities. The lens through which we view and analyse classroom interactions also allows us to appreciate how diversity may contribute to creativity. Keywords: Mathematical creativity, teachers’ beliefs, heterogeneous classes
INTRODUCTION What does it mean for students to be mathematically creative? Can we observe mathematical creativity in mixed-abilities classrooms? How can we support mathematical creativity at the primary school level? These are some of the questions that this plenary talk will address. As the focus of this talk is on creativity among school children, it is concerned with everyday creativity, focusing on students’ “novel and personally meaningful interpretation of experiences, actions, and events” (Kaufman and Beghetto, 2009, p. 3). It is not concerned with the creativity of a few eminent persons who have made a significant and lasting contribution to society. Similarly, Liljedahl and Sriraman (2006) differentiated between professional and school-level mathematical creativity. Professional mathematical creativity relates to work that significantly extends a body of mathematical knowledge and opens up new directions for other mathematicians. School-level mathematical creativity includes unusual and/or insightful solutions to a given problem or viewing an old problem from a new angle, raising new questions and possibilities. One of the hallmarks of creativity in general, and mathematical creativity specifically, is divergent thinking. Divergent thinking is often measured in terms of the fluency, flexibility, and originality of ideas produced. Silver (1997) referred to fluency as “the number of ideas generated in response to a prompt” (p. 76). Flexibility, according to Silver (1997) refers to “apparent shifts in approaches taken when generating responses to a prompt” (p. 76). Leikin (2009) evaluated flexibility by assessing if different solutions employ strategies based on different representations (e.g., algebraic and graphical representations), properties, or branches of mathematics. Flexibility may also be thought of as the opposite of fixation. In problem solving, fixation is related to mental rigidity (Haylock, 1997).
Tel Aviv University and Kibbutzim College of Education, Israel; e-mail: [email protected]
42 Overcoming fixation and breaking away from stereotypes are signs of flexible thinking. Novelty and originality are also related to mathematical creativity. In the classroom, this aspect of creativity may manifest itself when a student examines many solutions to a problem, methods or answers, and then generates another that is different (Silver, 1997). In this case, a novel solution infers novelty to the student or to the classroom participants. Originality may also be measured by the level of insight or conventionality with respect to the learning history of the students (Leiken, 2009). At the primary level, most children learn in mixed-abilities classrooms. Yet, mathematical creativity is often associated with giftedness. Taking a look at the title of some books, such as Creativity, giftedness, and talent development in mathematics (Sriraman and Lfixaee, 2011) and Creativity in mathematics and the education of gifted students (Leikin, Berman and Koichu, 2009), as well as conferences such as the International mathematical creativity and giftedness conference, one may get the impression from the pairing of creativity and giftedness that mathematical creativity is not for everyone. Equity, however, is about access. It rests on “beliefs and practices that empower all students to participate meaningfully in learning mathematics and to achieve outcomes in mathematics that are not predicted by or correlated with student characteristics” (NCTM, 2014, p. 60). In this talk, I adopt the view that mathematical creativity is “an orientation or disposition toward mathematical activity that can be fostered broadly in the general school population” (Silver, 1997, p. 75). As such, it is up to educators and teachers to provide all students access to learning experiences that may occasion mathematical creativity. The rest of this talk will be divided into two parts. In the first part, I look at primary school teachers’ perspectives related to promoting mathematical creativity in heterogeneous classes. In the second part, I look at actual classroom episodes. TEACHERS’ PERSPECTIVES This section describes results from different interactions with teachers during professional development. The first section is based on a study conducted with teachers of grades 1-6 (Levenson, 2012). The second section is based on interactions with participants of a graduate course with teachers of grades k-8 (Levenson, 2013) Teachers as learners – solving a multiple solution task In an effort to enhance primary school teachers’ familiarity with multiple-solution tasks and the role these tasks may have in occasioning mathematical creativity, the following problem was posed to 17 teachers participating in a professional 2 development course: Solve :6 in as many ways as possible. 5
43 The teachers were requested to work alone without discussing their solutions amongst themselves. As the teachers worked quietly, the instructor walked around encouraging the teachers to continue thinking of additional ways to solve the problem. What follows are a few lines of comments by the instructor (I) and some of the teachers as they worked at the problem. Time Speaker Comments 05:14 I (Talking to one of the teachers who has stopped working) Do some more. 05:16 Tina I don’t have more than four [solutions]. 05:19 I Four? Four. Four. Wonderful. More. 05:45 I (to Yona) How many methods do you have? 06:19 Yona Two. (Yona laughs a little.) 06:21 I (to Yona) She (pointing to teacher Tina) has four [methods]. 07:54 Shanna Forget it. I’m stuck with two. 08:32 I (To Betty) How many did you write? 09:00 Betty Six. 09:02 Carol I also found six. 09:10 Gerry (Sounding surprised) Six? Six? 09:12 I She found six. 11:31 Sonia Ah! I have another way.
As seen from the above lines, encouragement may raise motivation – a positive factor. On the other hand, Yona’s laugh may indicate nervousness, a more negative factor. When discussing the activity after the work was handed in, several teachers expressed feeling frustrated. Shanna said, “it was as if I was stuck on one method and I couldn’t succeed and when I heard (the others calling out) six, seven, eight solutions, it made me feel even worse.” The teachers then discussed if and how they might implement a multiple-solution task in their classrooms. Tammy: Let’s say we implement it in class…There is great danger of (students feeling) great frustration. Yona: First, I would tell everyone to solve [the problem], and then for children who I know are advanced and can look for more…Not to everyone… because, if I tell a child who with difficulty finds one solution, to look for more, it would frustrate him.
44 Renee: You can say – whoever solves it one way, try to think of a second way, it’s not pressure. It’s if you finish, then you can do more. Tina: You have to challenge the students. In the above discussion, Tammy, who had listed four solution methods on her handout, raises her concern that students might become frustrated from this task. Picking up on this thought, Yona and Renee offer ways in which the task might be implemented and yet avoid feelings of frustrations. Yona claims that she would only tell the advanced students to look for additional solutions. Renee, who had previously revealed only positive feelings, agrees that students might not be able to deal with the pressure and that it is best avoided. Tina raises the issue of challenging the students. To summarize, teachers’ emotions, as well as other affective constructs, may influence teachers’ choice of tasks to be implemented in the classroom as well as how they choose to mediate the implementation of the task. As several teachers pointed out, it was not only the task which elicited these emotions but the instructor’s mediation of the task. Thus, an important implication from this study is the need to consider affect and how emotions experienced during professional development may influence teachers’ decisions related to implementing tasks that may occasion creativity in their classrooms. Promoting mathematical creativity – for all students? In the above study we saw how teachers reacted to engaging with a multiple- solution task and how this engagement might contribute to beliefs regarding which types of tasks they choose to implement in their classes. I now turn towards participants of a graduate course entitled Creativity in mathematics education, who were requested to find a task that has potential for occasioning mathematical creativity and to explain why they chose that task. In addition to mentioning task features and cognitive demands, several teachers raised the issue of accessibility. For example, one primary school teacher wrote about her choice, “…there isn’t a student who cannot participate in this activity, even special-needs students [can participate].’’ Another teacher wrote, ‘‘Every student can find his own unique solution method.” Another value mentioned by several teachers was encouraging individuality. One teacher chose a certain task because “every student can look at the given [data] in a different way and can create a different formula from the others.” Another explicitly related to open tasks and multiple-solution tasks, “The task contains an open question and is worked in such a way so that the student can give an answer that is his own.” Finally, one teacher focused on multiple representations, “The task promotes the use of … graphs, algebra, numbers, and words and does not
45 limit the solution to a specific media thus allowing students the possibility of expressing themselves in the area where they are strongest.” The above statements may hint at an implicit belief by teachers that all children should have access to tasks that can promote creativity and furthermore, that such tasks are beneficial, specifically when taking into account the diversity of the classroom. In an effort to investigate these beliefs explicitly, in the last two years, the following question was added to the same homework assignment described above: There are those who say that mathematical creativity is related to excellence in mathematics. What is your opinion? Forty-one participants responded to this question, 21 who had chosen a task for grades K-5, 20 who chose a task for grades 6-8. First analysis showed that 83% of all participants claimed that there is a relationship, 12% said there was no relationship, and the rest (2 participants) said it depended on other factors. At first glance, these results seem at odds with the results above which pointed to teachers’ beliefs in providing tasks that are accessible to all students and that may occasion mathematical creativity for all. A second, finer analysis, proved otherwise. Two general trends revealed themselves. Some teachers claimed that in order to be creative, one needs excellent mathematics knowledge. For example, a sixth grade teacher wrote, “A student who is excellent in mathematics has the tools and knowledge that will help him think of connections between familiar topics and a new problem and be creative.” Other teachers, however, argued that it is the promotion of mathematical creativity which leads to excellence in mathematics. A second grade teacher wrote, “When you sit with a student and discuss different solutions to the same problem, it helps him develop different strategies and to excel in mathematics.” The mixed beliefs have great implications for teaching. If a teacher believes that creativity can promote excellence in mathematics, it is likely that he or she will attempt to occasion creativity amongst all students. If, however, a teacher believes that mathematical excellence is a prerequisite for mathematical creativity, then that teacher might only engage excellent students with tasks that may occasion mathematical creativity. This research is still in its early stages. After discussing teachers’ professed beliefs, we now turn to the classroom. OCCASIONING CREATIVITY IN THE CLASSROOM This section describes two classroom episodes from different studies. The first takes place in a classroom where the explicit aim of the lesson was to review multiplication of decimal numbers (Levenson, 2014). The second takes place in two classrooms, where the explicit aim was to promote mathematical creativity (Swisa, 2104). Different theories are used to examine the classroom interactions. The first study emphasizes how creativity may arise in a diverse group of participants. The second study emphasizes the roles of the teacher in promoting mathematical creativity.
46 Viewing the classroom through the lens of complexity theory A Complex Adaptive System is a system “in which many players are all adapting to each other and where the emerging future is very hard to predict" (Axelrod and Cohen, 1999, p. xi). The classroom may be viewed as a complex system because the components of the system—the teacher and the individual students—may have different goals and driving forces, yet each individual is highly connected with the other; the decisions and actions of one may affect the decisions and actions of others (Hurford, 2010). In addition, the knowledge and insights that are shared, and the creativity that emerges, can hardly be predicted at the onset of the lesson. Davis and Simmt (2003), suggest five features or conditions that must be met in order for systems to arise and maintain their ability to adapt and learn: (1) internal diversity, (2) redundancy, (3) decentralized control, (4) organized randomness, and (5) neighbour interactions. Taking into consideration the theme of the conference, “Equity and diversity in elementary mathematics education,” I focus on the first two features. Internal diversity refers to the different ways members of the community contribute to finding solutions to a given problem. On the other hand, in order to communicate, members must share some similarities such as background, language, and purpose. Redundancy refers to the similarities that enable the system to cope with stress and allow for different members to compensate for others’ failings. The following episode was taken from a sixth grade class consisting of 28 students, 16 girls and 12 boys. The teacher had 14 years experience teaching elementary school mathematics. At the time of the study, the class had already been introduced to multiplication of fractions, and had practiced the procedure during previous lessons. The teacher put the following problem on the board, __ × __ = 0.18, and asked the class, “What could the missing numbers possibly be?” Many children raised their hands and the teacher commented, “There are many possibilities.” She then called on one at a time: Gil: 0.9 times 0.2. Teacher: Another way. There are many ways. Lolly: 0.6 times 0.3. Teacher: More. Tammy: 0.90 times 0.20. Teacher: Would you agree with me that 0.2 and 0.9 is the same [as 0.90 and 0.20]? I want different. Miri: I’m not sure. 9 times 0.02. Teacher: Nice. Can someone explain what she did?
47 Note that although Gil and Lolly gave different answers, both may be considered similar in that they consisted of two numbers with one digit after the decimal point. Tammy broke the mould by using numbers with two digits after the decimal point. From a mathematical point of view, as noted by the teacher, Tammy’s answer is the same as Gil’s. On the other hand, from a student’s point of view, 0.9 may be very different from 0.90. In addition, as will be shown later on, different representation of the same number may afford different possibilities and thus representing 0.9 as 0.90 may not only be acceptable but even preferable. Tom: What about 0.18 times 0.1? Tad: No. Teacher: Why not? Mark: [The answer would be] 0.018 because there would be three digits after the decimal point. Teacher: Ah. Ok. Thank you. We want a number (a solution) with two digits after the decimal point. Gad: 0.18 times 1. Ben: And 1 times 0.18. Teacher: You’re using the commutative property of multiplication. But, it’s really the same as Gad’s answer. Toby: 18 times 0.1? Many students: 18 times 0.01. All together, the class produced five different correct solutions, which may be considered the fluency for the class. Regarding flexibility, the second solution, 0.6×0.3 followed more or less the same strategy as the first solution 0.9×0.2. The last three solutions, 9×0.02, 0.18×1, and 18×0.01, differ from the first two solutions, but may be considered similar to each other. Each example consists of one factor that is a whole number and a second factor that is a decimal fraction with two digits after the decimal point. In addition, there was one attempt to find a solution that included a factor with one digit after the decimal point and a second factor with two digits after the decimal point. Viewing the episode through the lens of complexity theory, what stands out is the balance between internal diversity and redundancy that allows the students to not only raise different suggestions but to evaluate each other’s suggestions as well. Diversity was specifically promoted by the teacher who encouraged the students to find different solutions. Diversity may also be found in the various solutions, acceptable and unacceptable, that arose during the episode. Diversity may also be seen in the different ways in which students contributed. Some students hesitantly offered solutions implicitly seeking confirmation. Others boldly stated their solution. Still others took the role of evaluators. Yet, the students were also able
48 to compensate for each other’s deficiencies. There were three instances where one student put forth an incorrect or unacceptable solution and others, building on the idea, corrected the situation. When the teacher does not accept Tammy’s solution of 0.90 times 0.20, Miri is able to build on Tammy’s idea of changing the place of the decimal point and come up with an acceptable solution. When Tom changes both the factors and the place of the decimal point, he comes up with an incorrect solution. Gad, using the same factors, comes up with a correct solution. When Toby suggests the incorrect solution of 18 times 0.01, many students chime in with the correct solution without the teacher intervening. In other words, the dynamic interactions among participants allow the system to right itself. This analysis, viewing the classroom through the lens of complexity theory, allows us to see not only that mathematical creativity can be for all, but that diversity can actually contribute to the promotion of mathematical creativity in heterogeneous classrooms. Promoting creativity with and without digital technology In this last part of the talk, I present results of a recent study (Swisa, 2014) which compared students’ mathematical creativity in two different classroom environments. Three geometry tasks were given to two fifth-grade classes learning with the same teacher in the same school. In this section, I focus on the second task. In one class, students were given grid paper and pencil, and in the second class, students worked with a computer applet which showed a grid on the screen and included a drawing tool that allowed students to connect dots with straight lines or to drag solid yellow squares to fit within the gridlines on the screen. Students were told to create (with pencil or with the applet) as many different polygons as they could with an area of 15 square units. In both classes, students worked individually, either with their paper or on their own screen. Each student received a score for fluency, flexibility, and originality. Solutions were categorized according to different methods for solving the task. A student’s fluency score was the number of correct solutions that student found. The flexibility score reflected the number of different methods utilized by the student, and the originality score reflected the number of original solutions that student had, compared to other students in that class. Results indicated that fluency, flexibility, and originality scores were higher in the class where students worked with paper and pencil. In order to gain some insight into these results, the teacher’s actions in each class were analysed based on Drijvers’ (2012) theory of instrumental orchestration and Levenson’s (2011) theory of collective mathematical creativity. Although it was the same teacher in each class, several important differences were found. First, in the class with technology, after the initial introduction to the task, the teacher spent 26 minutes walking around and helping individual students, answering questions and offering helpful hints. In the class without technology, the teacher
49 also walked around and offered individual help. However, from time to time she would call out how many polygons one or another student had found, saying, for example, “Who already found 11 solutions?” In the technology class, during the last seven minutes of the class, the teacher projected on the board different solutions which students had found. In the non-technology class, demonstrating solutions occurred during the lesson and at the end of the lesson. After letting students work for a few minutes, the teacher paused to show an example of one student’s solution saying, “Look here a minute. Here is shape that looks like a dog or a duck, but it’s a polygon.” After another few minutes, she did this again, and again let them continue working. Finally, at the end of the lesson in the non- technological class, the teacher not only requested from different students to display their solutions on the board, but held a discussion with students comparing the difference between solutions. It seems that it was not necessarily the difference in environments that caused the differences in fluency, flexibility, and originality, but the differences in the teacher’s actions in each environment. CONCLUSION Three questions were presented at the beginning of this paper: What does it mean for students to be mathematically creative? Can we observe mathematical creativity in mixed-abilities classrooms? How can we support mathematical creativity at the primary school level? The first question was addressed by referring to theories (e.g., Silver, 1997). The second question was answered by bringing examples from studies (e.g., Levenson, 2014). I now address the third question and tie it into the theme of this conference. The onus for supporting mathematical creativity in the classroom falls first and foremost on the classroom teacher. Many mathematics curricula today provide tasks, activities, and guidance that support this aim. However, it is the teacher who implements the curricula, and it is the teacher’s beliefs, along with his or her mathematics knowledge, that can influence the way a curriculum is implemented. One way to support teachers is through professional development. A first step might be to educate teachers regarding the possibilities for occasioning mathematical creativity in the classroom. This could include presenting theories to teachers that relate to everyday creativity, and having them engage in classroom tasks that have the potential to occasion mathematical creativity. In addition to discussing the cognitive affordances and constraints of each task, it is important to also discuss affective affordances and constraints. Competition can be felt as pressure, as the teachers expressed during the professional development course, or as motivation, as seemed to occur among the students who worked in the non-technological classroom. This, as we saw, can lead to differences in how teachers implement tasks. Do we ensure that all students have access to developing mathematical creativity, or only those who are excellent in mathematics? Helping additional teachers see, through the eyes of complexity
50 theory, that diversity may actually be an affordance when it comes to occasioning mathematical creativity, and not a hindrance, may lead additional teachers to believe that providing access to mathematical creativity for all students, may support excellent mathematical learning for all. References Axelrod, R., Cohen, M. D. (1999). Harnessing complexity: Organizational implications of a scientific frontier. New York, NY: Free Press. Davis, B., Simmt, E. (2003). Understanding learning systems: Mathematics Education and Complexity Science. Journal for Research in Mathematics Education, 34, 137–167. Drijvers, P. (2012). Teachers transforming resources into orchestrations. In G. Gueudet, B. Pepin, L. Trouche (Eds.), From text to ‘lived’ resources: Mathematics curriculum materials and teacher development (pp. 265–281). New York/Berlin: Springer. Haylock, D. (1997). Recognizing mathematical creativity in schoolchildren. ZDM Mathematics Education, 27, 68–74. Kaufman, J., Beghetto, R. (2009). Beyond big and little: The four C model of creativity, Review of General Psychology, 13, 1–12. Leikin, R. (2009). Exploring mathematical creativity using multiple solution tasks. In R. Leikin, A. Berman, B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129–135). Rotterdam: Sense Publishers. Leikin, R., Berman, A., Koichu, B. (Eds.). (2009). Creativity in mathematics and the education of gifted students. Rotterdam: Sense Publishers. Liljedahl, P., Sriraman, B. (2006). Musings on mathematical creativity. For the Learning of Mathematics, 26, 17–19. Levenson, E. (2011). Exploring collective mathematical creativity in elementary school. The Journal of Creative Behavior, 45, 215–234. Levenson, E. (2012). Affective issues associated with multiple-solution tasks: Elementary school teachers speak out. ICME12, Korea. Retrieved April 23, 2013 from http://icme12.org/upload/UpFile2/TSG/0140.pdf Levenson, E. (2013). Tasks that may occasion mathematical creativity: Teachers’ choices. Journal of Mathematics Teacher Education, 16(4), 269–291. Levenson, E. (2014). Investigating mathematical creativity in elementary school through the lens of complexity theory. In D. Ambrose, B. Sriraman, K. M. Pierce (Eds.), A Critique of Creativity and Complexity- Deconstructing Clichés (pp. 35–52). Rotterdam: Sense Publishers. National Council of Teachers of Mathematics (2014). Principles to action: Ensuring mathematical success for all. Reston, Virginia: NCTM. Silver, E. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM Mathematics Education, 3, 75–80. Sriraman, B. & Lee, K. H. (Eds.). (2011). The elements of creativity and giftedness in mathematics. Rotterdam: Sense Publishers.
51 Swisa, R. (2014). Mathematical creativity in both a technological and non- technological environment: Students’ work and a teacher’s actions. (Unpublished master’s thesis). Tel Aviv University: Israel.
52
R ESEARCH R EPORTS
TWO PERSPECTIVES ON DIVERSITY BASED ON THE PEDAGOGICAL CONSULTANT’S WORK ON PROBLEM-SOLVING IN A TEACHING CONTEXT Lily Bacon,Nadine Bednarz, Caroline Lajoie, Jean-François Maheux and Mireille Saboya
Abstract Our presentation addresses diversity, one of the themes of SEMT 2017, through the lens of the pedagogical consultant (PC) profession. This diversity is considered, on one hand, from the angle of the utilisation of in-class problem-solving and what it requires from teachers. On the other hand, we also look at it from the angle of PCs, these upstream actors that accompany teachers in the development of their practice in relation to problem-solving. Keywords: Collaborative research, pedagogical consultant, problem solving, reflective practice, classroom context
Introduction Problem-solving (PS) is a key issue of mathematics teaching throughout the world as evidenced by the important part it plays in curricula (see ZDM thematic issue, 2007). Many studies have contributed to informing this notion of problem, its characteristics in relation to different intentions, its potential, as well as the complexity of in-class problem management (Brousseau, 1983; Douady, 1987;
Université du Québec à Montréal, Canada; e-mail: [email protected]
53 Schoenfeld, 2007; Coppé and Houdement, 2009; Lajoie and Bednarz, 2012, 2014, 2015; Arsac et al., 1988; Charnay, 1992-93; Tanner and Jones, 1994; Grenier and Payan, 2003; Adjiage and Rauscher, 2013; Oval-Soto and Oliveira, 2012). In this context, pedagogical consultants (PCs) are as key actors as they are called upon to assist teachers in situ in relation to in-class PS. A collaborative research project conducted with 8 PCs from different school boards and working with elementary level teachers has sought to better understand the more specific issues they face with regard to PS in a teaching context, and the ways to tackle them. In this presentation, we focus on issues regarding both the diversity faced by the teacher in a classroom teaching situation and the diversity characterising the PC’s teacher assistance work. Before discussing some of the elements emerging from the analysis, we will come back to the role of these PCs and the reasons that lead us to investigate these issues. The PC’s role within the Quebec education system PCs have been part of the Quebec education system for over 30 years. They work with teachers as “expert advisers” in pedagogy and intervention (Héon, 2004): they provide information, support and training to teachers. They also collaborate in implementing educational programs and policies, and are therefore considered by school managers and teachers as “resources” for their implementation or for innovation development (Houle and Pratte, 2003). The PC is therefore one of the actors influencing the teachers’ work and, by extension, what is being implemented by the latter with students in the classroom. A number of studies on the PC profession mention the importance of developing a professional credibility with teachers (Little, 1985; Kent, 1985; Nunes, 2011), highlighting the sensitive nature of the relation with ministerial authority in this relationship (Draelants, 2007; Lessard, 2008;Duchesne, 2013). Between teachers who sometimes “resist” the institution’s directives and an institutional agenda carried by the PCs, the relation to authority is a source of tensions (Duchesne, 2013), leading PCs to distance themselves from ministerial prescription, a distance they need to base their action with teachers (Lessard, 2008). A tensioned identity appears crucial to understand the difficulty of working with teachers (Draelants, 2007): as they themselves come from the teaching profession, and generally recruited on the basis of their experience of innovative practices, PCs are, according to Draelants (2007), an “elite segment” of the teaching profession, while defining themselves as teachers serving other teachers. Thus, PCs seek to assert themselves by building on both dimensions of their identity, and seeking the right balance between the peer and the expert. This analysis, and others, underlines the complexity and subtlety of working with teachers and the skills it requires: use of a common language, use of relevant in-class situations, reciprocity with special attention given to teachers’ knowledge and experience, without imposing what the work should be (Little, 1985; Nunes, 2011).
54 Exploration of problem-solving issues in a teaching context The present collaborative research project (Desgagné, 1997; Bednarz, 2013) was organised around pedagogical consultants assigned with a mathematics task in elementary education as well as researchers in didactics of mathematics to explore the issues related to problem-solving in a teaching context. For PCs, this subject poses a real challenge as we have been able to observe in national meetings. The difficulties experienced by teachers in relation to the utilisation of problems in the classroom, and their evaluation, impacted on the requests they submitted to PCs, who were not always in a position to respond, or who had questions about them. Of course, PCs have developed ways to answer these requests, but they felt the need to distance themselves from these spontaneous practices, and to refine their understanding of the issues and questions in order to support their interventions. Thus, the need to clarify what problem-solving entails in a teaching context, to identify the related issues and potential approaches, represented a significant challenge for PCs. Incidentally, their questioning concurred with our concerns as researchers. Indeed, we had developed an interest in these issues following an historical analysis of Quebec’s official documents from 1900 to this day, an analysis that allowed measuring the extent of this challenge: by highlighting the increasingly ambitious nature of the PS-related functions (Lajoie and Bednarz, 2012, 2015) and the almost nonexistent information provided to teachers to address problem-solving (Lajoie and Bednarz, 2014). These two considerations have confirmed the importance of furthering research by combining researchers’ and PCs’ knowledge and experience with a view to clarifying the issues being raised in relation to this subject. The following questions guided our investigation: What characterises the issues faced by pedagogical consultants in relation to problem-solving in a teaching context? What is their significance in the PCs’ work context? What characterises the methods that were developed to take these issues into account? What insight can this provide us on their work, in return? The analysis we present is part of this more comprehensive research project. It focuses on discussions that took place between PCs and researchers as part of the group’s reflective sharing sessions. The participants were actively involved in various explorative and analytical tasks considered as a means to trigger discussion and clarification of questions and points of view. A first aspect emerging from the analysis: Diversity related to the work of the PC who reflects upon PS in a teaching and teacher assistance context In the period used for analysis purposes, the different participants ̶ PCs and researchers ̶ were grouped into teams of 3 or 4 persons. Each team addressed the “Giant’s foot” problem (see statement below) without numerical data, and tried to find a) what it allowed to work on, b) how the problem could be worked out in class, and c) as PCs, how it would be possible for them to use it with teachers.
55 The discussed “Giant’s foot” problem is presented in the form of a picture with the following text: This picture was shot in an amusement park in England. It shows part of a giant’s leg. What is the giant’s height?1 In the plenary session that followed, the teams shared their exploration of the problem2: R1: … I realise that we did not begin with “what it allowed to work on”. We really got into it [thinking that] we would resolve it. […] we [in our team] all worked with the idea of ratios-proportions, but not the same objects were put in ratios and proportions. That was the difference. This team’s participants have all undertaken the reflection and exploration of the “Giant’s foot” problem by first positioning themselves as problem solvers. Each one has resolved the problem on his own and the sharing of solutions brought them to identify the reasoning they used: some of them set a ratio between the length of the man’s foot and the length of the giant’s foot; this allowed them to consider the ratio of the man’s height with that of the giant by keeping the same proportion. Others have rather set a ratio between the lengths of the man’s foot over its height, and then considered the length of the giant’s foot and his height in the same proportion. Sharing the solutions brought up questions on concepts and processes used to resolve the problem. The participants then considered the problem from a mathematical point of view: R1: We realised that we were in an estimation process, [and] a question quickly arose because our two processes produced a variance [in the answers obtained], so we said since we are in estimation, yes a variance is possible, but what is the significant variance that will tell us that our reasoning is inappropriate or that [the variance] can be acceptable. PC3: the variance also lead us to go back and review the data we had used […] when we measure something, where do we start from, to what extent do we go, from what angle did you measure it […] also to see the incidence of our initial measuring yardstick, assuming we transfer it to obtain the giant’s height, which is seven times our own, finally the small difference between the first unit we had, i.e. seven times, it creates a significant variance that we had to […] take into account. On one hand, the discussion addressed the idea of the meaning of estimation, which entails a measure that presents a certain variance with an accurate measure.
1Rauscher Jean-Claude and Adjiage Robert. (2012). Espaces de travail et résolution d’un problème de modélisation. Proceedings of the symposium Espace de Travail Mathématique 3. Montréal, 24, 25, 26 October 2012. 2The pedagogical consultants are identified PC1, PC2, etc., and researchers are R1, R2, etc.
56 On the other hand, it also focused on different elements referring to the measuring process such as the end points and the point of view (perspective, angle) considered since the exercise was done from a picture, the selected yardstick and its impact on the result. This idea of variance was also considered from the students’ perspective, as a source of discussion with them to validate the reasoning and processes used. Thus, the participants adopted a didactic point of view centered on learning and potential developments provided by the problem: PC5: […] you arrived at different numbers and then you questioned yourselves on this, right? R1: Well, that difference did not bother us, we wondered to what extent / PC5: we tolerate it R1: The difference is acceptable. What tells us that our estimation was right? […] Perhaps we didn’t use a proper reasoning or a good process because the difference is too big or something like that. We did ask ourselves questions about that […] we said “this might be an interesting discussion topic in class”. Another team said they addressed the problem in a similar way and noted a potential discussion topic with students. They also sought to see how such discussion could be conducted in class: PC1: We had a similar discussion on what do we do with a student or a group of students who would address the problem from a wrong angle, you know, they see it as additive: the height is that much more than […] rather than that many times more than. Then we came out with the idea that we could present all the different answers and have the group explain what is acceptable and what is not. It somehow goes back to the discussion you had on accuracy. When the participants dwelled on the utilisation of the problem with students, they adopted a new perspective which is that of the teacher in class. For the participants, steering the resolution of a problem with a group of students is fairly complex: PC3: we pursued our reflection this way… Now, one of the difficulties is planning, but R3 said “it’s also the capacity to propose problems in the context and on the spot”, but we noted that it took fairly high aptitudes to manage this type of situation, to catch the ball on the fly with an answer that would be wrong, […] to question the students in order to readjust their reasoning or to see where they made the error, and not going in all directions, and then [as teachers] being taken by surprise. PC3: R1 [talked about] the variance between our two answers [and] we asked “how to explain this variance?” […] Ok, we took the length of the man’s foot over the length of the “giant’s foot”, and at the same time PC4 was writing the mathematical equation that […] illustrated […] how to solve the problem using the two different ways, but I’m not sure that all teachers would have the skill to do this, to go over a
57 student’s reasoning, formulate it in a mathematical equation form, to isolate a variable, and then say “Ah! This is what explains the error!” The participants’ reflection on the problem at hand then changed its focus. The discussions shifted towards the perspective of assisting teachers with the challenges posed by PS in a teaching context. The “Giant’s foot problem” was then approached in terms of what it allows to develop in teachers: R1: … when we clarified our position “Ok, we are PCs and we want to work that out with teachers”, PC4 said […] “I would proceed to bring teachers to accept fuzziness management” […] there was that idea to use this type of problem with teachers to bring them to develop the capacity to manage fuzzes […] we found that idea of fuzziness management interesting. Finally, other PCs mentioned the discomfort felt by teachers towards the utilisation of such problems without numerical data in class. Indeed, they consider them quite different from the problems usually worked on, and which are found in the teaching material and also used in provincial evaluations. Anticipating from the outset the teachers’ reluctance, these PCs will propose to add a numerical datum to the “Giant’s foot” problem (e.g.: the measurement of the fence height) in order to provide a more familiar point of entry into the problem. The preferred perspective is then that of the problem designer: the focus is on the problem as it is designed, on its characteristics and the possibilities to modify it, to change it with a view to making it more “sellable” to teachers, for instance. Findings from this analysis: In the course of his practice, the PC conducts his/her reflection by adopting several different points of entry or perspectives on a problem: problem-solver, teacher, pedagogical consultant, problem designer, mathematical and didactical points of view. His/her activity is thus organised in an initial movement of deployment around a diversity of perspectives on a same mathematical problem. It is characterised by a second movement of tensioning, of negotiation between the different issues emerging from these entry points. For example, a potential discussion with students and the complexity of its management by the teacher; or between a problem that is difficult for teachers to manage and the potential of such problem in the assistance work with teachers. This clearly highlights the complexity of the PC’s work due to the diverse perspectives to be used and coordinated as well as to their interwoven and dynamic nature. To be able to do this, PCs tap into three types of knowledge (Lessard, 2008). First we have “field knowledge”, which notably from their experience as teachers, but also from in situ observations. We see this for example when they refer to the difficulty for the teacher to manage this discussion, or to react to an error, or again the discomfort felt towards such a problem that has no numerical data, a problem that they do not know exactly how to resolve in class. Field experience provides a set of practical data that they must consider, what Lessard (2008) calls “field intelligence”; Second, comes the (2) “theoretical knowledge” underlying their
58 interventions and argumentations (highlighted reasoning processes, concepts and processes at play, analysis in terms of the data they rely on and that may explain the variance, etc.); Finally, we have (3) “counselling knowledge”, which can be relational and/or adult training knowledge (for example, in the way they consider modifying the problem to involve teachers in the situation). A second aspect emerging from the analysis: diversity related to the utilisation of the problem in class As discussed above, utilising PS in a mathematics teaching context represents a significant challenge for teachers at the planning stage, but also and more importantly when it comes to manage problem-solving with students. R2 asked: “Are teachers interested in doing problem-solving or are they not?” PCs mentioned that the management complexity required for PS in class makes teachers reluctant to utilise problems with their students. As PC3 said above, “We noted that it took fairly high aptitudes to manage this type of situation”. The studies conducted in Quebec on PS in a teaching context confirm the difficulties posed by PS management in class: about managing the validation process (Barry, 2009; Saboya, 2010), taking solutions into account (Oliveira, 2008), including erroneous ones, implementing a research culture (Barry, 2009) and using mathematical complex problems (Maheux, 2007). In the analysed episode, still regarding the “Giant’s foot” problem, a PC/researcher team noted that this type of activity lead with students required from teachers what they call “fuzziness management”: R1: And then we said, there are two potential fuzzes. One concerning … We’re not faced with an estimation problem [explicitly, but] but we can work it out in that sense. So we will get different answers and this leads to good discussions as we just mentioned. So there is this fuzziness in terms of … of answers, but also fuzziness in terms of different entry points to the problem, different strategies and so we said “both aren’t easy to manage”. You need to be comfortable with that, you have to be able to tackle that. Thus, part of the complexity of this type of management is due to the fact that the diversity of the students’ answers to the problem can be related to the estimation process. And also to the diversity of entry points to the problem and the strategies used by the students, for example. There is also a diversity of mathematical notions that can be utilised, whether they are identified from the outset or emerge during the problem-solving process, or during the discussion with the students. Therefore, this “fuzziness management” appears as an important in-class PS issue for PCs with regard to the question of diversity: taking into account the diversity of answers, strategies, different entry points to the problem to further student’s learning, to accept uncertainty in the discussion and what may emerge from it. “Fuzziness management” represents part of the complexity of the teacher’s work in class and will therefore become for the PC a matter of teacher assistance. An
59 issue all the more important because the official Quebec documents give teachers very limited, not to say almost nonexistent information to address this type of problem-solving (Lajoie and Bednarz, 2012, 2014, 2015). These analyses of the PC’s work and the official documents underline the importance of furthering research by sharing researchers and practitioners’ perspectives to clarify and take into account the issues raised by the utilisation of PS in a teaching context, and those related to the teacher assistance provided by PCs. Conclusion Diversity as it is seen here through the lens of the pedagogical consultant’s explorations around problem-solving puts into light adopting various points of view when thinking about people involved in one’s work. Although this might be tempting, we think these perspectives cannot be simply reduced to that of a student, a teacher, a mathematician or a researcher for example. Students and teachers can both be, at time, problem solvers, problem composers, and even in some way share the pedagogical and the didactical configuration of the classroom (e.g. Roth and Radford, 2011). Navigating the fuzziness of thinking about classroom problem-solving in a PC context also highlight the potentially conflicting nature of these perspectives: something the image of different roles or positions does not necessarily captures best. To further our joint inquiries into the issues faced by pedagogical consultants in relation to problem-solving in a teaching context, one way to think about this diversity we would like to explore in the future draws on Mikhaïl Bakhtin’s (1993) notion of polyphony. For Bakhtin, attenting to the diversity found in the multivoicedness of speech and thought is fundamental to understanding people’s actual lives and experiences. His work mostly focuses on how utterances create signification (as opposed to “meaning”) by responding to and being responded to by other utterances. While explaining this, Mayen (1999) also refers to Bakhtin’s notion of polyphony as a “potential vector of development” (p.75). In the context of a collaborative research such as ours, we thus readily envision two ways in which polyphony could be a powerful concept to explore: 1) to examine how different voices populate (and even saturate) PC’s work around PS; and 2) to observe how the research context itself allowed these voices to be heard, respond to one another, and thus contribute to the collaborative saturation of signification around PS. This might offer a powerful way to deepen our conceptualisation of the complexity of PC’s work, and support it. References Adjiage, R., Rauscher, J.C. (2013). Résolution d’un problème de modélisation et pratique écrite de l’écrit. Recherches en didactique des mathématiques, 33(1), 9–43. Arsac, G., Germain, G., Mante, M. (1988). Problème ouvert et situation-problème. Lyon: IREM de Lyon. Bakhtin, M. M. (1993). Problems of Dostoevsky's poetics. U of Minnesota Press.
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62 ASSESSMENT ISSUES THAT TROUBLE MATHEMATICS TEACHERS
Yaniv Biton, Sara Hershkovitz, Maureen Hoch, Bilha Ben-David and Osnat Fellus Abstract This research investigated issues in assessment that concern mathematics teachers. The research comprised four phases: identification of assessment-related issues; categorisation of these issues into themes—assessment validity, assessment reliability, and pedagogical knowledge; construction of a questionnaire; and dissemination of the questionnaire to teachers. Eighty-four teachers answered the questionnaire. Findings suggest that teachers are dissatisfied with the traditional method of assessment that boils down to a numerical grade based on exam results and that they need more support in alternative assessment methods. Keywords: Mathematics teaching, assessment
INTRODUCTION Assessment is one of the important issues in teaching that has received much attention in recent years, and especially in the last decade. The aim of this present research is to explore—and clarify—assessment issues that concern and trouble mathematics teachers. Assessment in mathematics plays a part inseparable from the teaching and learning of the subject. Teachers, who are required to assess their students in a valid and reliable manner, while at the same time to encourage them to continue learning, face many and various difficulties in this endeavor. In order to better understand how practices of assessment are experienced by teachers, we consider two types of assessment: traditional assessment and alternative assessment. The prevalent distinction in literature is “summative” assessment (of learning) and “formative” assessment (for learning) (Black and Wiliam, 2006). In this present context, traditional assessment references practices that are based mainly on exams whose aim is to quantify the level of a student’s knowledge in the learned subject. Much research points to the need for a change in the method of traditional assessment (Simpson, 2004) and suggests moving to a method of assessment that is multi-dimensional, integrated into the curriculum, reliable, and flexible, as well as one that develops learners’ problem solving skills (Earl, 2012). These alternative methods of assessment, combined with the traditional exam, enable teachers to expand their teaching beyond the focus on learning for an exam and to get to know their students in a personal manner. Many researchers who have worked in the field of alternative assessment claim that these methods— including, for example, self-assessment, reflecting on the learning process, and involving the learner in decisions about assessment categories—can lead to students taking more responsibility for their learning (Bedford and Legg, 2007;
Tel Aviv University, Israel; e-mail: [email protected]
63 Birenbaum et al., 2006; Black and Wiliam, 2006; Bolte, 1999; Dori, 2003; Sullivan, 1997; Tal, Dori, Lazarowitz, 2000). Sullivan (1997) emphasizes the need for a change in traditional assessment, and the need for a transition to methods of assessment that can help improve the learning of mathematics. Such methods allow students to not only know why and how they are assessed, but also to take charge of their own learning process. Alternative assessment could increase students’ responsibility while supplying them with immediate feedback. According to the Assessment Reform Group (2002), assessment for learning is part of effective planning; focuses on how students learn; is central to classroom practice; is a key professional skill; is sensitive and constructive; fosters motivation; promotes understanding of goals and criteria; helps learners to know how to improve; develops the capacity for self-assessment; and recognises all educational achievement—all goals that all students deserve to experience or achieve. The question we raised was: How do teachers view the subject of assessment and what is liable to prevent teachers from including alternative assessment in their teaching practices? Watt (2006) investigated the viewpoints of approximately 60 Australian mathematics teachers on applying alternative assessment and found that the teachers were satisfied with traditional exams, specifically due to their associated validity for measuring students’ achievements. Watt identified the following five central categories that emerged from the teachers’ responses and that expressed their opposition to integrating alternative assessment tasks: not enough time for implementation; difficult to implement; not suitable for mathematics teaching; unreliable; and lack of resources due to expense. In regard to the association between such preferences and number of years on the job, Watt as well as Velduis and van den Heuvel-Panhuizen (2014) found that experienced teachers tended to prefer traditional assessment over alternative assessment and that they were less likely to use these methods in comparison to novice teachers. In conclusion, the research literature encompasses many issues in alternative assessment emphasizing the positive influence of alternative assessment tasks on the learning of mathematics in general and on students’ achievements in particular. In light of these findings, and despite the lack of consensus on the extent of the influence (Briggs et al., 2012; Kingston, Nash, 2011, 2012), the National Council of Teachers of Mathematics has recently decided to encourage extensive use of alternative assessment in mathematics (NCTM, 2013). RESEARCH QUESTIONS As mentioned, assessment plays an inseparable part in the teaching and learning of mathematics, and teachers are required, and in fact want, to assess their students in an encouraging, valid, and reliable manner. In the process of this assessment, they meet with various difficulties. The aim of this research is to investigate what these difficulties are and also to what areas of assessment they are connected. An
64 additional aim was to also check whether the teachers’ difficulties were significant issues or technical issues—dubbed “root problems” and “leaf problems” as will be explained below. Clearly, every teacher is likely to be troubled by a different issue depending on various factors such as the age of the students, the level of the teachers’ training, their working conditions, and their teaching experience. We sought to identify assessment issues which concern most teachers to the greatest extent. The research questions are: 1. What assessment-related issues concern mathematics teachers? 2. Among these issues, are there some that are particularly troublesome to many teachers? If so, what are they? METHODOLOGY This research was undertaken in four main phases: face-to face meetings with teachers and prospective teachers to identify troublesome issues in assessment; categorisation of these issues; construction of a questionnaire for teachers; and dissemination of the questionnaire. In the first phase of the research, 113 mathematics teachers participated: 43 in-service teachers (16 primary, 27 secondary) and 70 prospective teachers (23 primary, 47 secondary). Assessment- related statements were collected during three in-service professional development sessions and two courses in a mathematics education program. Each participant was asked to write on a sticky note one issue that troubles them about assessing students, and that they felt was important to find a solution for. Thereafter the participants were divided into groups of 4-6 participants and asked to classify each statement as either a “root problem” —defined as a significant assessment problem requiring external intervention or a “leaf problem” —defined as a technical problem not requiring external intervention. The teachers stuck their statements onto a picture of a tree (see Figure 1 for an example of how the teachers mapped the statements in one of the groups).
Figure 1: Mapping of root problems and leaf problems
65 The classification of the issues into root problems and leaf problems had two aims. One was to provide the teachers with tools to identify where they could implement alternative assessments. The second was to garner better understanding of the types of issues occurring in the field. In the second phase of the research, the teachers’ issues were initially classified by the researchers into 12 categories, which later yielded three main categories as follows. Assessment validity o What is being assessed o Meaning of grade o Differences between students o Influence of teacher’s content knowledge on grade o Student’s partial understanding o Students’ affective influence on their performance in mathematics Assessment reliability o Inconsistency of marking scheme o Inconsistency of assigning grades o Subjective assessment Pedagogical knowledge o Lack of encompassing assessment tools o Habitual considerations in assigning grades o How to provide effective feedback Following the first categorization, 25 statements (one or more statements from each category) were selected for the questionnaire. The criteria for selecting the statements were, in the main, the succinctness of the wording, its transparency, and how well it represented the collection of statements in the same category (see Table 1, in the Findings section). It is important to emphasize that these statements were not edited, in order to preserve the authenticity and the original intention of the responders who worded the issues they identified as they experienced them. In our translation we have attempted to remain as close as possible to the original. In the third phase of the research, the questionnaire was put together as an online questionnaire. The first part included general questions on the teachers’ education, their teaching experience in general and in teaching mathematics in particular, their background in the area of assessment, and the classes they taught. In the second part, the teachers were asked to rank to what extent they were concerned in regard to the assessment issues represented by the 25 statements (“extremely
66 troubled;” “troubled to a large degree;” “troubled;” “troubled only a little;” or “not troubled at all.”) At the end of the questionnaire the teachers were asked two open- ended questions: 1. Is there another/different issue in assessing students that troubles you as a mathematics teacher? If so, what is it? 2. If you could receive immediate help, support, or a solution to one of the issues that troubles you, which one would you choose? Please explain why. In the fourth phase of the research, the questionnaire was sent to 140 mathematics teachers (different from those in the first phase) who were asked to complete it anonymously. Eighty-four primary school teachers completed the questionnaire. FINDINGS To simplify the presentation and to ensure a clear description of the most troublesome issues we have grouped together those who answered “extremely troubled” and “troubled to a large degree” under “very troubled” and those who answered “troubled only a little” and “not troubled at all” under “hardly troubled.” In Table 1 we see the list of statements, in the order they appeared in the questionnaire, and the distribution of the teachers’ responses. The statement Very Troubled Hardly troubled troubled No. % No. % No. % 1 Children who are stressed and are taking an 49 58.33 25 29.76 10 11.90 exam can’t fully express their knowledge 2 How to assess a student without an exam? How 40 47.62 25 29.76 19 22.61 to assess when giving a thinking task in pairs or groups. 3 The student understands the material in the 55 65.48 17 20.24 12 14.29 class. He/she fails the exam or gets a low grade that is at odds with his/her knowledge 4 The gap between the report card that is based 46 54.76 23 27.38 15 17.86 on a numerical grade and the assessment that is not necessarily a numerical grade 5 How do we know that the students did the work 36 42.86 20 23.81 28 33.33 themselves? 6 Difficulty in seeing the thinking process that 48 57.14 27 32.14 9 10.71 led to the solution (whether it is correct or incorrect) and by extension, how to deal with difficulties or mistakes 7 A numerical grade does not give students a 37 44.05 27 32.14 20 23.81 chance to correct and improve their work because they don’t know or understand their mistakes.
67 8 How many points to deduct for a recurring 28 33.33 25 29.76 31 36.90 error? 9 Assessment on method as opposed to 40 47.62 28 33.33 16 19.05 assessment on result 10 Assessment does not check a student’s personal 30 35.71 27 32.14 27 32.14 progress 11 Indecision in how to mark a solution that is 37 44.05 28 33.33 16 19.05 incomplete even though it is clear that the student understands and knows the answer 12 Every student is different with respect to 35 41.67 24 28.57 25 29.76 knowledge, level, strengths, and background and so assessing achievements is not something certain and comprehensive 13 Questions do not match what was learned in 36 42.86 21 25.00 27 32.14 class 14 Difficulty in assessing a student with unclear 32 38.10 22 26.19 30 35.71 hand-writing 15 Assessing partial answers/answers full of 28 33.33 29 34.52 27 32.14 mistakes 16 In exams: equal marks for each question even 40 47.62 22 26.19 22 26.19 though there are different levels of difficulty (expectation: harder question should be worth more marks) 17 Exams mostly check final answer rather than 48 57.14 22 26.19 14 16.67 process. Sometimes an incorrect answer is marked as invalid even though the direction of thinking was correct. 18 The teacher’s difficulty in assessing 50 59.52 22 26.19 12 14.29 achievements rather than willingness, efforts, and ability: if students try hard and apply themselves, but don’t necessarily succeed in getting 100. If they get 70 they see that as a failure. 19 There is no description of the thinking process 61 72.62 18 21.43 5 5.95 - only a final answer. Students find it difficult to explain how they got the answer. 20 A teacher’s subjective assessment (influenced 36 42.86 19 22.62 29 34.52 by previous acquaintance with the student) 21 Students’ difficulties in understanding the 56 66.67 20 23.81 8 9.52 wording of exam questions (reading comprehension) 22 Difficult to give personal attention to a large (or 64 76.19 17 20.24 3 3.57 not so large) group of students 23 Difficulties in checking homework 46 54.76 21 25.00 17 20.24
68 24 Varying levels of mastery – while learning they 57 67.86 20 23.81 7 8.33 know it, but when they are busy with something else – they forget. 25 MEG1 – all the preparation aimed towards it – 66 78.57 12 14.29 6 7.14 does it really give a reliable picture? Table 1: Teachers’ grading of assessment issue statements From the findings in the table we see that the most troubling issue for the teachers (79%) was “MEG – all the preparation aimed towards it – does it really give a reliable picture?” (statement 25). Two other issues that most troubled the teachers (more than 70%) were: “Difficult to give personal attention to a large (or not so large) group of students” (statement 22); and “There is no description of the thinking process - only a final answer. Students find it difficult to explain how they got the answer” (statement 19). The issue that the greatest number of teachers (37%) found the least troubling was “How many points to deduct for a recurring error?” (statement 8), closely followed (36%) by “Difficulty in assessing a student with unclear hand-writing” (statement 14). These two issues may be considered to be less troubling not because they are unimportant but rather because they touch on specific points in relation to the more general topic of alternative assessment. From the collection of issues that more than half of the teachers considered to be very troublesome, we can see that the subject of exams, whether external or internal, troubles the teachers very much. They wonder how much the MEG gives a reliable picture of their class. In addition, they raise the fact that an exam does not reflect their students’ thinking processes. Additional elements that trouble teachers pertains to assessing effort, ability, willingness, and application. Teachers do want to assess these parameters but they have no adequate knowledge of how to do that. As expected, the answers to the open-ended questions were worded in a non- uniform manner. Some of the statements contain references to more than one issue and thus it is difficult to group them under one category. The first question was: “Is there another/different issue in assessing students that troubles you as a mathematics teacher? If so, what is it?” We identified the main issues as follows: variance in the classroom; exams (extent of exam, exam anxiety, and timing of exam); MEG; and the discrepancy between doing the task in the classroom and the results in the exam. The second question was: “If you could receive immediate help, support or a solution to one of the issues that troubles you, which one would you choose? Please explain why.” We identified the main issues as follows: class size; knowledge gaps; discrepancy between the student’s exam grade and their
1 Measure of Efficiency and Growth within schools. This is an external, national exam whose aim is to provide information for principals and pedagogical staff about each school’s performance in core subjects – to compare it to that of other schools, and to its own performance in previous years
69 ability; questioning the need for external assessment; and the reliability of the assessment. DISCUSSION AND CONCLUSION This research investigated assessment issues that concern mathematics teachers. We saw that teachers are troubled by issues connected with issues of validity, reliability, and pedagogical knowledge. Issues relating to valid and reliable assessments dealt with such questions as: What are we trying to assess? How can we assess satisfactorily? How do we know if our assessment is a reliable reflection of the reality we are interested in assessing? What should we include in our assessment? Issues relating to pedagogical knowledge dealt with questions that pertain to coping with the varying abilities of students in the classroom; assessing individual progress of each student; identifying the need for more knowledge and tools to deal with struggling students; and so on. Assessment issues relating to pedagogical knowledge were not as pronounced in the first phase of the research. These issues arose indirectly through other issues relating to one or other of the above-mentioned topics. So, for example, when teachers ponder over the question of the gap between the knowledge that a student displays in class and the knowledge revealed by their performance in the exam, this indirectly raises the issue of lack of pedagogical knowledge and of professional tools that can effectively address these gaps. Similarly, when the responders indicated that a numerical grade does not allow students to improve and correct themselves because they cannot understand their mistakes from the grade alone, this indirectly raises the need for an alternative assessment tool. The issues that most teachers claimed they needed an immediate help with were size of class, variance between students, and methods of alternative assessment. The research findings suggest that teachers are in need of an answer to the practical questions that bother them when assessing students. The findings could inform teacher education programs with respect to building a focused and pedagogically practical program. Such programs may provide relevant answers to the difficulties teachers are facing in the classroom. Perhaps the statements produced by the participants of this research could springboard a workshop where teachers would take an active part in creating alternative assessment tools. It could be hypothesized that teachers who used assessment tools that they themselves have created would be more willing to use these tools and more likely to use them effectively. Perhaps our next research project will examine this hypothesis. References Assessment Reform Group (2002). Research-based principles to guide classroom practice. https://www.aaia.org.uk/content/uploads/2010/06/Assessment-for- Learning-10-principles.pdf. Bedford, S., Legg, S. (2007). Formative peer and self-feedback as a catalyst for change within science teaching. Chemistry Education Research and Practice, 8(1), 80–92.
70 Birenbaum, M., Breuer, K., Cascallar, E., Dochy, F., Dori, Y., Ridgway, J., Wiesemes, R., Nickmans, G. (2006). A learning integrated assessment system. Educational Research Review, 1, 61–67. Black, P., Wiliam, D. (2006). Assessment for learning in the classroom. In: J. Gardner (Ed.), Assessment and Learning (pp. 9–25). London: Sage. Bolte, L. (1999). Enhancing and assessing preservice teachers' integration and expression of mathematical knowledge. Journal of Mathematics Teacher Education, 2, 167–185. Briggs, D. C., Ruiz, Primo, M. A., Furtak, E., Shepard, L., Yin, Y. (2012). Meta-analytic methodology and inferences about the efficacy of formative assessment. Educational Measurement: Issues and Practice, 31(4), 13–17. Dori. Y. J. (2003). From nationwide standardized testing to school-based alternative embedded assessment in Israel: Students’ performance in the “Matriculation 2000” Project. Journal of Research in Science Teaching, 40(1), 34–52. Earl, L. M. (2012). Assessment as learning: Using classroom assessment to maximize student learning. Corwin Press. Kingston, N., Nash, B. (2011). Formative assessment: A meta-analysis and a call for research. Educational Measurement: Issues and Practice, 30(4), 28–37. Kingston, N., Nash, B. (2012). How many formative assessment angles can dance on the head of a meta-analytic pin? Educational Measurement: Issues and Practice, 31(4), 18–19. National Council of Teachers of Mathematics. (2013). Formative Assessment: A Position of the National Council of Teachers of Mathematics. Retrieved from https://www.nctm.org/uploadedFiles/Standards_and_Positions/Position_Statements/ Formative%20Assessment1.pdf. Simpson, N. J. (2004). Alternative assessment in a mathematics course. New Directions for Teaching and Learning, 100, 43–53. Sullivan, P. (1997). More teaching and less assessment. The Primary Educator, 3(4), 1–-6. Tal, R. T., Dori, Y. J., Lazarowitz, R. (2000). A project-based alternative assessment system. Studies in Educational Evaluation, 26, 171–191. Veldhuis, M., van den Heuvel-Panhuizen, M. (2014). Exploring the feasibility and effectiveness of assessment techniques to improve student learning in primary mathematics education. In Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education (Vol. 5, pp. 329–336). PME. Watt, M. G. (2006). Attitudes to the use of alternative assessment methods in mathematics: A study with secondary mathematics teachers in Sydney, Australia. Educational Studies in Mathematics, 58(1), 21–44.
71 KINESTHETIC & CREATIVE APPROACHES TO PRE-K SPATIAL GEOMETRY LEARNING: A QUALITATIVE CASE STUDY Natalie Blundell, Kathryn Bentley, Barbara Ann Temple and David K. Pugalee
Abstract Accessing creative segments of the brain through dance allows Pre-K students to learn Math concepts like spatial awareness more easily, providing them with critical Math knowledge prior to formal assessment in elementary school. This early foundation in Math learning through the arts creates educational equity by granting these students more than a month of additional Math learning compared to their non-arts residency peers (Ludwig, Marklein and Song, 2016). Arts-integrated Math learning engages both hemispheres of the brain and all types of students, especially English Language Learners (ELL) and those of low socioeconomic status. Math and movement residencies bring equity to the classroom by preparing students with a critical foundational knowledge in Math prior to encountering formal Math concepts in Kindergarten. Keywords: Spatial awareness, dance & movement, math literacy
Introduction and Aims of Research Experiential learning in and through the arts benefit young students in Pre-K classrooms by engaging both creative right-brain activity with more rational functions of the left brain. The average classroom teacher (CT) does not possess the knowledge to engage the brain holistically and simultaneously in this way. These strategies are not traditionally taught in public school teacher training, thereby creating disparities in equity and access for students. This inequity can be mitigated through arts-infused learning. Wolf Trap (WT) Institute for Early Learning Through the Arts administers seven-week arts- integrated residencies that provide Pre-K teachers with skills that work to engage students wholistically. Professional Teaching Artists (TA) work with CTs to acquire arts strategies to be used throughout their careers. The aim of the research is to answer these questions: How does learning improve when the performing arts are used to teach Math? What happens to Pre-K students’ acquisition of spatial awareness skills when they experience Math literacy through dance and movement? Research & Theory Theorists and early education scholars advocated for interdisciplinary learning in the traditional classroom. John Dewey (1902) promoted the importance of active and experience-based learning while psychologist Howard Gardner (1999)
Arts and Science Council Institution, USA; e-mail: [email protected], [email protected], [email protected] University of North Carolina at Charlotte, USA, e-mail: [email protected]
72 suggested the existence of multiple intelligences within the human intellect. These two assertions suggest that students of diverse abilities and backgrounds need interdisciplinary experiences in order to create equity in every classroom. Research reveals that children from a lower socioeconomic status often start Kindergarten significantly behind their peers (O’Brien and Dervarics, 2007). This opportunity gap between students of poverty and students of medium or high socioeconomic status is narrowed through Pre-K programs in public schools that allow more students to perform at grade level when entering Elementary School. The social constructivist approach to Wolf Trap learning, exercised by both teachers and students, establishes a foundation for continued success in Math learning. As students engage in the social activity of dance during Math lessons, they learn more from engaging with each other and with the teacher than when learning independently. This situated cognition of learning spatial geometry through participating in dance is strengthened by students’ social interactions with their peers (Vygotsky, 1978). As Pre-K students first learn about the notion of space and how they exist within the space around them, they begin learning spatial geometry. When young children discover new pathways in space that represent early spatial geometry, they gain self-confidence because the arts empower them to be actively engaged in their own learning (Pugalee, Harbaugh and Quach, 2009; Temple, 2007; Holloway, 2001; Burton, Horowitz and Abeles, 1999; Catterall, Chapleau and Iwanaga, 1999; Eisner, 1998). Success in STEAM disciplines (Science, Technology, Engineering, Arts and Mathematics) is closely linked to spatial abilities. A study at the University of Chicago found that “Improving children’s spatial skills may have positive impacts on their future success in [STEM]...by enhancing numerical skills that are critical for achievement in all STEM fields,” (Gunderson et al., 2012). Specifically, acquiring spatial awareness will help students “mentally manipulate objects” which will serve them in the future when reading graphs and understanding diagrams (Gunderson et al., 2012). These studies indicate that these skills, required of any engineer or scientist, must be taught as early as possible. WT Math residencies translate to higher scores in Math assessment. Through a grant from the U.S. Department of Education, the study on improved Math performance in WT residencies revealed students acquired an additional 26 days of Math learning (Ludwig et al., 2016). In a similar study, the mean gain score for residency classrooms in Mathematics was two times higher than non-residency classrooms (Klayman, 2004). What does it mean to be spatially aware? Spatial awareness can be defined as "an awareness of the body in space, and the child's relationship to the objects in the space," (Hohmann, Weikart and Epstein, 2008). As children’s brains develop, so too does their understanding of their body’s relation to the world around them. At first it is difficult for young children
73 to understand early spatial concepts such as “on,” “under,” “over,” and “behind.” Children develop spatial awareness through whole-body activities. Once children understand early spatial concepts, they can then advance to more challenging Math concepts in the future. For many preschool children, an understanding of spatial concepts predicts later success in Math, reading, and following directions. Why is spatial awareness important to Math development? The National Council of Teachers of Mathematics (1989) recognizes the importance of geometry and spatial sense in its publication Curriculum and Evaluation Standards for School Mathematics: “Spatial understandings are necessary for interpreting [and] understanding our inherently geometric world...Children who develop a strong sense of spatial relationships and who master the...language of geometry are better prepared to learn...advanced Mathematical topics” (p. 48). These spatial skills are essential not only for students who will enter into a STEM field but also for students to function in everyday life. How does engaging in dance promote increased spatial awareness? Dance and movement are key to the development of spatial abilities in children. Physical movement is a child's natural first approach to learning. Dance elements such as body awareness, pathways, and creating shapes with the body all build spatial awareness in young children. Integrating these dance elements into daily Math lessons allows possibilities for exploring locomotor and non-locomotor movements. During dance or movement exercises, a child's understanding of spatial language enables him/her to move one's body through space. Background and Ideology of Wolf Trap Residencies WT residencies place a TA with an early childhood CT to provide professional development for CTs while helping children learn through the arts. TAs are instructed in early childhood development, twenty-first century skills, lesson design, and coaching. WT has an exponential positive effect on students: teachers benefit during their three years of residencies and continue to benefit their students for the rest of their teaching careers. Case Study Setting Mecklenburg County, North Carolina has a population just over 1 million and demographic data reports that just under half the county’s population is a non- White minority (United States Census Bureau, 2015). In the local Charlotte- Mecklenburg School system (CMS), 77 of the total 170 public schools are categorized as Title I (high poverty) (Charlotte-Mecklenburg Schools, 2016). Creekside Elementary School houses 39 Kindergarten through 5th grade classrooms, and 16 Pre-K classrooms. Socioeconomic data revealed 98.89 percent of four-person families whose children attend Creekside earn an annual income of $24,600 or less (Department of Health and Human Services, 2017).
74 Case Study Methodology The authors used ethnographic techniques of interview and observation to collect teacher narratives on student Math learning in a high poverty school. The authors used these techniques to understand how the arts specifically created equity within Pre-K classrooms. Data Collection Interviews. The authors interviewed two Pre-K teachers directly involved in a Math-based WT Residency in Dance and their two corresponding TAs for a 7- week residency in their respective classrooms. For the purpose of this particular case, we focused on the concept of spatial awareness in Geometry lessons. Both Pre-K teachers were asked the same questions; for example, What specific spatial awareness words, positions, and exercises did your students learn through dance/movement and why are they important? What are the specific components of dance/movement that have helped your students increase their spatial awareness? We conducted the interviews at the conclusion of the 7-week residency and both teachers were present during the interview period. Participants The WT STEM (Math) residencies in Dance and Movement were conducted in two Pre-K classrooms of 18 students each. Ms. Jones’ class consisted of four native English speakers and 14 non-English speakers. WT movement TA, Elizabeth, worked with Ms. Jones’ classroom for this Math residency and together they identified Math learning goals and arts strategies for the residency. First, students would understand and describe spatial relationships and shapes through body strength, balance, flexibility and stamina. Second, students would increase their ability to move their bodies in space (running, jumping, and skipping). Ms. Graham taught the second Pre-K classroom consisting of three native English speakers and 15 non-English speakers. WT movement TA Stacy worked with Ms. Graham’s Pre-K students. Based on needs of Ms. Graham’s students, the two agreed on Math learning goals and arts strategies for the residency. First, students should be able to understand and describe spatial relationships, shapes, and patterns through axial, locomotor, non-locomotor, and body shapes. Second, students should be able to understand and describe spatial relationships, shapes, and patterns by moving in time to different patterns of beat, rhythm, and music. Data Analysis Through qualitative data, the authors uncovered fascinating findings that ultimately helped answer the two main research questions. Below, the interview responses from CTs and TAs are categorized to answer these two research questions. 1. What happens to Pre-K students’ acquisition of spatial awareness when Math lessons are integrated with dance and movement? The kinesthetic act of
75 movement through dance creates access for Pre-K students to acquire spatial awareness. Dance and movement are key to the development of spatial abilities in children and result in active engagement, attention to focus, and increased self- confidence. Integrating dance into daily curriculum supports acquisition of spatial understanding and increased self-awareness. Elizabeth: Being spatially aware allows the children to move in a group setting with other bodies without safety issues...They learned how to control their bodies. When Math and dance are combined to teach students about spatial awareness, the classroom becomes more safe, focused, and accident-free. Fewer physical accidents in the classroom create a healthier learning environment where students are more capable of sustained learning for long periods of time. Stacy: Dance inherently requires and develops constant spatial awareness. Dancing in groups allows children to gain a general understanding of how much space their body takes up and how to move safely next to their friends and objects in the room. This newly acquired sense of personal space can then be transferred to tasks that students must complete as a group. The improvement of spatial awareness through Math and dance allows entire classrooms to complete simple, daily tasks more efficiently. Ms. Jones: The exercises [where students] walk on different shapes and lines taped to the floor help them walk in a straight line [as a group] in the hallway, for example. These seemingly simple tasks of walking in a straight line as a classroom to the cafeteria are profoundly difficult for a Pre-K child with little to no spatial awareness. When children learn these skills, teachers see major improvements in the efficiency of moving Pre-K students from point A to point B. In addition to these tasks, crossing the midline of the body helps build pathways in the brain and is an important skill required for the appropriate development of motor and cognitive skills. Many Pre-K activities require a child’s ability to cross the midline of the body. By crossing a hand, arm, foot, or leg to the opposite side of the body, children build increased brain functions to perform more advanced tasks. This movement is a type of bilateral movement. Children who are unable to develop this bilateral skill often have difficulty learning to read. Similar to bilateral body movements, reading requires the eyes to move from left to right across the page and across the midline of the brain. When young children lack practice in bilateral movements, this task can be taxing for the eyes, resulting in the inability to understand the meaning of the text. Bilateral skills also assist in the development of handwriting skills. There are many dance and movement activities that help develop these cross- lateral motions. As children move along different pathways, repeat dance patterns, or create body shapes, they connect these movements with a growing understanding of geometric concepts.
76 2. What happens when Pre-K students learn the concept of spatial sense through the performing art of dance (and movement)? The performing art of dance, when used in Math lessons, helps students feel more comfortable breaking from more traditional learning, thereby gaining comfort in risk and newness. Stacy: As students’ spatial awareness develops, they are able to make more exciting choices and begin to take risks and problem solve. The benefits of practicing and developing these skills in movement experiences positively affect the ways in which children interact with space in the classroom and at home (ex: following directions, fine and gross motor skills, playing safely with others) Ms. Graham: I enjoy seeing my students make connections between the Wolf Trap residency and their real life. We use position words when we’re out on the playground. I try to extend the lessons and draw these connections by saying “you’re going down the slide” or “you’re under the slide.” They really use these words now. By increasing children’s spatial awareness through dance and Math in the classroom, the teachers are also increasing students’ opportunities to expand their learning beyond the walls of the classroom. Learning the concept of spatial awareness through dance and repetition contributed to the development of self-efficacy in the Pre-K students. Ms. Jones: I saw their confidence grow with identifying and describing shapes. Toward the beginning the students went directly to the station with the shape they knew best and felt most comfortable describing. But toward the end, I saw them feel more comfortable going to a shape that was newer to them. Ms. Jones’ observations of her students’ increased self-confidence in experiencing shapes through movement is important for understanding Math learning. Something equally as important was taking place within the students: they were taking risks and developing self-efficacy as they believed they could attempt to describe a shape unfamiliar to them. Their active participation in learning the concept of shapes through dance and movement enabled them to experiment with new learning in a safe environment. Ms. Graham: I saw a confidence boost when we were learning shapes. Toward the beginning, I saw the students copying each other when they were asked to make shapes with their bodies. But toward the end, it was amazing to see the variety of shapes. They were being creative. Everybody finally felt comfortable to make their own shape. The student-centered approach to Math literacy learning through the arts fostered a socially-constructed new knowledge and understanding of spatial awareness. Notice that as the students’ self-confidence was growing, their creativity was also increasing throughout the residency. Within this participatory classroom, students not only learned new shapes through teacher instruction but also learned new shapes from one another (Vygotsky, 1978). This social constructivist approach to Math literacy learning through the arts empowered students to become leaders
77 within the collaborative classroom. In that leadership role, students became active participants in their own learning. Ms. Jones: I see a big difference in participation [now that the residency is complete]. They weren’t so sure and didn’t want to join in during the first few sessions. But toward the end, they have to lead the song, and you can see that they get it, you can see they gained confidence. In this case study, the arts cultivated access to a new language of Math. Regardless of the linguistic capital each student brought to the classroom at the start of the residency, the arts removed barriers in learning new Math concepts. By creating a safe space and “moments of inclusion,” new Math skills were developed through social interaction more easily (Temple, 2007; Holloway, 2001; Bourdieu and Passeron, 1977). Ms. Graham: The residencies help reinforce [position] words that normally pose a challenge like “behind,” “over,” “under.” We used the snowman from the story, so they got in front of the snowman, behind the snowman. At the end of the residency… they were finally getting the hang of it. Together, the Pre-K students developed a common language centered on spatial awareness using position words within their Math lesson. Students finally understood their own body’s relation to space and objects in space. Analysis of Results/Conclusions This qualitative case study supports the existing, related research. The case study reveals that as a result of WT dance residencies in Math lessons, Pre-K students are acquiring strong foundational geometry skills, resulting in increased confidence, participation, and formation of real-life Math connections outside the classroom. The following are the results of the qualitative case study centered on Pre-K students at Creekside Elementary School who experienced movement residencies in Math experiences: Increased knowledge in spatial awareness and early geometry concepts including: Recognize and describe the attributes of common shapes; Understand spatial relationships; Use position words that identify different positions in space; Understand positional words in order to follow specific directions during transitions; Use the body to make physical representations of the four common shapes; Apply movement activities to reinforce concepts of shapes Increased confidence when speaking in front of peers Increased willingness to participate Increased connections to Math concepts outside classroom From these results, we can conclude that WT residencies are meeting and exceeding local and national classroom standards. The teachers’ new creative
78 abilities acquired from the residency allowed them to engage all students in Math lessons, regardless of language limitations, not only within the classroom but also outside the walls of the classroom. The students’ increased confidence and willingness to participate in Math-focused movement residencies display a positive relationship to Math learning, suggesting that these students will likely approach Kindergarten Math lessons with willingness, excitement, and a solid foundation in Math learning. Implications Pre-K learning prepares students for learning in Elementary Education, contributing to their readiness for future learning and success later in life. Formal assessments which identify students for accelerated learning programs identify “spatial intelligence” as one of four identified areas (West Windsor-Plainsboro, 2015). By demonstrating a strong spatial intelligence, paired with talents in other academic arenas, students have an increased chance of being chosen for an accelerated academic program. Students who are exposed to and gain spatial awareness skills as early as Pre-K have more experience with these processes and are therefore more likely to meet, if not exceed, Math learning standards for their age. The Partnership for 21st Century Learning (2016) research coalition created a framework which identifies skills that will prepare this next generation of students for future complex work environments: creativity, critical thinking, communication, and collaboration. The WT residencies outlined in this case provide the perfect foundational learning environment for Pre-K students to begin an educational journey directly aligned with our nation’s goals in mind. The knowledge of spatial sense acquired through dance and movement during Math lessons created opportunities for continued learning and reinforcement in other instructional arenas during the school day. Additionally, spatial thinking and geometry learning at the Pre-K level “reduces differences related to gender and socioeconomic status that may impede full participation in a technological society,” (Get Set for School, 2011). The student acquisition of spatial awareness through dance created an equitable opportunity for Pre-K students to be better prepared for Kindergarten and for a successful life in a complexifying society. References Bourdieu, P., Passeron, J. C. (1977). Reproduction: In education, society, and culture. London: SAGE Publications. Burton, J., Horowitz, R., Abeles, H. (1999). Learning in and through the arts: Curriculum Implications. In E. B. Fiske (Ed.), Champions of Change: The impact of the arts on learning (pp. 36–46). Washington, DC: President’s Committee on the Arts and Humanities. Catterall, J.S., Chapleau, R., Iwanaga, J. (1999). Involvement in the arts and human development: Extending an analysis of general associations and introducing the
79 special cases of intensive involvement in music and theatre arts. (Americans for the Arts Monograph No. 11). Los Angeles, CA: University of California at Los Angeles, Graduate School of Education and Information Studies. Charlotte-Mecklenburg Schools (2016). All school SES level 2016. Charlotte, NC. Department of Health and Human Services [DHHS] (2017). Annual Update of the HHS Poverty Guidelines. (FR Doc. 2017-02076.) Dewey, J. (1902). The Child and the Curriculum. Chicago, Illinois: The University of Chicago Press. Eisner, E. W. (1998). The kind of schools we need: Personal essays. Portsmouth, NH: Heinemann. Gardner, H. (1999). Intelligence Reframed: Multiple Intelligences for the 21st Century (32). New York, NY: Basic Books. Get Set for School (2011). Get Set for Readiness, Writing, Language, and Math: A Pre- K Roadmap for School Success. Gaithersburg, MD. Gunderson, E., Ramirez, G., Beilock, S., Levine, S. (2012). The relation between spatial skill and early number knowledge: The role of the linear number line. Developmental Psychology, 48(5), 1229–41. The University of Chicago: Department of Psychology and Committee on Education. Hohmann, M., Weikart, D., Epstein, A. (2008). Educating Young Children: Active Learning Practices for Preschool and Child Care Programs (428). Ypsilanti, MI: High Scope Press. Holloway, D. L. (2001). Authoring identity and agency in the arts. Dissertation Abstracts International, 62(02), 426. (UMI No. 3005059.) Klayman, D. (2004). Executive summary of the final evaluation report of fairfax pages professional development project: An effective strategy for improving school readiness. Potomac, MD: Social Dynamics LLC. Ludwig, M. J., Marklein, M. B., Song, M. (2016). Arts Integration: A Promising Approach to Improving Early Learning. Washington, DC: American Institutes for Research. National Council of Teachers of Mathematics, Inc. [NCTM] (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA: Reys, B. O’Brien, E. M., Dervarics, C. (2007). Pre-Kindergarten: What the research shows. Alexandria, VA: Center for Public Education. Pugalee, D. K., Harbaugh, A., Quach, L. H. (2009). The human graph project: Giving students Mathematical power through differentiated instruction. In M.W. Ellis (Ed.), Mathematics for Every Student: Responding to Diversity, Grades 6-8. Partnership for 21st Century Learning [P21]. (2016). Framework for 21st Century Learning. Washington, DC: P21. Temple, B. A. (2007). Creating studios of literacy learning through the arts: A narrative case study in arts integration for urban high school education. Charlotte, NC: The University of North Carolina at Charlotte. United States Census Bureau (2015). QuickFacts: Mecklenburg County, North Carolina. Retrieved from www.census.gov.
80 Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge: Harvard University Press. West Windsor-Plainsboro Regional School District (2015). Gifted and Talented Program Review Internal Rnalysis of two invalsi tasks on power properties.eport. Plainsboro, NJ.
HIGHLIGHTS ON GENDER GAP FROM ITALIAN STANDARDIZED ASSESSMENT IN MATHEMATICS Giorgio Bolondi, Clelia Cascella and Chiara Giberti
Abstract In this paper we present a study in progress on math gender-gap, based on the analysis of Italian standardized tests. The methodology explained is designed to study gender gap in Maths not only in the whole test but more deeply on specific items. The analysis of Item Characteristic Curves (ICCs) plotted both in male and female group on the basis of Rasch estimates, is useful to examine tests of various grades and also to compare similar items in different grades. In this paper we focus on primary students and we provide two examples of items analysis from grade 5 and 6 integrating statistical enquiry and didactical interpretation of students’ responses, for the purpose of revealing the potentiality of this approach in studying gender differences in math. Keywords: Gender gap in mathematics, decimal numbers, misconception, standardized test, differential item functioning, measurement bias
Introduction The issue of gender gap in math education is deeply studied since the last century and, in recent years, the increasingly important role of national and international assessments has given new emphasis on this theme (Leder and Forgasz, 2008). The results of international investigations such as PISA 2015 and TIMSS 2015 confirm that in mathematics males and females have different performances: boys outperform girls at all school levels and in almost all the countries (OECD, 2016; Mullis et al., 2016). In both these tests administered in 2015, Italy is one of the countries in which the gender gap is more remarkable and this gap is also confirmed by results of national standardized tests, called INVALSI tests (INVALSI, 2016; Di Tommaso et al., 2016). Gender differences in math performances have been debated in several studies and plenty of researches has focused on the determinants of gender-gap (Forgasz, 2010). As reported by Winkelmann et al. (2008), we identify both external and internal factors that can be a cause of gender differences in maths achievement. Internal factors include biological variables considered responsible for gender gap in math in some studies, but this hypothesis needs to be overcome because
University of Trento, Italy; e-mail: [email protected]
81 international surveys have revealed that gender gap in math differs enormously across countries (Di Tommaso et al., 2016; OECD, 2016). Possible, yet-to-be- confirmed, biological, internal factors must be accompanied with other explanations connected to social and cultural factors, related to the context in which the students live. In this perspective, many researchers highlight the importance of social and cultural reasons, evidencing that in more gender-equal cultures this gap disappears (Guiso et al., 2008; OECD, 2015; Cascella, 2017). Furthermore, beliefs of teachers and parents about boys and girls math abilities, parental expectations and gender stereotypes play an important role in students’ self-perception and then have a huge influence on their performances (Jacobs and Bleeker, 2004; Riegle-Crumb, 2005; Freyer ND Levitt, 2010). For instance, many studies show that the role of the mother in the family and in the society is strictly related with female performances in math tests (Freyer and Levitt, 2010; Jacobs and Eccles, 1992; González de San Román and De La Rica, 2012). Several studies have also shown differences between boys and girls in metacognitive aspects related to maths that have a negative impact on their performances: for instance, girls tend to be more influenced by math anxiety and display less math self- efficacy (Cargnelutti et al., 2016; OECD, 2016; Pajares, 2005). At last, also factors strictly related to the school context seems to be a possible explanation of the gender gap in math. Leder (1992), in order to explain gender differences reflects also on “curriculum variables, like content areas of mathematics, types of the items and method of assessment and instruction” and more recent studies have shown that not only mathematics curriculum but also classroom practices and assessment practices, educational methods have a huge impact on gender gap in math (Leder and Forgasz, 2008; OECD, 2016; Giberti et al., 2016). In this perspective, many studies have shown that males and females use different strategies in problem solving activities: girls usually prefer routine procedures, well known algorithm and conventional strategies, whereas boys are less afraid of making mistake and try new methods and unconventional approaches (Bell and Norwood, 2007; Gallagher et al., 2000; Gould, 1996; Fennema and Carpenter, 1998). Also in this case, the reasons of differences in problem solving strategies seems to be not so much related to biological differences between males and females but mostly to stereotypes, such as ‘good girls follow the rules’ (Langer, 1997) and to pedagogy, school system and classroom practice (Boaler, 1997). As we have shown there is not just one possible explanation of gender gap in mathematics, but there are numerous factors that influence this issue. In this paper, we endorse the idea that gender gap in math achievement is particularly influenced by cultural and social causes and also micro-social factors related to the mileu habits and classroom practice. This hypothesis, in opposition to biological causes, is also supported by the fact that this gap is not present at the early stage of school but it raises during the school years (Fryer and Levitt, 2010; Di Tommaso et al., 2016).
82 The issue of gender gap in maths is deeply studied in mathematics education from both a qualitative and a quantitative point of view. Many studies focus on small group of students and try to understand gender differences on specific tasks thought interviews but most of the recent researches on gender gap in maths are based on national and international standardize tests, such as PISA and TIMMS. Indeed, using the results of these tests, it is possible to analyse gender gap in several ways, for example many studies compare the relevance of the gap in different countries and evidence where this gap is more remarkable. Standardized assessments give the possibility to study this issue on a large population of students and from a statistical point of view. In this paper we study gender differences in Italian standardized tests (INVALSI tests) using particular statistical tools to evidence gender-gap not only in the whole test but also evidencing the items in which this gap is more remarkable and focusing the analysis on these specific items. In the INVALSI reports (INVALSI, 2016) we observe gender differences on the basis of total medium score on the whole test and in the same way the gender-gap is studied in PISA reports (OECD, 2016). This information is important to signal the existence of a gap between male and female performances in maths and this is also useful to make comparison between different geographical areas and to analyse the relation between gender gap and social and cultural factors. Nevertheless, if we want to make assumption on the origins of this gap and analyse it with the lenses of the maths education theories, it is essential to analyse this gap on the single items of the test. Are males better than females in all mathematical items? Are there some items in which the gender gap is more remarkable than in the others? In this case, it could be interesting to focus on items in which the gender gap is more evident and make a deeper analysis from both a statistical and a didactical perspective. This kind of analysis seems to be able to provide at least two main research products: 1) an insight into the classrooms in order to ascertain possible links between didactical protocols and skills development in males and females; and, 2) individuation of recurring items’ features that cause item bias in sub-groups of students clustered by gender. Moreover, the latter is undoubtedly be useful in order to construct standardized achievement test that, as well known, in accordance to Item Response Theory approach frequently used in education research, must guarantee measurement invariance. Although undoubtedly a lot of different factors that might affect gender gap exist, exploring the relationship between gender gap in Math and/or in STEM and socio-cultural and economic environment within which students grow up is an ongoing issue timely topic in EU agenda but it is outside the scope of this work. Nevertheless, we claim that the observation of gender gap for different ability levels and the exploration of possible links between didactical protocols and differences in males and females test scores can be absolutely useful in order to formulate new research hypotheses and to provide new answers to the questions that our results might get emergent.
83 Methodology Our analysis is conducted using answers given by Italian students to Math achievement test developed by the Italian National Institute for the evaluation of Educational Systems. We analysed several of the 1400 items of that make up the INVALSI database but in this paper we give two examples of item analysis. Questions in Mathematics test cover the subareas of “space and shape” (roughly geometry), “change and relationship” (algebra), “quantity” (arithmetic) and “uncertainty” (probability), in a range of difficulty from those that require simple mathematical operations to those that require complex thinking. The mathematics scores have been estimated by using the Rasch model and have been scaled in an empirical range equal to [-4; +4], i.e. the latent trait along which both items and persons are scaled depending on their difficulty and their ability respectively. The wide use of standardized achievement tests in Social and Human Sciences has put the spotlight of psychometrical and educational literature on the treatment of item bias. The term bias, when used to describe mental tests, has a specific technical meaning: it is defined as a systematic error in measurement process that can alter the expected items’ psychometrical properties and cause a consistent distortion of a statistic, i.e. for achievement tests, unfair person and items estimates. In other words, item bias refers to differences in the way a test item functions across sub-groups of persons matched on the attribute measured by the test (e.g. ability) and stratified by gender, socio-cultural background and/or any other variables not explicitly hypothesized by the statistical model used to analyse data (Osterlind, 1983; Camilli, 2006; Camilli and Shepard, 1994; Holland and Wainer, 1993; Embretson and Reise, 2000; Penfield and Camilli, 2007; Osterlind, 2009). Therefore, when the same item shows different behaviour in different sub- groups of persons, it is biased and it is a grave violation of Item Response Theory (IRT) models, frequently used to assess students’ ability. In particular, the Italian National Institute for the Evaluation of Educational System employs the Rasch model, according to which the probability of a correct answer depends on students’ relative ability, i.e. his/her ability compared to item’s difficulty: no other variables (such as for example students’ socio-demographical characteristics) can affect this probability. Therefore, according to the mathematical mechanism underlying the model, an achievement test developed to be analysed by the Rasch model has to be constructed to avoid any possible effects exerted by each other possible variable. In the sectorial literature, term bias refers to the whole student’s test scores and thus it has a more general meaning compered to Differential Item Functioning (DIF). DIF refers to each single item and to item behaviour in sub- group of students matched on ability. In order to control item functionality, a lot of different techniques to detect DIF have been proposed (Osterlind, 2009). Many studies carried out on both real and simulated data have showed the efficacy of these techniques to detect DIF in small or medium sample. For larger dataset, unfortunately, the number of useful techniques decreases significantly. In the
84 present study, we carried out a DIF analysis within Item Response Theory (IRT) framework, based on the comparison of the Items’ Response Function (IRF) that links the probability of a correct answer to student’s ability. In our study, we have two IRFs, one for males and one for females. Comparing them, we can compare item functionality in both male and female sub-group of students scaled along the same latent trait, i.e. in each sub-group, we have the same number of students with the same ability level. Therefore, we can observe three different situations: 1. Not biased item (IRFs in both group have the same behaviour); 2. Uniform DIF between students in each group (item functionality is different in each group because the probability of a correct answer is higher in one group compared to the other ones, for each ability level); and, finally, 3. Not uniform DIF (the probability of correct answer is higher in one group relative to the other one only for specific ability level). The DIF analysis based on the comparison of IRFs provides very useful information: 1. It identifies advantaged versus disadvantaged sub-group of students; and, 2. It locates along the latent trait students’ probability of a correct answer in each sub-group. Unfortunately, many statistical pakages do not plot also distractors’ behaviour in each sub-group of students. Distractor plot is a graph plotted only for multiple choice item, that shows the functionality of an items’ distractors. For this reason, we carried out three analysis: first at all, we analyzed male and female answers all togheter (DIF analysis); then we analyzed them separately and compared distractor graphs plotted for males and females. As regards measurement bias, the first way to identify the entity of the gender gap on each item is to compute the difference between the percentage of correct answer of male and the percentage of correct answers of female. In this way we observe which items contribute most to creating the gender gap on the whole test: in fact if the measure of gender gap on the test is on the bases of total medium score, it derives directly from the raw scores and hence to the percentage of correct answers. The problem of considering the gender gap of an item in this way is that a difference of 10 point percentage on an easy item in which the percentage of correct answer is around 80% is the same of a difference of 10% on a very difficult item in which only the 30% of students choose the right answer. To take into account also the difficulty of the item, we create a specific index to evidence the size of gender gap on each item. In which: Mk is the percentage of correct answer for males to the item k. Fk is the percentage of correct answer for females to the item k. Pk is the percentage of the whole population to the item k. In this way we can observe that the items with a positive value of the index are those in which male outperform female and the items with a negative index are those in which female results are better than male’s ones. The index gives also an evaluation of the magnitude of the gap considering also the difficulty of the item
85 analysed and items selected in this way can be than studied in deeper using DIF and the other procedures described above. Results In our research we analyse INVALSI tests using the statistical tools described in the previous section. In particular, we focus our attention on four INVALSI test belonging to different levels and different years: level 2 of 2009, level 5 of 2012, level 6 of 2013 and level 8 of 2015. These tests are particularly interesting because they were administered to different sample of students but representative of the same population (approximately 30000 students), growing during the years. In this way it is possible also to observe the changing of a specific population of students at different stages of their schooling. First, we used gender-gap-index to point out which items highlight mostly gender gap. In this way it is possible to notice that the gender gap is not distributed on all the items of a test, but it’s most marked on some of the items and there are also some items with no gender differences in favour of male and few items in which female have better results. It is interesting to compare the results of this first analysis with the results of DIF analysis compute on the same tests. DIF analysis point out not merely which items create a higher gap, but also if this gap is more remarkable for particular ability levels of the students. In the next section we focus on two tasks belonging to tests mentioned above. Both tasks refer to the same math content, that is comparison between decimal numbers, but the kind of task, the context and the level of the test are different. The first task belongs to the INVALSI test of level 5 administered in 2012.
Figure 1: Items 21a-21b from the grade 5 INVALSI test administered in 2012 and results of the items (9-10 years old students) This question is composed by two items and in both of them we observe a significant gender-gap favouring boys in terms of percentage of correct answer. At a first glance, both items require to order decimal numbers but the first item is much more difficult than the second one and moreover the first item points out a
86 wider gender gap. There are many studies concerning decimal numbers that evidence misconceptions related to this issue and, in particular, to comparison between decimal numbers. Misconceptions concerning decimal numbers are frequent and often arise during the transition between natural numbers and decimal numbers: many students tend to continue applying procedure and properties valid for natural numbers also when they operate with decimals (Sbaragli, 2012). To solve the first item, students have, in particular, to compare two decimals with the same integer part: the countries with better results are Italy (80.12 points) and United States (80.2 points). Many students in this case compare the decimal part of the two numbers and since 12 > 2, they concluded that 80.12 > 80.2. Observing the percentages for the first item we note that less than 40% of students give the right answer and this may confirm that this is a widespread misconception. It is addition, girls are more influenced than males: only the 32% of female answer correctly compared to the 43% of males.
Figure 2: Item D21a administered by INVALSI to pupils attending the 5th grade of primary school, in 2012 (9-10 years old) Results of DIF analysis confirm the existence of a gender-gap in item 21a that is null in the lower decile but increases with the ability of students (Fig. 4). This means that the different influence of misconception between male and female is deeper for high ability levels: with increasing ability-levels male are better at getting over barriers given by misconceptions, while girls struggle most. Even if the question of the second item seems similar to first one, we observe that the second item is less difficult (67% of correct answer, INDEX = 11%), the gender- gap in terms of percentage of correct answer still exists but we can see that the distribution of this gap is strictly different. In fact, DIF analysis shows that gender- gap is focused on medium and low ability levels and there is no gender differences for higher ability levels. This different behaviour of the two items analysed can be explained observing that both items ask to compare decimal numbers but, in the second case, it is possible to get around this difficulty and relative misconceptions: students of high ability levels do not consider decimal numbers but only their integer part, because they realize that it’s enough to find the fourth ranked. In this way, the procedure applied to answer is changed and, since the misconception existing in the first item no longer influences, also the gender-gap
87 is filled. Furthermore, analysing the fit of the first item, we notice that the model overestimates lower levels and underestimates higher ones and this misfit of the item might be related with the influence of the misconception. This hypothesis becomes even more interesting analysing the fit of the second item: we observe that for lower ability levels the model overestimates the results but for medium and higher levels, when we assume a change of procedure, the fit is optimal, the gender-gap disappears. The other task analysed belongs to INVALSI test of level 6 administered in 2013 and, even there, students have to compare decimal numbers.
Figure 3: Item 23 from the grade 6 INVALSI test administered in 2013 and results of the item The results point out that difficulty related to this content still exist also at level 6 and this item (also compared with other items belonging to other tests) give us an endorsement of the fact that girls tends to have more difficulty than male in comparing decimal numbers and are more influenced by the misconception explained before. Comparing the percentages of each answer we can observe that females are more attracted than males by answer B: in this case the first three numbers are ordered following the misconception by which 3.5 < 3.28 < 3.124 because 5 < 28 < 124.
Figure 1: Male (left plot) and female (right plot) distractor graphs plotted for the item D23 administered by INVALSI to students of grade 6 in 2013
Deeper information not merely on the right answer but also on the trends of the incorrect ones can be sourced using DIF-distractor plots: we observe that the difference in selecting option B is particularly due to the lowest ability level in
88 which females have a probability of choosing this answer slightly less than 50%. Moreover, also in this case, the model overestimates lower levels and overestimates the higher ones. Further perspectives In this paper we have reported examples of the methodological strategy we are adopting to analyse around 1400 Math items administered from 2008 up to now. Focusing on two items of grade 5 and 6, we used this method to observe gender differences related to a specific misconception in comparing decimal numbers. We expect that the analysis carried out on the entire database will allow us to identify recurring features of Maths items influencing gender gap and will give us the possibility to produce a first mapping of these items’ characteristics (type, mathematical contents and context). We believe that, by combining statistical analysis and didactical interpretation of students’ cognitive processes, it will be also possible to point out some possible causes of gender differences in Maths. References Bell, K. N., Norwood, K. (2007). Gender equity intersects with mathematics and technology: Problem-solving education for changing times. Gender in the classroom, 225–258. Boaler, J. (1997). Reclaiming school mathematics: The girls fight back. Gender and Education, 9(3), 285–305. Cargnelutti, E., Tomasetto, C., Passolunghi, M. C. (2016). How is anxiety related to math performance in young students? A longitudinal study of Grade 2 to Grade 3 children. Cognition and Emotion, 1–10. Cascella, C. (2017). Male and Female social roles in daily file, pupils perceptions and gender gap in Math test scores. Some empirical evidences from Italian primary school. (Forthcoming.) Che, M., Wiegert, E., Threlkeld, K. (2012). Problem solving strategies of girls and boys in single-sex mathematics classrooms. Educational Studies in Mathematics, 79(2), 311–326. Di Tommaso, M. L., Mendolia, S., Contini, D. (2016). The Gender Gap in Mathematics Achievement: Evidence from Italian Data. IZA Discussion paper, n. 10053, Bonn. Embretson, S. E., Reise, S. P. (2000). Item Response Theory for Psychologists. Mahwah, NJ: Lawrence Erlbaum Associates. Fennema, E., Carpenter, T. P. (1998). New perspectives on gender differences in mathematics: an introduction. Educational Researcher, 27(5), 4–5. Forgasz, H. J. (2010). International perspectives on gender and mathematics education. IAP. Fryer, R. G., Levitt, S. D. (2010). An empirical analysis of the gender gap in mathematics. American Economic Journal: Applied Economics, 2(2), 210–240.
89 Gallagher, A. M., De Lisi, R., Holst, P. C., McGillicuddy-De Lisi, A. V., Morely, M., Cahalan, C. (2000). Gender differences in advanced mathematical problem solving. Journal of experimental child psychology, 75(3), 165–190. Giberti, C., Zivelonghi, A., Bolondi, G. (2016). Gender differences and didactic contract: Analysis of two INVALSI tasks on power properties. In PME 2016 proceedings (p. 275). PME. González de San Román, A., De La Rica, S. (2012). Gender gaps in PISA test scores: The impact of social norms and the mother's transmission of role attitudes. IZA Discussion Paper, 6338, Institute for the Study of Labor. Gould, S. L. (1996). Strategies used by secondary school students in learning new concepts which require spatial visualization. Unpublished doctoral dissertation, Teachers College, Columbia University, New York. Guiso, L., Monte, F., Sapienza, P., Zingales, L. (2008). Culture, gender, and math. Science – New York then Washington, 320(5880), 1164. INVALSI (2016). Rilevazione nazionale degli apprendimenti 2015-2016. Le rilevazioni degli apprendimenti. Retrieved March, 2017, from http://www.invalsi.it/ invalsi/doc_evidenza/2016/07_Rapporto_Prove_INVALSI_2016.pdf Jacobs, J. E., Bleeker, M. M. (2004). Girls' and boys' developing interests in math and science: Do parents matter. New directions for child and adolescent development, 2004(106), 5–21. Jacobs, J. E., Eccles, J. S. (1992). The impact of mothers’ gender-role stereotypic beliefs on mothers’ and children`s ability perceptions. Journal of personality and Social Psychology, 63(6), 932–944. Langer, E. J. (1997). The power of mindful learning. Cambridge, Ma.: Perseus Books. Leder, G., Forgasz, H. (2008). Mathematics education: new perspectives on gender. ZDM – The International Journal on Mathematics Education, 40(4), 513–518. Mullis, I. V. S., Martin, M. O., Foy, P., Hooper, M. (2016). TIMSS 2015 International Results in Mathematics. TIMSS & PIRLS International Study. OECD (2015). The ABC of Gender Equality in Education: Aptitude, Behaviour, Confidence. PISA, OECD Publishing. OECD (2016). PISA 2015 Results (Volume I): Excellence and Equity in Education. Paris: OECD Publishing. Pajares, F. (2005). Gender differences in mathematics self-efficacy beliefs. In Gender differences in mathematics: An integrative psychological approach, 294–315. Riegle-Crumb, C. (2005). The cross-national context of the gender gap in math and science. The social organization of schooling, 227–243. Sbaragli, S. (2012). Il ruolo delle misconcezioni nella didattica della matematica. Bolondi B., Fandiño Pinilla MI (2012). I quaderni della didattica. Metodi e strumenti per l’insegnamento e l’apprendimento della matematica, 121–139. Winkelmann, H., van den Heuvel-Panhuizen, M., Robitzsch, A. (2008). Gender differences in the mathematics achievements of German primary school students: Results from a German large-scale study. ZDM, 40(4), 601–616.
90 THE USE OF TEXTBOOKS IN DIFFERENT APPROACHES – HEJNY METHOD, MONTESSORI AND TRADITIONAL APPROACH IN THE CZECH REPUBLIC Miroslava Brožová
Abstract The present study reports on the use of (or lack thereof) textbooks in three different elementary level mathematics teaching methods – the Hejny method, the Montessori method, and the traditional approach. The aim of this study was to find out if the teachers, using the different teaching methods, use textbooks while teaching mathematics and if the textbooks play a key role in their classrooms. Analysis of interviews with six teachers showed that using textbooks depends on the teaching method. With the Hejny method and the traditional approach, textbooks (workbooks) are followed. On the other hand, teachers see them as the source of inspiration in the Montessori method. Keywords: teaching method, elementary, mathematics, Hejny, Montessori Introduction The majority of educators have been using textbooks as a significant resource for teaching mathematics. A lot of activities in teaching and learning mathematics are associated with a textbook (Rezat, 2006). The question is to what extent the teachers use textbooks and how it influences their teaching and the students´ learning. Japanese teachers claim that there is a difference between “teaching the textbook” and “using the textbook to teach mathematics” (Takahashi, 2016). Textbooks are designed for the purpose of helping teachers organize their teaching. “There is a good deal of evidence that many teachers like the security and freedom from responsibility that a text series provides” (Love and Pimm, 1996, p. 384). To use textbooks effectively, teachers need to evaluate both the limitations of the textbook and the potential in order to use them as a support in a way that corresponds to their pedagogical intentions. Teachers in the Czech Republic can choose from variety of textbooks according to their teaching style, program and the purpose of using it, or the approach by which the textbook was written. They are not usually governed or limited in the choice of textbooks (the commercial criteria can be a factor) but in many cases we can find the same textbook used throughout a whole school. It is the responsibility of each individual teacher to choose the right textbook that most suits their situation. In reality it is more practical to choose a textbook together and discuss possible usage with colleagues. Johansson (2006) listed several aspects of instructions that textbooks rarely determine, for instance, time allocation and standards of students’ achievement. The individual teacher can often decide:
Charles University, Czech Republic; e-mail: [email protected]
91 a. Which textbooks to use, b. which sections of the textbook to use, c. when and where the textbook is to be used, d. what topics to teach, e. sequencing of topics in the textbook, f. how much time to spend on each topic, g. the way in which pupils engage with the text, h. the level and type of teacher intervention between pupil and text. (Johansson, 2006, p. 57) The main purpose of this study is to describe specific aspects of three teaching methods or approaches in which the author of this paper is interested. Firstly, it was intended to do a textbooks analysis but it was found that not all the teachers use textbooks. Therefore the following aim was set up. The aim of the study is to look for answers to the following questions. 1. Do the teachers, using the Hejny method, the Montessori method, and the traditional approach, use textbooks while teaching mathematics? Do the textbooks play a key role in their classrooms? For what purposes do they use them? 2. How does the textbook control or influence teaching and learning? How are the textbooks used for individualization or differentiation of students? Theoretical Framework The data from the textbooks´ analysis part of TIMSS study (Valverde et al., 2002, p. 17) shows that textbooks around the world are not similar. They exhibit substantial differences in presenting and structuring pedagogical situations and these differences are systematically related to country, grade level, and subject matter differences. In her study, Moraova (2014) looks at the textbooks of mathematics as texts with some content. A mathematics textbook can be also seen as a product of a given society, representing its values, beliefs and opinions on what it means to be doing mathematics; to be doing it at school, to be working with a textbook, and to be learning. We can find various methods to present the same mathematical topic in different textbooks. An author can make an effort to cover all the topics from a certain curriculum in one book and the teachers still look for other sources. Connections between topics can be illustrated in different ways and the topics can be organized differently. Freeman and Porter indicated that teachers do not always defer to the authority of their textbooks when deciding about the topic, how much time to spend on each topic, or the order in which topics are presented. (Freeman and Porter, 1989). Textbooks usually offer the content and the pedagogical approach that is implemented in the curriculum. (Reys et al., 2003). For studying about textbook use, it is essential to study the teaching method first. Significant points of the three methods will be described in the following paragraphs.
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The Hejny Method This method is based on scheme-oriented education. “…it is necessary that a pupil is intellectually autonomous in the sense that he/she discovers new ideas or gets to them by communicating with classmates or takes them over from the classmates. It follows that a teacher’s role is indispensable for scheme-oriented education...” (Hejný, 2012, p.47) From the perspective of individualization and differentiation in this approach, exercise sets in the textbooks and workbooks are graded according to their level of difficulty. It is also extremely common, for example, in Swedish mathematics textbooks (Johansson, 2006). This kind of textbook could be seen as one solution to the problem: How to manage a nonhomogeneous group of students so that each individual student can work according to his/her skills and needs. Johansson (2006) says that if problems in the book are graded by level of difficulty, it facilitates the individual work and progress of the students. According to Vališová and Kasíková (2007), there are two basic principles of individualization: Principle-managed learning and the principle of continuous learning progress. The first principle can be interpreted so that each learner will have the chance to achieve his educational goal independently and in a different way. For instance, a slower learner can have more time to reach the goal. Teachers use alternative means of teaching, adapted to the learner´s abilities and learning style. Correction is offered, if necessary (Vališová and Kasíková, 2007, p. 155). The method has been adopted by more than 750 of the 4 100 Czech schools at the elementary and lower-secondary level. Hejny’s edition of textbooks for elementary schools has been approved by the Czech Ministry of Education in 2007. (www.h-mat.cz) The Montessori Method According to Ryniker and Shoho (2001), the Montessori approach is based on the tenet that children learn most effectively when information is developmentally appropriate. Central to this approach is the notion that children’s natural tendencies “unfold” in specially designed multi-age environments that contain manipulative self-correcting materials (North American Montessori Teachers’ Association, 2003). In this approach, children learn at their own pace through manipulation of objects. As such, personal independence, self-discipline, and initiative are essential for learning and motivation, with motivation purportedly fostered through interactions in the environment (Kendall, 1993). Harris and Callender (1995) contend that the emphasis on these aspects leads to “inner discipline.” Furthermore, Montessori is distinct in that it does not use textbooks, worksheets, tests, grades, punishments, or rewards (Haines, 1995).
93 The second principle from Vališová and Kasíková (2007), the idea of continuous learning progress, is consistent with the Montessori method. Two learners of the same age can work on different tasks, solve different problems, or read on different pages in the book. Moreover, the students can work in different subjects at the same time. The teacher respects the needs and interests of each learner and encourages them in their overall development to reach their potential. “This principle can be interpreted so that each learner should still move to the new curriculum requirements in order to achieve all that he is able to achieve in a certain time and under certain conditions. For example, we would not expect faster and more motivated learners to wait for the others, and learners should not lose time repeating tasks already mastered.” (Vališová and Kasíková, 2007, p. 155). The Traditional Approach The traditional transmissive approach relies on the teachers where the responsibility for the students´ achievement always falls on them. The pupils need to practice and repeat math facts and usually do not get a chance to think, discover, or simply understand independently. In this study author’s opinion, a truly traditional approach is not common in the Czech schools. Teachers attend many teaching development courses where they are introduced to different methods but it does not change their fundamental beliefs. The instructions of teachers do not look traditional because the teachers integrate projects and explorative activities but the essence of their work is still the same – they are always correct and the solving method they introduced (or let the children explore) is the only one. Although the traditional approach cannot be considered as a precise term, in the interest of clarity in this study, the approach where the teachers did not define any specific method for teaching mathematics, will be called the traditional approach. Methodology The Grounded theory method was used for analyzing the data in this study. This method was developed by Glaser and Strauss (1997) and has been used for generating higher level understanding that was derived from a systematic analysis of data. Participants Six teachers, teaching mathematics at elementary level (first through fifth grade where pupils are 6-12 age), were identified for the method they use while teaching. Two of the respondents, who will be identified as A, B, were intentionally chosen for Hejny method. Two other respondents (C, D) were chosen from the schools working according to Montessori method and the other two (E, F) fell into the category of the traditional approach. The teachers work in Prague (A, B, C), Pardubice (D), Nepomuk (E), and Hostivice (F), Czech Republic. One other teacher from Prague was not included in this study for practical reasons, such as a lack of time and sickness. Six teachers of different
94 ages and length of experience (3.5 – 20 years) therefore participated in the study. Teachers were asked to an interview for half an hour. Data Collection The study used a semi-structured interview (Fylan, 2005) with a paper based interview guide. The guide included a list of 15 open-ended or closed-ended questions and also topics that needed to be covered during the conversation, in a particular order, if possible. All participants were informed about the background of the study before the interviews started. Two pilot interviews with other elementary teachers were carried out in the first phase. The interviewer followed the guide, but was able to follow topical trajectories in the conversation. Questions were focused on using textbooks during lessons (for instance Do you use textbooks while teaching?). Interviews took place in February 2017 and lasted from 12 to 44 minutes (average 27). The longest interview was about textbooks used in teaching style, which was unfamiliar for the interviewer, and it caused additional questions and prolonged the time of interview. Recordings took place in a relaxed atmosphere, mostly in a separate room with a minimum of external influences; only two of them were outside the schools. The interviews were recorded with the approval of the teachers on the voice recorder. Data analysis Audio recordings were transcribed verbatim and have been downloaded to the Atlas.ti. Each of the answers was labeled (A1, A2). The data was coded, using the techniques of grounded theory, in three levels (Strauss and Corbin, 1997). In the first level (open coding), the attention was focused primarily on statements related to using textbooks and workbooks, however, communication describing the teachers’ beliefs and teaching style and other comments that could affect the examined area were taken in account. The axial and selective coding of the data aimed at specifying the core categories. The teachers’ statements were translated by the author of this study. Findings The study shows that the textbook influences what kinds of concepts of mathematics are introduced and how they are introduced. Some textbooks are written purposefully for teaching method and the textbook plays a key role in the teaching method, for instance within the Hejny method. For better understanding of the methods, respondents said about Hejny method during our interviews: A5 It is discovering method. Mathematics is taught the way that children discover by themselves…, I often hear: Aha, yes, it is that… There is a feeling that I found out something. It is open approach for discussion, we discuss very often during the whole lesson (45minutes). Not all the children achieve the same goals.
95 B11 The most important is constructivist approach.1 Children explore and explore and then create the abstract knowledge. Children enjoy learning, not in a sense that it is colorful but because the problems are tricky and children have a chance to explore. Respondents say about Montessori method: C9 There is an effort of individualization in a sense that each child is considered individually. Another important thing is a prepared environment where we can find a prepared teacher and didactical material which is very specific. D9 It is an incredibly sophisticated system of mathematics, but not in relation to a teacher but in relation to the child. The system is based primarily on the amazing teaching aids… using those aids the children explore a lot of important things. Those aids have, in fact, completely different meaning than in the traditional school. In a traditional school, the didactical aid serves as a tool to make children better understand a problem, or a situation, but in Montessori the aid serves as a tool for exploration… Teachers (E, F) say that they do not use any specific method of teaching mathematics (E7, F7). There is a good deal of evidence that many teachers depend upon textbooks (Love and Pimm, 1996). To answer the first of the author´s questions in this study, the teachers confess that they follow the textbooks (A, B, E, F). Some teachers think that they need textbooks or workbooks as a key source for their teaching: A13 – It is pleasant to have a workbook because I know the sequence and the work is prepared. B21 – It is kind of relief if the book is good and the teacher has the experience with it. It is pleasant because he doesn’t need to have a pile of books and look for other resources. It is also easier for the parents because they have it if the child is unable to attend school. E13 I do not need the books in the first and second grade. Subject matter is more complicated when the children are older and I also verify if the method I use is appropriate. I also use the exercises. The other teachers do not use the textbooks in the same way. Both Montessori teachers claim that they teach without a textbook and use it only as inspiration sources: C14 We do not use the textbooks in our method the way they are used in the traditional schools. The pupils never use them. The only book the pupils in my classroom had was the little workbook where they practiced math facts. D18 Textbooks do not exist in Montessori method because every child works
1 Hejný uses the term constructivist education in the sense of Noddings, 1990 (Hejný, 2011)
96 on something different in the same time; everyone works according to what he needs and does not have to go through everything. One child understands when he just looks at it and the other child needs to spend weeks before he realized a principle… Every child needs something different and textbooks basically do not serve it. It is not working. It is fundamentally impossible… C26 Homework is not assigned on a regular basis. However, when I do give assignments, students are supposed to work with tasks from a chosen textbook or a copied page. Takahashi (2016) put the stress on teachers´ knowledge: “In order to provide a better learning experience for their students, all the teachers should be able to use the textbook to teach mathematics effectively using their knowledge and expertise for teaching mathematics.” (Takahashi, 2016, p. 318) Four out of six respondents in this study admitted that they needed to study specifically the method (C, D) or the method and using the textbooks (A, B). D57 Manuals for each phenomenon are provided during the Montessori teachers’ training. They accurately describe each phenomenon because isolation of phenomenon, which means that every single thing needs to be explained separately, is a very important principle in Montessori. Teachers use different parts of the textbooks. Some of them say that they use all the parts and pupils complete almost all the exercises in the workbooks (A, B, E). Sometimes they use additional sources because they miss the important topics in the books (F). A28, 32 Sometimes I include technical drawings from other sources because we cannot find it until upper grades of elementary program in our textbooks and workbooks. B15 Textbooks and the method include open-ended questions for guiding students to develop discussion and understanding, instead of explanations of the concepts and the procedures. C44 I need to concede that sometimes I am very critical. My biggest complaints are about cartoonlike characters which move away the attention from providing clear explanations of the concepts and procedures or from simple practicing. Textbooks are usually written for one subject. One of the respondents (A) agreed (A41) with the statement that in her classroom they use the method (concretely discussion) in other subjects and the method from one subject can change the way teachers teach mathematics. For individualized teaching, the textbook is especially helpful if the tasks are graded by level of difficulty. A7 I write several exercises on the black board and tell the children to choose what they want to do, what they feel like doing. They can work at their pace. When they are ready they get extra work which is graded by level of difficulty. They choose easier or more difficult work according to their interest.
97 For other teachers, the textbook is used for individualized looking for the information. D10, 11 Textbooks are not necessary as in the traditional schools. They are excellent to have one or two copies in the school library because it is kind of summary, it is short and simple and understandable source of information especially for younger pupils... Conclusion and Discussion The use of textbooks is not a consistent decision. Teachers are not forced to use or not to use the book in any teaching method in the Czech Republic. The choice of textbook can be determined by directors of school, school methodologist, the price of the book, or the method of teaching. The respondents, teaching according to the Hejny method, use the textbooks written specially for this method.2 They follow it closely and emphasize the positive aspects of the use such as carefully invented and constructed tasks already thoroughly arranged in the book or awareness of the parents who can work with workbooks at home. In teachers´ opinions, following the textbook closely prevents them from skipping important topics out of an appropriate sequence. In the Montessori method, where textbooks and workbooks are not followed, they do not have a book per child but rather can choose to borrow and research independently from the school library. Students in Montessori have their own little books (or worksheets) where they can work on one isolated concept. Those little books are usually constructed on the wooden Montessori material base (pupils work with wooden material and write the results in the little books or work with the book abstractly). Teachers use the regular sets of textbooks and workbooks mainly as sources of inspiration. In the schools with traditional approach, the regular textbooks and workbooks are followed closely. These books are used for the purposes in the sense that Johansson (2006) described in her study. The textbooks: a. are artifacts that preserve and transmit knowledge in the educational systems, b. facilitate the daily work of the teachers, c. can be seen as some kind of guarantee that the students have the necessary basic knowledge and training for the next level in the school system, d. can be regarded as tools to accomplish uniformity and consistency within the school system, for example with respect to a curriculum, f. seem to reduce both freedom and responsibility of the teachers (Johansson, 2006, p. 28) Even if the textbook plays the key role for the teaching, it does not decide all the activities for the lesson. Teachers need guidelines from the authors or, in some cases, special education, but do not always have the access to them.
2 Textbooks used in Hejny method have been analyzed by Sovič (2016)
98 Textbooks and authors´ guidelines are extremely supportive for the teachers who are not confident in the teaching methods or theoretical aspects of mathematics. Textbooks can influence the method of teaching and the experience from another subject can change the way teachers teach mathematics. For individualized teaching, the textbook is especially helpful if the tasks are graded by level of difficulty or they are used individually as a source of inspiration. References Freeman, D. J., Porter, A. C. (1989). Do textbooks dictate the content of mathematics instruction in elementary schools? American Educational Research Journal, 26(3), 403–421. Fylan, F. (2005). Semi-structured interviewing. A handbook of research methods for clinical and health psychology, 65–78. Haines, A. M. (1995). Montessori and assessment: Some issues of assessment and curriculum reform. The NAMTA Journal, 20(2), 116–130. Harris, I., Callender, A. (1995). Comparative study of peace education approaches and their effectiveness. The NAMTA Journal, 20(2), 133–144. Hejny, M. (2012) Exploring the Cognitive Dimension of Teaching Mathematics through Scheme-oriented Approach. Orbis Scholae, 6(2), 48. Hejny method. (n.d.). Retrieved from http://www.h-mat.cz/en/hejny-method. Johansson, M. (2006). Teaching mathematics with textbooks: a classroom and curricular perspective (ed.). (Doctoral dissertation.) Paper presented at Luleå: Luleå tekniska universitet. Kasíková, H., Dittrich, P., Valenta, J. (2007). Individualizace a diferenciace ve škole. In A. Vališová, H., Kasíková (Eds.), Pedagogika pro učitele (pp.153–164). Praha: Grada Publishing. Kendall, S.D. (1993). The development of autonomy in children: An examination of the Montessori educational model. The NAMTA Journal, 18, 64–83. Love, E., Pimm, D. (1996). This is so´: a text on texts. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, C. Laborde (Eds.), International handbook of mathematics education (Vol. 1, pp. 371–409). Dordrecht: Kluwer. Moraová, H. (2014). Pre-service and in-service teachers' preference when selecting mathematics textbooks In K. Jones, Ch. Bokhove, G. Howson, L. Fan (Eds.), International Conference on Mathematics Textbook Research and Development 2014 (ICMT-2014) (pp. 351–356). Southampton: University of Southampton. North American Montessori Teachers’ Association (2003). Introduction to Montessori education. Retrieved from ww.montessorinamta.org/NAMTA/geninfo/whatismont.html. Reys, R., Reys, B., Lapan, R., Holliday, G., Wasman, D. (2003). Assessing the impact of standards-based middle grades mathematics curriculum materials on student achievement. Journal for Research in Mathematics Education, 34(1), 74–95.
99 Rezat, S. (2006). A model of textbook use. Paper presented at the PME 30, Mathematics in the centre, Prague, Czech Republic. Ryniker, D. H., Shoho, A. R. (2001). Student perceptions of their elementary classrooms: Montessori vs. traditional environments. Montessori Life, Winter, 45–48. Sovic, P. (2016). Podnětná výuka obsahu trojúhelníku a rovnoběžníku ve dvou třídách s odlišnou zkušeností s výukou matematiky. (Diploma thesis.) Prague: Charles University, Faculty of Education. Strauss, A., Corbin, J. M. (1997). Grounded theory in practice. London: Sage. Takahashi, A. (2016). Recent Trends in Japanese Mathematics Textbooks for Elementary Grades: Supporting Teachers to Teach Mathematics through Problem Solving. Universal Journal Of Educational Research, 4(2), 313–319. Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., Houang, R. T. (2002). According to the Book. Using TIMSS to investigate the translation of policy into practice through the world of textbooks. Dordrecht: Kluwer Academic Publishers.
PARAMETERS INFLUENCING DIFFICULTY OF WORD PROBLEMS - A CASE OF FOUR WORD PROBLEMS IN GRADE 4 Jiří Bruna and Radka Havlíčková
Abstract This paper reports on an inquiry into parameters which influence difficulty of a word problem. The study that we present is a part of a larger research project and focuses on a particular case where groups of fourth graders have been given four versions of one word problem which differed in two parameters – the representation of a numerical value and a number of message levels. Although no substantial difference in success rate has been observed between the versions, we give a brief account of differences in mistakes pupils made and contrast them with mistakes of third graders who have been asked to solve one version of the problem as well. Keywords: word problems, parameters of word problems, representation of a numerical value, message levels, difficulty of word problems, mistakes, grade 4
Introduction In this paper we describe partial results of the research focused on identifying aspects of word problems which make them difficult for pupils to solve. It is a collaborative effort between the Department of Mathematics and Mathematical Education, the Department of Czech Language and the Department of Psychology of the Faculty of Education at Charles University, Prague. This research can be seen as a continuation of previous research project (Vondrová, Rendl et al., 2016) aim of which was to identify areas problematic for pupils of elementary and lower secondary level of Czech schools. The main idea behind the project is to analyse
Charles University, Czech Republic; e-mail: [email protected], [email protected]
100 word problems mainly in terms of mathematical, linguistic and psychological parameters, vary individual word problems using these parameters and observe how change in parameters influences pupils’ solution of these problems. In this paper we focus on results obtained during the first phase of testing in schools concerning a single quadruplet of word problems. Following paragraphs describe theoretical underpinning and methodology related to this phase of testing, as well as the results themselves and the discussion. Theoretical framework The topic of parameters influencing the difficulty of word problems has been already covered in literature to some degree. For comparison see for instance (Verschaffel and De Corte, 1993), (Martin and Bassok, 2005) or (Daroczy et al., 2015). However, no comprehensive study has been carried out in the Czech Republic. Previous research (Vondrová, Rendl et al., 2016) has identified word problems as one of the problematic areas for Czech pupils. Natural extension of these findings is to look for the roots of difficulties in word problems themselves. Their structure, language used and contextual factors. The main aim of this section will be to make clear what we understand by word problem and what the parameters we focus on in this paper are. In research literature there is no singular definition of a word problem. We choose to continue in line with the afore mentioned research in Czech school environment and define word problems as problems which include some context within which some numerical data are given and a question, or possibly multiple questions, is posed for pupils to solve using their mathematical knowledge and out-of-school experience. Context of the word problem can be real, real-like or imaginary. The research project as a whole seeks to look into multitude of parameters which potentially influence the difficulty of a word problem. This paper, however, focuses only on two of them. The first one is the way a numerical value is represented in the formulation of a word problem. We distinguish two basic categories – a number and a number word. For better understanding of this distinction consider following two sentences. Homework from mathematics is on page 26 in the workbook. Homework from mathematics is on page twenty-six in the workbook. Clearly, the first sentence uses a number to express the value associated with page address, whereas the second one uses a number word. As far as research literature goes (Daroczy et al., 2015), for instance, in their review refer to several studies documenting the effect of linguistic aspects of number words on numerical cognition but at the moment we are not aware of any literature studying the effect of value representation on difficulty of a word problem. Our assumption is that representing a value using a number word(s) may be more difficult for pupils as they first have to process the word and convert it into a number for the sake of the solution process.
101 The second parameter featured in this paper is a linguistic parameter called message levels. It was first introduced by our esteemed colleague Milada Hirschová, a team member, for the purposes of this particular research. It is a term that builds on the idea that from the point of view of pragmatics a word problem is a message and as such is structured into layers, or levels, hence message levels. Let us illustrate this using following sentences. Homework from mathematics is on page 26 in the workbook. Pupils learned that homework from mathematics is on page 26 in the workbook. As you can see there is an additional level present in the second sentence that encapsulates the original message by expressing a perspective of a protagonist, in this case pupils. It is also important to point out that in word problems which we study this additional level does not contain any new information required to solve the word problem. On one hand, it is reasonable to assume that removing such a level would simplify the task for the pupils. On the other hand, although the level does not provide pupils with crucial information it often contains narrative elements that may motivate pupils to solve the problem. This research tries to establish which of these two assumptions is more relevant in the case of the four word problems.
Methodology The purpose of the section on methodology is threefold. Firstly, it describes the sample of pupils, namely the criteria for selection and its size. Secondly, it discusses the way the particular quadruplet of word problems used in this research was created and the way individual word problems were distributed among the pupils. Lastly, it looks at the process of pupils’ solution analysis in terms of solution strategies and scoring. As far as the sample is concerned, the pupils who participate in the testing come from four primary schools in the area of outer Prague. These schools have been selected based on following criteria. Firstly, no specialisation, as for example on gifted pupils or pupils coming from disadvantaged background. Secondly, sufficient size of at least two classes per grade. Thirdly, we required that, if possible, pupils should come from immediate surrounding of the school with percentage of pupils from abroad not exceeding the national average due to language concerns. Into the research we included all classes of grades 3 through to 9, securing the sample of roughly 2100 pupils – 2101 to be precise but due to fluctuations in the number of pupils attending the schools and their absence for testing, the sample size may slightly vary for each phase of the testing (see below). For the purposes of this paper, however, we consider only pupils of grade 3 and 4 as the word problems which are the focus of the paper appear only in tests for these grades. That amounts to a reduced sample size of 759 pupils.
102 After the schools were selected, pupils of respective grades were given two initial tests. One focused on mathematical knowledge of the pupils, the other was aimed at knowledge of the Czech language. Both were grade specific and reflected the educational programmes of the schools. The test on mathematical knowledge was distributed to pupils of grades 3 up through to 9 while the Czech language test only to grades 4 up through to 9. Furthermore, the test on mathematical knowledge was divided into the numerical/calculative part and the part containing word problems. The purpose of these tests was to gather data based on which the pupils in the individual classes would be divided into eight comparable groups in terms of size and knowledge of the subjects. In order to do so following aspects were considered. The average score – by score we mean the percentage of the maximal number of points for each part that the pupil obtained. The second aspect was the standard deviation of the score and the third one the highest and lowest achieved score. First the pupils were sorted by their score (average of the scores of the two tests), the class and school they attend, which created an ordered list of all pupils. Each pupil was then assigned one of the following versions of the test (see below for more information) 1A, 2B, 3A, 4B, 1B, 2A, 3B, 4A – in this particular order. By doing so every class was divided into eight groups of pupils. These groups were further adjusted by moving individual students so that the three aspects mentioned above were balanced across the groups. Although learning disorders and mother tongue other than Czech of some pupils were taken into account, no further adjustment was felt necessary due to these factors. This distribution produced typical difference in average score between the groups of no more than 2 %, 4 % in extreme cases, standard deviations in the range of 20 % to 25 % and range of scores in each group between 5 % and 95 % – again between 0 % and 100 % in extreme cases. For the main testing a test has been created in eight versions, as mentioned above. The idea behind this test was to include between four to six word problems, depending on the grade, and for each word problem create four versions of this problem by varying a specific parameter or a couple of parameters. Each of the four versions of one word problem was included in two versions of the test. These two versions differed only in the order in which the problems were presented to pupils. This was due to the concern that pupils sitting next to one another would otherwise be tempted to copy the solution as the word problems looked visually almost identical. In order not to discourage pupils, first word problem in every test was a problem with a higher success rate. The estimated rate of success had been established by a pilot. The pilot had also helped to establish the appropriate number of word problems in a test for each grade based on the time it took students to solve it, and pointed out potential pitfalls in the text of the problems. The following schematic shows the distribution of four versions of word problems into eight versions of the test for the case of four word problems A, B, C and D, where A1 to A4, B1 to B4 etc. stand for individual versions of these problems.
103 T1A T1B T2A T2B T3A T3B T4A T4B Task 1 A1 D2 A2 D3 A3 D4 A4 D1 Task 2 B4 C3 B1 C4 B2 C1 B3 C2 Task 3 C3 B4 C4 B1 C1 B2 C2 B3 Task 4 D2 A1 D3 A2 D4 A3 D1 A4 In order to minimise copying even further pupils were asked to follow the seating arrangement shown in the table below. Every row represents a pair of pupils seated next to one another. T1A T3B T2A T4B T3A T1B T4A T2B The tests were also designed so that every pair of consecutive grades shared a word problem, referred to as an anchoring problem. Every anchoring problem is present in four versions in a particular grade but only one version of this problem appears in the lower grade. This provides further possibilities in statistical analysis of obtained data for all seven grades, The four word problems that we look into in this paper are in fact versions of the anchoring problem used in grade 3, with one version present, and grade 4, with all four versions present. Tables below show the composition of tests for grades 3 and 4. The word problems in question are marked 4A1 to 4A4. Version 4A3 is the one included in grade 3. The last column shows expected success rate based on the pilot. Grade 3 T1A T2A T3A T4A Success rate Task 1 3A1 3A2 3A3 3A4 65% Task 2 3B4 3B1 3B2 3B3 40% Task 3 4A3 4A3 4A3 4A3 45% Task 4 3C3 3C4 3C1 3C2 ??
Grade 4 T1A T2A T3A T4A Success rate Task 1 4B4 4B1 4B2 4B3 70% Task 2 4A1 4A2 4A3 4A4 50% Task 3 4C3 4C4 4C1 4C2 50% Task 4 5B2 5B2 5B2 5B2 70% Task 5 4D2 4D3 4D4 4D1 70% As mentioned above, the quadruplet of word problems has been generated by varying certain parameters of the problem, in this case a representation of a value and message levels. The base problem used numbers for expressing values and had no level expressing other person’s perspective. It was labelled 4A1. It is also
104 noteworthy that numbers used in the problem represented not only quantities but also addresses. What follows is the translation of the base problem into English. (4A1) Homework from mathematics is on page 26 in the workbook. That is exactly 7 pages past the half of the workbook. How many pages are there in the workbook? The other versions have been derived by exchanging numbers for number words to represent values (4A2), by adding the message level expressing a personal perspective (4A3) and finally by combining both procedures (4A4). What follows are translations of these versions. (4A2) Homework from mathematics is on page twenty-six in the workbook. That is exactly seven pages past the half of the workbook. How many pages are there in the workbook? (4A3) Pupils learned that homework from mathematics is on page 26 in the workbook. Fanda said: “That is exactly 7 pages past the half of the workbook.” How many pages are there in the workbook? (4A4) Pupils learned that homework from mathematics is on page twenty-six in the workbook. Fanda said: “That is exactly seven pages past the half of the workbook.” How many pages are there in the workbook? As the evaluation of the main testing is at the time of writing this paper in an early phase we attempt only a rudimentary analysis of pupils’ solution. We combine basic quantitative analysis, with the goal of describing the differences in success rates between individual versions of the word problem, with qualitative analysis regarding the differences in pupils’ mistakes. Following paragraphs provide a better description of the two aspects of the analysis. With regard to the quantitative part, solution of each pupil has been assigned a certain number of points. Three points have been given to correct solutions, irrespective of the approach a pupil chose. Two points have been given to correct solutions which contained some error in calculation. One point has been given to a partially correct solution, that is a solution in which some correct steps were taken but the pupil was not able to finish the solution and arrive at the correct answer. Zero points were given to incorrect solutions and in instances where the solution was missing. In this part of analysis we are mostly concerned with the total number of pupils as well as percentage of those who scored three, two, one and zero points respectively to assess differences in success rates between the versions of the problem. As far as the qualitative analysis is concerned, we are interested in two kinds of phenomena. Those which generally apply to word problems, such as the presence of a graphic legend, presence of a verbal legend, legend represented as a table, algebraic legend, solution process written down and/or answer given in a sentence, and those which apply directly to the four word problems, most
105 notably steps in expected solution and anticipated mistakes. These were arrived at through analysis apriori and data from the pilot. The table below shows the list of these phenomena and coding used in the research. 4Ap1 Determining half the number of the pages (26 – 7) 4Ap2 Determining the number of all the pages (times 2) 4Ap3 A different method of solution 4Ach1 Mistaking additive relation for multiplicative one (e.g. 26 × 7) 4Ach2 Solution ends at the first step (that is 26 – 7 or incorrect 26 + 7) 4Ach3 Pupil calculates 26 +7 (presumably due to a counter-signal word 4Ach4 Pupil does not deal correctly with “half” (either ignoring it or giving incorrect calculation) 4Ach5 Misreading the numbers (where expressed by words) 4Ach6 A different kind of mistake
Results and discussion Out of 759 pupils only 650 took the test. The rest was absent on the day of testing and as of the time of writing this paper we have not been able to procure and analyse their solutions. Of these 650 pupils 307 were fourth graders and 343 were third graders. The tables below show the distribution of points pupils of grade 4 received for their solution. First in absolute numbers, then in percentage of pupils who took that particular version of the word problem. Percentages are rounded to one decimal place. Grade 4 – absolute numbers 4A1 4A2 4A3 4A4 0 points 20 22 18 19 1 point 13 9 12 12 2 points 0 1 2 3 3 points 43 46 46 41 Total 76 78 78 75
Grade 4 – percentages 4A1 4A2 4A3 4A4 0 points 26.3% 28.2% 23.1% 25.3% 1 point 17.1% 11.5% 15.4% 16% 2 points 0% 1.3% 2.6% 4% 3 points 56.6% 59% 59% 54.7%
As you can see the number of pupils for each version is virtually the same. Likewise, it appears, is the distribution of points. All the differences between versions for a given number of points are contained within a 5-percentage-point
106 interval, with a singular exception, where 17.1 % scored one point for version 4A1 while only 11.5 % scored the same amount of points for version 4A2. Based on this observation we conclude that the influence of parameters in question on the success rate is very limited. As for the representation of a numerical value, misreading the number word was not a problem for pupils, as only 5 pupils out of 307 made this kind of mistake. We state in the theoretical framework section that we expect representation of a numerical value by a number word to be more problematic for pupils due to the need to interpret the word and then translate it into a number. In the case of these four word problems this does not seem to hold true. We speculate that this may be so due to the size of the numbers, or more precisely, due to the shortness of number words. As for the parameter of message levels we see two possible explanations of its limited influence. First, the necessary abstraction away from protagonist’s perspective is not, in this case, demanding enough to become a hurdle in a solving process. Second, the need of abstraction and narrative aspects of the added message level, which go in opposite directions when it comes to difficulty, cancel out. As stated before, 343 third graders were given version 4A3 of the word problem. For the sake of comparison, we provide their results – distribution of points both in absolute numbers and percentages – in the table below.
Grade 3 – version 4A3 Absolute numbers Percentages 0 points 168 49% 1 point 44 12.8% 2 points 10 2.9% 3 points 121 35.3%
When compared with the results of fourth graders who took the same version of the problem, the differences are quite pronounced with respect to those who scored zero and three points. The percentage of those who scored the maximum amount of points is roughly 25 percent points lower than in fourth graders. On the other hand, the percentage of those who scored zero points, that is those with no correct steps in solution or with missing solution altogether, is roughly 25 percent points higher. While the influence of the parameters on the distribution of points in fourth graders is limited, the versions of the word problem are not equal when it comes to mistakes. A striking difference has been noticed with respect to a phenomenon coded 4Ach2, that is the solution ends at the first step. For versions 4A1, 4A3 and 4A4 there were 11, 9 and 12 pupils respectively (14.5 %, 11.5 % and 16 % of those who took each respective version) who got stuck after making the first step.
107 For version 4A2 there were only three such pupils, amounting to 3.8 % of those who took this version. The reason for this discrepancy has not been studied. Last remark we make in this section concerns the difference in mistakes specific to third graders and those specific to fourth graders. With respect to third graders, by far the most frequent incorrect result was 33, at 79 out of the total number of 151 incorrect results, or 52.3 %. This result is also present in the solutions of the fourth graders but its rate of incidence is much lower – 27.8 % of all the incorrect results in grade 4 and 20 % of incorrect results of fourth graders who took version 4A3. This result is obtained by adding 26 and 7 which means that pupils chose incorrect operation, possibly due to counter-signal word “past”, and their solution does not reflect “the half” in the formulation of the problem. On the other hand, no third grader arrived at the result 182, which is the product of 26 and 7, while 11 fourth graders, amounting to 13,9 % of all incorrect results in grade 4, did. Of those fourth graders who took version 4A3 three arrived at this result, which amounts to 15 % of all incorrect results in this group. One possible explanation of this discrepancy lies in the fact that multiplication of two-digit numbers is not readily available to third graders as it is to fourth graders. Concluding remarks In this paper we studied the influence of two parameters – a representation of numerical value and number of message levels – on the difficulty of a word problem. At this point it seems too early to derive solid conclusions based on analysis presented above. After all, the similarity of the distribution of points between respective versions of the problem is not mirrored by the same degree of similarity when it comes to mistakes pupils made. This discrepancy is poorly understood as of yet and may serve as a point of departure for subsequent research. With regard to subsequent research, it seems appropriate at this point to clarify the link between the presented research and the topic of the conference, that is equity and diversity in elementary mathematics education. Firstly, although the differences in success rates for the four versions of the word problem are very similar, there are still discrepancies. One possible source of these discrepancies is the way specific groups of pupils react to particular values of the parameters. Interviews with pupils, which are to be conducted as a part of the overarching research project, will look, among other things, into how variations in parameters help or hinder the solution process in pupils with differing abilities and backgrounds. Secondly, the project as a whole builds on a premise that the success in dealing with mathematical content, and by extension the experience of a pupil of being successful in dealing with it, is also dependent on parameters of the presentation of this content which are non-mathematical in nature. In other words, the diversity of a formulation of a word problem results in diversity in success in groups of pupils which are equal or comparable from the point of view of mathematical proficiency. Even though the parameters studied in this paper seem
108 to show limited significance in this respect, as the research project progresses, more parameters will be put under more careful scrutiny, both quantitative and qualitative, with the aim to inform educational research and classroom practice. Acknowledgment The research described in this paper is supported by Czech Science Foundation in the project called Context problems as a key to the application and understanding of mathematical concepts, registration number 16-06134S. We would also like to thank our esteemed colleagues, and team members, Martin Chvál from Institute of Research and Development of Education, at the Faculty of Education, who provided us with materials that inspired our methodology section, and Milada Hirschová from the Department of Czech Language at Faculty of Education for kindly clarifying the idea of the parameter of message levels to us. References Daroczy, G., Wolska, M., Meurers, W. D., Nuerk, H. C. (2015). Word problems: a review of linguistic and numerical factors contributing to their difficulty. Frontiers in psychology, 6. Martin, S. A., Bassok, M. (2005). Effects of semantic cues on mathematical modeling: Evidence from word-problem solving and equation construction tasks. Memory & cognition, 33(3), 471–478. Verschaffel, L., De Corte, E. (1993). A decade of research on word problem solving in Leuven: Theoretical, methodological, and practical outcomes. Educational Psychology Review, 5(3), 239–256. Vondrová, N., Rendl, M. (2015). Kritická mı́sta matematiky zá kladnı́ š koly v ř eš enı́ch ž á ků . Praha: Univerzita Karlova, Karolinum.
PROGRESSIVE DEVELOPMENT OF PERCEPTION OF THE CONCEPT OF A SQUARE BY ELEMENTARY SCHOOL PUPILS Irena Budínová
Abstract Developing geometric thinking is a long-term path and requires an informal encounter of pupils with geometric concepts and their properties. The first years of education are really essential because intuitive perceiving of the world is developed. Geometric shapes are one of the first subjects in mathematics education during the early stage of elementary school. Children form conceptions about the concepts on the base of their experiences. Some of these early conceptions about geometric shapes might be incorrect, which might negatively impact children’s further understanding of geometric shapes. In this study, we present outcomes of a research that dealt with ability of pupils to think about plane figures, and we show possibilities of developing geometric thinking of pupils by tasks which are stimulating enough for them.
Masaryk University, Czech Republic; e-mail: [email protected]
109 Keywords: Development of geometric thinking, geometric concepts, cognitive process in geometry
Introduction One of the first geometrical topics dealt with by children in early school or preschool education are geometric figures. Already before starting school attendance, children obtain basic information on geometric figures from the world around them. Some of this early information may be wrong and can have a negative impact on the child’s future understanding of geometric figures. In this contribution, there are shown outcomes of a study that dealt with ability of pupils to think about plane figures. Misconceptions in pupil´s concepts are also included. Further on, there is shown an activity which can develop children’s understanding the geometric concepts. Theoretical background Concerning the recognition of geometric figures, two traditional approaches are mentioned: Piaget’s approach to developing geometric thinking in childhood and van Hiele’s approach. Piaget shows that the development of geometric thinking in childhood takes place in two stages. This approach explains recognizing the environment and figures in childhood via topology. According to Piaget, children in the first stage are able to recognize familiar figures; this recognition includes Euclidean figures. Piaget believes that children in this stage adopt topological knowledge, such as whether figures are open or closed, which they recognize by means of sensory-motor activities, and can distinguish figures according to topological properties. Piaget (Piaget and Inhelder, 1967) claims children in the second stage manage to distinguish Euclidean figures, such as circle, square, triangle, rectangle, and are able to distinguish one from another. On the other hand, van Hiele (1986) argues that the development of geometric thinking does not take place in two stages, as is asserted by Piaget, but in five separate levels. These levels are as follows (Tipps, Jahnson and Kennedy, 2011, Žilková, 2013): Level 0: Visualisation – recognizing and naming the figures. Level 1: Analysis – describing the properties of the figures. Level 2: Informal deduction – classification and sorting figures by their properties. Level 3: Deduction – developing proofs using postulates and definitions. Level 4: Rigor mathematics – working in various geometric systems. The first three levels (zero to two) occur during elementary school; levels three and four usually come later. Visualisation starts in early childhood and continues in the first four years of elementary school. Analysis should continue in the upper
110 grades of primary school and should stem from children’s repeated manipulation with the given objects when they start to intuitively understand the attributes of the given object. At the end of elementary school, children should be at the level of informal deduction. According to some research, students at the age of 11 to 12 often stay on the first level of van Hiele’s scale. Another research by Pavlovičová and Barcíková (2013), which tested pupils of this age, shows that approximately 67% of pupils of this age were on the first level of van Hiele’s scale. Moreover, 18% of the pupils did not manage it even to the first level. For van Hiele (1986), Level 0 of geometric thinking is a visual level. At this level, children perceive figures as a whole and classify them by comparing them with a prototype. This level comprises the first two years of primary education. At this level, children do not pay attention to defining the attributes of the figure, such as sides or vertices. According to van Hiele’s theory, when child starts to define geometric figures according to their attributes, such as the number of sides or vertices, it finds itself in the Level 1 of geometric thinking, which is analysis (Hannibal and Clements, 2000). Children achieve this stage in the 3rd and 4th year of primary education (Aktas Arnas and Altun, 2010). Giaquinto (2007) sees the formation of geometric concepts from the perspective of perceiving various objects by an individual. In his opinion, our initial geometric concepts depend on how we perceive the given figures. He distinguishes between the perceptual concept and the concept itself. The ability to recognize e.g. squares is different from the ability to recognize perceptively “something like a square”. Before the geometric concept of a square is formed, the perceptual concept of a square is formed. De Villiers (1994) mentions that the classification of any set of concepts does not take place independently of the process of defining. For pupils at Van Hiele Level 2 it is useful to use hierarchical classification. De Villiers (1994) also points out that students should not be given ready-made definitions and classifications, but that they should actively participate in the process of defining and classifying, and critically comparing the alternatives. However, pupils in the Czech Republic are usually given ready-made partitional definitions. Methodology I wanted to find out the most frequent mistakes in pupil’s cognitive process in early geometry and frequent misconceptions. In the first phase, I aimed at children in the fourth grade who should have passed from level of visualisation to level of analysis according to van Hiele’s theory. Eleven classes were included in the study and 226 pupils of the fourth grade (age 9 – 10). The children were given a geometric test, which was aimed at the level of development of geometrical thinking and at the ways of formation of conceptions of geometrical notions. The test was formed by my colleagues from Catholic University in Ružomberok, Slovakia (Kopáčová and Žilková, 2015). Children filled in the tests with their math teachers in their regular mathematics lesson. The teachers were given
111 detailed instructions not to help pupils with terms and not to let them communicate with each other. Teachers were also asked to answer the questions about frequency of their geometry lessons and about the tools they use in geometry lessons. On the base of outcomes, I modified the test using the previous results and commissioned it to the fourth, sixth and eighth-graders. The test was completed by 129 pupils of elementary school, 39 fourth- graders (age 9 – 10), 64 sixth- graders (age 11 – 12) and 26 eighth- graders (13 – 14). The main aim of this part of research was to discover how the understanding of the concept of a square is changing throughout these years. The main difference between the two tests was that if the pupils were asked to decide whether the object is a square for example, they were also asked to justify their choice. I was interested in the following questions: What are misconceptions about square (and rhombus) in children at single grades? What attributes of the square (laterality of sides, same length of sides, right angles) do pupils consider? Are pupils able to define the square in their own words? After testing, an activity was prepared for pupils of the eighth grade. The activity made pupils to think about the notion of square and its properties. The aim of the activity was to find out if pupils have ability to formulate the definition. Common mistakes in fourth-graders’ cognitive process It was typical for the pupils that they perceived the shapes of the objects as prototypes, often according to the turn or untypical shape. In case of a square, pupils marked as a square a square standing on a base (99%), a little turned square (96 %). However, 40 % of the pupils marked the square turned by 45 degrees as a rhombus. This phenomenon has been described many times before, e.g. Giaquinto (2007). Many pupils think that we get a rhombus by turning a square by 45 degrees, which indicates that they know that it is the same object, but they call it differently. In hierarchical classification, the square is a special case of rhombus. However, in the Czech curriculum children meet partial classification, in which various subsets of concepts are considered to be disjoint from one another (de Villiers, 1994). According to the results of the test, pupils did not think about a square and its properties. There seemed to be a group of pupils who understood a square to be a figure with “sides of the same length”. It has been proved form the results of the test that the process of establishing geometric concepts is much dependent on the way of going through the subject matter in school mathematics. The pupils whose teachers indicated that they integrate geometry in their lessons at least once a week were more successful than the pupils who encounter geometry less often. Another important factor of the
112 success rate was using various teaching aids in lessons as the pupils who had the chance to work with teaching aids and manipulate with different figures had more fixed concepts. Thinking about the results of the test, I started to consider how the pupils´ conceptions of the concept square are developing during the years of the education. I drew my attention to the two most often misconceptions: 1. A square has got four sides. 2. By turning a square by 45 degrees we get a rhombus. Outcomes of the modified test I modified the test and decided to commission it to fourth-, sixth- and eighth- graders. If the pupils were asked to decide whether the object was a square, they were also asked to justify their choice. Pupils were asked to decide if the shape on the Picture 1 was a square. It has been proved that the number of students who have mistaken a square turned by 45 degrees for a rhombus has been slightly decreasing for elder pupils. Even some eighth-graders worked with this wrong concept.
Answer The fourth- The sixth- The eighth- grade grade grade The figure in the picture is 9 / 39 21 / 64 15 / 26 a square Figure 1 The figure in the picture is 24 / 39 41 / 64 10 / 26 a rhombus Other or no answer 6 / 39 2 / 64 1 / 26
However, if the figure was not turned by whole 45 degrees, pupils were able to think about it differently, see the next table, where the differences between the particular grades are more visible.
Answer The fourth- The sixth- The eighth- grade grade grade The figure in the picture is 27 / 39 57 / 64 26 Figure 2 a square
The figure in the picture is 10 / 39 6 / 64 0 not a square No answer 2 / 39 1 / 64 0
113 Pupils were asked to justify their choice. The following are examples of answers of the fourth-graders: It is turned and that is why it is not a square (26%). Another example is: It is a rhombus, but by turning it I get a square. The length of the sides played an important role for the sixth- graders: The length of the sides is the same, so it is a square (41%). The pupils also mentioned the process of turning a figure: It is only a bit turned, but it is a square (8 %), It is turned, so it is not a square (9 %). Even the same length of the sides was mentioned (5%) and one pupil stated that the lengths of the sides are the same and there are right angles. The eighth- graders mostly stated that the sides are of the same length (46 %) and that the angles are right and the sides are of the same length (27%). We can deduct from the results that the concepts of the term square are changing in time, however, the misunderstanding of the term rhombus remains even in higher grades. Moreover, not even the eighth-graders take into consideration all the important aspects of the figure. Another task for the pupils was to decide if the shape in the Picture 3 was a square. The position of the rhombus was quite untypical for the pupils as they are used to matching a rhombus to a figure standing on a vertex.
Answer The fourth- The sixth- The eighth- grade grade grade Figure 3 The figure in the picture is a 12 / 39 15 / 64 8 / 26 square
The figure in the picture is 23 / 39 49 / 64 18 / 26 not a square No answer 4 / 39 0 0
Some of the explanations of the fourth- graders why the figure is not a square are as follows: It is inclined (10%), the sides are not of the same length. The second statement is wrong and pupils did not verify it as they relied on their visual perception. The eighth-graders mentioned some of these reasons: It is inclined, so it is not a square (19%), there are no right angles (16%), the lengths of the sides are not the same, so it cannot be a square (14%), the lengths of the sides are the same, so it is a square (9%), it is only inclined, it is a square (5%). And also the eighth-graders stated some reasons: the angles are not right (27%), it is a rhombus (23 %), the lengths of the sides are the same, so it is a square (19%). We can see that the concepts are becoming more specific as the pupils solve different mathematical problems in that the term of a square is mentioned. The eighth-graders are taking into consideration even the angles, not only the lengths of the sides. Nevertheless, not even at this grade is the concept totally correct.
114 Students were asked to complete the sentence: A square is … The most common answers are in the following table: Answer The The The fourth- sixth- eighth- grade grade grade Without answer 15 / 39 12 / 64 4 / 26 A four- sided figure 8 / 39 6 /64 3 / 26 A figure with four vertexes 6 / 39 3 / 64 0 A figure whose sides are of the same length 6 / 39 23 / 64 7 / 26
A figure with four sides of the same length 0 10 / 64 0 A figure whose sides are of the same length and right 0 1 / 64 5 / 26 angles
Up to the eighth-grade, the number of aspects that the students are considering is rising. At first, they are interested only in the length of the side or the number of sides, later pupils connect these two aspects and they add new one: information about the right angles. We only have seen five examples of thorough specification in the work of the eighth-graders.
Activities which help children develop geometric concepts Not only abroad (Arnas Aktas and Aslan, 2010, Dindyal, 2015, Duval, 1999, Hannibal and Clements, 2000), but also in the Czech Republic (Budínová, 2015), teachers are recommended to give certain freedom to pupils at the beginning of their cognitive process in geometry and let them use autonomous language. Natural language can help students express their ideas and study mathematics. Vighi (2015) has conducted an experiment that proved that pupils who were asked to use their natural language and metaphors while discovering geometric rules (namely parallelism and perpendicularity of a straight line), had better results from a long term point of view that the pupils who were given the mathematical theory. Some authors have found it very useful to allow pupils firstly to formulate, compare and choose their own definitions and classifications of squares and rhombi (de Villiers, 1994). Many of pupils spontaneously prefer partitioning. DeVilliers (1994) mentions that a hierarchical class inclusion is often useful during problem solving; in particular for proofs. There are several activities we can offer pupils, so that they can discover properties of geometric shapes. One of the activities is to ask pupils to describe to their classmates a geometric shape in their own words.
115 One pupil gets a picture of a square and his/her task is to describe the figure to his peers while not using the word “square”. A strong procedural orientation of the pupils can be observed in this activity. Procedural knowledge is made up of two distinct parts. One part is composed of the formal language, or symbol representation system, of mathematics. The second part consists of rules, algorithms, or procedures used to solve mathematical tasks. (Hiebert and Lefevre, 1986). Conceptual knowledge is a connected web of knowledge. Unit of conceptual knowledge cannot be an isolated piece of information. (Hiebert and Lefevre, 1986). Here is a particular example. A eight-grader was describing a square to her peers: “Rule a line that is parallel with the edge of the exercise book. Make there a point and on the right side from it make another one. Name these points: the left one A, the right one B. From the points make a perpendicular line to the sides. There make a new point that of the same distance from the two points. The last thing is the point D that you make on the second perpendicular line. Now connect C and D.” This description contains some inaccuracies. , e.g. “a perpendicular to the both sides”, the pupil does not mention from which point we measure the distance to the point C. Other pupils had to ask additional questions for a better understanding. We can see that for the pupil is very important the position of the square (“rule a line that is parallel with the edge of the exercise book”) and she also focuses on the naming of the points in a set way: A, B, C, D, though the model did not contain any letters. She is procedural oriented. One of her classmates even told her: “Describe what you see. You have to say what it is, not how to draw it.” In the end, pupils were able to describe the square in a conceptual way: “A flat geometrical figure whose four sides are of the same length and two neighbouring sides are clenching a right angle”. There were even discussions whether the side is straight, line segment, or if it can be bent.
Conclusion Some frequent mistakes in the geometric concepts can be seen in the first part of the study. These are the mistakes that pupils make at the first four grades and they are mainly misconceptions in pupil’s concepts of geometric terms such as interchanging of a square with a rhombus. Pupils also do not know the terminology properly and they stay at the level of visualisation according to van Hiele´s theory. Even the fourth graders often decide only according to visual aspects of a figure without considering its properties. In some cases, pupils are partly interested in properties of shapes, which refers to analysis level. According to the van Hiele theory (van Hiele, 1986), depending on their geometrical thinking level children use either the visual attributes (visual level) or the property attribute
116 (analysis level) when working on a classification of a shape. Hannibal and Clements (2000) oppose this classification and claim that there is not a clear transition from the visual level to the analysis level. Clements also claims that there is a period in which children use both the visual and property attributes simultaneously. Our findings also support Clemets’ claim. It seems pupils sometimes make a decision according to visual attributes of the shape, and sometimes use properties (or some of properties) of the shape. Some pupils used both the visual and property attributes at the same time, which correspond with findings of Aktas Arnas and Altun (2010). If pupils are not offered any stimuli in the field of geometry, the concept-building process stagnates. On a sample of 129 pupils from the fourth-, sixth- and the eighth- grade, we have seen that if we do not purposefully focus on the term making process, pupils form their own conceptions of the concepts. They do so through various activities and they progressively re-evaluate their conceptions. Nevertheless, it is generally known that is difficult to change the wrong conception. That is why it is a shame that many of pupils at elementary school do not distinguish between a square and a rhombus, although according to Czech curriculum these concepts are considered to be different. Since pupils are not used to think of properties of geometric shapes, their definitions in the test were inaccurate, often naive. Most of pupils have not started to consider properties of a square that are needed for clear defining. The most important property of a square is for pupils the same length of the sides, 7 of the 26 eighth-graders identified a square by this. The eight-graders should be on the level of informal deduction according to van Hiele and they should be able to consider figures according to their properties. According to de Villiers, it is useful for pupils at van Hiele Level 2 to use hierarchical classification. De Villiers however mentions that many studies on the van Hiele theory over the past number of years have clearly shown that many students have problems with the hierarchical classification of quadrilaterals (de Villiers, 1994). During the activity of describing the square, pupils were able to construct sensible definition of a square, which corresponds with literature (de Villiers, 1994). During the concept-building process in children of an early age, it is necessary to observe two basic aspects: 1) accuracy of the terminology formed, 2) forming correct conceptions about the concept. When teaching geometry, it is apt to acquaint pupils with as many models related to the given concept as possible. Non-models are very important as well (Jirotková, 2010). Pupils should not only watch or draw shapes, but also manipulate with them.
117 References Arnas Aktas, Y., Aslan, A. G. D. (2010). Children’s Classification Of Geometric Shapes. Çukurova Üniversitesi Sosyal Bilimler Enstitüsü Dergisi. Budínová, I. (2015). Development of communicating about notions in geometry at primary school. In Studia Scientifica Facultatis Paedagogicae (pp. 43–47). Ružobmerok: Verbum, Universitas Catholica Ružomberok. De Villiers, M. (1994). The role and function of a hierarchical classification of quadrilaterals. For the Learning of Mathematics, 20(1), 11–18. Dindyal, J. (2015). Geometry in the early years: a commentary. ZDM Mathematical Education. FIZ Karlsruhe. Duval, R. (1999). Representation, vision and visualisation: cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt, M. Santos (Eds.), Proceedings of the 21st annual meeting of the North American chapter of the International Group for the Psychology of Mathematic Education (pp. 3–26). Columbus: ERIC Clearinghouse for Science, Mathematics, and Enviromental Education. Giaquinto, M. (2007). Visual Thinking in Mathematics. An epistemological study. Oxford: University Press. Hannibal, M. A. Z., Clements, D. H. (2000). Young children’s understanding of basic geometric shapes. National Science Foundation, Grant number: ESI-8954644. Hiebert, J., Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Lawrence Erlbaum Associates. Jirotková, D. (2010). Cesty ke zkvalitňování geometrie. Praha: Charles University, Faculty of Education. Kopáčová, J., Žilková, K. (2015). Developing Children’s Language and Reasoning about Geometrical Shapes – a Case Study. In J. Novotná, H. Moraová (Eds.), Proceedings of SEMT ´15. Developing mathematical language and reasoning (pp. 184–192). Praha: Charles University, Faculty of Education. Pavlovičová, G., Barcíková, E. (2013). Investigation in Geometrical Thinking of Pupil at the Age of 11 to 12 through Solving Tasks. In Novotná, J., Moraová, H.: Tasks and tools in elementary mathematics. Prague: Charles Univerity, Faculty of Education. Piaget, J., Inhelder, B. (1967). The child’s conception of space. New York: W. W. Norton. Tipps, S., Johnson, A., Kennedy, L. M. (2011). Guiding Children’s Learning of Mathematics. Wadsworth, Cengage Learning. van Hiele, P. M. (1986). Structure and insight: a theory of mathematics education. Orlando: Academic Press. Vighi, P. (2015). Language for Learning: Spontaneous vs Specific Geometrical Language. In J. Novotná, H. Moraová (Eds.), Proceedings of SEMT ´15. Developing mathematical language and reasoning (pp. 341–349). Praha: Charles University, Faculty of Education. Žilková, K. (2013). Teória a prax geometrických manipulácií v primárnom vzdelávání. Praha: Powerprint.
118 DIVERSITY OF TEACHERS’ BELIEFS ON MODELLING THROUGH A FRENCH-SPANISH COMPARISON Richard Cabassut
Abstract We present an exploratory research on the beliefs of actors of the primary school (trainees, teachers, trainers ...) in relation to the teaching of modelling. A questionnaire was answered by French and Spanish teachers on different variables (institutional conditions, experience, training, teaching conditions, mathematical and modelling design, teaching difficulties ...). The results show that teachers’ diversity on modelling can be represented by four teachers’ types, which explains the difficulties encountered in this teaching in relation with time, assessment, lesson organization, context, student’s involvement, and resources. Consequences are drawn for teachers’ training, resources and research. Keywords: diversity, teacher, beliefs, modelling, teaching, France, Spain
Statement and aim of the research about teachers’ beliefs on modelling International research revealed the importance of teachers’ beliefs on modelling teaching (Kaiser et al. 2006; Mischo et al. 2013). International comparison has proved the diversity of teachers’ beliefs on modelling (Cabassut and Villette 2011). In France and Spain, the official texts and textbooks about mathematics in primary school deal with problems in relation to the real world (Cabassut and Wagner 2011; Cabassut and Ferrando 2014). The 2016 mathematics curriculum for French compulsory education presents, from 1st grade on, "modelling" as one of the seven main skills worked in mathematics. But we didn’t find research about teachers’ beliefs on modelling in France and Spain and the difficulties they can come across when teaching modelling (Cabassut and Ferrando 2015). Therefore we have started an exploratory research to answer the following questions: What are the teachers’ beliefs about modelling in France and Spain? What difficulties do they run into teaching modelling? Based on this ongoing research from primary to tertiary education we will illustrate here the first results concerning primary school. As this research is an exploratory one (as it will be explicated in the methodological part of this paper) there is no statement (hypothesis) to be confirmed. Let us argue now with more details on the theoretical frame work that sustains this research. Theoretical frameworks about teachers’ beliefs on modelling and their diversity We retain four domains to investigate these beliefs on modelling, based on current literature. The first domain is related to teachers’ biographical context: Borromeo and Blum (2013) about the differences between teachers studying mathematics as
Strasbourg University, France; e-mail: [email protected]
119 a subject or not; Cabassut and Villette (2011) about the role of country, age and type of school; Kuntze (2011) and Borromeo Ferri and Blum (2013) on the influence of experience in teaching; Dorier and García (2013) about the importance of initial training. The second domain concerns the role of mathematical beliefs and practice. Kaiser (2006) shows that German teachers’ beliefs about mathematics are an essential reason for the low involvement of modelling and its applications in mathematics education. Maass and Gurlitt (2009) demonstrated the influence of beliefs in mathematics on the design, selection, implementation and evaluation of modelling tasks. Lee (2012) noted the impact of mathematical knowledge and beliefs on the interpretation and implementation of the program. The third domain deals with conceptions and practice about modelling. Borromeo and Blum (2013) showed the influence of experiences in teaching mathematical modelling concerning barriers and motivations. They point also the importance of training on modelling, which can only be learned effectively if there are teachers that have competencies in this field. Interviewees from a modelling training (Cabassut et al. 2009) express needs to create a lot of modelling situations to be used during teaching.The fourth domain relates to difficulties with teaching of modelling. Schmidt (2011) and Borromeo and Blum (2013) studied the difficulties of teaching modelling in German elementary school teachers: time, assessment and resources are the three main areas of difficulty. Cabassut and Villette (2011) found variables about difficulties in lesson organisation (for example, to design the lesson or to help student). Sometimes “the resistance does not only come from teachers but also from students or maybe even parents or the society as a whole” (Dorier and García, 2013). Difficulties about students’ involvement are pointed by Borromeo and Blum (2013). The diversity of national contexts could influence the teaching of modelling and the diversity on national bases (Garcia & al. 2007). But we conjecture that diversity of teachers’ beliefs can be represented by different type of teachers among different national spaces as shown in (Cabassut, Villette. 2011). Every type is present in every national space and is a common type of these different national spaces. This type can be understood as a transcultural type and is present over the different national cultures. This is why we adopt a comparative approach between France and Spain in order to determine transcultural types. We will argue now how the exploratory approach through a questionnaire a particularly well adapted methodology to analyze this diversity in the framework of transculturality. Methodology of exploratory data analysis of questionnaire From a methodological point of view, we review the literature of the last few years of some journals (Educational Studies in Mathematics, Zentralblatt für Didaktik der Mathematik, Research in Mathematics Didactics, ICMI Studies) and conference proceedings (CERME, ICTMA, SEMT, Copirelem) as well as
120 references revealed by previous sources. Two researches based on questionnaires on the teaching of modelling have been used (Borromeo and Blum 2013; Cabassut and Villette 2011). The main results of this review of the literature (mentioned in the previous part on theoretical framework) are used to construct an online questionnaire for students, professors, trainers, researchers and members of the noosphere (inspectors, pedagogical advisers, authors of resources) related to the teaching of mathematics in France and Spain. The questionnaire1 on difficulties on modelling is composed of 93 questions, mostly multiple choice questions, with sometimes a scale with four or five degrees. Sometimes a single open-ended question enables to explicit the answer. There are 8 questions on biography, 23 on mathematics’ conception and practice, 25 on modelling conception and practice, and 37 on modelling difficulties split into six parts: time, assessment, lesson organization, context, student's involvement, and resources. It is an on-line questionnaire, advertised through different national networks, which was completed between February and March 2015 by 231 people, including 124 French and 107 Spanish people. This sample was not constructed on a representative basis but on an exploratory one: people answered the questionnaire voluntarily. An exploratory approach, according to Tukey (1977), means a representative sample is not needed. Our sample includes pre-service teachers, teachers, mathematics education researchers, educations inspectors, resources’ writers and teachers’ trainers. The analysis of the answers to the questionnaire is done with the SPAD software: flat sorting, analysis in classes (Cabassut and Villette 2012), textual analysis. Variables are the different answers to multiple choice questions. Cluster analysis is done by taking the variables related to the difficulties in teaching of modelling. Once the clusters are constituted, we observe to which cluster the other variables, called the illustrative variables (biographical mathematical conceptions and practice, modelling conceptions and practice) are connected. Results The first result shows the heterogeneity of the answers, especially those of primary school teachers. The statistical results are extracted from (Cabassut and Ferrando 2017). We will point out the place of primary school teachers in this results. Frequency analysis From 231 answers to questionnaire, 23% are primary school teachers, 54% are secondary ones and 23% are in tertiary education. Let us point out the main difficulties (more than 50% of respondents). For 70% of the respondents indicated it is difficult to estimate how long it takes
1 https://docs.google.com/forms/d/1wbNG64gq- U_3QhhGfhzuJ7X06DLKVc7HvJMO3DRS_Mw/edit#
121 to solve a modelling task. For 58%, it took too much time to prepare modelling tasks for teaching. In addition, 54% of respondents thought that most students do not know how to work out modelling tasks. For 55%, modelling tasks required a lot of extra things that teachers could obtain only at great expense. Lastly, 51% of respondents thought that teachers do not have enough materials for modelling tasks. It was also possible to identify positive aspects through the frequency table analysis. Firstly, 77% thought that modelling tasks promote greatly students’ autonomy. Secondly, 57% felt able to support students in developing competencies in arguing related to modelling tasks. Thirdly, 56% felt able to use students’ mistakes to facilitate their learning in modelling. Fourthly, 50% of respondents considered that modelling tasks promote, at the same time, both low achievers and high achievers. In short, the main difficulties are related to time and resources for modelling tasks. Main positive aspects are related to competencies developed by students (autonomy, arguing) or by teachers (adaptation to students ' difficulty or diversity). Let us present now the cluster analysis that produces the four following clusters. Cluster analysis First cluster consist of 74 individuals (32%) In this cluster, much more than in the whole population, they did not feel able to design modelling lessons that could help students overcome difficulties in all modelling steps, to support students in developing competencies in arguing related to modelling tasks, to develop detailed criteria for assessing and grading students’ solutions to modelling tasks, to effectively assess students’ progress as they worked on modelling tasks, to use students’ mistakes to facilitate their learning in modelling, to design their own modelling tasks, and to adapt tasks and situations in text books to provide realistic open problems. Much more than in the whole population, they agreed with the following statements about modelling tasks: it is difficult to assess the presentation of a solution; it is difficult to differentiate what is correct from what is not correct; assessment takes too much time; it is difficult to assess group work, the solutions found by the pupils or the students are not comparable; the presentation of the solutions is complex; these require complex operations that primary school children cannot cope with; most students do not know what to work out by modelling tasks; it is difficult to manage group work; and modelling lessons are unpredictable. Primary teachers are less represented (15% in this class compared to 23% in the total population). Is it because at the elementary level one is more used to open activities on everyday life and other disciplines? Future interviews will attempt to clarify these elements. In short the first cluster gathers respondents who are negative towards modelling and who meet difficulties in mathematics and in modelling. Second cluster consists of 46 individuals (20%). In this cluster, much more than in the whole population, they disagreed that, in a modelling task, it is difficult to
122 assess group work; that most students do not know how to work out modelling tasks; that when teaching modelling, not enough time is left for other learning content; that it is difficult to manage group work by modelling task; that the pupils or students are hard to discipline during modelling activities; that in a modelling task it is difficult to assess the presentation of a solution of a modelling task; that when pupils or students work on a modelling problem, the environment in the class becomes harder; that modelling tasks require complex operations which primary school children cannot cope with; that it takes too much time to assess modelling tasks; that the lessons are unpredictable by modelling; that in a modelling task it is difficult to differentiate what is correct from what is not correct; that the presentation of the solutions is complex; and that working on modelling tasks in the classroom is very time consuming. They felt able to design modelling lessons that help students overcome difficulties in all modelling steps, to develop detailed criteria (related to the modelling process) for assessing and grading students’ solutions to modelling tasks, to effectively assess students’ progress as they work on modelling tasks, to support students in developing competencies in arguing in relation to modelling tasks, to design their own modelling tasks as teachers, and to adapt tasks and situations in text books to provide realistic open problems. Primary school teachers are 19% of this cluster (compared with 23% in the total population). Is it because there are more used to modelling (because there are multi-subject teachers using a lot the project pedagogy or the relations with other subjects)? The fact that they are more used to modelling could explain that they could be less enthusiast and more neutral. Additional interviews could look for institutional or cultural factors that could explain these different characteristics. In short the second cluster brings together respondents without difficulty with modelling and positive about modelling. Third cluster consists of 85 individuals (37%). In this cluster, much more than in the whole population, they felt able to use students’ mistakes to facilitate their learning in modelling, to support students in developing competencies in arguing in relation to modelling tasks, and they agreed that modelling tasks promote greatly students' autonomy, that students acquire a lot of knowledge about the use of mathematics in modelling tasks. Much more than in the whole population, they were neutral to feeling able to design modelling lessons that help students overcome difficulties in all modelling steps (e.g., problems in validating), to adapt tasks and situations from text books to provide realistic open problems, to develop detailed criteria (related to the modelling process) for assessing and grading students’ solutions to modelling tasks. Alternatively, they were neutral about the following statements: most students do not know how to work out modelling tasks; it takes too much time to assess modelling tasks; the lessons are unpredictable with modelling.
123 We observe that primary school teachers are 28% in this class (compared to 23% in the total population). Connected to our interpretation in the second cluster is it because primary school teachers are more used to practice problems related to everyday life or to other subjects. In this case they will be more confident and positive about modelling. Their neutral position on difficulties could be related to the fact that they are conscious about difficulties but, as they are used to manage them, they are more confident and by consequence neutral on difficulties. Additional interview will clarify our conjectures. In short, respondents of third cluster are positive about modelling and neutral on difficulties. Fourth cluster consists of 24 individuals (11%). Much more than in the whole population, these were neutral on the following statements: modelling tasks promote greatly students' autonomy; most students do not know what to work out by modelling tasks; students acquire a lot of knowledge about the use of mathematics in modelling tasks; I feel able to support students in developing competencies in arguing related to modelling tasks; students recognize that often there is not only one right solution; the pupils or students are hard to discipline during modelling activities; students have difficulty with the fact that there are many different solutions for modelling tasks; modelling tasks promote at the same time both less powerful and more powerful students; I feel able to use students’ mistakes to facilitate their learning in modelling; in a modelling task it is difficult to assess the presentation of a solution of a modelling task; when pupils or students work on a modelling problem, the environment in the class becomes harder; in a modelling task, it is difficult to differentiate what is correct from what it is not correct; when teaching modelling, I am not left enough time for other learning content; I feel able to design modelling lessons that help students overcome difficulties in all modelling steps; the solutions found by the pupils or the students are not comparable; in a modelling task, it is difficult to assess group work; students can use the openness of the tasks to handle them well; and it is difficult to manage group work for modelling tasks. Primary school teachers are over represented (46% in the cluster versus 23% in the whole population). Their neutrality is more surprising. Is it because they do not know the concept of modelling? There are more used to the expression “problems related to everyday life. Is it because they are less used to have feedback on such practices? Future interviews will help to explain this. To try to answer some of the previous questions we could look for conjectures from Chi- squared Test. In short the fourth cluster brings together respondents who are neutral on modelling and its difficulties.
Conjectures from Chi-squared Test Chi-squared helped to identify whether there was a significant relation between two variables. Clearly, this allowed us to find some differences related to country or gender. We observed that women felt less able than men to design modelling
124 lessons that help students overcome difficulties in all modelling steps. Regarding the country, Spanish people felt more able than French people to design modelling lessons that helped students overcome difficulties in all modelling steps. In the LEMA cluster analysis of Cabassut and Villette (2011) about teachers attending a in-service teacher training course, we found also that Spanish teachers were more positive about modelling than other teachers. This cultural fact is difficult to explain. There was no significant difference between countries and biographical variables. We also observed some differences in relation to modelling difficulties: people who had difficulties in their mathematics teaching with an inquiry based approach also had difficulties with modelling in relation to time, evaluation, lesson organization and resources; people who had difficulties in their mathematics teaching with small groups also have difficulties with modelling in relation to evaluation, resources and students involvement ; people who have difficulties in their mathematics teaching with open problems also have difficulties with modelling in relation to students’ involvement. At the level of primary school teachers, we do not find significant dependence with other variables. We observe that there are more women, young people and less experienced teachers than among the total population of respondents (especially secondary teachers). Concerning conceptions and practices, unlike high school teachers, a majority of primary school teachers find the investigation process easy, often proposes modelling problems, rarely uses colleagues to find modelling problems. Concerning difficulties, unlike high school teachers, a majority of primary school teachers do not agree that solving modelling problems takes too long, or that most students do not know what to work in modelling problems; a majority of primary school teachers are neutral on the fact that modelling does not allow enough time for other learning, or that it is difficult to assess the presentation of a problem solution, or on being able to design modelling lessons that help students to overcome difficulties in all stages of modelling. The fact that we did not find significant dependence between to be a primary school teacher and other variables illustrate the diversity of teachers’ beliefs among primary school teachers. But we have to be cautious with our conjectures because we are in an exploratory phase to help to express our conjectures. We need a confirmatory phase, with a representative sample, to confirm our conjectures our by qualitative interview making explicit the relations. In short Spanish people felt more able than French people to design modelling tasks solvable by students. Difficulties in the teaching of modelling are related to difficulties in the teaching of mathematics. Textual analysis Cluster analysis allows the SPAD software to determine the best paragons for each cluster. Semi-structured interviews with these paragons are planned to explain the difficulties and explanations of these difficulties. Let us give, by way of illustration, three examples of extracts from answers to open-ended questions:
125 “How is the teaching of modelling easy or difficult? Why?” The first example deals with the interview of a French primary school teacher: the official syllabus brings too much constraints with not enough time for the teaching of modelling. He mentions that class with two grades2 makes this teaching uneasy. Here the constraints concern levels of the mathematic discipline (syllabus) and school organization (two grades class). The second example concerns the interview of a French part-time trainer for pre-service primary school teachers: he is not sufficiently trained on modelling training and has not enough resources to offer for the class. Here the constraints are at the level of trainers’ training and available resources on modelling. The last example is related to the interview of an educational adviser: the teaching of modelling requires a level of abstraction that is acquired only gradually at primary school. Here the constraints is at a cultural level. These excerpts illustrate the value of open-ended questions that will complement the analysis of closed-ended questions. The SPAD software allows a textual analysis of open questions. For the previous question "How is the teaching of modelling easy or difficult? Why ?” the following characteristic words or segments appear: for the members of the noosphere (inspectors, pedagogical advisers ...) "competence" for French and "resource" for Spanish, for primary school teachers "implementation" for French and "time" for Spanish, secondary teachers "pupils" for the French, and "examination to enter university" for Spanish, for higher education "students" for French and "difficulties" for Spanish. Textual analysis of interviews of clusters paragons will help to better describe the different clusters. It is one of the next aim of this ongoing research. Conclusion Our exploratory research has illustrated diversity by teachers’ beliefs on modelling. Four types are proposed: a type negative towards modelling and with difficulties in mathematics and in modelling, a type positive and without difficulty in modelling, a type positive about modelling and neutral on difficulties, and a type neutral on modelling and its difficulties. A majority of teachers seems to be positive or neutral towards modelling. In the frequency analysis we observed that time was the main domain of difficulties: time necessary to prepare teaching of modelling, yearly time planning of this teaching, time for modelling assessment, time management of modelling activities for teachers and for students. This means that for resources available to teachers (by considering teachers ‘training as one of these resources) time is one of the topics to be investigated. These results could help to develop professional development. A pre-questionnaire before training courses could help to identify what type is a teacher. For teachers’ training collaboration groups mixing different type of teachers could help to change teachers’ beliefs by moderating their beliefs: reducing the intensity of negativity or positivity towards modelling and reducing difficulties. A
2 It is frequent in France that a primary school teacher has in charge a class group where two grades are present (for example grade 3 and grade 4 pupils).
126 differentiation could be offered by resources or training: to overcome difficulties for the type with difficulties and to develop positiveness toward modelling for other types. Different types are present in the three educational degree: primary, secondary and tertiary. It encourages inter-degree training on modelling to improve transition between degrees. The ongoing research has to clarify by interviews or by confirmatory analysis the conjecture on the dependences of variables in order to better adapt training and resources. From the chi square study, particularly the roles of inquiry based approach, small groups work, open problem solving and assessment has to be made explicit by modelling beliefs. The positiveness toward modelling associated to Spain has to be investigated. The participation of other countries could possibly contrast the role of institutional conditions, which could not be clearly demonstrated in this exploratory research. Interested colleagues are invited to contact the author. References Borromeo Ferri, R., Blum, W. (2013). Barriers and motivations of primary teachers implementing modelling in mathematical lessons. In B. Ubuz, Ç. Haser, M. A. Mariotti (Eds.), Proceedings of the Eighth Congress of the European Society for Research in Mathematics Education (pp. 1000–1010). Ankara: Middle East Technical University. Cabassut, R., Villette, J.-P. (2011). Exploratory data analysis of a European teacher training course on modelling. In M. Pytlak, E. Swoboda, T. Rowland (Eds.) Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (pp. 1565–1575). Rzeszów, Poland: University of Rzeszów. Cabassut, R., Ferrando I. (2014) Comparaison franco-espagnole de ressources sur l'enseignement de la modélisation. In Actes du 40ème colloque COPIRELEM. IREM de Nantes. Cabassut, R., Ferrando I. (2015) Conceptions in France about mathematical modelling: exploratory research with design of semi-structured interviews. In K. Krainer, N. Vondrová (Eds.), Proceedings of 9th Congress of European society for research in mathematics education (pp. 827–833). Prague: Charles University. Cabassut, R., Ferrando, I. (2017). Difficulties to teach modelling: a French-Spanish exploration. In Proceedings of the 17th Conference of the International Community of Teachers of Mathematical Modelling and Applications (ICTMA). University of Nottingham: July 2015. Cabassut, R., Wagner, A. (2011), Modelling at Primary School through a French-German Comparison of Curricula and Textbooks. In G. Kaiser, W. Blum, R. Borromeo-Ferri, G. Stillman (Eds.), Trends in Teaching and Learning of Mathematical Modelling (pp. 559–568). Dordrecht: Springer. Dorier, J.-L., Garcia, J.G. (2013). Challenges and opportunities for the implementation of inquiry-based learning in day-to-day teaching. ZDM Mathematics Education, 45(6), 837–849. Garcia, F.J., Wake, G., Maaß, K. (2007). Theory meets practice: working pragmatically within different cultures and traditions. Paper presented at the 13th International Conference on the Teaching of Mathematical Modelling and Applications. University of Hamburg.
127 Kaiser, G., Shriraman, B. (2006). A global survey of international perspectives on modelling in mathematics education. ZDM, 38(3). Kuntze, S. (2011). In-service and prospective teachers’ views about modelling tasks in the mathematics classroom – results of a quantitative empirical study. In G. Kaiser, W. Blum, R. Borromeo-Ferri, G. Stillman (Eds.), Trends in teaching and learning of mathematical modelling (pp. 279–288). Dordrecht: Springer. Lee, J.-E. (2012). Prospective elementary teachers’ perceptions of real-life connections reflected in posing and evaluating story problems. Journal of Mathematics Teacher Education, 15(6), 429–452. Maaβ, K., Gurlitt J. (2009). Designing a teacher-questionnaire to evaluate professional development about modelling. In V. Durand-Guerrier, S. Soury-Lavergne, F. Arzarello (Eds.), Proceedings of 6th Congress of European society for research in mathematics education (pp. 2056–2065). Lyon: Institut National de Recherche Pédagogique and ERME. Mischo, C., Maaß, K. (2013). The effect of teacher beliefs on student competence in mathematical modelling-An intervention study. Journal of Education and Training Studies, 1(1), 19–38. Tukey, J. (1977). Exploratory data analysis. Reading, MA: Addison-Wesley.
CHANGES IN THE TEACHING OF NUMBERS AND RATIOS IN THE PRIMARY CURRICULUM Christine Chambris Abstract This paper tackles the issue of changes in the teaching of numbers and rations in the primary curriculum in France, especially the crucial changes that occurred during the New Math. It investigates specifically the role of mathematical reference knowledge. It shows how these changes still impact the teaching of numbers and ratios. Keywords: Numbers, ratios, New Math, reference knowledge, didactic transposition
INTRODUCTION In the 1960s and 1970s, in many countries throughout the world, the New Math Reform aimed at renewing the teaching of mathematics from the primary school up to the university, and at increasing mathematics proficiency in the general public in the context of the cold war (Furinghetti et al., 2008). One of the major concerns of the New Math was the teaching of some “new” math. This enables to make the assumption that this period showed major changes in the mathematical foundations of basic arithmetic. The case of the teaching of set theory in primary schools in these years is a famous example. Another concern of the reform was to take into account some psychological features related to the learning or to the child development. The famous psychologists Piaget and Bruner contributed at
Université de Cergy-Pontoise, France; e-mail: [email protected]
128 least indirectly but significantly to the implementation of this reform. It has often been said that New Math failed, however “In no country did school mathematics return to where it had been before the new math movement began: The pendulum is not a suitable metaphor for curriculum change.” (Kilpatrick, 2012, p.569). The aim of this paper is thus to identify some of the changes promoted by the New Math, and their long term effects. It will focus on the teaching of numbers, operations, and ratio in the primary curriculum in the French curriculum1. Hence, I aim at identifying possible changes in mathematical foundations for numbers and ratios surrounding the New Math, and possible effects of these changes on the curriculum up to nowadays. METHOD Theoretical frame The Theory of Didactic Transposition (TDT) (Chevallard, 1985) considers school mathematics as a reconstruction by the educational institutions from the mathematical knowledge produced by academic scholars. Four steps are distinguished (fig. 1): scholarly knowledge, syllabi and textbooks, teachers, then students. The Anthropological Theory of Didactics (ATD) (e.g. Bosch and Gascon, 2006 for a reference in English) extends the TDT. It postulates that practicing mathematics, as any human practice, can be described with the model of praxeology. It is constituted by four components: a type of tasks -a set of similar problems-, a technique -a “way of doing” for all the tasks of the type-, a technology justifies the technique, and a theory legitimates the technology. In this model, the reference knowledge is the scholarly knowledge.
Figure 1: The process of didactic transposition (op. cite) The ATD introduces the “ecology of knowledge” metaphor to describe the evolution of teaching objects. For a given object at a given moment, it defines: its habitats (where it lives) and ecological niches (its roles in each of its habitats). It also considers “trophic chains”: A needs (to eat) B to live. Thus, the ATD asserts the following paradox: To be eaten is a reason to live in the teaching system.
1 In this note I indicate some specific uses that may be helpful to better understand some of the data presented after. In France, at present time, there is no use of comma to write great numbers, a space is required. It seems a century ago, at least some people were using a point. Moreover, for writing decimal numbers – with the positional notation- a comma is required to indicate the unit. Hence, 30 000,4 means thirty thousand and four tenth. With regard to multiplication, the sign x is used between two numerals: 13 x 15. In algebra, “x”, “.” or no sign can be used to indicate multiplication between two letters or between a numeral and a letter.
129 Academic knowledge In the history of mathematics, the development of numbers has been closely connected to the measurement of quantities, especially that of continuous ones (e.g. Book V by Euclid). But, since 1870 approximately, in academic mathematics, numbers have no longer been elaborated from quantities but from whole numbers, then from sets (Bourbaki, 1984). Did such a change impact the primary curriculum? To what extent has present academic knowledge impacted the primary curriculum during the New Math, and still does? Background Bronner (1997) highlights five main periods in the teaching of numbers in the French system: the stable classical period 1870-1950, the New Math highly turbulent period 1970-1980, the continuously evolving contemporary period 1995-2010, and two transitional ones before and after the New Math. Treatises by Bezout (1779), then Reynaud (1821) appear as reference books for primary teacher education, and textbooks by 1940. Research questions What are the habitat and niche of quantities in the different praxeologies during the different periods? Are there traces of changes of “reference knowledge” for numbers, operations, ratios and quantities during the different periods, especially around the New Math? Data The studied period is huge. In order to get an overview of the curricula, data have been first restricted to the national French syllabi (1882, 1923, 1945, 1970, 1977- 1980, 1985, 1995, 2002, 2008) for grades 1 to 5 and to previous works studying teaching of arithmetic throughout the 19th or 20th century. Several 2nd and 3rd grade textbooks of have been included after. Mathematical treatises and books for teacher education inform us about scholarly or reference knowledge. Methodology First, an extensive analysis of syllabi is conducted in order to identify each occurrence of discrete and continuous quantity. This enables to locate their habitats, and then the role of quantity is interpreted in term of praxeology (niche). Similar analysis is conducted with treatises, textbooks analysis enables to refine, infirm, or confirm the previous analyses on specific topics (addition and multiplication). According to the collected data – i. e. syllabi and textbooks, only the two first steps of the didactic transposition (Fig. 1) is considered. FINDINGS Before the New Math (1870-1970) The syllabi from 2nd grade (1923) even 1st grade (1945) gather, in the same paragraphs, discrete and continuous quantities: indeed both types of measurement
130 are involved for whole numbers, operations, and place value -which is clearly connected with the metric system. Moreover, in the 4th grade, in 1945, fractions are fractions of quantities and decimals are introduced using the metric system. Before 1940, the teaching of arithmetic is much inspired by the treatises by Bezout (1779) then Reynaud (1821). This influence remains up to 1970. Both treatises present an arithmetical theory relying on measurement of quantities for numbers and ratios. In order to highlight the didactic transposition, I present excerpts of the beginning of Reynaud’s treatise and a page of a textbook (fig. 2) of the beginning of the 2nd grade. First chapter. Numeration, Addition, Subtraction, Multiplication, and Division of whole numbers. 1. Whatever is capable of increase or diminution is called quantity. When one thinks about the nature of quantities, one feels it would be impossible to have an exact idea of magnitude of quantities of the same kind without considering one of these magnitudes as a term of comparison; this magnitude is called unit; combining several units of the same magnitude forms a number. The manner of forming, expressing, and writing numbers is the object of numeration; and the science which teaches how to perform operations upon numbers is called Arithmetic. 2. Numeration (…) 3. A number is abstract or concrete when no particular denomination is mentioned to which its units belong or not. Thus, 3 and 5 times are abstract numbers; 3 toises and 5 leagues are concrete numbers (…). Within whole Arithmetic, as the nature of the units is known in advance, one only has to find their number; this leads to operate on abstract numbers. (Reynaud 1821, p. 1 and p. 6, my translation).
3. Addition of numbers. Be they pears or meters, three and two are five. Counting 3 and 2 are 5: this is doing an addition. Instead of 3 and 2 are 5, one writes: 3 + 2 = 5, what one reads: 3 plus 2 equals 5.
Figure 2: (Marijon et al., grade 2, lesson 4, 1947, p. 14, my translation)
131 In the treatise, numbers emerge from the measurement of quantities. Within the textbook’s page, the words “concrete” and “abstract” are not used. Whatever, three cases are taken for talking about addition. Putting together firstly discrete quantities (1. collections of pears), secondly continuous quantities (2. adding length through two pieces of a fishing rod) provide two cases with concrete numbers. A third case using abstract numbers is elaborated from the two first cases (fig. 2). On figure 3 – a page of the beginning of the 3rd-grade textbook, published in 1932 –, the themes of the word problems may seem old-fashioned. I don’t know whether they were familiar or not to the children of the 1930s. The structure of the lesson is common for the period: the rule is established using “solved problems”, and then it is stated. Exercises – not shown – are proposed after: simple and complex word problems involving multiplication the rule. This excerpt reveals several clues related to the underlying reference knowledge. They may not be easily visible for an unaware reader. Multiplying by several tens or hundreds Multiplying an amount of tens. Problem A box of quills is 60 g heavy. How heavy are 72 boxes? The weight is 72 times 6 tens of g.; 6 tens x 72 = 432 tens; 432 tens= 4.320 g. Solution. The total weight of 72 boxes is: 60 g x 72 = 4.320 g. Multiplying by an amount of tens. Problem How much do 60 barrels of 72 l. each contain? A ten of barrels contains: 72 l. x 10 = 720 l. 6 tens of barrels contains: 720 l. x 6 = 4.320 l. Solution. The total capacity of the barrels is: 72 l. x 60 = 4.320 l. Rule and practical disposition. To multiply an amount of tens 60 by 72, or to multiply 72 by the amount of tens 60, one multiplies 72 by 6 and one put a zero at the right side of the product. [column computation] Practically, one writes first the zero, then the product one the left. Figure 3: (Châtelet et al., 1932, p.160, my translation) First, I go back to the treatise: 6. The purpose of the multiplication is to calculate a number which is called product. The latter is formed with a known number called multiplicand, similarly as a given number -called multiplier- is formed with the unit. So, to obtain the product, one just has to operate on the multiplicand the same operations one would have operated on the unit to form the multiplier. Multiplicand and multiplier are the
132 factors of the product. For instance, for multiplying 5 by 3, one can observe that the multiplier is formed with three times the unit. So the product must be formed with three times the multiplicand 5; this product is thus 5 plus 5 plus 5 or 15. Generally, when the multiplier is a whole number, multiplication is simply repeating the multiplicand, as many times as there are units in the multiplier. The multiplier is always an abstract number because it indicates how many times the multiplicand has to be taken. The product is of the same nature as the multiplicand; indeed it expresses the addition of several numbers that are equal to the multiplicand. (…) (Reynaud, 1821, p. 10, my translation) The definition given by Reynaud may seem very complex. Actually, the formulation allows to gather in a same approach multiplication by whole numbers, and by fractions. With whole numbers, it goes back to repeated addition of the multiplicand. Bezout (1779) gives a shorter definition, though in the same vein: “Multiplying a number by another; this means: taking the first one as many times as there are units in the other one. Multiplying 4 by 3; this means: taking three times the number 4” (p.28, my translation). The reference knowledge is thus related to the external multiplication of quantities by a number. For instance, this implies that multiplication is not commutative. Within the textbook’s page, a major consequence of the definition is the distinction between multiplicand and multiplier. It has several implications: e.g. 1) 60 g. x 72 is read 72 times 60 g or 60 g multiplied by 72; 2) multiplier is written on the right, multiplicand on the left; 3) two cases are needed for elaborating the rule: multiplying tens and being multiplied par tens. Some textbooks elaborate the possibility of changing the order of abstract numbers within a multiplication before, so they refer to it and develop only one case; 4) actually this definition constraints but also allows to refer to quantities for elaborating the rule. Last, another point is the use properties of numeration, especially linked with units (Chambris, 2015). This hides the use of associativity. The New Math reform curriculum (1970) The introduction of the reformed syllabus asserts the curriculum is not a new one but a “different writing” of the previous one from 1945, a huge ecological reorganization I add. The main visible change is the creation of a new domain: “measurement: practical exercises”. Previously, there were only two domains: arithmetic and geometry. Now, they are three: numeric, geometry, measurement. An acute look at the text shows that: 1) The new measurement domain gathers the study of continuous quantities which was previously in the arithmetical part - measurement and computation with length, capacity, mass, time and corresponding metric units-, and that of the geometrical quantities which was previously in the geometrical part –computation of area and volume, and the corresponding metric units-. 2) The “numeric” domain is devoted to measurement (i.e. counting) and computation of discrete quantities, with many quotations of the
133 set theory. 3) The only exception to this is that of scales problems which moved from geometry to “numeric”, and joined there the old “rule of three” problems under the new “proportionality” label. Last, computation involving units’ names are forbidden: “Sentences like “8 apples + 7 apples = 15 apples” do not belong to mathematical language neither to ordinary language” (Instructions for syllabus, 1970, my translation). Decimals and fractions are studied in both domains: numeric and measurement with the same techniques and technologies. It is noteworthy that they do not involve fractioning of quantities. I interpret this as the transposition of the academic construction of the real numbers with whole numbers. The “numeric” domain contains many tables of numbers with operators. They support the new way to study ratios: with linear properties and coefficient, that is a trace of the transposition of the theory of linear application. These are traces of new reference knowledge. All computations are computations between numbers; and proportionality is no longer a relation between two quantities: it became a linear numerical relation. To sum up: The reorganization of the New Math syllabus is the sign of the implementation of new reference knowledge for numbers and ratios, of academic knowledge. Paradoxically, the birth of the measurement domain is the visible side of the disappearance of quantities as the roots of arithmetic. From the 1980s From 1970, the measurement domain has remained with several changes in each new syllabus, but more and more continuous quantities get into the “numeric” domain (fractioning of quantities for fractions and decimals from 1980, number line for whole numbers and fractions from 1980). This can come from the influence of several academic works in mathematics education: e.g. (Perrin- Glorian, 1992). Yet, despite the assertion that fractions are measurement of quantities, there are no fractions of quantities – such as: 3/4 of a length or of 300 g – in the syllabus. Moreover, most of mathematical definitions of quantities (e.g. Euclid Book I, Rouche, 1992, Griesel, 2007) involve a basic relation between order and addition: a < b if it exists c, a + c = b; yet, addition is never linked with order relation in the curriculum. Related to proportionality, in 2002, the syllabus indicates examples of reasoning (technologies) with quantities in the form of computation with units: One needs 400 g of fruit with 80 g of sugar to prepare a fruit salad. How much sugar is required with 1000 g of fruit? Reasoning can be of the following types: - For 800 g of fruit (twice more than 400), 160 g of sugar (twice more than 80) are required […]. For 1000 g (800g + 200 g) of fruit, 200 g (160 g + 40 g) of sugar are required; - The mass of sugar required is five times less than the mass of fruit; 200 g of sugar (1 000 : 5 = 200) are required. (Instruction for syllabus 2002, 3rd-to-5th grade, my translation)
134 Looking closely at these rationales, a phenomenon becomes visible: operations with units are used in discursive or in uncomplete arithmetical ways (the only equal sign is used between numbers). Operations with units seem not to belong to mathematics, but seem to fulfil pedagogical needs. My interpretation is that quantities come back for didactical needs, but the underlying mathematical reference knowledge has not changed from the reform. This brings to the fore the need for new reference knowledge: an adequate theorization of quantities for the teaching of numbers and ratios. Figure 4 displays an excerpt of a present 3rd-grade textbook: the topic is roughly the same as in 1932. According to the teacher’s guide, through the “discovery” activities, the students attempt first to compute 8 x 40 then 60 x 30 by their own means, then the focus –within the bubble- is on the implicit use of the associativity, mixed with commutativity of multiplication. It is supposed to be drawn on students’ attempts – and, if need be, on a “solved problem”: Leila’s method. There is no unit, no quantities. I specify that in the previous lesson – multiplying by ten –, no link had been established between multiplying tens and converting ones into tens (numeration). Last, current curricula do not propose a specific way for writing the numbers within multiplication (which number on the right?). Some examples of commutativity for multiplication are often given at the very beginning of the curricula. Then 3 x 5 or 5 x 3 are used indistinctly. There seems to be very little further investigation about this property. Multiplying by multiples of 10, of 100, of 1000 Goals: understanding the rule of multiplication by multiples of 10, of 100, of 1000. Discovery: 1. Compute: 8 x 40 then 60 x 30 2. See how Leila (a character of the book) has computed 8 x 40 then 60 x 30: […] Compare with your own computation. Within the bubble: In a product, one can choose the order in which multiplications are performed. Example: 6 x 10 x 3 x 10 or 6 x 3 x 10 x 10. 3. Compute: 7 x 50 and 30 x 20 using Leila’s method. Exercises: 1. Compute […] 2. Complete […] 3. Copy down the correct answer. a. One CD costs 12 €, 10 CD cost: 1 200 € 22 € 120 € 1 210 € b. One bike costs: 100 €, 5 bikes cost: 150 € 500 € 105 € 5 000 € 4. Copy down the table and complete it. […] Figure 4: (Peltier et al., 2010 p. 89, my translation), 3rd grade, middle of the year
135 A “learning" theory rules the design of the activity: building on students’ knowledge. The mathematical reference knowledge seems to be that of operations within the set of whole numbers –N-. Yet, within the ancient texts, quantities provide rationales for the multiplicative properties. It is not the case within the present excerpt. The rationale seems rather to be a matter of writing. Does this involve learning difficulties? Whatever, depending on the topic and on teachers’ choices, quantities can be solicited or not: e.g. some textbooks refer to “word problems” for this given topic. This fosters my previous interpretation: quantities have come back for didactical needs, especially when continuous quantities or ratio are involved. This come back is less clear for whole numbers.
DISCUSSION To begin with, the grain of the current study is rough and further investigation is needed to better characterize even the written curriculum. However, it is clear that the measurement domain cut many old trophic chains in the 1970s. Some new trophic chains were planned – e.g. the reorganization of proportionality - but it is not sure they “compensate” the broken links with quantities. Indeed, more and more continuous quantities have been added in the “numeric” domain after the reform, as if it were a pedagogical need. There is still a measurement domain which role is poorly defined, especially its contribution to the understanding of operations on quantities –e.g. addition-, involved for instance in the learning of fractions. The New Math is an international phenomenon (Kilpatrick, 2012) that impacted several western countries in the 1960s and 1970s, among them the US and Germany. Ma (2013) is very critic on the current written math curricula of the US, and connects its “disorganization” to the New Math and to the disappearance of a classical theory of arithmetic. Singaporean and Chinese curricula are built on such theory. Griesel (2007) identifies different “New reference knowledge” for fractions in West and East Germany during the New Math: with quantities, with sets or whole numbers. CONCLUSION This work helps to better understand some effects, even long term effects of the New Math in France. Quantities disappeared in reference knowledge for numbers and ratios surrounding the New Math. Yet, they seem to have partly come back - more and more in recent years- for pedagogical needs. It shows the lack of adequate reference scholar knowledge for numbers and ratios, connected to quantities despite relevant scholar works (e.g. Griesel, 2007; Rouche, 1992). Such theory notably enables to formulate basic arithmetical rules in basic terms. The increasing role of learning theory based on students’ activity that has been promoted from the New Math may hide –or compensate- this lack. Present study
136 leads to ask whether the results could be similar to other countries that were close (in term of curriculum) or not to France. Before the New Math, whole numbers, place value; and operations were taught with both discrete and continuous quantities. The creation of the “measurement” domain is the visible side of the transposition of the set theory and the elimination of continuous quantities in the reference knowledge for numbers, operations and ratios. Yet, continuous quantities seem to be a key input for conceptualization of numbers and ratios (Barrett et al., 2011); and the epistemology (Artigue, 1991) of numbers also fosters the desirable approaches on continuous quantities for numbers and ratios. Another question is: to which extent the roles of quantities in basic arithmetic impact teachers’ perceptions of Mathematics, especially the idea of rigor, and that of the relation between math and everyday life. References Artigue, M. (1991). Epistémologie et didactique. [Epistemology and didactics] Recherches en didactique des mathématiques, 10(2-3), 241–286. Barrett J., Cullen C., Sarama J., Clements, D.H., Klanderman, D., Miller A.L., Rumseyl, C. (2011). Children’s unit concepts in measurement: a teaching experiment spanning grades 2 through 5. ZDM, 43, 637–650. Bezout, E. (1779). Cours de mathématiques à l'usage des gardes du pavillon et de la marine, Eléments d'arithmétique (1). Ed: Musier fils. Bosch, M., Gascón, J. (2006). 25 years of didactic transposition. ICMI Bulletin, 58, 51–64. Bourbaki (1984). Eléments d'histoire des mathématiques. Masson. Bronner A. (1997). Étude didactique des nombres réels. Idécimalité et racine carrée. Thèse. Grenoble: Université de Grenoble I. Chambris, C. (2015). Mathematical foundations for place value throughout one century of teaching in France. In X. Sun, B. Kaur, J. Novotná (Eds.), Conference Proceedings of ICMI Study 23 (pp.52–59). Macau: University of Macau. Châtelet, A., Condevaux, G., Blanchet, L. (1932). Arithmétique. Cours élémentaire. Paris: Bourrelier-Chimènes. Chevallard, Y. (1985). La transposition didactique. Grenoble: La pensée sauvage. Furinghetti, F., Menghini, M., Arzarello, F., Giacardi, L. (2008). ICMI Renaissance: The emergence of new issues in mathematics education. In M. Menghini, et al. (Eds.), The first century of the International Commission on Mathematical Instruction (1908-2008). Reflecting and shaping the world of mathematics education (pp. 131– 147). Rome: Istituto della Enciclopedia Italiana. Griesel, H. (2007). Reform of the construction of the number system with reference to Gottlob Frege. ZDM, 39(1-2), 31–38. Kilpatrick, J. (2012). The new math as an international phenomenon. ZDM, 44(4), 563–571. Ma, L. (2013). A Critique of the Structure of U.S. Elementary School Mathematics. Notices of the American Mathematical Society, 60(10), 1282–1296.
137 Marijon, A., Masseron, R., Delaunay, E. (1947). Cours d’arithmétique. Le calcul à l’école primaire. Cours élémentaire. Paris: Hatier. Peltier, M.-L., Briand, J., Ngono, B., Vergnes, D. (2010). Euro Maths. CE2. Paris: Hatier. Perrin-Glorian, M. J. (1992). Aires de surfaces planes et nombres décimaux. Doctoral dissertation, Université Paris VII. Reynaud, A. A. L. (1821). Notes sur l’arithmétique, à l’usage de l’école polytechnique et de l’école spéciale militaire (9e édition). Paris: Librairie pour les sciences. Rouche, N. (1992). Le sens de la mesure. Bruxelles: Didier Hatier.
PROSPECTIVE ELEMENTARY SCHOOL TEACHERS’ LEARNING TRAJECTORY OF THE NUMERATION SYSTEM Olive Chapman
Abstract This paper reports on a study that investigated an intervention consisting of an inquiry task to facilitate prospective elementary school teachers (PTs) understanding of the numeration system to inform their teaching in helping all students to learn meaning of whole numbers and whole number operations. The focus is on the learning trajectory of the PTs during the intervention and the knowledge they developed. Findings indicate the challenges PTs are likely to encounter in their learning and possibilities of how these challenges can be meaningfully addressed. Keywords: Prospective elementary teachers, numeration system, learning trajectory One of the challenges teachers experience in the mathematics classroom is dealing with students of different mathematics ability or different interests in learning mathematics. Some teachers deal with this by giving students they consider to be weak/slower in learning mathematics simpler tasks that do not require students to work to their potential or be challenged in a meaningful way. Some give students they consider to be “good at mathematics” extra tasks that may not be of high cognitive demand and be demotivating in learning mathematics meaningfully. But as the NCTM’s (2000) equity principle states, “excellence in mathematics education requires equity--high expectations and strong support for all students” (p. 12). To achieve this, teachers need to hold mathematics knowledge for teaching that enables them to differentiate tasks meaningfully to engage all students to their potential. This provides one way of contributing to equity in the classroom. This paper reports on a study with prospective elementary school teachers (PTs) that investigated an intervention to facilitate their understanding of the numeration system and how it could inform their teaching in helping all students to learn meaning of whole numbers and whole number operations. The focus is on the
University of Calgary, Canada; e-mail: [email protected]
138 learning trajectory of the PTs during the intervention and the knowledge they developed. Related Literature It is well established that PTs’ mathematics content knowledge for teaching based on their learning of school mathematics is likely to be problematic. Llinares and Krainer (2006) and Ponte and Chapman (2006) referenced studies over a 30-year period that identified several issues with teachers’ knowledge in relation to what is adequate to teach mathematics. Ponte and Chapman (2008, 2016) showed that studies continue to identify limitations with, or raising concerns about, prospective teachers’ mathematics knowledge. These issues include misconceptions or lack of understanding for different topics of school mathematics, such as, numbers and number theory. Based on Ponte and Chapman’s (2008) survey, this includes procedural attachments that inhibit development of a deeper understanding of concepts related to the multiplicative structure of whole numbers; influence of primitive, behavioral models for multiplication and division; inadequate conceptual knowledge of division and sparse connections between the two; inability to establish connections among different representations of rational numbers; difficulties in the language of fractions; and distorted definitions and images of rational numbers. Early studies tended to focus on whole number operations and word problems. For example, Tirosh, Graeber and Glover (1986) explored PTs’ choice of operations for solving multiplication and division word problems based on the notion of primitive models in which multiplication is seen as repeated addition and division as partitive. Findings indicated that the teachers were influenced by the primitive models and their errors increased when faced with problems that did not satisfy these models. Greer and Mangan (1986) also found that for single- operation verbal problems involving multiplication and division, primitive operations affected the PTs’ interpretation of multiplicative situations. Recent studies tended to focus on rational numbers. For example, Lo and Luo (2012) studied the knowledge of fraction division of Taiwanese PTs and found that tasks of representing fraction division through word problems and diagrams proved to be challenging even for proficient prospective teachers. Tobias (2013) examined how PTs in the US developed an understanding of language use for defining the whole and found that even when important mathematical ideas were appropriated some participants still had difficulty in distinguishing among the terms ‘of a’, ‘of one’, ‘of the’, and ‘of each’. Stacey et al. (2001) in their investigation of Australian elementary teachers’ understanding of decimals, concluded that some participants lacked good content knowledge of decimals. While studies on PTs’ knowledge draw attention to concerns that should be addressed in teacher education, it is also important to view such studies from the perspective of what PTs can do and how to build on it to support their growth in
139 understanding of the mathematics concepts they are required to teach. In this study the focus is on their growth trajectory in understanding the numeration system. Theoretical Perspective The theoretical perspectives framing this study are related to mathematics knowledge for teaching and inquiry-based learning. Mathematics Knowledge for Teaching [MKT]. Ball, Thames, and Phelps’ (2008) perspective of MKT has received much attention in studies on the mathematics teacher and was also adopted in this study. Their perspective suggests that there is an important distinction between knowing how to do mathematics problems and knowing mathematics in ways that enable its use in teaching. Teachers need to know both, reflected in their model of MKT, which consists of six components of knowledge. The focus here is on three of them: specialized content knowledge, knowledge of content and students, and knowledge of content and teaching. Specialised content knowledge enables teachers to accurately represent mathematic-al ideas, provide mathematical explanations for common rules and procedures, and examine and understand unusual solution methods to problems or patterns in student errors. Knowledge of content and students is “content knowledge intertwined with knowledge of how students think about, know, or learn this particular content” (Hill et al., 2008, p. 375). This includes knowledge associated with teachers having to anticipate student errors and common misconceptions, interpret students’ incomplete thinking, and predict what students are likely to do with specific tasks and what they will find interesting or challenging. Knowledge of content and teaching is knowledge about how to build on students’ thinking and how to address student errors effective-ly. This includes knowledge associated with teachers having to sequence content for instruction, recognize instructional pros and cons of difficult representations, and size up mathematical issues in responding to students’ novel approaches. Also related to this study is Ma’s (1999) perspective that teachers should have “profound under- standing of fundamental mathematics” which is attuned to and usable in teaching. This understanding includes connectedness, multiple perspectives, fundamental ideas, and longitudinal coherence. These theoretical perspectives were considered in the design and goal of the learning activities used in this study. Inquiry-based learning. Inquiry as a basis of learning is well established in the literature (Dewey, 1938; Wells, 1999). Based on Dewey’s (1938) notions of inquiry, the process begins with the learner encountering a puzzling situation that initiates the generation of questions, which leads the learner to notice certain features of the situation that results in the formulation of possible lines of action. Recent perspectives of inquiry include Wells’ (1999) dialogical inquiry defined as: “A willingness to wonder, to ask questions, and to seek to understand by collaborating with others in the attempt to make answers to them” (p. 121).
140 Inquiry-based learning, then, is a cyclical process with the following components: pose questions/problems, create conjectures/predictions, investigate/explore, create new knowledge, and communicate and evaluate. This leads to new questions and further investigation through a new cycle of inquiry. Inquiry-based teaching allows students’ questions and curiosities to drive curriculum, honours previous experience and knowledge, makes use of multiple ways of knowing, and allows for creation or adoption of new perspectives when exploring issues, content, and questions. Students are given opportunity to direct their own explorations and find their own answers. In this study, PTs were provided with the initial task to then engage in inquiry. Research Method The research methodology is a case study (Stake, 1995) grounded in a naturalistic paradigm (Lincoln and Guba, 1985) focused on the experiences of the participants in a natural setting to understand their realities by identifying significant patterns/ themes in their thinking and actions while participating in the educational activities. The case consists of one course on mathematics education. Participants were 24 PTs in a course on mathematics for elementary teachers. They were in their final semester of a 4 semester, 2-year B.Ed. program. They all held a Bachelor of Arts degree with no post-secondary mathematics or science related courses. They all expressed fear of mathematics and weak backgrounds in it. The course focused mainly on numbers with consideration of NCTM (2000) standards for grades 3-5 numbers that include: understand numbers, ways of representing numbers, relationships among numbers, and number systems; understand meanings of operations and how they relate to one another. Specific topics covered in the course consisted of: relationships among sets of numbers; numeration systems; for each set of whole numbers, integers and rational numbers: number properties, meanings of numbers, meanings of operations, meanings of algorithms/procedures, modes of representation, models of representing meanings, applications/contextual situations, history (culture), students’ errors/misconceptions/alternative conception. The 24 PTs worked in groups of 4 with inquiry tasks to explore these topics. The focus here is only on the PTs’ engagement with tasks on the numeration system. The Numeration System Task (Table 1) evolved over two years based on its use with two classes of PTs. The first version of the task was: “Build a numeration system for a society of aliens with 3 fingers. The system must follow the same structure as our numeration system. Represent each of the first 30 numbers with the unifix cubes.” After clarifying the requirement with the PTs, their approach was to replace the symbol with materials, using different colors of the cubes to represent different place values and interpreting the three positions to represent the base 3 system as in figure 1 (e.g., 123 in base 3).
141
Figure 1: PTs’ base 3 system It was difficult to get them to shift from this way of thinking through inquiry. The task was also tried with secondary mathematics prospective teachers who did the same thing, tried to find patterns in the symbols or recalling what they learned about bases with little success, but were also not focusing on meaning and structure based on concrete representation. The revised task allowed the next class of PTs to engage in an interesting learning trajectory that became the focus of this study.
Assume you are a mathematics teacher hired as a consultant to a science fiction screen writer who is creating a civilization of people with the unique feature of a total of 3 fingers for both hands. They can count only in a maximum of threes. You are asked to build a numeration system for them so they can count and record large quantities efficiently. You decide to build a system that uses the structure of the Hindu-Arabic numeration system. However, to avoid confusion, you choose not to use the symbols or their names for the numerals of the Hindu-Arabic system. Instead, you decide to use the English alphabet (beginning with A) for numerals. 1. Plan and develop the system. 2. Describe the system by representing each of the first 30 numbers in sequence in terms of its quantity with concrete materials using unifix cubes of only one colour. [Note: You are to work in a concrete mode only. So no writing is allowed here.] 3. Describe the system by recording each of the first 30 numbers in sequence symbolically and with its pictorial representation.
Table 1: Numeration System Task
Data sources consisted of interviews, classroom observations and the PTs’ written group work of all related activities. The researcher/instructor and research assistants made field notes of group discussions and work with the manipulatives. Related whole-class discussions were recorded and transcribed. A PT from each of the 6 groups was randomly selected for interview on the groups’ processes and learning.
142 Analysis focused on the PTs’ learning trajectories and learning from the experience. The researcher and trained research assistants independently coded the data to high-light key aspects of the PTs’ learning trajectories and knowledge developed. This included identifying: turning points; instructor’s role; PTs’ role; key activities; when, how, and why the PTs got stuck; and what happened to get out of stuck during the trajectories. The theoretical perspective of MKT guided the coding of the knowledge the PTs developed. The coded information was grouped by emerging themes and validated through an iterative process of identification and constant comparison. Findings Findings focus on how the learning trajectory unfolded in the PTs’ development of MKT whole numbers and the knowledge they developed. While the PTs’ learning progressed at different rates, they passed through similar key points in the trajectory. I report only on these key points, which were barriers/challenges they had to over-come and needed help to shift their thinking to do so. I refer to the system they were required to create as ‘Alien System’ and the Hindu-Arabic system as the H-A system. Learning Trajectory. The PTs’ initial interpretation of the task was to represent the numbers in the Alien System in terms of their understanding of place value in the H-A system. This involved modelling numbers as in figure 1. Because they could use only one color of unifix cubes, they used different orientations of them to differentiate the three columns. Their questions and confusion resulted in a discussion about why this approach was not appropriate. They were prompted to think of another way to build the numbers in sequence and represent the meaning/quantity of each number concretely as they would do in the H-A system. Some groups did the following (figure 2) and named each number A, B, C, D, E, … (i.e., 1, 2, 3, 4, 5, …). Others tried to make groups of 3 in random ways, and naming them A, B, C, D, E, ….
….
Figure 2: PTs’ Alien system #1
143 They were prompted to think of how many numerals should be in the Alien System based on how many were in the H-A system. This resulted in a discussion of the meaning of numerals, how many are in the H-A system, the relationship to number of fingers and the base of the H-A system. They finally agreed that there are 10 numerals in the H-A system, so there should be 3 in the Alien System and it should be base 3. Their challenge became how to represent base 3 numbers. The following two versions (figures 3 and 4) eventually emerged from their group inquiry.
Figure 3: PTs’ Alien System #2
Figure 4: PT’s Alien System #3
Discussion among the groups resulted in agreement that figure 4 was the correct representation. However, when they were asked to count in their Alien Systems, they said: “A, B, C, D, E, F, G, H, …” [i.e., 1, 2, 3, 4, 5, 6, 7, 8, …] in both cases. They were reminded to think of the number of numerals in the system. They decided it should be A, B, C (i.e., 1, 2, 3) and now counted as in figure 5.
A B C CA CB CC CCA CCB
Figure 5: PTs Alien System #4
They were prompted to think of what was missing in this system compared to the H-A system regarding the numbers associated with the numerals. They realized it was zero. Prior to this task, they were given examples of numeration systems without 0 to explore. So, there was a discussion comparing their non-zero system to these, which led to connections regarding the evolution of the H-A system and how it got its name. Returning to building the Alien System with the knowledge that it needed zero, some groups debated whether they should have A, B, C, D as their numerals. Their thinking was that A would represent zero as a placeholder
144 and not a number. They now counted as in figure 6, where x was a different orientation of a cube.
x A B C D DB DC DD DDB DDC ….
Figure 6: PTs’ Alien System #5 They were again reminded to think of the number of numerals in the H-A system. Discussion among groups resulted agreement that it should be A = 0, B = 1, C = 2. However, they continued to use a cube to represent zero to start the sequence and to not use it again in the sequence as in figure 6. This led to a discussion of the meaning of zero as a number in the H-A system and how to model it with materials. This was followed by discussion of the challenge for kids to understand zero as a concept/ number since it cannot be represented concretely as a whole number. They then removed the cube for zero for the first number and the D, and renamed the numbers in figure 6 as variations of A, B, C, BC, BBC, BCC, 2BC, 2BCB, 2BCC, …. They were prompted to consider how zero was used throughout the H-A system compared to their Alien system. They realized that zero was still missing from their system and started exploring where the zeros should occur. The biggest challenge for many of them was the renaming of the groups of 3 as one. At this point it was suggested that they use the H-A numeral (0, 1, 2) of letters if it is more helpful to them. Once they were able to see the connection between renaming in the two systems and the pattern in their system in terms of cycles, they were able to name the numbers as in figure 7. This led to a discussion of the relationship between the cycles, the base and place values regarding what they noticed.
0 1 2 10 11 12 20 21 22 …
Figure 7: PTs’ Alien System #6 The PTs tested, practiced and applied their understanding with tasks, which they did with no help or prompting by the instructor, that included repeating the task for bases 4 and 6; counting orally in bases 3, 4, 6 or others; counting backwards in these bases; completing a counting worksheet that included bases 11 and 12; adding, subtracting, multiplying, and dividing multi-digit numbers in bases
145 between 2 and 9; interpreting numerical exponents; converting to/from base 10; interpreting other numerations systems (e.g., Mayan and oriental); exploring arithmetic algorithms with base 10 blocks; and interpreting students’ thinking/algorithms of whole number operations Finally, the class reflected on the learning trajectory and compared it to the path of humans’ in developing the numeration system and how needs of society influenced its development at different points in time. They also discussed how students are likely to go through a similar learning trajectory for base 10 and how their experience with the Alien System could inform how they support students through it. Knowledge developed The PTs not only developed a deep understanding of the numeration system, but also learned about where their students could encounter challenges in learning the base 10 system, what kind of challenges, and how to provide help to support their learning. PTs’ feedback and follow up activities, including lesson planning and task selection or creation for teaching base 10 concepts, indicated that they understood how to use manipulatives to represent and help students to interpret meaning of the number concepts, create or better select appropriate tasks to meaningfully engage students to learn the number concepts, and differentiate tasks to address diverse learners. They understood challenges students could experience in learning the base 10 system and explained how they would be empathetic to their students in such situations because of their experience with the base 3 system. The experience helped them to understand what it means to learn the base 10 concepts and to engage in making and testing/ justifying conjectures and noticing patterns and relationships. Implications and conclusions Teachers’ understanding of the numeration system is fundamental to provide appropriate and useful support for students to learn the base 10 system and whole number concepts in the elementary school curriculum. This paper suggests that PTs with an instrumental understanding of the numeration system can be helped to develop conceptual understanding of it through an inquiry task that enables them to experience conflict, puzzlement, curiosity that lead to questioning and exploration to achieve this shift. The paper provides an example of PTs’ learning trajectory in developing understanding of the numeration system that could inform teacher education regarding the challenges PTs are likely to encounter in their learning and possibilities of how these challenges can be meaningfully addressed. It draws attention to the role of their prior knowledge, both an asset in supporting what they noticed and a liability in limiting what they noticed. They required a shift in their attention to see a familiar situation from a different perspective. This shift required prompting by the instructor to point them to a different direction but to allow them to get there on their own and ‘to see’ for
146 themselves. It is important to both challenge and build on what they know so they could see it in new ways instead of as replacing a wrong way with a right way, which could be demoralizing instead of empowering. The study also shows that the pathway of the learning trajectory is not linear and cannot be predetermined. It was influenced by the barriers the PTs experienced that required detours or different lenses to get to the destination. It was these unexpected barriers, detours, and lenses that made the journey educative in relation to their MKT. The trajectory enabled them to gain insights of possibly key stages in the historical development of the base 10 numeration system and students’ learning of it. References Ball, D. L., Thames, M. H., Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407. Dewey, J. (1938/1997). Experience and education. New York: Simon and Schuster. Greer, B., Mangan, C. (1986). Choice of operations. In Univ. of London Inst. of Educ. (Eds.), Proceedings of the 10th PME International Conference, 1 (pp. 25–30). London. Hill, H., Ball, D. L., Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400. Lincoln, Y. S., Guba, E. G. (1985). Naturalistic inquiry. Beverly Hills, CA: Sage. Llinares, S., Krainer, K. (2006). Mathematics (student) teachers and teacher educators as learners. In A. Gutierrez, P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 429–460). Rotherdam: Sense. Lo, J.-J., Luo, F. (2012). Prospective elementary teachers’ knowledge of fraction division. Journal of Mathematics Teacher Education, 15(6), 481–500. Ma, L. (1999). Knowing and teaching elementary mathematics. Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, New Jersey: Lawrence Erlbaum Associates, Inc. National Council of Teachers of Mathematics (2000). Principals and Standards for School Mathematics. Reston, VA: NCTM. Ponte, J. P., Chapman, O. (2006). Mathematics teachers knowledge and practices. In A. Gutierrez, P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present and future (pp. 461–494). Rotterdam, The Netherlands: Sense. Ponte, J. P., Chapman, O. (2008). Preservice mathematics teachers’ knowledge and development. In L. D. English (Ed.) Handbook of international research in mathematics education: Directions for the 21st century (2nd Ed, pp. 225–263). New York: Routledge. Ponte, O., Chapman, O. (2016). Prospective mathematics teachers’ learning and knowledge for teaching. In L. English, D. Kirshner (Eds.) Handbook of International
147 Research in Mathematics Education (3rd Ed, pp. 275–296). New York: Taylor & Francis. Stacey, K, Helme, S., Steinle, Y., Baturo, A., Irwin, K., Bana, J. (2001). Preservice teachers’ knowledge of difficulties in decimal numeration. Journal of Mathematics Teacher Education, 4(3), 205–225. Stake, R. E. (1995). The art of case study research. Thousand Oaks, CA: Sage. Tirosh, D., Graeber A., Glover R. (1986). Pre-service teachers’ choice of operation for multiplication and division word problems. In Univ. of London Inst. of Educ. (Eds.), Proceedings of the 10th PME International Conference, 1 (pp. 57–62). London. Tobias, J. M. (2013). Prospective elementary teachers’ development of fraction language for defining the whole. Journal of Mathematics Teacher Education, 16(2), 85–103. Wells, G. (1999). Dialogic Inquiry: Toward a Sociocultural Practice and Theory of Education. Cambridge: Cambridge University Press.
IMPLEMENTING AN INSTRUCTIONAL SEQUENCE FOR SOLVING NET WORTH PROBLEMS Victor Cifarelli,Michelle Stephan, David K. Pugalee and Chuang Wang Abstract This paper reports on a study of students solving net worth problems, a class of real- world problems that model the operations of addition and subtraction and thus represent familiar mathematical situations for elementary grades students. Using episodes drawn from classroom work, the analysis focuses on 1. Examining the concepts involved in solving these types of problems and 2. The teacher’s implementation of an instructional sequence designed to support students in making connections between the underlying concepts and formal integer addition and subtraction operations. Keywords: Mathematics learning, problem solving, instructional design Among mathematics educators there appears to be agreement that reform-based approaches for teaching mathematics should include problem solving that gives students opportunities to explore real-life mathematical situations (Ball, 1993; Felton, Simic-Muller, Menéndez. 2012; Lesh and Lamon, 2013; NCTM 2000; Sáenz-Ludlow, 2004). Investigations and Connected Mathematics are examples of popular US reform-based curriculum that include a prominent role for real-life mathematical applications. Despite this agreement that real-life applications should be included in the mathematics curriculum, formal standards and recommendations typically state
University of North Carolina in Charlotte, USA; e-mail: [email protected], [email protected], [email protected]
148 the need but offer little in the way of detail of the conceptual benefits of this approach. If for example, we consider 6th grade Common Core State for Mathematics (Common Core State Standards Initiative, 2011) relating to addition and subtraction of rational numbers:
CCSS.MATH.CONTENT.6.NS.C.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge). Use positive and negative numbers to represent quantities in real-world contexts and also explain the meaning of 0 in each situation.
While the first sentence states the primary goal of the standard, the importance of understanding positive and negative numbers, as quantities having opposite directions on a number line, the second sentence citing the importance of connecting operations with positive and negative numbers to real-world contexts seems like an afterthought with no specificity of why it might be useful to students’ conceptual development. Against this backdrop, Whitacre et al. (2017) state that unlike positive integers, the notion of a negative number as being less than zero has few examples in real-life situations. These issues present at least two challenges for primary classroom teachers: 1. Finding appropriate real-life tasks for students to solve that that model addition and subtraction, and 2. Designing a sequence of instructional activities that help students make conceptual connections between the real-life problem situations and corresponding mathematical operations. The second of these challenges is particularly problematic given the ways that most textbooks typically introduce formal arithmetic operations with integers. As noted by Stephan (2009), “Traditional textbooks tend to introduce integer operations abstractly through additive inverses and maneuvers on a number line” (Stephan, 2009, p. 16). The current paper addresses these issues by examining classroom episodes that demonstrate how students’ solution of real-world problems can connect with the formal operations of addition and subtraction. GOALS AND PURPOSES The purpose of the study is to examine how students’ problem solving in familiar mathematical situations can help provide the needed conceptual foundation leading to formal addition and subtraction operations with integers. Our framework combines aspects of Realistic Mathematics Education (RME) with a constructivist view of learning. RME is based on the idea that mathematics should not be viewed as a closed system but rather as a human activity in which the mathematization of real situations is the key constructive process (Freudenthal, 1973). In addition to RME, our study includes a constructivist focus on learning
149 including the view that students can further their mathematical knowledge through self-generated problematic situations that can serve as learning opportunities. The study focused on students’ solutions of problem tasks involving a person’s net worth. Considering a person’s net worth is a familiar context for students in the upper elementary grades. These problems allow for various transactions such as assets and debts that can be modeled as using arithmetic operations of addition and subtraction.
METHODOLOGY
Participants The students came from a 7th grade classroom in a public middle school (Grades 6-8) in the southeast United States and were participants in a classroom teaching experiment conducted by the second author (Stephan and Akyuz, 2012). The class consisted of 13 boys and 7 girls. While not elementary grades students, we thought it appropriate to examine this population for the following reasons. First, the concepts of algebra and other secondary grades topics can be viewed as involving a transition from arithmetic; hence we can learn about the foundations of algebra by observing student solving problems that involve the arithmetic operations learned in the elementary grades (Ball, 1993). Second, we noted that three of the boys were identified as having learning disabilities and over half the class was performing below grade level; hence we believe that our analysis sheds light on apparent gaps’ in elementary students’ understanding of integer operations. Finally, 7th grade is not that far removed from elementary and this study informs conceptions of net worth in elementary mathematics. For these reasons, we believe that the study is of interest to the SEMT audience. Measures and instrumentation We wanted to examine the sequence of tasks used by the teacher to have students explore and solve net worth problems. Consistent with the goals of RME, the teacher had a hypothetical learning trajectory (HLT) for the concept of net worth and thus, at least a preliminary idea of a sequence of tasks that would aid the students’ development of conceptual understanding. That is not to say that the tasks were pre-determined prior to classroom instruction; rather they are consistent with an initial assessment by the teacher, identified areas of potential difficulty for students and become actualized by the teacher and students based on how classroom activity unfolds. Hence, in developing the tasks that were used (Table 1), the teacher had an explicit problem in mind to present to students (Task 1) that formed an initial problem solving opportunity for students, the solution of which guided the teacher to pose additional problems for students to solve (Tasks 2-4).
150 Task 1: Solve a Target Net Worth Problem Dylan has a net worth of $1000. He has to pay a fine of $3000. What is his net worth? Task 2: Solve net worth problem to address concept “adding asset” is equivalent to “taking away debt” Task 3: Solve simpler related problem to solidify the concept “adding asset” is equivalent to “taking away debt” Task 4: Students make up and solve a related problem
Table 1: Tasks Used in the Study The teacher presented Task 1 in order to engage students in problems having negative net worth. Based on the ways students solved Task 1, the teacher posed Tasks 2 and 3 in order to 1): Address any difficulties the students experienced in solving the task and thus promote further discussion of the underlying concepts; and 2): Isolate and focus on a particular aspect of the concept of net worth, that the transaction of “adding an asset” is equivalent to “taking away debt”. Finally, with Task 4 the teacher provided an opportunity for students to make up and solve their own net worth problems to see how well the students understood the underlying concepts. Data collection procedures The classroom episodes presented in this paper came from two class periods of a course in which the second author served as a co-teacher. The teacher presented the problem tasks to students who were then given adequate time (usually 2-3 minutes) to develop their solutions. Upon completion of work on a task, the teacher asked students to share and explain their solutions with the class. Students volunteering solutions used a white board to present and explain their solutions. The classes were videotaped and a transcription of classroom verbal responses was prepared for use in the analysis. The episodes reported in this paper are numbered so that we can refer to them in our analysis. Data analysis procedures To analyze data, we examined videotapes from two classes to document 1): the problem solutions volunteered by individual students, identifying particular questions and issues raised by other students in making sense of the solution; and 2): the teacher’s instructional actions, particularly her ongoing assessments of student work and how these lead her to pose to new questions to the class. RESULTS In summarizing the results, we note that the students at the time of the classroom observations had solved simple net worth problems involving 1): What it means to add and/or take away assets and debts; 2): Whether or not particular
151 transactions represent good or bad decisions regarding one’ net worth; and 3): How mathematicians would write the transaction using mathematical symbols. In addition, students had some experience representing these quantities on an open vertical number line (VNL). Examples of these problems are included in Table 2. Examples Math Symbols Bradley added an asset of (+$3000) to his +(+3000) net worth statement Ann took away an asset of (+$200) from her -(+200) net worth statement Devon added a debt of (-$650) to his net +(-650) worth statement Ernie took away a debt of (-$5400) from his -(-5400) net worth statement Table 2: Examples of Simple Transactions and Impact on Net Worth Perusing the entries of Table 2, we might say for example that mathematicians would write Ann’s transaction as –(+200), with the first sign indicating an action “take away” or “add” and the second sign within the parenthesis indicating whether it is an asset or a debit. In the results that follow, we will summarize classroom episodes from two classes. Due to space limitations, we will focus on Tasks 1 and 2 only. For each of these, we will present the task, relevant episodes of student and teacher, and our analytical comments. Task 1 In completing Task 1, students must solve a problem that has a negative net worth solution. Noteworthy in these episodes are 1): The initial solution presented by student Sergio; 2): The questions and issues raised by other students in making sense of Sergio’s solution; and 3): The teacher’s role in orchestrating classroom discussion. 1. TEACHER: Let’s say that Dylan starts with net worth $1000 and he has to pay a fine of $3000 (Figure 1). I want you to do this mentally and see how you use a number line to find out what happened to his net worth.
Figure 1: Dylan’s Net Worth (Task 1)
152 After a few minutes, Sergio volunteers to share his solution. 2. Sergio: Remember how we bring it down to zero, that is what I did. I wrote minus one thousand, I got to zero and then minus rest of the 2000 (Figure 2).
Figure 2: Sergio’s Solution 3. Sergio: I took away 1000 from this number and I just subtracted 1000 from it. Since they are both negatives then it comes to -2000. I split up two chunks. That is what we did before (Figure 3).
Figure 3: Sergio’s “Chunking” Strategy
4. Melissa: Do you write minus 1000 to find out what you left? 5. TEACHER: I am going to pull together what Sergio said and what Melissa said all in one explanation. See if they are connected. Sergio says, “I have got to go down total chunk of 3000 dollars” you got that part, right? He says because we are going through zero we are changing from positive to negative, black area to red area. His first jump is 1000. Sergio, how did you choose to do that first 1000 chunk? 6. Sergio: What I did was 1000 minus 1000 is zero. 7. TEACHER: So you get down to zero. How much did he go down totally? The teacher drew an arrow on Sergio’s diagram and writes 3000 (Figure 4).
153
Figure 4: Teacher’s Annotation of Sergio’s Diagram
8. TEACHER: He only did little chunk of 1000 and he got to the zero. What is the rest left over? I already jumped 1000 that means I had better jump 2000 more. I have only gone debt of 1000 and I have 2000 more to go. So that is going to set him 2000 dollars net worth, negative. Did I clearly represent two points of view?
Figure 5: The Teacher’s Further Annotation of Sergio’s Diagram Students: Yes! 9. TEACHER: I know some of you guys really struggle with this number line thing. But I think it is helpful for us to see why you do subtraction or why you add when you add. Other questions? Andrew you struggle with his or you got it? 10. Andrew: I got it 11. TEACHER: Samantha? Maria? No other questions for Sergio? Anybody do anything differently? Some of you may have done this in your mind without drawing number line. Now you have a way to show it. Thanks Sergio. Analytical Comments on Task 1 Sergio’s solution (21, Figure 2) was quite sophisticated: he used both a VNL as well as a vertical stacking of numbers to show and explain his ‘jumps’ on the
1 Numbers refer to particular episodes.
154 number line (Figure 3); further, he was able to go back and forth between the two representations in explaining his solution when he answered questions posed by other students (3, 6). The teacher’s interventions included re-casting (5, 7, 8) and extending the discussion through annotations of student diagrams (Figures 4 and 5). Task 2 In task 2 the Teacher looked to vary the situation by having students explore what it means to add an asset to a negative worth situation. In so doing, her goal was to introduce the idea that reducing a debt could be viewed equivalently as adding an asset to one’s net worth. 12. TEACHER: Let me try another one. Brandon, his net worth is -10,000. What transaction could happen? He wins bowling competition. How about this transaction +(+5000). What is his new net worth? (Figure 6)
Figure 6: Brandon’s Net Worth (Task 2) 13. Jennifer: What happens to him? He gains 5000, right? He goes up. 14. Matthew: What I did first this was positive number so I did -10,000 minus 5,0000. Why I did that, to see how much asset would help him. In episode 14, Matthew’s comment “-10,000 minus 5,0000 …to see how much asset would help him” is an indication he sees that adding the asset is equivalent to reducing the debt. The teacher then seizes upon this opportunity to draw the other students into a discussion of what it means to add an asset to a debt. 15. TEACHER: I like that language he said I wanted to see how much new asset was going to help him. You all agree it is going to help? 16. Students: Yes! 17. TEACHER: (Matthew presents solution on white board) Can you move side so that people can see. 18. Matthew: I added 5000 to -10,000 and I got -5,000. (Figure 7) 19. Teanna: I do not agree with that. You are adding an asset.
155
Figure 7: Matthew’s Solution 20. Melissa: Yes, you are adding an asset but you have greater debt than that. 21. TEACHER: We start with net worth and we have arrow either going up or down. What number goes on to arrow? 22. Dakota: Assets or debts that are being added. 23. TEACHER: All right. The transaction whatever asset or debts being added or taken away. This is the transaction we said that goes into here. Is that clear everybody? Because I know some people struggle with that. Transaction is whatever happens and we are going to go down or up. And then this is the new net worth where we land. Matthew went short of zero. Should we go up to zero? 24. Students: No! 25. Matthew: You go above zero if you pay off whole debt. 26. TEACHER: I do not know if you heard what he said, it is really nice. You could go above zero if you could pay off all 10,000 debt. Did he pay all off? He only paid off actually, how much of it? The teacher then annotated Matthew’s VNL diagram, using an arrow to indicate the transaction -10000-(-5000) explaining that “he took away 5000 of the debt” (Figure 8).
Figure 8: Teacher’s Annotation of Matthew’s Diagram 27. Students: 5000.
156 Analytical Comments on Task 2 Matthew’s solution (14, 17, Figure 7) introduced the idea that adding the asset was helpful to Brandon’s overall situation by eliminating part of the debt. The teacher then had the students explore Matthew’s assertion by performing corresponding jumps on the number line (18-20). Finally, the teacher promotes discussion of the students’ findings (26, Figure 8). Though unable to include results from the students solving Tasks 3 and 4 in this paper, we nevertheless emphasize that in posing these tasks, the teacher looked to solidify the students’ understanding of the transaction by having them solve a related problem (Task 3) and then make-up a problem that was similar (Task 4). CONCLUSIONS We must be careful not to conclude too much from these findings but some observations are noteworthy. First, the view that real-life problems can be used to aid students’ development of understanding of formal operations with integers appears warranted and should be considered as a useful resource for exploring concepts. The students could personally relate to the various transactions of net worth problems and through their problem solving actions, make some useful connections to the more formal operations of integer addition and subtraction. Second, the teacher’s role of stimulating discussion with the various interventions would have been more difficult if she did not already have an initial sequence (HLT) in mind for the concepts underlying net worth problems. Hence, we see the approach as a useful strategy for teachers to implement when teaching particular concepts. Third, the classroom climate was one where students were not at all afraid to volunteer their thoughts or defend their ideas. That the students were comfortable to receive and send constructive comments to and from peers was a testament to the teacher’s establishment of a positive classroom environment in which everyone’s ideas were valued. Finally, the results remind us that how students express their mathematical knowledge including the language they use is fraught with much subjectivity. The teacher’s role to orchestrate of discussion then becomes challenging. She needs to be able to pay particular attention to the language and the mathematical symbols students use in expressing themselves to others, and be prepared to regularly recast and comment upon the actions students perform. References Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93(4), 373–397. Common Core State Standards Initiative. (2011). Common Core state standards for mathematics. Felton, M. D., Simic-Muller, K., Menéndez, J., M. (2012). ‘Math Isn’t Just Numbers or Algorithms’: Mathematics for Social Justice in Preservice K–8 Content Courses. In L. J. Jacobsen, J. Mistele, B. Sriraman (Eds.), Mathematics Teacher Education in the
157 Public Interest: Equity and Social Justice (pp. 231–52). Charlotte, NC: Information Age Publishing. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, the Netherlands: D. Reidel. Lesh, R. A., Lamon, S. J. (Eds.). (2013). Assessment of authentic performance in school mathematics. Routledge. Sáenz-Ludlow, A. (2004). Metaphor and numerical diagrams in the arithmetical activity of a fourth-grade class. Journal for Research in Mathematics Education, 1(35), 34–56. Stephan, M. (2009). What are you worth? Mathematics Teaching in the Middle School, 15, 16–23. Stephan, M., Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education 43 (4), 428–464. Whitacre, I., Azuz, B., Lamb, L., Bishop, J. P., Schappelle, B. P., Philipp, R. A. (2017). Integer comparisons across the grades: Students’ justifications and ways of reasoning. The Journal of Mathematical Behavior, 45, 47–62.
THE CONTENT KNOWLEDGE ABOUT RHOMBUS OF TURKISH AND SLOVAK PRE-SERVICE ELEMENTARY TEACHERS Asuman Duatepe-Paksu, Ľubomír Rybanský and Katarína Žilková Abstract This particular research is a comparative study where the similarities and differences of geometry content knowledge between Slovak and Turkish pre-service teachers focusing on rhombus are examined. This research was conducted on total 144 preservice teachers with a quadrilateral test. The paper summarizes the similarities and differences in terminology and categorization of quadrilaterals in Slovak and Turkish curriculum. Subsequently the analysis is presented focusing on rhombus and the results are discussed. Moreover, a few significant differences within two groups were indicated by the results such as decreased ability to distinguish relationships between rhombus and square. Furthermore, an important finding in this research suggests that Slovak preservice teachers consider these shapes as a two entirely disjoint sets compared to Turkish preservice teachers for whom the differences are much more identifiable. Keywords: Pre-service elementary teacher, quadrilaterals, comparative research, rhombus
INTRODUCTION Mathematical education of pre-service elementary teachers should involve integration of at least two components. The answer to the question “what to
Pamukkale University, Turkey; email: [email protected] Comenius University in Bratislava, Slovakia; e-mail: [email protected]
158 teach?” (content) provides an explanation of a content of the first component and the question “how to teach?” (methods) reveals the answer that stands for the second component. Shulman (1986) described pedagogical knowledge and content knowledge and stated that these two fundamentals form together pedagogical content knowledge (PCK). This knowledge of the subject and its organizing structures is named as teacher content knowledge in the literature (Shulman, 1986) and Shulman and his colleagues (Grossman, Wilson and Shulman, 1989; Shulman, 1987) highlighted the importance of the content knowledge in teaching. Even the fact that both elements are considered to be determining, for this research the math content knowledge of pre-service elementary teachers is crucial. In terms of teaching geometry Levenson, Tirosh and Tsamir (2011) stated that “teachers’ knowledge must be sufficient in order to specifically teach geometrical concepts as well as to use geometrical concepts in order to reach more global mathematical aims“. Nonetheless, the literature reveals that the level of geometry content knowledge of prospective teachers is not as high as expected and often insufficient (Gunčaga, Kopáčová and Duatepe-Paksu, 2013; Partová and Židek, 1996; Umay, Duatepe and Akkus, 2005; Žilková, Gunčaga and Kopáčová, 2015). When we look at content knowledge of quadrilaterals with the particular focus on rhombus, studies revealed that pre-service teachers do not have the necessary content knowledge about this shape. Duatepe-Paksu, Pakmak and Iymen (2012) interviewed forty-five elementary pre- service teachers to determine how well they identify the necessary and sufficient conditions of rhombus. They revealed that some of the participants noticed many properties of rhombus, but they did not understand the relationships between the attributes of rhombus. They indicated that pre-service teachers could not reduce the list of properties to a concise definition with necessary and sufficient conditions. Similar results were described by Žilková (2013) where she analysed knowledge of 159 pre-service elementary teachers about quadrilaterals. The author found out that the concepts about quadrilaterals are very stable and strong but often incorrect. In the results she stated that the connection between a square and a rhombus belongs to the most difficult relationships among the quadrilaterals. Furthermore, Žilková claimed that their own mental figural representation of rhombus plays an important role in the success rate of respondents. Another study investigated rhombus, was carried out by Pickreign (2007) who asked 40 pre-service teachers to define rhombus. He found out that only one of them gave an adequate definition of rhombus. Participants defined rhombus as “a tilted square with all sides not equal” or “a rectangle with two slanted sides”. These visual definitions implied that pre-service teachers did not consider square as a rhombus.
159 The aim of the study was to investigate and compare elementary pre-service teachers' geometry content knowledge from two different countries with the focus on common misconceptions of rhombus. We are interested in whether different terminology approaches in introducing geometrical concepts can be related to the formation of correct ideas, respectively misconceptions of elementary pre-service teachers'. Therefore, we collected data from two different countries, namely Slovakia and Turkey. THEORETICAL FRAMEWORK In different countries, there are different approaches and definitions of introducing geometry concepts. The consequences of these differences may create different ideas of students about these concepts. A common manifestation of the problem resides with the students understanding of inclusive relationships between geometrical shapes (in our case, quadrilaterals). Therefore, the examination and comparison of results between countries is not easy. The National Educational Program (2015a) in Slovakia declares that a student of the 4th grade in the elementary school should be able to identify square and oblong; identify its sides and vertices; distinguish between opposite and adjacent sides; mark the diagonals and determine their perimeter (graphically and also numerically). Nonetheless, it is very important to point out the fact that in Slovakia the elementary schools differentiate three categories of quadrilaterals, which are disjoint: squares, oblongs (rectangles that are not squares) and other quadrilaterals (Fig. 1, Fig. 2). The term rectangle for these students is not known as a term for squares and oblongs. In the lower secondary education (lasts 5 years) the terminology expands form a set of quadrilaterals to rhombus, parallelogram and trapezoid. These findings are according to National Educational Program (2015b) in Slovakia not included in the lower secondary education (8th year of the education) until 4th year. After seeking into the curriculum of the higher secondary education we found out that rectangle and kite are not placed into the prescribed national documents.
Figure 1: Quadrilaterals in the Slovak Figure 2: Rectangles in the Slovak primary education (Šedivý and Križalkovič, math terminology 1990)
160 As in our research we focus on rhombus, as an example, we provide some definitions of a square, a rhombus and a oblong in Slovakia according to mathematical terminology (Medek et al., 1975): "square is a regular quadrilateral"; "rhombus is a parallelogram with all equal sides in length, which is not a square"; or "oblong is a parallelogram, where all internal angles are right, and that is not square." In another words the definition of rhombus is accepted in non-inclusive way. That means square is not considered as rhombus. The main problem is that many other concepts are introduced in a way that it is impossible to expect from the student to perceive the common characteristics of different groups of quadrilaterals. The terminology introduced in this way causes problems for students to rethink of their own concepts of quadrilaterals and they have difficulties to understand some quadrilaterals as a subset of other types of quadrilaterals. Fortunately some new mathematics textbooks in Slovakia take into account a number of classifying criteria and reflect on foreign approaches to the introduction of defined geometric concepts recently. But the generation of our students, that completed the introduction of quadrilaterals without any deeper analysis of all their properties of individual types of quadrilaterals, has great difficulty to understand that, for example square is a subset of rhombus. Other problems caused understanding of logical implication, namely, that in reverse order inclusive relationship is not correct. For this reason, we include in elementary education teachers´ training activities and tasks for the refinement of geometrical concepts. Experiences have shown, that established concepts of geometrical terms in students are usually connected to the mental figural concept and the aspect of introducing terms without reasoning the inclusive relationships make impossible for students to progress in van Hiele levels. The idea to imagine a square to be a special case of a rhombus is for these students as difficult as idea to imagine two parallels in non-Euclidean geometry. On the other hand in Turkey, 4 + 4 + 4 system (4 elementary + 4 middle school + 4 high school years) is used. When we look at the Turkish curriculum, quadrilaterals first appear in the first 4 years. Quadrilateral namely square and rectangle are seen as early as grade 1 (MEB, 2015). According to grade 1 curriculum, students are able to classify square, rectangle, circle and triangle in terms of number of sides in this grade. In grade 3, students are expected to realize square and rectangle has more than one symmetry axis. In grade 4, students are supposed to name the vertices and side of square and rectangle and identify the attributes related with sides. In this grade, students are also expected to comprehend the relationship between side length and perimeter of square and rectangle. Students also investigate the relations between area of square and rectangle and the addition and subtraction. In the middle school years (grade 5, 6, 7 and 8) students are introduced to the other quadrilaterals in addition to the square and rectangle (MEB, 2013). In grade
161 5, they first meet with rhombus. Students are expected to understand fundamental attributes of rectangle, parallelogram, rhombus and trapezoid in this grade. Here fundamental attributes are limited to angle, side and diagonals. Grade 5 is a benchmark in terms of hierarchical classification idea. In this grade student are first introduced with class inclusion. Square is presented as a special case of rectangle. However, no other class inclusion is visited in this grade. Students continue to investigate these quadrilaterals in grade 7, they formulate the area of rhombus and work on problems related with the area of rhombus. Students of this grade are required to understand more hierarchical relations. Square is defined as special case of rhombus as well as rectangle. Moreover, rectangle and rhombus are defined as special cases of parallelogram. On top of that, rectangle, rhombus and parallelogram are analysed as special cases of trapezoid. Therefore, one can deduce that square is also special case of trapezoid. Briefly, at the end of the grade 7, students are supposed to be able to comprehend the class inclusion between quadrilaterals given in Figure 3.
Figure 3: Hierarchical classification in 2013 Turkish mathematics curriculum (MEB, 2013) METHOD A. Participants The sample consisted of final year students of the elementary teachers training department. These are the preservice teachers who no longer have any geometry course for the completion of their studies. Therefore, we examined their output knowledge of the quadrilaterals, particularly rhombus. In total about 144 pre- service primary teachers in Slovakia and Turkey participated in the study. 84 of them were pre-service teachers from Slovakia (25 from the Catholic University in Ružomberok and 59 from the Comenius University in Bratislava) and 60 of them were pre-service teachers from the Pamukkale University in Turkey. We are aware that research file is not representative, but the data and the conclusions, which we have obtained, may serve as a basis for further research.
162 B. Measuring Tool and Data Analysis The research tool was a test with multiple-choice questions (one or more than one answers could be correct), which contained a section on quadrilaterals (16 test questions). Rhombus appeared in six test questions (Figure 4). Each question, which contained a picture (question no. 4, 5, 13, 14), was analysed through every alternative (in particular part of a), b), c), d)). Hence we created four new items i4a, i4b, i4c, i4d (analogous in questions 5, 13, 14). Questions 10 and 12 we coded dichotomously; right and wrong. Thus we categorized data into 18 items. Each answer was coded binary (number 1 for the correct answer, 0 for incorrect answer).
4. Which words describe the shape? 5. Which words describe the shape? a. rhombus a. rhombus
b. square b. quadrilateral
c. quadrilateral c. parallelogram
d. rectangle d. rectangle
10. Rhombus is ______a square. 12.Parallelogram is ______rectangle. a. always a. always
b. sometimes b. sometimes
c. never c. never
13. Which words describe the shape? 14. Which words describe the shape? a. quadrilateral a. square b. parallelogram b. kite c. square c. parallelogram d. rectangle d. rectangle
Figure 4: The part of test containing questions about rhombus Cronbach's alpha value of the whole test was 0.863. This is an indicator of good internal consistency of the test. The reduction of test to only 18 items containing rhombus (questions 4, 5, 10, 12, 13, 14) caused decrease of Cronbach alpha to the value 0.565.
163 Item SVK (n = 84) TUR (n = 60) P i4a 92% 43% 40,15 < 0,001 i4b 90% 93% 0,37 0,541 i4c 86% 58% 13,74 < 0,001 i4d 95% 92% 0,76 0,383 i5a 4% 57% 51,68 < 0,001 i5b 94% 42% 47,87 < 0,001 i5c 75% 17% 47,65 < 0,001 i5d 87% 13% 76,98 < 0,001 i10 20% 40% 6,71 0,010 i12 57% 30% 10,39 0,001 i13a 83% 25% 49,24 < 0,001 i13b 86% 82% 0,43 0,513 i13c 90% 98% 3,69 0,055 i13d 92% 98% 2,97 0,086 i14a 88% 90% 0,13 0,720 i14b 61% 55% 0,47 0,493 i14c 76% 43% 16,12 < 0,001 i14d 99% 95% 1,88 0,172 Table 1: The success rate in items and results of chi-square test of independence To verify the assumption of the equal percentage success of Turkish and Slovak preservice teachers in individual items is possible when using the z-test for equality of two proportions (individually for each item). In the case of pair categorical variables with two levels is this test equivalent to chi-square test of independence in 2x2 table. This is true for our case as the first variable “country” has two levels (TUR, SVK) and also the second variable, which describes the answer to the test item, has two levels (1 for correct and 0 for incorrect answer). We used an alpha level of 0.05 for all statistical analyses. The success rate of Turkish and Slovak preservice teachers in individual items along with the result of chi-square test of independence for each item is stated in the Table 1. C. Findings and Interpretation The relationships between variables were significant in items i4a-i4c-i5b-i5c- i5d-i12-i13a-i14c, in which Slovak preservice teachers achieved a higher percentage of success and in items i5a and i10, in which Turkish preservice teachers achieved a significantly higher percentage. In other items the difference between Turkish and Slovak preservice teachers was not statistically significant. The Table 1 shows that in the tasks where the Slovak pre-service got significantly better results, the Turkish sample have much higher tendency to name the shapes without considering any bigger sets. For example in the question number 13, a rhombus was exposed to the participants with a question to select the alternative(s) to describe this figure. The alternatives were (a) quadrilateral, (b) parallelogram, (c) square and (d) rectangle (see Figure 4). The correct answer,
164 here are both (a) quadrilateral, and (b) parallelogram. When we examine the results, there were not significant difference shown for the alternatives (b) parallelogram, (c) square, and (d) rectangle between Slovak and Turkish sample. Therefore, we can conclude, that the number of participants with knowledge, that this figure is parallelogram and not square or rectangle is comparable in both groups. However, the number of Turkish participants who considered this shape as quadrilateral was significantly less compared to the Slovakian participants. Although, the question was not whether a parallelogram is a quadrilateral or not in this test, we can simply say that subjects were with very high probability aware of this fact. However, they did not select this alternative, which gives bigger set of parallelogram. When the responses are more closely examined, the results are showing that 82 % of Turkish sample selected the correct answer of (b) but the percentage of the participants who selected both (a) and (b) is only 11,6. Thus, we can say that Turkish sample showed a higher tendency to name the shapes by the smallest set as possible and the participants generally did not select more than one alternative. This phenomenon is similar to some other items, as well. For example, participants were given a figure of square and subsequently they were asked to decide which word describes this figure in the question 5. All the alternatives of (a) rhombus, (b) quadrilateral, (c) parallelogram and (d) rectangle are correct (see Figure 4). While the Turkish sample got significantly higher score for (a) rhombus, Slovak sample got significantly higher score for (b) quadrilateral, (c) parallelogram and (d) rectangle. Evidently, Turkish sample considered this shape as a rhombus and they did not consider naming the shape by any other term. Obviously we can expect that if they selected (a) rhombus, they should automatically select the alternative of (b) quadrilateral. Surprisingly, percentage of both selecting (a) rhombus and (b) quadrilateral was only 15 in Turkish sample. Furthermore, selecting all alternative was merely 1%. Related to the question 5, we can make another interpretation. From the results of i5a items (percentage 4%) and i10 (percentage 20%), we can conclude that Slovak preservice teachers are not able to perceive square as a subset of rhombus, despite the fact that geometric attributes of rhombus and square were covered in their university courses. Almost 49% of Slovak preservice teachers marked in the item i10 that rhombus is never a square, 31% of them marked that rhombus is always a square and only 20% of them chose the correct answer “sometimes“. It is thus evident that the process of defining rhombus, with which the preservice teachers met in secondary education, had a significant impact on the formation of their ideas about concept rhombus, and the preservice teachers were unable to review them. We note that the main problem for Slovak preservice teachers was a relationship between square and rhombus. This finding is consistent with the research results of Žilková (2013) and Pickreign (2007).
165 The questions i4 and i13 have very similar character – both of them contain a picture of rhombus in standard position. The most of items in these two tasks aimed to investigate the same relationship: i4b and i13c (rhombus vs. square), i4c and i13a (rhombus vs. quadrilateral), i4d and i13d (rhombus vs. rectangle). According to the results we can conclude that preservice teachers of both countries decided correctly that the picture of i4 and i13 is not a model of square and it is not a model of rectangle, either. Their decision was made following the picture of rhombus, not their figural mental imagination. Reasoning about these relationships was comparable in both groups and the differences were not statistically significant. Similar results are seen for preservice teachers of both countries in items where they should choose that rhombus is a model of parallelogram (i13b) based on the picture of rhombus. We can say that Slovak and Turkish preservice teachers had no problem to correctly identify the relationship between rhombus and parallelogram as shown (task no. 13b), when a rhombus was displayed in a standard position (one pair of sides in horizontal position). However, in similar task i14 is the rhombus pictured in different position and the success of identification of the relationship rhombus and parallelogram (i14c) decreased in both groups and the differences between the groups were statistically significant. No statistically significant differences were detected between results of the two groups’ correct percentage for this item. CONCLUSIONS This study revealed that participants from both countries have lack of understanding in class inclusion related with rhombus. We can conclude, that since they do not have a sound understanding of relations between quadrilaterals they do not have abilities typical for second level of van Hiele Geometric Thinking Model (1986). Even though, that the score of the participants differed on different task, we can say that they all of them had difficulties when considering the hierarchical relations between smaller groups or bigger groups of quadrilaterals. In another words, we can say that participants are not as in the expected level on considering relations of rhombus with other quadrilaterals. This result is consistent with the results provided by Duatepe-Paksu, Pakmak and Iymen (2012), Žilková (2013), Pickreign (2007), who revealed, pre-service elementary teachers’ content knowledge on rhombus is inadequate. One might challenge the research by pointing out the limitation of this research such as the fact, that participants could not be sure whether the sides are parallel or congruent in our test. However by preparation of that type of test where the items are placed on a squared or dotted paper we can avoid this misleading factor because participants would consider the size of angles and sides more precisely. Nevertheless, we believe that the idea about abstract geometric concept (e.g. rhombus) would be correct in the case with syncretism of form of element shape (personal mental figural concept) and its basic characteristics of geometrical
166 shape. Because of that, it is necessary to provide different models and non-models of geometrical shapes to students for observation followed by identification of the characteristic attributes of that particular shape. The students then have opportunity to classify whether these characteristic attributes are definitorical and unambiguously definable by its shape and by understanding that then later develop the ability to classify quadrilaterals according to various criteria. Acknowledgement This study was supported by VEGA 1/0440/15 „Geometric conceptions and misconceptions of pre-school and school age children“ and by PAU-ADEP. References Duatepe-Paksu, A., Pakmak, G. S., Iymen, E. (2012). Preservice Elementary Teachers’ Identification of Necessary and Sufficient Conditions for a Rhombus. Procedia- Social and Behavioral Sciences, 46, 3249–3253. Grossman, P. L., Wilson, S. M., Shulman, L. S. (1989). Teachers of substance: Subject matter knowledge for teaching. In M. Reynolds (Ed.), The Knowledge Base for Beginning Teachers (pp. 23–36). New York: Pergamon. Gunčaga, J. Kopáčová, J., Duatepe-Paksu, A. (2013). A comparative Study: Turkish and Slovak Preservice Primary Mathematics Teachers’ Skills about Symmetry. In J. Novotná, H. Moraová (Eds.), Proceedings of SEMT ´13 – Tasks and tools in elementary mathematics (pp. 99–107). Prague: Charles University. Levenson, E., Tirosh, D., Tsamir, P. (2001). Preschool Geometry. Theory, Research, and Practical Perspectives. Rotterdam: Sense Publishers, 2001. ISBN 978-94- 6091-600-7. Ministry of National Education of Turkey (MEB). (2013). Ortaokul Matematik Dersi (5, 6, 7 ve 8. Sınıflar) Öğretim Programı. Ministry of National Education of Turkey (MEB). (2015). İlkokul Matematik Dersi (1, 2, 3 ve 4. Sınıflar) Öğretim Programı. Medek, V. et al. (1975). Matematická terminológia. Bratislava: SPN. Partová, E., Židek, O. (1996). Elementary geometry teaching and teacher preparation in Slovakia. In ICMI Study: Perspectives on the teaching of geometry for the 21st century. Proceedings. University of Catania. Pickering, J. (2007). Rectangles and Rhombi: How Well Do Preservice Teachers Know Them? IUMPST: The Journal, Vol 1, 1–7. The National Institute for Education (2015a). Inovovaný Štátny vzdelávací program pre 1. stupeň ZŠ. Matematika a práca s informáciami. (In Slovak.). The National Institute for Education (2015b). Štátny vzdelávací program pre nižšie stredné vzdelávanie – 2. stupeň základnej školy. Matematika a práca s informáciami. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1–22.
167 Umay, A., Duatepe, A., Akkus-Çikla, O. (2005). Readiness on Content of New Mathematics Curriculum. In XIV. Procedings of National Science and Mathematics Education Congress (pp. 456–458). Šedivý, O., Križalkovič, K. (1990). Didaktika matematiky pre štúdium učiteľstva I. stupňa ZŠ. Bratislava: SPN. Van Hiele, P. M. (1986). Structure and insight. New York, NY: Academic Press. Žilková, K., Gunčaga, J., Kopáčová, J. (2015). (Mis)Conceptions about geometric shapes in pre-service primary teachers. Acta Didactica Napocensia, 8(1), 27–35. Žilková, K. (2013). Teória a prax geometrických manipulácií v primárnom vzdelávaní. Praha: Powerprint.
MIRROR CURVES AND BASIC ARITHMETICS Franco Favilli Abstract The paper aim is to uncover a few of the several arithmetical properties that can be associated to the analysis of (p,q) grids of points and easily introduced in a 6th grade class. In particular, besides the introduction of a non-standard way to compute the greatest common divisor of two positive integer numbers, a geometrical version of the Euclidean algorithm and an unconventional algorithm to verify whether a positive integer number is prime are presented. The algorithms are introduced through different dynamic software, that make use of mirror curves, closed lines including points of the given grid. Keywords: Greatest common divisor, prime numbers, mirror curves, technology
The aim of the research The greatest common divisor (hereafter: gcd) of two natural numbers is generally, for students of the first class of secondary school level in Italy (6th school grade), a concept of easy learning with respect to its meaning closely related to the language, but certainly of greater commitment in terms of calculation methods and its use outside the scope concerning the reduction of a fraction to its lo west terms. The educational objective of the teaching proposal described in this article and of its piloting is to create the conditions so that the gcd concept, of strictly arithmetical nature, is introduced from a problematic situation apparently of geometric nature, being connected to the design of lines to be drawn in accordance with certain rules assigned. The task of tracing the lines, initially done in manual form by students, is then proposed to be entrusted to a simple software that allows students to calculate the gcd. The introduction of this software in the classroom helps create the conditions for the realization of a germinal micro-world, where technology assists and
University of Pisa, Italy; e-mail: [email protected]
168 supports the emergence and development of students’ mathematical thinking. In fact, using this software creates, in turn, the conditions for the emergence of the classroom reflections, questions and demands concerning the gcd concept for which the introduction of additional software may represent the teaching instrument that fosters the consolidation of this concept. Theoretical framework The theoretical reference of this paper is basically on two strands: studies conducted on mathematics associated with Sona designs, belonging to the cultural heritage of the Angolan people; studies related to the introduction of new technologies in the mathematics classroom, with particular reference to the creation of micro-worlds. The Sona sand drawings With his studies of Sona drawings, made in the sand by storytellers belonging to the indigenous cultures (Figure 1), Paulus Gerdes (1999, pp. 156-205) shows that some elements of the school mathematics knowledge are inherent and identifiable in the tracking of these drawings. The most elementary Sona are represented by lines drawn around points arranged in rows and columns, so as to form a grid. The rules for drawing these lines can be derived from the examination of Figures 2.
Figure 1 Figures 2
Gerdes shows how the number of closed lines that is necessary to trace to enclose all the points of a grid (p, q) is the gcd(p,q). This result has already provided the opportunity to Favilli and Maffei (2006) to design an educational activity that allows students to get to the same result in an experimental way, while introducing technology in the classroom: identification of the rules to draw the Sona from the request to complete a path partially drawn in a grid (5,4) of points; drawing, with paper and pencil, of a few (p,q) Sona, with p and q natural numbers greater than 1 and less than 20;
169 identifying, for each pair (p,q) used, the number n of closed polygonal lines (mirror curves) that have been necessary to enclose all the points in the grid; introduction of Sona_Polygonal_1.1 software (Figures 3) to draw some different (p,q) Sona with p,q < 100 and count the number of the necessary polygonal lines; construction of a table showing the different (p,q) used and the resulting n; analysis of the table to get to find the relationship n=gcd(p,q); introduction of the Sona_GCD_1.0 software (Figures 4) to compute the gcd(p,q) with p,q < 100.
170
Figures 3
Figures 4
171 The requirement p,q < 100 is only suggested to make visible the tracks and the number of polygonals. In fact, the two software will calculate the number of polygonals and the gcd(p,q) even without such limitation. (Fig. 5) The activities outlined above are carried out first individually and then are discussed and confronted in the classroom, the teacher taking on the role of moderator and facilitator.
Figure 5 Micro-worlds and technologies in mathematics education For the introduction of the notion of micro-world and its use in mathematics education we refer to Balacheff and Kaput (1996, p. 471) A micro-world consists of the following interrelated essential features: i. a set of primitive objects, elementary operations on these objects, and rules expressing the ways the operations can be performed and associated - which is the usual structure of a formal system in the mathematical sense. ii. a domain of phenomenology that relates objects and actions on the underlying objects to phenomena at the 'surface of the screen'. This domain of phenomenology determines the type of feedback the microworld produces as a consequence of user actions and decisions. In a micro-world students are stimulated by genuine problem-solving activities, which drive them to formulate and test mathematical ideas without the constraint of explicit formal presentations, to develop and use mathematical ideas in the solution of a problem. The didactic proposal and its methodology The didactic proposal object of experimentation aims to create a micro-world in the mathematics classroom, where technologies are to support the introduction
172 and development of the semiotic concept of greatest common divisor of two non- zero natural numbers. The teaching unit requires the availability of a personal computer for each student. After the introduction of the concept of gcd in the above described way, the teacher shows another algorithm which allows the calculation of the gcd without the need to use the concept of prime number, which definition may then be postponed: the Euclidean algorithm. This algorithm, introduced by Euclid as the solution to the Proposition VII.2 in the Elements, can be described as follows: if p Figure 6 This second step of exploration and consolidation of the concept of gcd, terminated not being necessary to precede it by the definition of prime numbers in view of the use of the common algorithm for the calculation of the gcd between two natural numbers p,q > 0, which requires their previous prime factorization. The chosen teaching path for the introduction of the gcd and the use of two software can then push the teacher to reverse the order in which prime numbers 173 and the gcd are usually introduced: first, the gcd and then prime numbers, instead of first prime numbers and then the gcd. In fact we can get to say that a natural number p is a prime number if gcd(p,q)=1 for every natural number q 174 Figures 7 175 The results analysis The piloting of the above described teaching unit has proved to foster, in the classroom, the construction of mathematical ideas from particular situations that affect students’ argumentation and expression of ideas. The technological setting showed to promote the development of a collaborative environment in the classroom, where the advancement of the students’ strategies and the development of their mathematical knowledge can be better identified and analysed. Furthermore, the different software (artifacts) were for the students effective instruments that allowed them to associate schemes and techniques to the instruments, when undertaking the given tasks. (Rabardel, 2002) As far as students’ affect and attitudes are concerned, a few meaningful comments of theirs can show the positive impact of the piloting: a. We could work relaxed and freely, and it was something fun and unusual. b. I loved all lessons on Sona. I liked everything in these beautiful activities. c. This teaching unit has helped me better understand the mathematical laws (as they are, what are they for). Conclusions The students-technologies interaction helps create a learning environment where the communication (student-student, student-teacher, student-technology) plays a key role, the discovery of the software features and their underpinning algorithms favour their conjecturing and investigating mathematical ideas, different semiotic representations of mathematical concepts are promoted. From the teacher’s side the innovative way for the introduction of the greatest common divisor between two natural numbers and for the identification of prime numbers can be seen and experienced as a challenge in the direction of significant and meaningful methodological change. Further piloting and feedback analysis are to be considered and studied for an actual and real validation of the proposed teaching unit as to its middle-long term efficacy in the development of students’ mathematical knowledge. References Balacheff, N., Kaput, J. J. (1996). Computer-Based Learning Environments in Mathematics. In A. J. Bishop, et al. (Eds.), International Handbook of mathematics education (pp. 469–501). Kluwer. Maffei, L., Favilli, F. (2006). Piloting the Software SonaPolygonals_1.0: A Didactical Proposal for the GCD. In F. Favilli (Ed.), Ethnomathematics and Mathematics Education – Proceedings of the ICME10-DG15 (pp. 118–125). Pisa, Italy: TEP. Gerdes, P. (1998). Geometry from Africa: Mathematical and Educational Explorations. Washington, DC: The Mathematical Association of America. 176 Laborde, C. (2002). Integration of Technology in the Design of Geometry Tasks with Cabri-Geometry. International Journal of Computers for Mathematical Learning, 6(3), 283–317. Rabardel, P. (2002). People and technology: a cognitive approach to contemporary instruments. Université Paris 8. LANGUAGE- AND MATHEMATICS-INTEGRATED INTERVENTION FOR UNDERSTANDING DIVISION AND DIVISIBILITY Daniela Götze Abstract About 20 % of the 15-year-old pupils in Germany have failed to develop an understanding of the four basic arithmetical operations in the course of their schooling – and of division least of all. The study presented in this paper affords an insight into the conception and evaluation of an intervention project involving 45 third- and fourth- graders from primary schools whose catchment areas have a low sociographic status. Individual misconceptions of division serve as a basis for initiating a sustainable understanding of factors amongst the children in a language-sensitive manner. Keywords: Division, divisibility, conceptual understanding, language support Theoretical framework Primary school children are meant to acquire a confident and workable idea of all the four basic arithmetical operations in their first years at school. The arithmetical operation that seems to be the most difficult to learn is meanwhile demonstrably division, with children especially experiencing mathematical difficulties, not acquiring any concept of this arithmetical operation at all (Cawley et al., 2001; Ehlert et al., 2013; Moser Opitz, 2013; Robinson et al., 2006). While there is a relatively large number of studies already delving into the operational ideas of addition, subtraction and also multiplication amongst children who are underachieving in arithmetic, division-related studies tend to be rare (Ehlert et al., 2013; Robinson et al., 2006). The research findings currently in hand are meanwhile largely focussed on two central aspects: specific strategies children use to solve division problems, or the frequency of solutions amongst children who have learning difficulties in mathematics, in particular. According to Robinson et al. (2006), typical strategies pursued by fourth- to seventh-graders to solve division problems include: factual knowledge, recourse to multiplication, partitive division, recourse to repeated addition and/or subtraction in the sense of quotative division, and derived facts. In addition to this, solutions are also guessed, inappropriate strategies are pursued, or strategies are not named by the children (ibid.). Only the strategies of factual knowledge, Technical University of Dortmund, Germany; e-mail: [email protected] 177 recourse to multiplication and recourse to repeated addition are regularly found in all grades (ibid.). The strategy of repeated addition appears particularly dominant in the lower grades (ibid.). Downton (2009) resorts to a similar strategy system. She uses the additional strategies of building up as a recital of multiplication tables, as well as doubling and halving (Downton, 2009). It is moreover sufficiently well-known that correct solutions of division problems are scarcer, particularly amongst children who have difficulties in mathematics. Robinson und LeFevre (2012) show, for example, that sixth- to eighth-graders who are weak in arithmetic fail to understand the connection between multiplication and division as each other’s reverse operation. Cawley et al. (2001) furthermore show that the understanding of division amongst eighth-graders with mild disabilities tends to equal those of fifth-graders. Specifically for Germany, this discrepancy appears to be even greater. The division concept of eighth- graders with difficulties in mathematics is below the level of fifth-graders in mainstream schools (Moser Optiz, 2013). The understanding of division and the place value system are meanwhile significant predictors for eighth-grade performance in arithmetic (ibid.). Cawley et al. (2001) therefore refer to the introduction of division as a “cut-off” point in maths teaching for many children. Without a confident understanding of division, all children lack central foundations for an understanding of divisors, and hence for elementary number theory, but also arithmetic learning contents (Feldmann, 2012). Insofar a conceptual understanding of divisibility is a central learning target for all children (Moser Optiz, 2013). An action-oriented introduction and stronger networking of multiplication and division are therefore unanimously demanded to advance the understanding of division (Moser Optiz, 2013; Robinson and LeFevre, 2012). Just as important, although the subject of little scientific attention yet, is the linguistic sensibility (Anghileri, 1995), as division, in particular, is distinguished by specific language structures, owed to the two basic concepts of partitive and quotative division, amongst other factors (ibid.). The children are demonstrably overtaxed by decoding the verbal information, by differentiating between everyday words and technical terms, and by recognizing the connection between mathematical symbols and language (ibid.). This problem is in particular faced by children with mild disabilities as their language competences are still less differentiated than those of mainstream schoolchildren (Cawley et al., 2001). “In summary, it is clear that children find these ideas difficult and progressing or development in understanding takes time” (Roche and Clarke, 2013, p. 264). That language can become an obstacle to learning in maths teaching has not only been known in Germany since the refugee movements of recent years. This phenomenon is described as a lack of equal opportunity in the German school system. It affects children and adolescents from families with little access to education as much as the steadily growing group from migration backgrounds (Gebhardt et al., 2013), currently making up around 30% of the pupils. Empirical 178 studies have shown that the connection between social background and arithmetical performance is largely mediated by their competence in the German language: What turns out to be the key background factor for maths performance is not the multilingualism, migration background or socioeconomic status, but the language skills in German (Prediger et al., 2013, p. 55): “If social disadvantages are linguistically mediated, this finding offers a good access point to enhance equity by fostering students’ academic language proficiency. We therefore need enormous efforts of designing language-sensitive teaching strategies and materials (Thü rmann, Vollmer and Pieper, 2010), especially with a focus on expressing connections (Prediger and Wessel, 2013)”. What can be said in summary is that we need to know more about whether and how the children’s skills in this regard, after the introduction of division, can be expanded or misconceptions dismantled, respectively. The support provided for this should meanwhile focus on the connections between multiplication and division and on deepening the understanding of division, but also on an understanding of factors as a preparation for future learning contents, and all that in a language-sensitive design. This is exactly the approach taken by the research project described below, which is dedicated to the research question: To what extent can the understanding of division be addressed, misconceptions reduced and an understanding of divisors initiated amongst socially disadvantaged children with heterogeneous mathematical skills by way of a support concept with a language-sensitive design? Methodology Design experiments as a data collection method As a basis for this support, a design experiment was developed for the rich learning task ‘partition tree’ (fig. 1) for the third and fourth grade. In a partition tree the starting number a is progressively partitioned into its natural divisors. The trivial factors 1 and the original number itself are meanwhile left out as the partition tree would never come to an end otherwise. 24 24 Divisors: Multiplicative 6 4 12 2 building blocks of 3 2 2 2 3 4 the natural numbers 2 2 Primes: Smallest multiplicative building blocks of the natural numbers Figure 1: Partition trees for 24 179 This rich learning task permits the connections between multiplication and division to be worked out by way of discovery: calculating from the top to the bottom shows the division, and from the bottom to the top multiplication. In case of content problems, the children can work out the individual divisions for themselves with the help of mathematical manipulative. The support was provided at four primary schools in major cities of the Ruhr area Partition Tree to a sum total of 45 third- and fourth- graders in assignments for partners or starting number small teams (ca. 40 – 45 minutes per session). All the schools are located in the first level districts with a very low socioeconomic status. the end numbers Four preservice elementary teachers the prime numbers the primes were trained and intensive consulted to serve as resource teachers (RT). The 12 is partitioned into 3 and 4. language-sensitive provision of the 3 times 4 makes 12. 12 divided by 4 makes 3. support was performed in keeping with 3 and 2 cannot be divided any further. the scaffolding approach: “(…) this 12 has the divisors/ the factors … functional theory provided a strong 3 is a divisor of 12. framework for the deliberate and explicit 4 is the co-divisor of 3, cause 3 times 4 makes 12. focus on teaching language, teaching through language, and teaching about Figure 2: Exemplary word list language…” (Hammond and Gibbons, 2005, p. 9). This differentiates between two essential levels of language support for technical learning processes: macro- (designed-in) and micro- (interactional) scaffolding (ibid.). Both language support levels were relied upon in the design experiment with the partition trees. On micro-level, the preservice teachers were trained to accompany the support units in a language-sensitive manner, i.e. by inviting the children to use more precise language, or by introducing technical terms and making consistent use of them. The shared language basis should thereby not only provide the children with access to the language of teaching and technical vocabulary, but demonstrably also support content learning (Prediger and Wessel, 2013). On macro-level, a word list that record and visualize specific technical terms and sentence phrases were drawn up in cooperation with the children (fig. 2). If the children wanted to verbalize their deliberations and are lacking the words to do so, this word list provided them with various forms of language support. Data analysis methods The data analysis showed that the existing strategy grids are insufficient for a qualitative assessment of the individual misconceptions, and most of all developments, of children who are underperforming in language and maths. A 180 new evaluation instrument was therefore developed by way of qualitative content analysis (Mayring, 2014). The strategies and notions, but also precursor strategies and misconceptions, of the 45 supported children relating to division on the one side and divisibility on the other were extracted from the data. At the same time, they were compared with the division strategies available in the literature already, and supplemented with new ones not found there yet. Especially the area of divisibility is not yet covered where primary school maths learning is concerned. The results of this broad analysis are shown in table 1. Division Divisibility (with a, b, c ∈ ℕ and a ≠ b ≠ c) d_1 Multiplication and division nt_1 A number can have various sets of have nothing to do with one divisors another nt_2 The larger a number, the more divisors will it have d_2 Multiplication is addition, nt_3 If b∙c=a applies, the divisors of b and c division is subtraction are not divisors of a d_3 Limitedness of multiplication nt_4 A number a where b∙c=a and b, c are and/ or division prime can potentially have other divisors but d_4 Focus on the results of b and c multiplication tables nt_5 All uneven numbers are prime numbers d_5 Doubling and halving is only and/ or only even numbers are divisible, interpreted as additive and not respectively multiplicative nt_6 To find all possible divisors of a, one d_6 Division is equated with needs to try out all the numbers from 1 to a halving, multiplication with nt_7 If a multi-digit number includes a prime as one of the digits, it is also a prime Misconceptions for ... strategies or precursor doubling D_1 Quotative strategies NT_1 Primes cannot be divided any further D_2 Partitive strategies NT_2 If c|b is true, then c|a∙b is also true D_3 Repeated addition or NT_3 All possible combinations of the subtraction prime factors lead to the divisors of a D_4 Using inversions number (multiplication) NT_4 If a number is repeatedly D_5 Using derived facts multiplicatively divided, the divisions will always end in the prime factors (modelled on Downton, 2009; NT_5 No matter which multiplicative Robinson et al., 2006) division one starts from, the division will always end with the same prime factors Sustainable notions or strategies for (Fundamental Theorem of Arithmetic) Table 1: Analysis grid (D, d stand for Division; NT, nt for early number theory) The individual (precursor) strategies and (mis-)conceptions do not always lend themselves to clear categorization as either strategy or notion because the context in which an argument is used is often highly dependent on the situation or child, as is demonstrated by the following analyses. 181 Empirical snapshots from the design experiments The following provides insights into the support lessons of the heterogeneous fourth-graders Abbas, Hamit und Xara. Abbas’ family is from Iraq originally. They have been living in Germany for a number of years already. His performance in arithmetic is middling. Hamit is from Syria. He has only been in Germany for a) At first I knew and then I saw b) this were multiplication tasks and then I wrote everything and I hope that is right. c) One simply needs to multiply because 2 ∙ 6 = 12 and 9 ∙ 2 = 18 and 12 + 18 = 30 this is how I got it. Simple multiplication and + calculations. Figure 3: Work done by Abbas (a), Hamit (b) and Xara (c) in the first remedial lesson (translated by the author) nine months at the time of the support. The school is advising him to repeat the fourth grade because his language and math skills are too weak. Xara comes from a family home with little access to education. She is receiving additional maths support. She is a very bright child, however, who loves playing an active part in discussions, as the following analyses show. In the first support unit, the children initially worked out the assignment format of the partition trees for themselves. Fig. 3 shows the results of this work. Abbas writes that he is calculating “multiplication tasks” and appears to have doubled the ten. He notes this doubling in an additive manner, but still describes it as “multiplication tasks” (d_5). He is possibly equating multiplication with doubling (d_6). He moreover fails to divide the ten any further, buttressing the impression that he is solving the partition tree from the bottom up. Hamit’s partition trees are correct. Any further diagnosis of his mathematical skills is elusive because he has selected easily divisible numbers (21 and ten) and not made any notes, probably because of his poor German. Xara appears to develop the partition tree multiplicatively from the bottom up (2 ∙ 6 = 12 and 9 ∙ 2 = 18). This perspective makes it difficult to see that she could have divided the six and nine further, naturally. What is interesting is that she is adding the numbers twelve and 18, not multiplying them. Perhaps she does not know how two-digit numbers are multiplied. Failing which there would be a need to see if she possibly thinks that multiplication is only limited to the integers from one to ten (d_3). 182 None of the three children’s documents provide any clues as to whether they perceive the inversion of multiplication and division in the partition tree. Once the assignment format had been clarified in the first remedial lesson and the word list elaborated (see fig. 2), the game “Who divides last?” was introduced in the second remedial session. In this game, the game master selects a starting number, the opponent divides it once, then the game master divides it again, etc. Whoever performs the last possible division is the winner. After a few rounds of the game, Xara suggests the use of 100 as the starting number. 313 X With the 100 one can calculate as long as one wants. Do you want me to divide now directly? 314 RT No, It’s Hamit’s turn now. 315 H (H notes 10 and 10. He passes the piece of paper to A) 316 A (A divides 10 into 5 and 2, points to the 2) Primes. 317 RT Mhm, and what about the 5? 318 A Do you want me to do that as well? Oh, I know, that is also a prime just like the 2. 319 X Can I now also divide the 10 into something else? 320 RT Yes. You are doing a new division now. 321 X Yippee! (…) er. OK. Let me see first what goes still? Ah! (X takes her fingers and counts loudly) 2, 4 (…) 2, 4, 6, 8, 10. 5, um, but that is the same as just now. (...) Fiver, twoer ahhh, the 2 and the 5 are the main numbers somehow. 2 times 5 makes 10. 322 RT Are there any other ways then? 323 X No, with the 3 I’d be at 9 then. The 1 would work but that isn’t supposed to. The 6 would be 2 too much. The 8 would be much too much in any case. No, there are no other ways, actually. Xara appears to assume at the beginning (l. 313) that the number 100, as a particularly large number, also has more partitions and hence more divisors (nt_2). From line 319 onward she starts thinking about whether the ten can possibly also be divided in any other ways than into the primes of two and five (nt_4). She is taking a consistently additive approach to this (D_3), but also considers numbers such as six and eight that are completely out of the question as divisors (nt_6). Her approach is therefore still marked by precursor concepts of divisibility. In the third support lesson, the children were invited to find all the partition trees of the numbers two to 25. The finding of all partition trees served to highlight further individual (mis-)conceptions of the children, e.g.: 451 X Yes. I Think the 9 doesn’t work either. 452 A The 9? The 4 can’t be divided any further and the 5 neither. Well, then the 9 is out. 453 X Yes, the 9 is out. 183 Abbas and Xara appear to assume that a divisible number needs to be divisible by two and, by implication, that all uneven numbers are primes (nt_ 5). Xara assumes repeatedly somewhat later that larger numbers can also have more divisors (nt_2). She finally discovers that her assumption is wrong when she sorts all the found partition trees (“...this actually contradicts that”, l. 546). 546 X Yes, right, here is the 4 (X points to the partition tree that starts with 4), that is very low. And that here is the 8 (X points to the partition tree that starts with 8), that is higher, but here is the 19 then already (X points to the partition tree that starts with 19) and the 24. Yes, but this actually contradicts that ‘cause here is the 25 and there the 24 (X points to the partition trees starting with 25 and 24). The finding and sorting of the partition trees and look at their commonalities and differences have turned out to be very fruitful activities for eroding misconceptions, as exemplarily reflected in the following sections: 558 RT What about the end numbers [of 24], do you notice something about them, too? 559 X Always a 3 and perhaps also twos. 560 A Only threes and twos. 561 X Yes, always only threes and twos. 562 RT (...) Why don’t you try and describe this as exactly as possible with our math words. (...) What you are saying isn’t all that clear to me yet. 563 X Well these are primes. First a 3 and then twos. Looking at the final numbers of their partition trees, the children discover that all partition trees of a number always end with the same primes (NT_4). Some minutes later, it is Abbas who increasingly realizes that it does not matter which division one starts from and that the prime factors will only be in a different order afterwards (l. 667, NT_5): 667 A But we have only put that (points to the primes) in a slightly different order here. And then (points at the first level of the tree) we have new ... divisors. Abbas insofar no longer appears to be viewing the partition trees of a special starting number in isolation, but realizes that they belong together. The upshot being that all possible combinations of prime factors, respectively, need to be considered to find all divisors of a special starting number (l. 667, NT_3). The children’s concept of divisors is expanding in this lesson in the sense of them taking all the numbers in all the partition trees into consideration as divisors of the starting number: 629 A Er, 8 and 3 are divisors of 24. 630 X The same as with the 12, the 4 and the 3. 631 A 8 and 3 are divisors of 24. 632 RT Hmhm (agreeing). And what about the 2 and the 4, then? (RT points to the divisors 2 and 4 in the partition tree). Are they also divisors? 184 633 A 2 and 4? 634 X Yes, I think they are, because those further down there are also divisors. While Abbas and Xara both initially only consider the numbers on the first level as divisors (l. 629-631, nt_3), an enquiry by the resource teacher (RT) appears to be followed by a rethink, so that two and four are also accepted as divisors, i.e. that the divisors of a divisor are also divisors of the number (NT_2). And refugee child Hamit, who has hardly actively participated in the discussions, also appears to have understood the mathematical core, as is evidenced by a written document at the end of the 16 is partitioned into 4 and 4 because 4 support lessons (fig. 4). He times 4 is 16. 4 is partitioned into 2 and 2 describes accurately how a because 2 times 2 is 4. partition tree needs to be 2 and 4 are divisors of 16. calculated, using the language of Fig. 4: Hamit’s individual description teaching and technical phrases. (translated by the author) His text reflects the connection between multiplication and division as an inversion as he arguments within the partition tree from the bottom up as well as top down (D_4). He furthermore correctly describes all the numbers in the partition tree as divisors (NT_2), but forgets that 16 can be factored different ways, and hence at least the additional divisor of 8. Discussion and Conclusions This short insight into the empirical data has shown that the deliberate support of the connection between multiplication and division appears to be a fruitful approach for developing an understanding of division and divisibility. Initial misconceptions (particularly concerning divisibility, but also division) are supplanted by sustainable ideas. Especially the math underachiever Xara and Hamit impress in their individual development. With how much stability these new concepts are invested remains unanswered at this juncture. The technical vocabulary supported in mathematical discourses appears to be a central requirement for success because misconceptions can only be rendered verbally diagnosable by the use of the technical terms and phrases in the first place. It was simultaneously possible to foster an understanding of division and divisors together because a potential language barrier could be deliberately avoided right from the start. A more detailed look at the effects of the language support would call for further analytical steps, however. Given the extensiveness of the empirical data, only a small insight could be provided here. Other misconceptions and developments will be observable amongst other children. The evaluation instrument newly developed for the analysis appears suitable for taking a closer look at the precursor concepts of division and understanding of divisors, and for highlighting developments. It would nonetheless need to be reviewed in the course of further analysis, possibly augmented with other aspects, and also operationalized via pre- and post-test, if possible. 185 References Anghileri, J. (1995). Language, arithmetic, and the negotiation of meaning. For the Learning of Mathematics, 15(3), 10–14. Cawley, J. F., Parmar, R. S., Foley, T. E., Salmon, S., Roy, S. (2001). Arithmetic performance of students. Implications for standards and programming. Exceptional Children, 67, 311–328. 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ANALYSES OF LEARNING SITUATIONS IN INCLUSIVE SETTINGS: A COEXISTING LEARNING SITUATION IN A GEOMETRICAL LEARNING ENVIRONMENT Kristina Hähn Abstract Offering the same mathematical content to all students and enabling cooperative learning situations is one approach of inclusive mathematics education. The following paper concentrates on the analysis of interactions, in detail on the analysis of participation and mathematical processes. This will be illustrated by a case example of a fourth-grader (10-year-old students) with learning disabilities. The content-related as well as social participation leads to empirical categories of learning situations of inclusive settings with respect to obstacles and opportunities for mathematical processes. The contribution focuses on a classroom situation, planned as a cooperative learning situation but realised in a different way. Keywords: Inclusion, learning disabilities, cooperation, interaction, student participation, substantial learning environment Introduction In several countries or contexts the terms »learning difficulties« or »learning disabilities« are used differently (for more details see Scherer et al., 2016). The school law of North Rhine-Westphalia/Germany defines children with learning disabilities as students with long-term and extensively impairments in contrast to partial impairments, like dyslexia or dyscalculia (cf. MSW, 2016). By the United Nations Convention on the Rights of Persons with Disabilities, the German Federal Government has committed itself to the development of an inclusive school system. The majority of students with learning disabilities on the primary level are now taught in regular schools and no longer in special schools for students with special needs (cf. Heimlich et al., 2016; Klemm, 2015). There is still lack of research in collective or cooperative learning situations of students with and without disabilities (cf. Peter-Koop and Rottmann, 2015). Learning in collective (inclusive) learning situations Howe (2009, p. 215) states: »Evidence exists that children´s understanding can be facilitated through collaborative group work with peers, but little is known about the underlying processes«. From an interactive-constructivist perspective (cf. University of Duisburg-Essen, Germany; e-mail: [email protected] 187 Sutter, 1994) collective learning processes (cf. Miller, 1986) are understood as constructions of individual knowledge, which are realised in processes of social knowledge construction. Taking into account special educational considerations, Feuser (1998) pleads for an inclusive education, in which disabled and non- disabled students collaborate on tasks with a common subject-matter, which means joint activities within a complex content. From the viewpoint of mathematics education one possibility is to realise Feuser’s requirement by the concept of Natural Differentiation (cf. Krauthausen et al., 2010) and the use of Substantial Learning Environments, which offer the same holistic and sufficiently complex mathematical content to the whole learning group. Those learning environments are mathematically substantial by representing central objectives, contents and principles of teaching mathematics. Low thresholds of input and rich opportunities for mathematical activities are offered to all learners. The need for discussion is fostered and social cooperation should be encouraged (cf. Krauthausen et al., 2010; Wittmann, 2001). A geometrical learning environment in an inclusive setting Aim of the research and analysis The aim of the research is to reconstruct the social and the content-related aspects of a collective learning situation by (a) identifying the participation of students with learning disabilities, (b) determining different collective learning situations in the interaction of partner work phases as well as (c) describing the mathematical processes (for a definition see Wittmann 2001, p. 12) in order to investigate relations between these three dimensions. Transcribed episodes, selected according to the research purpose are analysed by qualitative content analysis (cf. Mayring, 2004). The categories to analyse the learning situations have their origin in the theory of collective learning situations in inclusive settings (cf. Wocken, 1998) specified by the categories of the analysis of participation in processes of collective argumentation (cf. Krummheuer, 2015). The paper concentrates after a short overview about the main aspects of the categories on one of the learning situations to exemplify the analysis method by an episode of an interaction in a partner work. Wocken (1998) outlines in his theory, which is based on the anthropological assumption of equity and diversity of all human beings, four typical and incisive patterns to clarify the relationship between individual and collective learning processes within inclusive settings. He differentiates between the content level and the relationship level and considers if one is dominant and the other is nearly missing or if both occur (balanced or imbalanced). If content-related aspects are dominant and the focus is more on individual plans than on common ideas, Wocken speaks of a coexisting learning situation. In contrary, if content-related aspects are incidental, it is named a communicative learning situation. In subsidiary learning situations both aspects appear but in an imbalanced way: one 188 person helps the other intensively and one activity becomes dominant or the support is only marginal and both are focusing on an individual idea. Content- related and social aspects are balanced in a cooperative learning situation. For the project presented here, the theory will be described in a more differentiated way: e.g. a differentiation of the coexisting learning situation is necessary, which will be discussed in detail in this contribution. In a learning environment named »The Circle« (cf. below) the students work on the same mathematical context on the same task, but possibly in a coexisting situation, if different individual plans are dominant and the social interaction is recessive. Furthermore, it has to be distinguished what leads to the coexistence: (a) no or only parenthetic interactions with no relation to the mathematical content between the learning partners or (b) a rejection of an attempt to cooperate. To identify the participation of a person in collective learning situations, Krummheuer (2015) differentiates the responsibility and the originality of speakers´ utterances considering their functions according to the formulation and the content. He distinguishes four cases: 1. Author (responsible for new formulation and new content) 2. Relayer (neither responsible for formulation nor content) 3. Ghostee (responsible for new content, but using (almost) identical formulations prior utterances) 4. Spokesman (phrasing the idea of another person’s prior utterance) (Krummheuer 2015, 57ff.) Moreover, to analyse collective learning situations, it is necessary to describe the development of ideas within mathematical processes. It is important to distinguish e.g. (a) enhancement of the author´s idea by him-/herself or the learning partner, (b) rejection of the idea by the learning partner or cancelling the process by him- /herself. In this context the students do not only express ideas verbally within mathematical processes, but also by actions or gestures. Setting of the study In the research project the learning environment »The Circle« has been developed, which consists of four units. Each unit has a duration of 60 to 90 minutes. The empirical data consists of five videotaped realisations of the whole learning environment. In each of them, a primary mathematics teacher conducts the learning environment with a group of 6 to 8 students, including two students with learning disabilities. After the group work, the students with learning disabilities are interviewed individually to get the possibility to reconstruct the ideas and transfer the content. The interviews are semi-structured with elements of a problem-centred interview (cf. Witzel 2000) that should enable an analysis of the individual student´s 189 mathematical processes. In total the project´s data comprises 20 lessons, in which ten students with learning disabilities are observed together with their teacher and their classmates as well as 40 interviews. Development of the learning environment »The Circle« In the research project »good noses mathematics«, in which primary students visit the University of Duisburg-Essen (cf. Baltes et al., 2014), Substantial Learning Environments are designed, which follow the concept of Natural Differentiation. All students get the same holistic offer, referred to a complex mathematical content, which contains diverse levels of demands. In the context of this research project the learning environment »The Circle« was developed in a design research process. In the introductory unit »Properties and Construction of the Circle«, which has been piloted with about 260 students, learners explore different geometrical shapes and their characteristics. The exploration of the axes of symmetry in the circle (number, position and length) leads to the terms »centre«, »radius« and »diameter«. In a cooperative intended phase of a partner work, a pencil has to be combined with other materials (cf. the following section), so that a circle can be constructed. The cooperation is not initiated by methods of cooperative work but by the teachers´ instruction, the scarcity of the material (only one pencil per team) and necessity for assistance with some relevant actions. Later students’ discoveries should be documented and finally all produced tools and constructed circles are discussed with respect to the central terms »centre«, »radius« and »diameter« in a group reflection. In the following section the interaction of a partner work in this learning environment is analysed to characterise the learning situation with regard to the students’ participation. A collective learning situation: creating a tool for drawing a circle The teacher Mr. K. (T) and six students (including two students with learning disabilities) are sitting around the table. The central task of the partner work is creating a tool, with which they can construct a circle by combining a pencil with other available materials like little rods, cord, styrofoam, prick needles and plasticine. They also can use rulers and scissors. In this phase the teacher should take a passive role. His concrete instruction is: »Your task is now: How can you make an exact, an exact, exact is a strange word, draw a precise circle? You should combine the pencil with some material from those you have there, so that you can draw a circle with it. A kind of tool, or, yes, a circle constructing tool. In teams of two, you get a worksheet, on which is written ›A tool for circle construction‹. Construction is a word for production of a circle that adults use.« He further explains that the students should first experiment and later note their ideas on the worksheet. He answers some students´ questions, e.g. if they are allowed to cut the cord. Before the learners start, he emphasises that they have to come to an agreement so that they will have one solution per team. Firat (F) 190 (student with learning disabilities) and Aylin (A) work together (see transcription rules at the end of the paper): 31 A (takes a ball of cord) 32 F Wait, we need that (takes a pencil and a little rod, holds them like a compass in one hand) yes and we need that. 33 A (takes the scissors, shakes her head) You have to make a circle. You sly dog. 34 F Yes, sly dog. Do it! # You have to wrap and than I can do it like that ## (turns pencil and little rod around with an imitation of using it as a compass but without fixing one point) 35 A # (cuts a piece of cord) ## (shakes her head) 36 F Mr. K.? # Does it work like that? Mr. K.? 37 A # (puts the ball of cord into the middle of the table) 38 F ## Does it work like that? 39 A ## (arranges the piece of cord like a circle) 40 F ### Can I do it like that? (turns pencil and little rod around like a compass) Should be a circle? 41 A ### (cuts a piece of scotch tape) 42 T Discuss this with Aylin! 43 A (looks for a moment up to the teacher) 44 F Yes. [We] have to make a circle therefore [it must be like that]. The initiation of interaction is unilateral. Firat requests assistance (34) and a collective (subsidiary) learning situation could start. But his idea of building a compass is not accepted by Aylin (33, 35). Possibly, she cannot identify the circle construction in Firat’s activity, maybe because he performs the action not precisely enough without a fixed centre1. Her conception of what a circle is seems to be different: »You have to make a circle. You sly dog.« (33). With respect to both learners´ first results (see fig. 1a, b), Firat constructs a circle by using a radius and a rotation of a pencil and Aylin focuses on the circumference (39). Maybe the teacher´s instruction allows different interpretations: to combine the pencil with materials and construct the exact graphical representation of a circle on the one hand or to produce (like building) a circle on the other hand. Figure 1a, b: First ideas to construct a circle of Firat (left) and Aylin (right) 1 About ten minutes later, when Firat draws a circle with a little rod and a pencil stucked together with plasticine, she says to him and the teacher: »Oh, I see, you mean that. He wants to make a circle. I see, I know that, Mr. K.! There is thingummy. Thingummy. You turn this around (makes two rotary gestures with her right hand).« 191 Firat’s attempt that Aylin should help him to realise his idea, is rejected without a content-related concrete argument that could give him the chance to understand his partner´s doubts. As the consequence the possibility to cooperate failed and Firat addresses his teacher. But the teacher (42) does not give a content-related feedback, as he emphasises to work together with the learning partner. Maybe it would have been less likely that Firat addresses the teacher if he had not been present at the table. As Mr. K. recommends that the two students should discuss their content-related problems with each other, he tries not to influence their learning process. So Firat confirms his idea to himself (44). The production design shows two authors with different ideas and no or only parenthetic interaction between them with no respect to the mathematical content. The learning situation is coexisting. Both children pursue the individual idea, not considering the meaning of the other idea. At that moment, neither Firat’s suggestion is exactly explained nor Aylin’s idea becomes clear. Subsequently, Firat shelves his idea and works with the materials without concrete objectives: pricking two pieces of styrofoam at both ends of a little rod or kneading the plasticine, while Aylin is producing a circle out of a piece of cord and trying to place two little rods in the circle, but they are a bit longer than the circle´s diameter (see fig. 2). In that case only one creation can be observed. Since Firat does not take part while Aylin is developing her creation. It is an open question if a collective learning situation is still given or if Firat has left the learning situation. Figure 2: Development of Aylin´s idea Later on, Firat tries to cooperate with Aylin: 84 A Now. (takes a piece of cord, puts it away, takes two little rods, presses one of them in the plasticine, which fixes the other two little rods) 85 F Here. Sticking with scotch tape? (holds scotch tape in his hand) 86 A No, plasticine is better. (incomprehensible) (takes some plasticine and fixes the little rods in the middle) 87 F (puts scotch tape aside) (to Mr. K.) With. What do we do with that? (has a piece of styrofoam in each hand) Mr. K.? 88 A (presses the next little rod into the plasticine) 192 89 F I have an idea with that! 90 T Yes, then discuss this with Aylin! # Not with me! 91 F # (to A.) We can put this into it. F A 92 A (shakes her head) No. (puts some plasticine on the crossed little rods) 93 F Oh dear! (takes styrofoam away) You can also do it like this, Aylin. You take a circle # (unrolling a piece of cord, laying it in a rounding on the table, putting a piece of styrofoarm in the middle) and put this into it. Mr. K.! 94 A # (presses plasticine down firmly) 95 A How shall we put that together? (puts cord around the endpoints of the crossed little rods) 96 F Yes. Sticking with glue. 97 A But, how, glue? (takes hands away from the material) But we need [little rods otherwise it´s not correct]. … … 117 F Here (puts styrofoam into the area of the circle) I will do F so, than it is a circle. A Now Firat tries to participate in Aylin´s process. He makes several proposals to support the development of Aylin´s idea (85, 91, 93). Enhancing another person´s idea, Firat shows the participation role of an author. But the reasoning about different suggestions is missing and each proposal is rejected without an explanatory statement (86, 92) or by expressing a contraposition (97) in the participation role of a ghostee. A discussion about the content-related mathematical process, in which underlying suggestions can be reflected is not provoked. In this case, offering help does not lead to a subsidiary or cooperative learning situation. The coexisting learning situation persists. According to Firat´s activity with respect to different aspects of the mathematical object circle the sequ’ence can be divided into four sections: a. Consideration of the fixed centre of a circle (84-88) b. Objectification of the circle´s diameter as several objects with the same length (in symmetric arrangement) (89-92) c. Objectification of centre and circumference of a circle with cord and styrofoam (93ff.) d. Having a ›perfect‹ curvature through tessellating the area of the circle (93, 117) The second section (b.) can be interpreted differently: Firat puts the styrofoam between the diameters either to fix their distance, which may help to rectify the curvature of the circumference or to suggest tessellating the area of the circle (cf. d.). Aylin´s activity seems to follow the idea of fixing the centre with diameters 193 in a circumference. Her challenge seems to be the rectification of the circumference´s curvature. Firat’s efforts to take part in this collective learning situation are unsuccessful. With regard to his individual mathematical processes he has to adapt himself to Aylin´s ideas by interpreting her activities and difficulties. His proposals also show that he focuses on different aspects of the mathematical object circle. But as the two children are not reasoning or arguing mathematically about their ideas, it is unclear if Firat is focussing on diameter and centre (see b. and c.) or if he is considering for a longer time to tessellate the area of the circle (see d.) (89-117). In the collective reflection with the teacher and in the later interview, he explains, that the little rod (see fig. 1a) has to be fixed in the centre of the circle and the pencil has to be turned around it. He also considers that the circumference has to be a closed curve and that in his tool it is possible to change the size of the radius in contrast to other tools. Conclusions Firat probably knows what a compass is and how to use it for constructing a circle. On the one hand the rejection of assistance provokes that he cannot realise his idea. On the other hand, it gives him the chance to engage in Aylin´s creation. His offers for supporting her work open up different perspectives on the mathematical object circle for him. Maybe these perspectives enable him to have a more differentiated view on his tool, e.g. the possibility of changing the size of the radius or stating that the curve of a circle has to be closed. But as his proposals are not content-related discussed with Aylin, it cannot be concluded definitely that the participation in Aylin’s processes offers a deeper mathematical understanding to him. In this example of a partner work, initiating a collective learning situation seem to be more focused on the social element of taking-part than on the common subject-matter. Interaction about the common subject-matter would probably lead to a deeper understanding, e.g. in the attempt to persuade the learning partner of the advantages of the own idea or to argue about proposals, which could support the learning partner´s idea. The example illustrates that the analysis of the learners´ participation and mathematical processes can describe and categorise collective learning situations. The presented data show that an intended but failed cooperative learning situation could be both, an obstacle as well as an opportunity for learners with learning disabilities. In the research project children with learning disabilities often appear to be less active or take the role of a passive listener within a partner work. In further analyses, coexisting learning situations have to be differentiated: under which conditions is a learner´s attempt to enhance another one´s idea an obstacle or an opportunity for a deeper content-related understanding of the common subject-matter? How could a teacher´s support help to initiate a cooperative learning situation? »Access to mathematics is essential for equitable mathematics education« (AMTE 2017, 21). In Substantial Learning Environments like »The Circle« students with 194 learning disabilities do participate in collective learning situations. They have the opportunity to explore a holistic mathematical context, the level is not (pre-) determined by the teacher, but assigned by the students´ abilities themselves, influenced by the learning partner´s ideas. This could be an opportunity for mathematical learning for all students and not only for the better ones (cf. Krauthausen et al. 2010, p. 3). Transcription rules 31 single utterances or actions are consecutively numbered T teacher F Firat A Aylin , short stopping within an utterance what? identifiable question by reason of intonation yes. intonation at the end shows a lower voice #, ##, ### synchrony of utterances or actions [We] incomprehensible utterance with an assumption about the content (incomprehensible) incomprehensible utterance (shakes her head) action, (facial) expression, gesture, remarks … omission of lines of transcription References AMTS (Association of Mathematics Teacher Educators) (2017). Standards for Preparing Teachers of Mathematics. Retrieved from: amte.net/standards. Baltes, U., Rütten, C., Scherer, P., Weskamp, S. (2014). Mathe-Spürnasen – Grundschulklassen experimentieren an der Universität. In J. Roth, J. Ames (Eds.), Beiträge zum Mathematikunterricht (pp. 121–124). Münster: WTM-Verlag. Feuser, G. (1998). Gemeinsames Lernen am gemeinsamen Gegenstand. 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Forum Qualitative Sozialforschung / Forum: Qualitative Social Research, 1(1), Art. 22. Retrieved from: http://nbn-resolving.de/urn:nbn:de:0114-fqs0001228 [03.03.2017]. Wocken, H. (1998). Gemeinsame Lernsituationen. Eine Skizze zur Theorie des gemeinsamen Unterrichts. In A. Hildeschmidt, I. Schnell (Eds.), Integrationspädagogik. Auf dem Weg zu einer Schule für alle (pp. 37–52). Weinheim, München: Beltz Juventa. 196 ELEMENTARY MATHEMATICS TEACHERS CELEBRATING STUDENT VOICE: THE CLINICAL INTERVIEW AGAIN AND AGAIN Hanna Haydar Abstract This paper makes the case for the use of the Clinical Interview method by elementary mathematics teachers. It draws on theoretical considerations from developmental psychology and mathematics education and reports on a study of teachers’ learning to interview within the context of teacher education method and research courses. Findings confirmed that training teachers to use clinical interviews improve their questioning skills and help them bring back the voice of their students to lesson planning and teaching of mathematics lessons. Keywords: Clinical interview, mathematical thinking, teacher education, formative assessment Introduction This paper revisits the theoretical background supporting the use of the Clinical Interview method in elementary mathematics teaching. It reports on a study that examines beginning elementary teachers learning how to use clinical interviews in their mathematics instruction. It describes analysis of data from interviews conducted within graduate method courses and looks more closely on two case studies of teachers who took the interviewing further with action research to improve their lesson planning and differentiation practices. Theoretical background The Clinical Interview is Jean Piaget’s invaluable methodological gift for both researchers and practitioners in development and education. Piaget developed the method in the 1920s after the model of the interview used in psychiatric diagnosis, as an attempt to take full advantage of both testing and direct observation all when avoiding their disadvantages (Piaget, 1929; 1952). The strength was in the creation of a research method that depicts the child’s “natural mental inclination”, identify underlying thought processes and account for the larger “mental context” (Ginsburg, 1997). More specifically, a clinical interview can be defined as a “dialogue or conversation between an adult interviewer and a subject. The dialogue centers around a problem or task chosen to give the subject every opportunity to display behavior from which to infer which mental processes are used in thinking about that task or solving that problem.” (Hunting and Doig, 1997). Mathematics education witnessed in the last few decades a major shift from an earlier, nearly exclusive focus on rules, procedures, and algorithmic learning to a Brooklyn College of the City University of New York, USA; e-mail: [email protected] 197 categorical emphasis on conceptual understanding, complex problem-solving and children’s internal constructions of mathematical meaning. (CCSSM, 2010; Goldin, 2000). Part of this shift and because of the limitations of testing, teachers need to engage in “alternative” or “authentic” forms of assessment. (NCTM, 2000). Calls had been made by various mathematics educators, professional organizations and curriculum reformers (NCTM, 2000; Ginsburg, Lee and Pappas, 2016; Hunting, 1997; Long and Ben-Hur, 1991; Schorr and Lesh, 1998) for the use of clinical interview by teachers in their classroom instruction, as a powerful and necessary mean to help them meet with the current reform expectations. Steffe (1984) clearly saw the interview as the only way to understand the mathematical reality of children. Doverborg and Pamling (1993) affirmed that the clinical interview is the only way to link teachers’ thoughts, students’ development and instruction. Different studies developed analytic frameworks for evaluating teacher- conducted interviews (Arias, Schorr and Warner, 2010; Crespo and Nicol, 2003, Haydar, 2003, Jacobs and Ambrose, 2003, Moyer and Milewicz, 2002) Teachers learning the Clinical Interview Participants in this study are both pre-service and in-service teachers registered in a graduate program in childhood mathematics education. The backbone of the program consists of four courses in mathematics including number theory and algebra, geometry, history of mathematics, discrete mathematics and probability and statistics and a sequence of four mathematics education methods and action research courses. In this sequence teachers are exposed to clinical interviewing and trained to conduct them with their students. Training activities are inspired by Ginsburg, Jacobs, and Lopez (1998), Hunting and Doig (1997) and Haydar (2002). All four courses are very interactive, project-based and emphasize “action-reflection” (Duckworth, 1986). Students are guided to learn in and from their practice. Problem solving activities centered in mathematics, teaching and learning and drawing from Realistic Mathematics Education, Cognitive Guided Instruction, mathematical growth mindsets, equity and differentiation, mathematical educational technologies and others form the pillars for each cohort’s “community of practice”. Teachers progress in their experience using the clinical interview throughout the four methods and, or, research courses: In the first course they learn about the rationale, history and role of the method in research and education; they watch and analyze actual clips of interviewing; they practice the skills and sub-skills for preparing and conducting clinical interviews; they conduct their first clinical interview related to children’s number sense and they reflect about their experience and recommend steps for improvement. In the second course, teachers use what they learned and recommended above to conduct their second interview, this time about children’s algebraic or geometric thinking. They also analyze 198 various interview protocols from literature and videos about early algebra and geometry. The third course is an advanced methods and preparation to research in which teachers conduct a set of clinical interviews and learn how to collectively analyze the data of multiple interviews for research and teaching implications. In the fourth course, teachers conduct their action research, many of them choose to use clinical interviews as part of their data collection methods and some take it a step further and study how they can use clinical interviews to improve specific teaching practices. What teachers learned The analysis of questions in a random sample of 110 interviews from five different cohorts showed that clinical interview training helped teachers improve their questioning substantially. Teachers learned to ask less questions that “elicit information” and more questions that “press for reflection”. The results could also be interpreted to show that teachers developed the ability to move from using “observational interactions” to more “responsive interactions” in their interviewing (Ambrose et al., 2004). The analysis of the interviews and teacher reflections revealed also the following themes and findings: a) The challenge of listening and not teaching: Teachers became more aware of how challenging listening to students and interpreting their thinking could be: “We were instructed not to teach just probe without adjusting student response or helping them with their understanding of the concept. I found this task very daunting, every time the subject kept referring to the octagon and hexagon as the same shape I wanted to jump in and teach.” “Throughout the clinical interview, I realized how hard it is to not teach. Every time the subject makes a mistake or explains her thoughts inaccurately, I wanted to tell her the correct answer and how she needs to look at all the properties of a shape rather than just their sides.” b) The challenge of flexibility: The fluid nature of interviewing proved to be challenging to many teachers. The ability to be flexible and respond immediately to student’s behaviors need experience and solid content knowledge: “Although I was prepared with tasks and questions, materials, and manipulatives, being flexible in the interview became a bit challenging. I tried to modify my task according to the subject’s responses as best I could but this was challenging.” c) Comparing and contrasting types of questions: Analyzing one’s own clinical interviewing is a great opportunity to classify one’s own questions and their impact on eliciting students’ thinking: 199 “This clinical interview has shown me how the “why” and “how” questions play a huge part in evaluating a student’s level of understanding. We can simply ask a student what a square looks like but it is the “why” and “how” that get her to explain her thinking. Through her explanation on comparing shapes, I found that Jessica is really struggling in geometry.” “I also found that there were times when I couldn’t ask more questions in order to adequately test the subject’s knowledge.” “Next time I conduct a clinical math interview, I will narrow the focus of my topic to allow more time for follow-up questions.” “By far my biggest criticism is the types of questions asked. I certainly should have asked a lot more “why?” questions over “how?” While I did ask some decent questions that gave me insights into his thinking, a lot more why questions might have given me some more interesting results.” d) Implications for the classroom: Often clinical interviewing gives teachers signals to specific improvements that they need to incorporate in their teaching and lessons: “The clinical interview helps me as a teacher to be able to ask questions that my ideas were not interjected in. I believe that if I can practice this more in my classroom I will build confidence in my students because it fosters a relationship where they feel confident to experiment and to talk openly about the mathematical ideas they have, they will realize that there are different ways of coming up with the same answers and it is important that they hone in on what works individually for them and they will also become conscious that I value their opinion.” “As I listened to her struggle to describe what she knew about geometric shapes and their attributes, I realized that in the future I need to give my students more opportunities to practice describing and learning the names of geometric attributes such as vertices, edges, and angles.” e) Experiencing mathematics education theories first-hand: In some instances, teachers encounter interactions that echo theoretical backgrounds about the mathematical or pedagogical contexts: “I believe that Patty is capable of reasoning about shapes, but is not having experiences or conversations that would allow her concepts and vocabulary to develop. The van Hiele levels are highly dependent on instruction, not maturation. Although Patty currently demonstrates thinking primarily at the visualization level, I believe that she could transition to the analysis level fairly easily if she had the opportunity for more instruction. “Taking away the pressure of teaching or of leading the student towards the right answer freed my mind enough to notice patterns in the child’s problem solving strategies and general number sense.” f) Need and appreciation for Mathematical Knowledge for Teaching: A critical self-evaluation of clinical interview could reveal for teachers a lack 200 or gap in their “mathematical knowledge for teaching” and provide them a focus to fill this gap. “As a teacher, I have to evaluate whether there is a true understanding of the concept or are they just parroting a previously heard response. It is my job, as either a teacher or interviewer, to become very learned in the groundwork of mathematizing. I have to be comfortable and very proficient with all strategies so I could analyze how a student is thinking and how to guide him to reach his potential without forcing a strategy on him. Is he trying to draw a model? An array? Is he drawing to truly attempt to solve the problem or because he thinks that is what is expected of him? Is he attempting mental math? I have to become fluent in strategies I never used before, enabling me to ask questions which will bring me to the true level of the student's understanding. As a teacher, I need to be able to identify what strategy he is attempting to apply and gently guide him, only if needed, to see it through. I have to be able to identify his weaknesses and strengthen his foundations. I could only accomplish this goal if I myself have a strong base in number sense.” Taking Interviewing Further: Action Research For the last course in the sequence described above, teachers in our graduate program are required to engage in an action research and write their findings and reflections in a capstone thesis. Action research provides for teachers “a framework that guides [their] energies toward a better understanding of why, when, and how students become better learners (Miller, 2007). Teachers select a specific teaching practice that they want to improve and focus on a particular mathematical strand and design a responsive or proactive action research. Most of them use clinical interviewing as part of their data tools to help them evaluate the teaching practice in question. Few teachers however, decide to take the clinical interviewing one step further and choose it as the teaching practice to further experiment with and investigate. For example, one teacher asked how can he improve his understanding of student thinking through clinical interviews and thereby increase the effectiveness of his classroom questioning and lesson planning? Another teacher investigated how can she use Clinical Interviewing to create multilevel lessons for fractions. A first grade teacher wondered how can she use clinical interview to improve number sense in first graders? More specifically how can she effectively use the data from the clinical interviews to inform her teaching through lesson planning and small group instruction? To illustrate how the action research becomes a channel to fine-tune teachers’ interviewing skills and allow them to experiment with it in their regular classroom settings and for a relatively extended time, we will present here two cases: What Cecilia did: interviewing for differentiation Cecilia investigated the use of clinical interviewing as a tool for creating multilevel lessons for fractions. The subjects were 3 elementary students of varying mathematical ability levels. Each student was first observed in various 201 math centers and given an in-depth clinical interview for pre-assessment. The interview questions were chosen to cover multiple meanings of fractions that are appropriate for early elementary children, such as fractions as parts of wholes, fractions as parts of sets, and fractions as a result of dividing two numbers. In reflecting on her data analysis Cecilia wrote: “Interestingly, I discovered that even when children gave the same incorrect answer, they had different misunderstandings that led them to choose that answer. I also noticed that several times children gave correct answers, but didn’t understand or couldn’t explain the reason why it was the correct answer.” Based on the analysis of the interviews she developed a list of “misconceptions” and “difficulties” for each student noting some similarities, such as difficulties comparing fractions, and some differences, such as one student not being able to visually represent fractions. Through this list, she then created 6 lessons geared specifically towards these children’s misconceptions. The idea was that if these 3 students were struggling with certain concepts, many other children their age were too. The goal of each lesson was to accommodate the needs of the 3 multilevel learners, and this was done by adapting lessons to meet the needs of each student. Commenting on the nature of the lessons she developed Cecilia said: “Although I added activities for “advanced students”, the idea is that the lessons are multilevel in nature, so children of different ability levels could work on the same task. I achieved this by finding and creating problem-solving tasks that were open-ended, meaningful, purposeful, had multiple entry points, and visuals or manipulatives to go along with them. I chose lessons that would deepen the children’s conceptual understanding by requiring them to think, explore, and embrace a challenge.” Her lessons introduced students to various meanings and interpretations of fractions, such as fractions as parts of sets, fractions on a number line, and fractions in measurement. At the end of the study all students’ were given the original interview as a post- assessment. Based on the comparison of the pre and post assessments, all students’ improved their scores by at least 50%. The students’ answered nearly every question correctly and were able to give thorough explanations to support their answers. All of the difficulties and misconceptions students had originally (as noted from the results of the clinical interview) were not seen at all during the post assessment. In assessing her experience with clinical interviewing Cecilia commented: “After first learning about the clinical interview over a year ago, I became more aware of the types of questions I was asking to my students. My goal was to shift from closed-ended, short response questions to questions that required students to think, explain, and reflect. Similarly, the clinical interview questions I asked during my pre-assessment helped me to create problem-solving tasks and lessons that encouraged children to explore and explain their solutions and discoveries. Through this research experiment, I have discovered the clinical interview is a very powerful tool for both assessment and professional development.” 202 What Steven did: Incorporating students’ voice in the planning of lessons Steven conducted a series of interviews with a sample of upper elementary students who were at different levels and played different roles in the classroom. He selected a sample of 5 students varying from those who are engaged in the classroom activities on a daily basis, to students who were resistant learning and disruptive to class and to students who have low skill levels or were missing basic math skills. The first step in his action research process was to simply write a lesson plan as he would for any day of instruction. Each lesson plan was then supplemented with a protocol for a structured clinical interview to assess each selected student and another protocol for a flexible clinical interview to help him understand student’s thinking for both right and wrong answers. The results of the structured clinical interview helped guide and focus the flexible interview. All interviews were either video- or audio-taped. Analysis of the interviews and Steven’s reflective journals were used for him to rewrite and reorganize the lesson plans. The modified lessons were then taught and video-taped and analyzed. Special reflections on how the clinical interviews had an impact on the lesson were written. Steven gained a lot from the whole process. This double layer of structured and flexible interview helped him refine his interviewing skills and gave him room to work on his follow-up questions. Taking his findings to modify the lesson plans helped him ground his lessons on student’s thinking and needs. In his final reflection Steven came to a key realization about instruction with clinical interviewing that might influence his teaching for years to come, he wrote: “In some ways just the extra time I spent thinking about how to structure my interviews and ask questions made the lessons I implemented for this research successful, but overall the idea of listening to my students and making them part of the process of planning a lesson has changed the way I approach teaching. Understanding how your students think is incredibly important to effective teaching. It is a process that you must always build on and you will never reach a point where you understand every thought of every child and are able to anticipate everything they do. However, the clinical interview is a worthwhile tool that helps teachers to gain a general idea of how students think and gives teachers a better idea of how to formulate questions so their students will understand them. It creates a bridge of common ground where you are learning from the student and the student is learning from you.” Isn’t this what’s teaching/learning are all about? Conclusion This study confirmed that training teachers to use clinical interviews improve their questioning skills and help them ask more questions that shape understanding or press for reflection. The genius of Clinical Interview is that it is accessible and 203 can be presented as an assessment or teaching method but its implication and impact can prove transformative. It can easily become a way of thinking about teaching and celebrating students’ thinking. References Ambrose, R., Nicol, C., Crespo, S., Jacobs, V., Moyer, P., Haydar, H. (2004). Exploring the use of clinical interviews in teacher development. In D. E. McDougall, J. A. Ross (Eds.). Proceedings of the Twenty-Sixth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 89–91). 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New York: Penguin Books. INTRODUCTION OF FORMATIVE PEER ASSESSMENT IN PRIMARY MATHEMATICS FROM THE PUPILS’ PERSPECTIVE Alena Hošpesová, Iva Stuchlíková and Iva Žlábková Abstract The study focuses on pupils’ reflection on peer assessment in inquiry based mathematics lessons. Pupils participating in the study: (a) solved the problems, (b) provided oral or written feedback of a solution of the same problem solved by their classmates (in group discussion in the 2nd grade, in rubrics with written comments in 4th and 5th grade), (c) the solvers were allowed to use the peer feedback for revising their artefacts. After several lessons pupils were asked to reflect this experience. Keywords: mathematics education, primary school level, formative assessment, peer assessment Introduction Formative assessment and namely peer assessment can be considered as a feedback on pupils´ performance (Black and Williams, 2006) supporting the learning process by providing a check of the performance according to the criteria, accompanied by comments on strengths, weaknesses and/or tips for improvement (Falchikov, 1996). Learning benefits are supposed for pupils acting as assessor University of South Bohemia in České Budějovice, Czech Republic; e-mail: [email protected] 205 and assesse, as well; since assessing their peers can develop pupils’ judgement- making skills about what constitutes high-quality work and a self-reflection about their own understanding (Topping, 2013). This contribution reports the results of a research study on peer assessment carried out in the Czech Republic in the frame of EU-funded project ASSIST-ME aiming to develop formative assessment practices. The study is a continuation of the contribution presented during SEMT ´15 (Hošpesová, Žlábková and Stuchlíková, 2015) discussing inquiry-based approaches used in the project. Here we investigated how pupils perceive peer feedback they offer and they receive in the context of inquiry lessons. Peer assessment is conducted in mathematics lessons at primary school level, focusing on problem solving and investigational competences and is followed by semi-structured interviews with the pupils. Theoretical Background The idea of formative assessment (assessment for learning) emerged in Czech educational context in the nineties and since that time has been slowly introduced into the school practice (c. f. Žlábková and Rokos, 2013). The Czech Framework Educational Programme for Basic Education (2007), present curricular document on education of pupils aged 6 to 15, aims at gradually accomplishing changes in the assessment of pupils towards diagnostics of ongoing learning progression, assessment of pupils’ achievement history and a wider use of verbal assessment (compared to marks); respecting the individual need of each pupil. The practice of assessment ought to be driven by guidelines embedded in each school rules document (School Educational Programme). This document should describe principles and methods of assessment and self-assessment of learning outcomes and conduct of pupils, including the acquisition of data for evaluation and criteria for the evaluation. Self-assessment is thus explicitly stated as a necessary part of the school assessment practice, but explicit peer assessment is not. Peer assessment is seen rather as a particular method in cooperative teaching (Novotná and Krabsová, 2013). In recent years there has been great interest in upscaling the formative assessment as there is a need for change in learning culture (Santiago et al., 2012; Education at a Glance 2015: OECD Indicators Czech Republic, 2015). The problem is that though there are some examples of good practice (c. f. Košťálová and Straková, 2008; Kratochvílová, 2011; Slavík, 2003), they are not empirically studied and focus mostly on selected subjects and educational levels. Empirical research which would provide some evidence on the effectiveness of using various formative assessment methods is quite scarce (Novotná and Krabsová, 2013). Some types of formative assessment are seen as more or less embedded in common Czech teaching culture, like teacher provided on-the-fly assessment, or to a lesser degree written feedback (Lukášová, 2012; Košťálová et al., 2008; Laufková and Novotná, 2014; Novotná and Krabsová, 2013), and methodological 206 literature for teachers (e.g. Kratochvílová, 2011; Starý, 2006) pays more attention to these forms of formative assessment. In these materials, formative peer assessment is mentioned only as a supplementary option (e.g. Košťálová and Straková, 2008), probably also due to the fact that peer assessment as a form of classroom communication is not very frequent (Šeďová et al., 2012) and formative peer assessment was not yet empirically studied in Czech schools. Peer assessment is generally an educational arrangement for classmates to judge the level, value, or worth of the products or learning outcomes of their equal-status peers by offering written and/or oral feedback (Topping, 2013). There are several crucial aspects that complicated the use of peer-assessment in the classroom. Often pupils do not feel fully confident in their own or their peers’ knowledge as they are not expert in a subject area. They doubt their own and peers’ ability to assess and claim that it is the role of the teacher to be the assessor (Van Gennip et al., 2009). Nevertheless, Yang et al. (2006) showed that revision following the teacher’s feedback is less beneficial for pupils’ understanding than peers’ feedback. The authors argue that teacher’s feedback was accepted as such, often misinterpreted, and pupils often considered that no further corrections were expected whereas peer feedback leads to more discussions and checking for confirmation and consequently a deeper understanding. Another crucial aspect of peer assessment is the quality of peer feedback. Gielen et al. (2010) distinguished two opposite perspectives on peer feedback quality. The first perspective defines peer feedback quality as the degree to which a peer’s quantitative feedback (mark) matches that assigned by an expert assessor, where scoring validity is the leading concept (Falchikov and Goldfinch, 2000). The second perspective defines peer feedback quality in terms of content and/or style characteristics. Written comments on a specific piece of work/artefact could vary among peer assessors, because they might focus on different aspects of an assessee’s work (Topping, 1998). Objectives of the study and research questions The aim of this study is to investigate pupils’ views on introduction of formative peer assessment in inquiry based lessons of primary mathematics. The issues under investigation are: How did pupils perceive the peer assessment that they received and its value for their learning? Do they prefer peer or teacher assessment? For what reason? Methodology This study took place in six classrooms in which the teachers were willing to participate along with their pupils in the practical implementations of peer assessment included in the inquiry-based teaching sequence (six 60 minutes’ 207 lessons). The teaching experiments were conducted with three samples of pupils (2nd, 4th and 5th graders, two classrooms in each grade). We created conditions to implement oral (2nd graders), or written (4th and 5th graders) peer assessment while solving inquiry tasks oriented towards problem solving and empirical investigation competence development. Formative peer assessment was, as such, new method of assessment for both pupils and teachers. It was structured in oral form by teachers’ questions, in written form by questions or unfinished sentences in worksheets created for this purpose; for example: Is the process of solving the problem correct? Is it comprehensible? We do not understand the procedure because … Also the inquiry tasks were developed ad hoc for this study. (For example Triangle Problem for the 5th graders: Create an instruction for your classmates to determine the content of the triangle, that is the number of squares) drawn in a square grid.) In the first step, the pupils worked on tasks individually, in pairs (randomly matched) or small teams. Then in 2nd grade they presented the results in front of the whole class and they assessed the work of their peers under the guidance of the teacher. Older pupils (4th and 5th graders) gave their written artefacts to another individual/pair/team of pupils and they were asked to use rubrics with pre- specified criteria for providing feedback to their peers. Once the pupils completed the peer assessment, they exchanged their peer feedback and reviewed it individually/in pairs/in teams. Pupils were allowed to use the peer feedback for revising their artefacts. The pupils were assured that the study would not contribute to their final mark. Structured interviews contained a set of questions to prompt pupils’ reflection of their experience (questions relevant to the peer assessment are in Table 1) were carried out after the sequence of inquiry lessons by a member of the project team. The interviews were transcribed. The answers to questions were coded by two coders according to the same template. The differences in codes were discussed until consensus was reached. Questions Template codes 1. Have you been assessing the work of some of your peers? 2. Do you think that you did well in 1) yes or mostly yes, 2) no or mostly no, assessing the peer's work? 3) I do not know or other answer 3. Did you have any difficulties when 1) lack of knowledge or skills (related to assessing the work of your peers? What task, criteria)1, 2) social regards kind and why? (positive and negative)2, 3) would also 1 Examples of coded utterances: Firstly, we need to know what it is about, what should be done, and I am not sure what is correct; I did not know how to describe what the flower needs, the assessment itself, whether it is correct or not; I did not understand the picture, it was difficult. 2 It is difficult if you know that it is your friend; I did not want to assess him badly. 208 rather know other solutions3, 4) bad handwriting, not sure whether under- stood correctly, 5) no difficulties reported 4. Would you rather get the feedback on 1) from teachers, 2) from peer, 3) does your work from your peers or from the not matter teacher? Why so? 5. Do you think that there are any differences between the teachers’ and peers’ assessment of your work? What sort? 6. When you received the feedback from 1) yes or mostly yes, 2) no or mostly no, the peer, did it help you to improve your 3) I do not know or other answer solution? 7. When you received the feedback, what 1) mark, 2) comments to your work, 3) information interested you the most? both Table 1: Structured Interview Questions Data analysis We used deductive thematic coding with a template approach (Crabtree and Miller, 1999). An a priori template of coding categories in the form of a codebook was applied as a means of organizing the data for subsequent interpretation. When using a template, a researcher defines the template (or codebook) before commencing an in-depth analysis of the data. The codebook is sometimes based on a preliminary scan of the text, but for this study, the template was developed a priori, based on the research question. The codebook presented coding categories (Template codes in Table 1). A quantitative survey of received codes was then compiled. Results Although the quality of pupil feedback is not the focus of this study, we consider it necessary to show three examples of most frequent feedback. (1) The standard with which they compared their classmates' work was their own solution (Dealing with Triangle Problem: “You did it the same way as we did. Nicely written. It is understandable.”). (2) Similarly, some negative evaluations were also generous and did not provide significant help for the solver (“Description incomprehensible. Try to write it more understandable next time.”). (3) There has also been concrete recommendations given in the "child's language” that are understandable to the solvers in given context (“They wrote they added one. But what one, it is not sufficient, if we did not know the task we did not know what one.”). The participating pupils did not have any previous experience with peer- assessment and therefore we wanted to know how well they did in providing the 3 I did not know whether it was correct, I may have solved it in another way, I would rather see more solutions. 209 feedback on peers’ work. The summary of pupils´ answers shows that they felt relatively competent in assessing their peers (see Figure 1). Figure 1: Subjective perception of doing well in assessing the peers Nonetheless, the pupils reported some difficulties they encountered when providing the feedback. Figure 2 shows the most frequent areas of difficulty that the pupils mentioned. Figure 2: Percentage of reported difficulties when providing peers with feedback The most frequent and most consistently reported difficulty relates to the lack of knowledge or skills necessary for correct assessment, which was associated either with uncertainty about the solution of the inquiry task, or the criteria for the assessment. The pupils mentioned for example that: “It was difficult to decide whether it is correct or not”; “I did it differently and I am not sure that this could be realized” etc. Dealing with this uncertainty in the classroom is crucial for implementation of peer feedback in a broader context. What is further evident is 210 that when worked in small teams, the pupils were more sensitive to social regards when providing the feedback (e.g. worries of negative emotions of peers, bad own feelings when assessing the bad artefact of a good friend etc.). On the contrary, pupils did not mention the need to see alternative solutions or trouble with handwriting and one half of the pupils did not report any difficulties. An important issue emerging from the interviews with pupils (and teachers) was the preference of feedback either from the teacher or from peers. Before the start of experimental teaching, the teachers frequently expected that the fact that peer feedback is expressed in more accessible peer language could be a potential advantage of implementation of formative peer-assessment. We therefore asked the pupils about their preferences (Fig. 3) and reasons for them. Figure 3: Preferences of teacher feedback and peer feedback Figure 4: Pupils’ perception of formative impact of peer feedback 211 The pupils valued the ideas of their peer assessors as easily accessible and did not worry that much about their own mistakes when these are mentioned by peers, unlike when it is the teacher. What is considered as essential for formative assessment is the informative part of feedback which can foster further learning. For the question whether the feedback from peers was seen as valuable for further improvement of the artefact/solution the codes were summarized into three categories: positive statement, negative statement and uncertain standpoint (see Figure 4). Other crucial point is how the feedback helps the pupils to correct their work. Some pupils did not pay attention to the utilization of any feedback, regardless of its quality. What is even more important is the fact that pupils reported that the feedback was valuable for improving their solution, but did not actually use it, as could be seen from the working sheets. Discussion A student’s perspective on their own learning is an important resource of valuable information for teachers, especially when they are trying to implement new approaches to teaching. To understand the ways in which their practice influences student learning, they need to listen to pupils’ accounts of their learning experiences (Kane and Chimwayange, 2013). Pupils participating in this study experienced peer assessment in inquiry lessons for the first time. The results show that the pupils considered themselves as rather capable of providing peer assessment. Nonetheless they also felt some difficulties (a subjectively perceived lack of domain knowledge needed for evaluating correctness of the peer’s solution, or procedural knowledge related to providing proper hints or advice for peers on how to proceed further). Pupils also expressed that they lacked the opportunity to see other alternative solutions which made them uncertain in assessing the peer’s work. It seems that specification of proper criteria in rubrics is not enough and that pupils need some support in applying them, at least when they are not yet used to it. Results showed that pupils took into account the social context of peer assessment. It was visible when the pupils worked in small teams. Working in teams can make the pupils more aware of the socio-emotional context of assessment. Working in small teams led, on the contrary, to more opportunities to discuss the solution and the formulation of proper feedback before giving it to the other team. This could also be influenced by the fact that pupils in these teams frequently used the possibility to add a verbal explanation to what is written when problems in understanding arise, or provided feedback within group discussion (2nd graders) instead of in writing. The quality and perception of the peer assessment was influenced by the character of task. In case that the pupils (mostly in teams) tried to solve divergent problems 212 and had the opportunity and need to discuss deeply the process of problem solving, solution and criteria of their assessment they did not express the doubt about plausibility, correctness and completeness of the feedback from peers. Conclusions The pupils’ interviews provided important information on the main challenges and hindrances the pupils faced in providing peer assessment. The pupils seemed to prefer feedback from a teacher over the assessment of their peers because they see it as more reliable and relevant. But this preference is dependent on the organization of pupils’ work and peer-assessment. When pupils worked in small teams, where they had more opportunities to discuss the solution, this prevalence disappeared. A similar finding is that the work in small teams was accompanied by fewer difficulties encountered in providing the feedback. Discussions in small teams may alleviate the uncertainty associated with a lack of factual knowledge and with any socio-emotional consequences of assessment. Though the pupils reported that the feedback helped them to improve the solution of the task, they carried out these improvements only in small numbers (which was found in the analyses of revised protocols). This could be due to a lack of time, motivation or fatigue of pupils (lessons videodata provide evidence that it was the case in two inquiry tasks). It is an important message that teachers must pay attention to. Pupils also commented on the difficulties they had in assessing inquiry of their peers (uncertainty about correctness, about a considerate way of reporting mistakes, etc.) and advantages they saw in peer feedback over feedback from the teacher, although these issues are probably dependent on the organizational forms of instructions (e.g. individual vs. small group work). Pupils mostly appreciated the inquiry tasks and would like to extend the proportion of such learning to their everyday lessons. They perceived peer feedback as a new alternative to assessment, but having experienced it for the first time, they remained more in favour of feedback provided by a teacher. Acknowledgement This research was supported by the project 321428 ASSIST-ME (Assess Inquiry in Science, Technology and Mathematics Education, FP7, Capacity, Collaborative Project). References Black, P., Wiliam, D. (2006). Developing the theory of formative assessment. Educational Assessment, Evaluation and Accountability, 2(1), 5–13. Crabtree, B., Miller, W. (1999). A template approach to text analysis: Developing and using codebooks. In B. Crabtree, W. Miller (Eds.), Doing qualitative research (pp. 163–177). Newbury Park, CA: Sage. 213 Education at a Glance. OICD Indicators Czech Republic (2015). DOI: 10.1787/eag- 2015-51-en. Retrived from http://www.oecd-ilibrary.org/education/education-at-a- glance-2015/czech-republic_eag-2015-51-en. Falchikov, N. (1996). Improving learning through critical peer feedback and reflection. Paper presented at the HERDSA Conference 1996: Different approaches: Theory and practice in Higher Education, Perth, Australia. Falchikov, N., Goldfinch, J. (2000). Student peer assessment in higher education: a meta-analysis comparing peer and teacher marks. Review of Educational Research, 70, 287–322. Framework Educational Programme for Basic Education (2007). Retrieved from: www.msmt.cz/file/9481_1_1/. 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Pohledy na formativní a sumativní hodnocení žáka v českých publikacích [Formative and summative assessment in Czech publications]. Pedagogika, 58(3), 328–354. USING CULTURAL TASKS TO DEVELOP GROWING PATTERN GENERALISATIONS FOR YOUNG CULTURALLY DIVERSE STUDENTS Jodie Hunterand Jodie Miller Abstract Both Pāsifika and Maori cultures have a rich background of mathematics including a strong emphasis on patterns used within craft design (Finau and Stillman, 1995), however there have been limited studies which have investigated young Pāsifika and Maori students’ understanding of growing patterns. The aim of this study was to explore how contextual Pāsifika and Maori patterning tasks can potentially support young children to develop their understanding of algebraic growing patterns. A teaching experiment was undertaken with 27 Year 2 students (6-year-old) participating in 10 lessons focusing on growing patterns. Analysis of the data reveal that the contextual growing tasks assisted these young students to generalise growing patterns. Keywords: Early algebra, patterning, generalisation, elementary/primary school mathematics, Pāsifika and Maori students INTRODUCTION Algebra has been described as a mathematical gatekeeper for all students. There are two reasons for this; (a) it is an area of mathematics that has the potential to provide both economic opportunity and equitable citizenship for all (Satz, 2007) and, (b) it is an area of mathematics that provides access to higher education such as university. This gatekeeping status is vitally important for students in marginalised communities (Gonzalez, 2009). In New Zealand, students from both indigenous Maori and Pāsifika backgrounds are characterised by unenviable Massey University, New Zealand; e-mail: [email protected] 215 statistics in which a large percentage are under-achieving compared to their peers. The marginalisation these students experience impacts on students’ school outcomes, particularly in mathematics, and this has a snow-balling effect for latter school outcomes (e.g., access to university). As algebra provides one pathway for potential equality and opportunity, and a possible reduction in exacerbated inequalities between ethnic and socioeconomic groups (Greenes, 2008), it is important to consider the significance of early algebra for young students in marginalised communities (Gonzalez, 2009). Thus, it is imperative to develop an understanding of how to best support the teaching and learning of early algebra, and create space to develop understanding, for young Indigenous students in marginalised contexts. Early algebraic thinking leads to a deeper understanding of mathematical structures (Warren and Cooper, 2008). As such, it is the basis of latter algebraic understanding and the powerful and transportable forms of mathematics that underlie modern technology, problem solving and planning. Hence, there has been an international push to prepare students to be successful in the 21st century, and as such there have been proposed changes to curriculum areas such as mathematics. A key aspect of proposed changes is greater emphasis on the teaching and learning of early algebra in primary classrooms (Blanton and Kaput, 2005). In part, this emphasis has arisen in response to the growing recognition of the inadequate algebraic understandings many students develop during their schooling and the role this has in denying them access to prospective educational and employment opportunities (Knuth, et al., 2006). In response, some curricula including New Zealand (Ministry of Education (MoE), 2007) advocate teaching arithmetic and algebra as a unified strand across the mathematics curriculum. This approach focuses on using students’ informal knowledge and numerical reasoning to build early algebraic thinking. Tasks involving functions and numeric patterning activities offer an opportunity to integrate early algebraic reasoning into the existing mathematics curriculum. RESEARCH LITERATURE Studies with primary students demonstrate that engaging with early algebra assists students to develop a deeper understanding of mathematical structures that can lead to mathematical generalisations (Radford, 2010). One particular path for developing this thinking is through students working with growing patterns (Warren, 2005). Research studies have focused on fostering early algebraic thinking through the use of patterning activities, in particular geometric patterns (e.g., Warren, 2005). Growing patterns are characterised by the relationship between elements which increase or decrease by a constant difference. As a means to develop early algebraic thinking, students in the elementary school engage in activities that provide students with the opportunity to copy, continue, and extend growing patterns. Eventually, there is a need for the student to see the relationship 216 between the pattern and their position (stage). This relationship can be termed a generalisation. However, the majority of these studies have been conducted with students from mainstream classrooms and the tasks presented are often from a formal mathematical contexts (geometric growing patterns). There have been limited studies which have investigated young Pāsifika and Maori students’ understanding of growing patterns. Both Pāsifika and Maori cultures have a rich background of mathematics including a strong emphasis on patterns used within craft design (Finau and Stillman, 1995). This includes geometrical designs which are used in repeating and growing patterns. In this paper, we investigate student responses and generalisations to growing pattern tasks situated in Pāsifika contexts. Generalising mathematical concepts must go beyond just the act of noticing (Radford, 2010). Students must also develop the capacity to address and express concepts algebraically for all elements of the sequence (Radford, 2010). Underpinning this assertion is Radford’s (2010) ‘layers of generality’: factual, contextual and symbolic generalisations. Factual generalisation is an elementary level where students engage heavily in gestures, words and perceptual activities. On this level, students attend to the particular pattern rather than general elements across the pattern. Contextual generalisation requires students to reduce the signs for greater expression of meaning. “They are contextual in that they refer to contextual, embodied objects, like “the next figure” which supposes a privileged viewpoint from where the sequence is supposedly seen, making it thereby possible to talk about the figure and the next figure” (Radford, 2010, p.52). Finally, the symbolic level requires a further contraction, where students replace words with symbols such as letters to express the generality of the rule. While there is agreement that students move through different stages during the generalisation process, how students’ progress to the first stage identified by Radford, and shift through these stages as young students form generalisations, remains a largely unexplored realm. This study aims to explore the following research question: 1. How do young culturally diverse students generalise growing pattern tasks from culturally based tasks? THEORETICAL FRAMEWORK When considering the social and cultural perspectives of the context and how this impacts on the teaching and learning of mathematics, it is necessary to acknowledge the richness of Indigenous knowledge systems. Thus, a decolonised approach has been adopted for the study with a focus on valuing, reclaiming, and having a foreground for Indigenous voices (Denzin and Lincoln, 2008). The authors acknowledge the richness in the cultural backgrounds of students and this informed the selection of pattern tasks and lessons in this study. 217 Mathematics as a subject was long considered by many to be value and culture free (Presmeg, 2007). Despite this belief, in the past few decades researchers (e.g., Bishop, 1991; D’Ambrosio, 1985) have shown that mathematics is a cultural product. We take the perspective that the teaching and learning of mathematics cannot be decontextualised from the learner. In this view, the teaching and learning of mathematics is wholly cultural and is closely tied to the cultural identity of the learner. However, the mathematics commonly presented to Pāsifika and Maori students in New Zealand classrooms are odds with the student’s cultural ontology. The mismatch between the practices within the classroom and the cultural background of the students (Bills and Hunter, 2015), relates to the underachievement of specific groups of students (such as Pāsifika and Maori within New Zealand). One key aspect of developing a culturally responsive classroom is ensuring that mathematical tasks are set within the known and lived, social and cultural reality of the students. An example of this within the context of early algebra is drawing upon authentic patterns from Pāsifika and Maori culture for exploration in the mathematics classroom. RESEARCH DESIGN This research reports on one aspect of a larger study focusing on young culturally diverse students’ developing understanding of growing patterns. A three-week teaching experiment was conducted with Year 2 students in a low socio- economic, high poverty, urban school in New Zealand. Twenty-seven students (16 males and 11 female students) aged 6 years old participated in the study. The students were predominantly of Pāsifika descent (n = 22), with three students from an indigenous New Zealand Maori background, and two students from South East Asia. Students of a Pāsifika background are not from a single ethnicity, nationality, language or culture but are a diverse group including those born in New Zealand, those who have migrated from the Pacific Islands, or those who identify themselves with the Pacific Islands and culture (Coxon et al., 2002). New Zealand has the largest population of Pāsifika peoples in the world. The teaching experiment comprised of 10 x 30 minute lessons. All lessons drew on a cultural context relevant to the students’ background and patterns were drawn from Pāsifika and Maori culture. In this paper we present findings from three lessons from the teaching experiment which used patterns from Pāsifika culture. Students in this classroom had previously engaged with tasks involving repeating patterns but growing patterns were unfamiliar as this is not a curriculum expectation until Year Four (MoE, 2007). Table 1 displays the three lessons from the teaching experiment which are a focus in this paper including the lesson names, cultural context, and the pattern image shown to the students. 218 Lesson name Cultural Context Pattern (image) Lesson 1: A Tongan artist has been Tongan Tapa working on their tapa Cloth cloth design. They decided to use this pattern for the border. Lesson 2: At the Pasifika festival Samoan Sasa they perform a Samoan sasa. As part of the sasa, each person slaps their legs two times and claps once. Lesson 7: Cook A group of Mamas are Island Tivaevae working on a tivaevae design. Table 1: Lesson name, Cultural Context and Pattern Image provided for Three Lessons in the Teaching Experiment All lessons were video recorded which were then transcribed before data analysis. Two video cameras were used to collect data during each lesson of each teaching experiment, with one camera focused on the teacher and one on a group of students. These video-recordings were used for in-depth analysis by the authors. The systematic approach of constant comparative method was used to analyse the lesson data. The video footage of the lessons was wholly transcribed and analysed to identify themes. To manage these documents a coding system was utilised to determine how to examine, cluster, and integrate the emerging themes (Creswell, 2008). Researchers coded the data at each phase with respect to early algebraic thinking and task design and met to discuss their themes and recode any data. To ensure reliability of the coding, both researchers coded the data independently and then crosschecked the analysis. For cases where there were contradictions, an independent researcher coded the data so that a consensus was reached. Insights gained from the three lessons are presented in the following sections. RESULTS AND DISCUSSION Use of tasks involving a cultural context beyond that which is typically presented in Western mathematics contexts appeared to be effective in supporting young diverse students to engage in growing pattern generalisations. Acknowledging that students bring their own cultural knowledge to the classroom provides an 219 opportunity for culturally diverse students to make more meaningful connections to mathematics. Student initial generalisations This data draws from three teaching experiment lessons (see Table 1) and provides an insight into the types of generalisations students were providing for these tasks. The responses have been categorised using Radford’s (2010) layers of generality: factual; contextual; and, symbolic generalisations. The following sections draw on transcripts from in class discussion with the teacher and students. Pseudonyms have been used for all students. Lesson 1 Tongan Tapa Cloth: Teacher: So, if you were going to tell someone what the rule is… What would you say? Sima: You’re adding the four [striped triangles]. Teacher: You said something, what are you adding each time Sima? Sima: One more four. During this lesson, students provided factual generalisations (referring to a particular pattern term) and contextual generalisations (generalities across pattern terms). When asked to identify the pattern rule, the majority of students provided a generalisation that demonstrated recursive thinking (adding four). Students identified contextual generalisations, (e.g., 10th position) where the rule was defined by a recursive element of the previous patterns (e.g., adding 4 on to the 9th pattern), and were also able to do this for patterns using quasi-variables (e.g., 100th position). Past studies with older students have also found recursive thinking to be obvious when exploring growing patterns (Lannin, 2005). Thus, students have difficulties in developing or shifting to covariational thinking, that is, identifying the relationship between the pattern term (dependent variable) and pattern quantity (independent variable). An initial step in moving to covariational thinking is the students accessing or seeing the multiplicative structure of the pattern. It is difficult to say, but it appears in the example provided above that the student may be beginning to see the multiplicative structure as ‘four’, as his language has shifted from ‘adding four’ to ‘one more four’ as a unit. Lesson 2 Samoan Sasa: Teacher: So listen to Binh’s idea and see if you can work that out. Binh said if you heard 800 slaps you would hear 400 claps. [Teacher and students then spent some time acting out the pattern with lower numbers] Teacher: So Justice and Binh, you talked about if you hear ten slaps. If you heard ten slaps, how many claps would you hear? Justice: Five. Teacher: Because? Justice: Because... ten slaps and five claps 220 Teacher: If it was 100 slaps, how many claps would it be? Justice: 50 claps. [Teacher asks students to write a general rule] John: Our rule is a (number of people) times two equals slaps. Seini and Binh: Number of people times three equals number of slaps and claps. In this transcript there are three interesting points to raise; (i) students were able to work with quasi-variables but had difficulty explaining the relationship; (ii) students were often splitting the pattern (the claps, slaps and people) to generalise; and, (iii) students were attempting symbolic generalisations. It appears that use of quasi-variables assists young students to begin to generalise the growing pattern structures (Miller, 2016; Warren and Cooper, 2008). First, the quasi-variable (e.g., 800 slaps) pushed some students to see the structure of the pattern, as they often found it challenging and unproductive to apply an additive rule to a quasi-variable to determine the quantity. Students still found it challenging to articulate the general rule even though they could work with a quasi-variable. Second, students were often only generalising part of the pattern, for example generalising the number of slaps in relation to the number of people, or the number of claps in relation to the number of slaps. Despite there being three variables for the students to attend to (the number of people, the number of slaps and the number of claps), students often ignored one component of this when articulating the general rule. Third, it can be seen in the above transcript that Seini and Binh provided a contextual generalisation, while John had contracted the language of ‘number of people’ with the symbol ‘a’. Past research found young students can engage with algebraic concepts and use symbolic notation earlier than anticipated (Blanton, 2008; Warren and Cooper, 2008). However, scaffolding is required for young students to use alphanumeric notation as they express their generalisations (Blanton, 2008). In this present study with young culturally diverse students, the variable did not naturally appear in the case of position n. Specific teaching actions were required before students used alphanumeric notation. However, once introduced students did not have difficulty using the letter as a symbol (e.g., thinking the letter represented a number) Lesson 3 Cook Island Tivaevae Teacher: Ok so now Melissa, Sima and Seini, and I think you guys all did something else similar I want you to talk about what you did, what did you notice? Melissa: Every time you add on leaves you add eight. Teacher: So can I just check how many were in the middle on your one? Seini: Four. Teacher: It’s still four in the middle, you said, every time we add eight. Turn and tell your buddy why they’re saying they’ll always add eight. Teacher: So the first one’s around it, can you see that? So it’s like this, the patterns you’ve got your four in the middle and then it goes eight and then 221 eight, and then eight, can you see like that, it’s growing like that. (draws diagram – see Figure 1) Teacher: How would we write a rule for this one? [Students talk in pairs] Seini: Eight times plus four. Figure 1: Teacher draws a simplified structure of the Tivaevae pattern Students in this study were able to provide a contextual generalisation for the Tivaevae pattern (see Table 1). There are two points to raise with this example. First, students were shifting their thinking to attend to the multiplicative structure of the pattern rather than recursive thinking. Prior research suggests that an understanding of multiplicative thinking is fundamental for older students when generalising growing patterns (e.g., Rivera and Becker, 2011). It is conjectured that students were able to shift their thinking from additive to multiplicative because of the way this pattern was visually structured. Many growing patterns are presented in a way that draws students to attend to the additive nature of the pattern, and this can limit students’ awareness and accessibility to generalise structures (Moss and McNab, 2011). For example, with the Tapa cloth pattern the students can see that four triangles are added each time to the pattern. This occurs in a linear sequence. In contrast, the Tivaevae pattern grows in all directions. Second point to raise, is that there were explicit teaching actions required to ensure that students attended to the constant variable in the pattern (the 4 original leaves in the centre of the pattern). The teacher explicitly drew the students’ attention to the constant value (+4) represented by number 4 in the centre circle of the drawing (see Figure 1 in the transcript above) to signify its relationship to the rest of the growing pattern (Moss et al, 2008). It is challenging for young students to co- ordinate and attend to each of the variables in growing patterns with a constant variable. However, it is when students begin to co-ordinate between these variables that they shift from recursive thinking to co-variational thinking (Cooper and Warren, 2008) and form generalisations. CONCLUSION AND IMPLICATIONS Young Pāsifika and Maori students were able to provide factual, contextual, and symbolic generalisations. While it can be argued that some generalisations (i.e. symbolic generalisation, that is the use of alphanumeric notation) are more sophisticated than others (i.e. factual and contextual generalisations), it is 222 suggested that in the early years greater importance lies in the ability to initially determine contextual generalisation. This involves moving beyond the particular pattern figures and identifying a relationship between the pattern figures and pattern terms (Radford, 2010). It is apparent from this study, that there may potentially be additional layers to contextual generalisations as students work with more complex pattern types (e.g., Sasa pattern). This warrants further exploration. Findings from this study provide a positive story as to how young Pāsifika and Maori students generalise patterns situated in cultural contexts, a group of students that are currently underrepresented in this literature. It was apparent that drawing on both Pāsifika and Maori rich background of patterns used within craft design (Finau and Stillman, 1995) and dance, proved to be a successful avenue for these students to begin exploring growing patterns. Similarly, this has also been found in studies with young Australian Indigenous students exploring environmental growing patterns before traditional mathematical patterns (Miller, 2016). Studies which have examined early algebraic thinking (e.g., continuing patterns) with respect to the use of traditional mathematical patterns and patterns from cultural contexts (e.g., Pāsifika and Maori) have demonstrated that young Pāsifika and Maori students appear to have more success when working with patterns from their cultural context (Miller and Hunter, 2017). Thus, it is conjectured that these cultural patterns should be introduced to young students prior to formal mathematical contexts (geometric growing patterns). The results of this study indicate important implications for teaching and research, particularly in relation to how the context (link to the students known world) of the growing pattern potentially assists students to access structures to generalise. References Bills, T., Hunter, R. (2015). The role of cultural capital in creating equity for Pāsifika learners in mathematics. In M. Marshman, V. Geiger, A. Bennison (Eds.), Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia (pp. 109–116). Sunshine Coast: MERGA. Bishop, A. (1991). Mathematical enculturation: A cultural perspective on mathematics education. Netherlands: Springer. Blanton, M. L. (2008). Algebra and the elementary classroom: Transforming thinking, transforming practice. Portsmouth, NH: Heinemann. Blanton, M., Kaput, J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36, 412–446. Coxon, E., Anae, M., Mara, D., Wendt-Samu, T., Finau, C. (2002). Literature review on Pacific education issues. Wellington: Ministry of Education. Creswell, J. (2008). Research design: Qualitative, quantitative, and mixed methods approaches (3rd ed.). Thousand Oaks, CA: SAGE Publications. D’Ambrosio, U. (1985). Ethnomathematics and its place in the history and pedagogy of mathematics. For the Learning of Mathematics, 5(1), 44–48. 223 Denzin, N. K., Lincoln, Y. S. (2008). Critical methodologies and Indigenous inquiry. In N. K. Denzin, Y. S. Lincoln, L. T. Smith (Eds.), Handbook of critical and Indigenous methodologies (pp. 1–30). London, UK: SAGE Publications. Finau, K., Stillman, G. (1995, June). Geometrical skill behind the Tongan tapa designs. Paper presented at the International History and Pedagogy of Mathematics Conference on Ethnomathematics and the Australasian Region held at Cairns. Gonzalez, L. (2009). Teaching mathematics for social justice: Reflections on a community of practice for urban high school mathematics teachers. Journal of Urban Mathematics Education, 2(1), 22–51. Greenes, C. (2008). Preface. In C. Greenes, R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp. ix–xii). Reston, VA: National Council of Teachers of Mathematics. Knuth, E., Stephens, A., McNeil, N., Alibabi, M. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297–312. Lannin, J. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231–258. Miller, J. (2016). Young Indigenous students en route to generalising growing patterns. In B. White, M. Chinnappan, S. Trenholm (Eds.), Proceedings of the 39th MERGA conference (pp. 469–476). Adelaide, South Australia: MERGA. Miller, J., Hunter, J. (2017). Young diverse students in New Zealand and their initial understandings of growing patterns. In B. Kaur, W. K. Ho, T. L. Toh, B. H. Choy (Eds.), Proceedings of PME 41 (Vol. 3, pp. 249–256). Singapore: PME. Ministry of Education (2007). The New Zealand curriculum. Wellington: Learning Media. Moss, J., Beatty, R., Barkin, S., Shillolo, G. (2008). What is your theory? What is your rule? Fourth graders build an understanding of functions through patterns and generalizing problems. In C. E. Greenes, R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp. 155–68). Reston, VA: National Council of Teachers of Mathematics. Moss, J., McNab, S. (2011). An approach to geometric and numeric patterning that fosters second grade students’ reasoning and generalizing about functions and co- variation. In J. Cai, E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 277–301). Berlin, Heidelberg, Germany: Springer-Verlag. Presmeg, N. (2007). The role of culture in teaching and learning mathematics. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 435–458). Greenwich, CT: Information Age Publishing. Radford, L. (2010). Layers of generality and types of generalization in pattern activities. PNA, 4(2), 37–62. Rivera, F. D. (2010). Visual templates in pattern generalization activity. Educational Studies in Mathematics, 73(3), 297–328. Satz, D. (2007). Equality, adequacy, and education for citizenship. Ethics, 117(4), 623–648. 224 Warren, E. (2005). Young children’s ability to generalize the pattern rule for growing patterns. In H. Chick, J. Vincent (Eds.), Proceedings of PME (Vol. 4, pp. 305– 312). Melbourne, Vic.: University of Melbourne. Warren, E., Cooper, T. (2008). Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking. Educational Studies in Mathematics, 67(2), 171–85. THE POTENTIAL OF THE CHESS ENVIRONMENT IN MATHEMATICS EDUCATION – PRE-SERVICE TEACHERS’ PERSPECTIVE Antonín Jančařík Abstract The paper presents findings from evaluation of seminar works of pre-service teachers who finished the undergraduate course Chess. The course does not focus on chess as a game but on the potential of chess environment in teaching mathematics and for posing problems. The course is most often selected by students who have no prior experience with playing chess. The goal of the study is to verify whether these students are able to use the potential of the chess environment. At the same time the study wants to find out what areas of mathematics students focus on in the problems they pose. The study shows that the posed problems support development of problem-solving power, logical reasoning, ability to visualize in geometry and in some cases also to development of specific skills in the area of arithmetic, all this already at primary school level. Keywords: Chess, teacher training, problem posing Introduction Undergraduate students have had the chance to enrol into the selective course of Chess since 2013. The course is taught by an external lecturer – an international champion in chess. Originally the course was conceived as a preparatory course for extracurricular clubs at schools. However, the lecturer (herself an expert in teaching chess to very young children) managed to change the focus of the course. Currently the course focuses on the use of the environment of chess in development of mathematical and logical thinking. The goal of this paper is to show the extent to which pre-service teachers of mathematics that have finished in the course are able to work with the environment of chess and with the game of chess in general when posing problems. The question is whether the students merely reiterate problems that they came across in the course, or whether they are able to pose their own problems actively. That is why an analysis focusing on identification of the motives used in the posed problems was conducted. The results of this analysis are presented in the following text. Charles University, Czech Republic; e-mail: [email protected] 225 Chess and mathematics The impact of games on development of mathematical skills and their use in teaching mathematics at primary school level has been of interest of researchers for a long time (Petters (1998), Poirier, L., Novotna, J. and Godmaire, C. (2009)). Researchers pay attention to various games and study their impact in various areas. The game of chess has a unique position among board games. For the one thing it is the most widespread board game. Also there are many excellent mathematicians among chess players and vice versa. Milat (2015) states that chess players are often considered mathematically oriented and there are obvious similarities as chess is a game of problem solving, evaluation, critical thinking, intuition and planning — much like the study of mathematics. Studies have shown that students playing chess have increased problem solving skills over their peers. A study conducted by Kazemia, Yektayarb and Abada (2012) proved a relation between teaching chess and development of mathematical skills. Their research was conducted with 28 fifth graders (plus 29 fifth graders in the control group). Tests after six months of teaching chess showed improvement both in metacognitive skills (measured by metacognitive questionnaire of Panaoura, Philippou and Christou (2003), and in performance in a mathematical test. The areas that are developed through chess are primarily problem-solving power, but also logical thinking and ability to visualize in geometry (Margulies 1993; Ferguson, 1995). All these skills and abilities can be developed already at primary school level as the environment allows to pose problems interesting for and solvable by primary pupils. The findings of research resulted in inclusion of a special subject focusing on chess into teacher training at the Faculty of Education, Charles University. This paper presents results of a research study exploring how well pre-service teachers are to use chess motives in their teaching. Materials, Methods and Data collection tool The research study was conducted with 18 seminar works from 2014-2017. These works were preselected by the lecturer of the course and sent to the academic guarantor of the course (the guarantor supervises the academic quality of the course). Three from this set of seminar works were excluded for the needs of this research study. One of these works focuses on history of chess and two on using chess at upper secondary school level. Thus the analysis presented in this paper was conducted with 15 seminar works. The task for the pre-service teachers was to pose at least three mathematics problems in the environment of chess. The assignment was not further specified but majority of the posed problems were posed for primary pupils. There is only one seminar work with this minimum number of problems, other pre-service teachers posed more 226 problems voluntarily. The longest work contains 22 problems. In total pre- service teachers posed 109 problems, i.e. the average per seminar work was 7.27 problems, which is more than twice more than requested. This suggests the students enjoyed posing the problems and did not perceive the activity as mere meeting of criteria to get the credit for the course. Within the frame of this research study all problems were analysed. The analysis was conducted in three stages. In the first stage the basic statistical characteristics of the data set was created (number of problems, used chess pieces, level of difficulty of the posed problems) and basic characteristics of the problems studied. In the second stage, similarities in the motives used by the students were studied. At this stage two basic types of problems were identified: Paths and Threatening. Typical for the problem from the category Paths is looking for a passage through the chessboard with a selected chess piece. However, this motive was used in many different contexts. Problems of the type Threatening make use of different moves of chess pieces. However, instead of looking for the path of a chess piece, pupils are expected to analyse which fields are threatened and which are not. In both cases these are motives that were used in a number of problems the students came across in the seminars, although in a slightly different context. In the seminar works we also came across several problems that were so original that they did not fit into any category. Here in the paper we also present some of these problems for illustration. In the third step the problems in both identified categories were analysed with respect to the motives, difficulty and targeted areas of mathematics. The results of the study Paths The most frequently posed type of problem was “paths” – looking for the passage of a selected figure on the chessboard. This type of problems was used in fourteen out of fifteen seminar works (93.3 %). The total proportion of this type of problems was 85 (81.7 %). The assignments feature all normal chess pieces. The most frequently used pieces were the rook and the knight. These pieces were used in ten of the seminar works. The queen was used in the assignment by fewest students (4). The use of different pieces by individual students is shown in graph 1. If we now look at it not from the point of number of students but number of problems, the most popular chess piece was the knight. It was used in 24 problems (28.2%). Then it was the rook and the bishop (each of them in 16 problems – 18.8%). The least used figure was the queen (4 problems – 4.7%). 227 Chess pieces 60 50 40 30 20 10 0 Rook Knight Bishop Queen King Pawn Number of seminar papers Number of problems Graph 4: Chess pieces The reason why the queen was used so little is probably the fact that it can move in too many possible ways, which makes it hard to pose a solvable problem. The queen was used a lot in a different type of problems – problems based on threatening chess squares (see further). None of the students used any fairy chess pieces or unorthodox chess pieces, i. e. chess pieces with other than usual movement. The reason for this might be that the majority of students who enrol in the selective course are not active chess players and are likely not to have heard of these pieces before. Types of paths The simplest type of passage used by students in the problems they posed was the task to move on a chessboard following the marked points (see Figure 1). This was usually the introductory task in which pupils were expected to get acquainted with movement of the particular chess piece. A slightly different variant of this task was to find a that bypasses all obstacles. Or the shortest possible, e.g. as is shown in Figure 2. Pupils solved this task by finding all possible paths (in this case three) and by comparing their length. 228 Figure 5: Rook Maria felt like eating pears. Will you help Maria get all the pears in fewest possible movements? Figure 6: Move the rabbit to the carrot through the chessboard using fewest possible movements. The rabbit moves like a rook (in one movement it can pass through any number of squares in one direction). More demanding are problems in which there are more paths possible and pupils are asked to select such ways that meets the conditions from the assignment. Figure 3 shows a problem in which the pupil must find the way that corresponds 229 to a name that the pupils do not know. This means that in every step they have to explore all accessible squares and make a logical decision which of the squares fulfils the requirement from the assignment. In this case pupils cannot try all possible ways because their number would be too large. However, they can solve the problem because they can gradually exclude the possibilities that make no sense and do not allow to create a name. Figure 7: Will you help the dinosaur who moves as a bishop to pass through all letters of her name These types of problems develop pupils’ ability to solve problem tasks while using both geometrical imagination (visualization of moves of a chess piece) and logical reasoning (selecting a strategy, excluding unsuitable ways). One of the students managed to connect arithmetic and chess passages. He proposed two types of problems in which pupils were practicing multiplication at the same time. The picture shows the first assignment in which pupils are asked to move through the chessboard as a pawn (from the bottom to the top). The piece can only capture those numbers which are multiples of the square where the passage began. The assignment of the task was not specified by the student. There are two interpretations. If the piece cannot enter a square with a number in other way than capturing it, then the assignment is unique. However, pupils can be allowed even this procedure. Then more ways are possible. 230 Figure 8: Find such a way for a rook and a bishop from the initial point that the third square is the multiple of the first two move. Another posed type of problem for practicing of multiplication is in the figure 4. The task is to find such a way for a rook and a bishop from the initial point that the third square is the multiple of the first two moves. These problems develop not only the above mentioned skills but also arithmetic skills. Threatening Another type of problems posed by more students were problems based on movements of different pieces and on how they threaten or fail to threaten one another. This was used in 8 problems (7.3%) by 3 students (20%). The topic was handled in different ways. The simplest problems were modifications of the classical problem asking pupils to position chess pieces on the chessboard in a way that they do not threaten one another. The student used an untraditional array of chess pieces (4x rook and 4x bishop, or 4x rook, 1x bishop and 3x queen). These problems are a modification of the well-known problem how to place 8x queen on a chessboard in a way that they do not threaten one another. Each solution of the original problem is at the same time a solution of the assigned task. However, the newly posed problems are easier and have other original solutions (see figure 5). This type of problems develops pupils’ ability to solve problem tasks using the heuristic strategies trial-error or systematic experimenting. At the same time the problems develop the ability to visualize in geometry. The pupils have to visualize the fields which are threatened by the chess pieces and at the same time choose fields suitable for placing other chess pieces. These problems also allow extensions developing pupils’ arithmetic skills. 231 Figure 9: Sample solutions proposed by the author A very interesting assignment combines the properties of chess pieces and calculation with numbers on threatened squares. The task is to place 5 queens in such a way that the sum of squares each of the queens threatens is 40. None of the numbers can be used 2x and none of the queens can threaten two equal numbers (see figure). Figure 10: Place 5 queens in such a way that the sum of squares each of the queens threatens is 40 In one case the author used a combination of both above described problems and posed problems in which a pupil was to find the way with the condition that the piece must bypass squares on which it would threaten other chess pieces. Problems outside the category The assignments of other problems used various motives that share no common feature. In many problems the chessboard was a mere background and the relationship to the game of chess was minimal. An example of this use are various crosswords and puzzles. One of the students used part of the chessboard as a guitar fretboard for recording notes and chords. 232 Only one of the students used chess notation in their problems. Even though the problems often used known motives, we could come across assignments that resembled no known riddle. The following is one of these: Figure 11: Problem and its solution Join the squares diagonally, horizontally or vertically •The square always contains 4 coloured parts (2x one colour and 2x another). •The squares can only be connected if both contain the same colour. •If I used two connecting lines of the same colour, the colour cannot be used again (it is impossible to have three yellow lines from a yellow-red square, they must be 2 yellow and 2 red). •All colours of squares must be used. Difficulty of assignment An important aspect of the seminar works was the difficulty of the posed problems. Although there were no requirements in this respect in the assignment of the seminar work, some students worked with problem difficulty intuitively or even intentionally. 7 students (46.6%) used in their work sets of problems of different difficulty. Many of the problem were labelled with respect to the level of its difficulty. Most often the students worked with pairs of problems labelled as easy – difficult. In one case the student posed quartets of problems of an increasing difficulty. In some cases students tried to explain why they perceived the selected problems as more difficult: “The difficult variant of the problem is made more difficult by the huge amount of data and the fact that the multiples are not subsequent, which is in contrast to the simple problem.” or “The difficult variant is more difficult because the solver must make a decision whether to go to a white or a black square, not just in a spiral like in the easy one, it contains extra numbers.” 233 All this implies that students were aware of the fact that the posed problems are of a graded difficulty and tried especially on intuitive level, to work with difficulty. Conclusion The aim of the research was to find to what extent pre-service teachers who had attended the course Chess were able to pose problems rooted in the environment of chess. The study shows that although most students do not have deeper personal experience with the game (some of them did not even know the rules of the game at the start of the course), they are able to pose their own problems after the course Chess. Although majority of students posed one type of problems – chess passages, in most cases they posed also other, non-traditional problems. A very important proof of students’ interest in the topic is that the vast majority of them posed many more problems than was required for credit. The analysis of the posed problems also showed that most of the posed problems has the potential to develop problem- solving power, logical reasoning and ability to visualize in geometry already in primary school children. We also came across problems that target specific mathematical topics, e.g. divisibility of numbers or multiples. Further research, now conducted within the frame of work on one diploma thesis, focuses on the use of this type of problems with pupils at schools. The course Chess is not linked to students’ teaching practice and thus the posed problems cannot be tested at schools. However, most of the students claim they have tested the problems presented in their seminar works in some form. Acknowledgement The paper was supported by grant from Charles University in Prague entitled "Q17 – Teacher training and teaching profession in the context of science and research". References Ferguson, R. (1995). Chess in education research summary. Paper presented at the Chess in Education “A Wise Move” Conference, New York, NY. Kazemi, F., Yektayar, M., Abad, A. M. B. (2012). Investigation the impact of chess play on developing meta-cognitive ability and math problem-solving power of students at different levels of education. Procedia-Social and Behavioral Sciences, 32, 372–379. Margulies, S. (1993). The effect of chess on reading scores: District nine chess program second year report. (Article No. 5.) United States Chess Federation Scholastic Department. Milat, M. (2015). The role of chess in modern education. On-line: https://movesforlifeblog.wordpress.com/2015/02/24/the-role-of-chess-in-modern- education/ Panaoura, A., Philippou, G. I., Christou, C. (2003). Young pupil's metacognitive ability in mathematics. European Research in Mathematics Education, 3, pp. 1–9. 234 Peters, S. (1998). Playing games and learning mathematics: The results of two intervention studies. International Journal of Early Years Education, 6(1), 49–58. Poirier, L., Novotna, J., Godmaire, C. (2009). Games and learning mathematics. In J. Novotná, H. Moraová (Eds.), International Symposium Elementary Mathematics Teaching SEMT'09 (pp. 277–278). Prague: Charles University, Faculty of Education. DO TEACHERS UNDERSTAND THEIR PUPILS? Darina Jirotková and Jana Slezáková Abstract In mathematics lessons we often come across situations in which a teacher considers their pupil’s answer wrong if it deviates from the expected answer. In the best case the teacher admits they do not understand their pupil’s thinking and do not discard the answer as wrong. However, even in this case they refuse to communicate about it any further. The fear of misunderstanding pupils often becomes an obstacle to initiation of class discussions. The paper focuses on the phenomenon of misunderstanding in the teacher-experimenter – pupil interaction. The data come from two experiments. The analysis of the data clearly shows that the misunderstanding is not caused by the pupil but by the teacher in the role of the experimenter. This finding is a strong motivation to study the phenomenon of misunderstanding in classroom interaction in further research. Keywords: Qualitative analysis, communication, pupil-teacher, misunderstanding in communication Introduction When analysing protocols from some experiments conducted in various researches, mostly focusing on the description of pupils’ cognitive processes and on identification of cognitive and interactive phenomena, we came across situations in which there was a misunderstanding between the pupil and the teacher. In our first analyses our focus was predominantly on the pupil. A repeated deeper analysis in which we were looking for answers to why the pupil reacted in a particular way and in which we turned attention to analysis of our communication as teacher-experimenters, we realized there was some misunderstanding, for more details see (Jirotková and Kratochvílová, 2004). In this paper we will not discuss the more general aspects of communication between a pupil and a teacher. The focus will be only on the phenomenon of misunderstanding, namely misunderstanding caused by the experimenter who were unaware of having caused it. Theoretical background Introduction to new subject matter in transmissive education has the form of one- way communication (it usually has the form of audible verbal communication) Charles University, Czech Republic: [email protected], [email protected] 235 from the teacher to the pupil or pupils (Mareš and Křivohlavý, 1995). The teacher presents the content in a way that he or she understands best, very likely in a way they acquired the knowledge or how they were taught it. Communication (exposition of the subject matter) is very often planned in detail in advance, especially by a novice teacher. The teacher focuses on the content in their lesson plan. The feedback the teacher gets from their students is predominantly in the form of responses to questions such as “Do you understand? Who has not understood?” or from tests. A correct answer is likely to be interpreted as the sign of understanding of the topic and a wrong answer as the sign of a pupil’s failure to understand the subject matter. This information is taken by the teacher as the basis for assessment or for further explanation of the given subject matter. What often happens is that the teacher repeats the same explanation of the subject matter, which they sometimes supplement by more illustrative examples. Two-way communication between the teacher and the pupil/s usually does not take place. One of the important elements of constructivist approach to teaching is discussion, both in the form of a discussion of the teacher with the whole class or a discussion among pupils. The discussion is usually directed by the teacher in a way that it leads to pupils’ “discovery” of something new, to verification of a conjecture formulated by one of the pupils, to evaluation and comparison of different solutions of a given problem, to finding mistakes and uncovering their causes. Discussion is one of the two activities which according to the neurologist M. Stránský (2014) develop thinking, since in a discussion most centres in the brain are activated. However, it puts high demands on the teacher to provide enough space for meaningful whole-class discussions, since the big amount of communication and great diversity among pupils do not allow the teacher to prepare for it in advance. The teacher can prepare only in general terms and in the discussion often faces an unexpected situation they could not have anticipated and to which they have to react at that moment. If the teacher faces such situation and especially if they fail to solve it, it is crucially important that they get back to it, analyse it and take it as a source of experience for the future. General knowledge about communication at school can be found in many publications, e.g. (Mareš and Křivohlavý, 1995). Communication is discussed also in (Nührengbörger and Steinbring, 2009). Various aspects of communication in mathematics are studied in several researches in the CR and abroad (e.g. Boero et al. 1998, Brown, 1997, Bussi, 1998, Jirotková and Littler, 2003, Steinbring, 1998). Some ideas in the analysis reported in this paper are from (Pirie, 1998). S. Pirie (1998) classifies language used in mathematics education into six categories: 1. everyday language, 2. mathematical verbal language, 3. mathematical symbolic language, 4. visual language, 5. non-verbal language, 6. quasi-mathematical language. This classification helps to localise misunderstanding and find its causes. The extent to which pupils are able to listen to their teacher, how they interpret their statements, comments and questions and 236 also how the teacher interprets comments and statements of their pupils (often as they wish it to be) is studied by T. J. Cooney and K. Krainer (1996). The phenomena of misunderstanding as they were revealed through analyses of protocols of conducted experiments were already presented in (Jirotková and Swoboda, 2001, Jirotková, Kratochvílová and Swoboda, 2002, Kratochvílová and Swoboda, 2002, 2003a, Kratochvílová, 2002). Research methods Studying misunderstanding has always been linked to research focusing on another area, e.g. exploring pupils’ geometrical conceptions (Jirotková, 2010), structure of pupils’ geometrical knowledge from their communication (Jirotková and Littler, 2003), conceptualization processes in geometry (Swoboda, 1997), cognitive processes in combinatorics (Kratochvílová, 1995) or in non-traditional arithmetic structure (Kratochvílová, 2001). It is probably impossible to plan a research focusing merely on misunderstanding since misunderstanding is a phenomenon that cannot be planned, that occurs by the way. The phenomenon can be come across in any research activity. The two qualitative researches from which we take the illustrative examples for this paper used the same methodology. Partial clinical teaching experiments or lessons of the authors were audio and video recorded, the recordings were transcribed into written protocols and supplemented with all other relevant data. Then comments were added about non-verbal communication of the participants, about climate in which the experiment or lesson were conducted, and, if possible, by pupils’ written solutions of the given task. The tool used in the research study were problems solved by pupils either orally (illustration 1) or in writing (illustration 2). With respect to typology of problems proposed by (Mitchell; Carbone, 2011) they were problems asking for non-traditional procedures. The experiments were conducted with 3rd and 4th grade pupils. The experimental materials were processed using methods of qualitative research, e.g. grounded theory (Strauss, 1999) and comparative analysis. For the needs of exploring the interaction teacher-experimenter – pupil and of exploring the phenomenon of misunderstanding, J. Kratochvílová and E. Swoboda (2002, 2003.) developed the method of layered atomic analysis by the means of which they analysed layers of communication. The basis of layered atomic analysis is decomposition of the whole process into several layers (cognitive, language, social and emotional). This decomposition is done for each of the actors in the interaction individually. The different layers are first studied separately and then in their interrelations. Results In this study, misunderstanding is the partial result of research focusing on identification of communication phenomena that allow a description of mental processes of individual participants in communication. The description of misunderstanding is the result of these analyses. 237 In this paragraph we will present two episodes. In one (illustration 1) we describe communication pupil–pupil–experimenter, in the second (illustration 2) experimenter–pupil. There were no clear signals of misunderstanding or communication interference. Misunderstanding came to light only when we tried to describe the pupil’s thinking process and knowledge structure by analyzing the protocol. In both cases misunderstanding is the consequence of wrong interpretation of pupils’ statements by the experimenter. In the first episode the experimenter (the first author of this paper) is observing communication between pupils. There is no misunderstanding between the pupils. The intervention of the experimenter shows it is the experimenter who does not understand the communication between the pupils. Moreover, the point when communication interference that can result in misunderstanding occurs is not noticed by the experimenter. The misunderstanding is caused by different interpretation of verbal description of a concept, which is a consequence of different level of understanding of the concept. The second episode illustrates the experimenter’s (the latter of the authors of this paper) wrong interpretation of the pupil’s correct idea. The misunderstanding springs out from a different use of communication means. Illustration 1. Game YES-NO In this episode we observe communication between 4th graders in a Prague primary school. Communication is mediated through the game YES-NO (Jirotková, 2010). The objects of the game were fourteen models of solids whose size enabled the pupils to manipulate with them easily. They were: tetrahedron (1), regular pyramid (2), rectangular frustum (3), cube (4), triangular prism (5), rectangular prism (6), pentagonal nonconvex prism (7), pentagonal nonconvex pyramid (8), cone (9), truncated cone (10), cylinder (11), sphere (12), regular hexagonal prism (13), cuboid (14). Excerpt from the protocol The game was played by two boys, Jarda and Tomáš. One of the boys selected a solid in his mind and the other was meant to ask yes-no questions to find out what solid it was. Tomáš selected the solid and Jarda was asking questions. (Tomáš chose a cuboid – solid number 14.) Jr03 stands for the third turn of Jarda (and Tm – Tomáš, Ex – experimenter). The text in parentheses is the added experimenter’s comment. Jr01 “Is it round?” Tm01 “No.” Jr02 (a) “Is it like a rhombus?” (b) “It is angular, isn’t it?” Tm02 “No.” Ex01 “Is it not angular?” Tm03 “Yes, it is.” Jr03 “Does it have a hole inside?” Tm04 “No.” Jr04 “Is it high?” Tm05 “A bit high.” Jr05 “Does it taper when you go up?” Tm06 “No.” 238 Jr06 “Does it look like a triangle?” Tm07 “No.” Jr07 “And does it look like a rectangle?” Tm08 “Yes.” Jr08 “Is it this one?” (points at the correct solid 14) Analysis This communication game supports development of pupils’ geometrical world while two cognitive strands blend: 1. Objects are discovered by exploring and comparison. 2. Objects are grasped linguistically. The pupils were aware of the danger of misunderstanding and to decrease this risk they checked understanding by questions. Our experience proves that the strategy of précising and checking understanding is formed in a child already at the age of two or three and is very common in everyday life. This strategy might seem obscure and incomprehensible for a mathematician, who is used to unequivocally deterministic language. Experimenter’s intervention at the beginning of the game shows she failed to understand both Jarda (Jr02) and Tomáš (Tm02). At that point she was not able to separate the role of the experimenter form that of a teacher. As a teacher she felt obliged to set it straight, not to let her pupils use erroneous or unclear statements and to prevent their use as much as possible. To make the experimenter’s misunderstanding clear, let us analyse the beginning of the game in detail and interpret words used with a geometrical meaning. Jr01 “Is it round?” Neither the first question nor the first answer (Tm02) tell us anything about the two pupils’ idea of a round solid since they were not manipulating with the solid. However, our experience says that roundness is one of the dominant classification properties of 3D objects and is usually perceived by children as the property of a class of solids, i.e. as a differentiation phenomenon – round versus non-round (angular). By using this word Jarda is likely to have been referring to four solids – 9, 10, 11, 12. However, sometimes the word round is used as a differentiation property within the group of spherical solids: a sphere is more round than a cylinder, cone etc.., or a sphere is “all round” while other solids are just round. Then the word round refers to the quality of an object. All this implies that the meaning of the word depends on the set of solids at stake. Jr02 (a) “Is it like a rhombus?” . . . (b) “It is angular, isn’t it?” The first part of Jarda’s turn Jr02(a) is analytic. When using the term “like a rhombus”, he probably wants to refer to solids that are neither a cube nor rectangular prisms, i.e. solids that are somehow oblique or skew. The use of the word rhombus seems to be metaphorical. The second part of Jarda’s turn Jr02(b) is linked to the previous turn Tm01 and only confirms that the sought solid is angular, because it is not round. It also 239 confirms that Jarda perceives the phenomena of roundness and angularity as two polar differentiation phenomena. The words round and angular come from everyday life and their meaning is blurred in the world of geometry. Although Tomáš was confident with his first answer, which gave no grounds for fearing communication interference, Jarda was aware of the possibility of interpreting the experimenter in the following way: word round from the first question in more ways. Thus using the précising strategy he prevents possible communication conflict, misunderstanding and formulates a verification statement in which he describes the explored phenomenon in a different way (it is not round = it is angular). A similar thing happened once more. In Jr05 Jarda checks his vague expression from everyday life (“Does it taper when you go up?”) and formulates a more accurate alternative expression (“Does it look like a triangle?”) in Jr06, even though Tomáš’s answer was clear. In Tm02 Tomáš answers Jarda’s question whether the solid is “like a rhombus”, which is obvious from the following course of the dialogue. The experimenter intervenes in the conversation at this point as she is convinced there is a misunderstanding between the boys and wants to prevent failure in the game. Mistakenly she believes that Jarda uses the words “it is angular” to clarify and supplement the question “Is it like a rhombus?”. She does not realize that the second part of the question Jr02(b) is only marginal, is not connected to the first part of the question but to Tomáš’s answer “No” (Tm01). The experimenter reacts to Jarda’s use of précising strategy. The experimenter also fails to realize that Tomáš is answering Jr02(a), not Jr02(b). There was no interference in the boys’ communication. It was the experimenter’s intervention that could have caused communication interference. However, the boys did not let her confuse them and continued in the game. Tomáš (Tm03) answers the question Ex01 accurately. It could appear that there is a misunderstanding in interpretation of the statement “It is like a rhombus.” The course of the game does not say explicitly which solids look like a rhombus in the boys’ point of view. We are convinced that although Tomáš was able to answer without any hesitation, he would not be able to decide about each of the solids in the set whether they look or do not look like a rhombus. However, he linked the question only to the prism he had in mind, which had no characteristics of a rhombus. Since Jarda did not manipulate with the solids, the course of the game gives no idea about the information power of this question. After the first experimenter’s reaction we could expect another her attempt to clarify things. Obviously she was not aware of incomprehensibility of the used characteristics “it is like a rhombus” at that point because Jarda used a geometrical term which did not make the experimenter alert. In fact not even after finishing the game would she ask the boys to clarify which solids “look like a rhombus”. 240 Illustration 2 Ways The other episode comes from an experiment conducted in the quiet atmosphere of a teacher’s office. The participants of the experiment were the experimenter (Ex) and a nine- year-old 3rd grader Marek (Ma). The tool for the research was one Abrakadabra problem related to Figure 1. The problem was set by the experimenter in the following Figure 1 way: Ex: “In the picture you can see a town map. Find all possible ways from the bottom left corner (points) to the top right corner (points). You can only walk up and right (shows). If you go the same way twice, you will pay a fine. Find all ways so that you do not pay any fine.” Marek was given a sheet of paper with six copies of Figure 1 in which he was allowed to plot the different ways. The experimenter conducted the experiment with the goal of finding the pupil’s solving strategy. Excerpt from the protocol Ma01 (At first, Marek plots 4 ways (1) rruu, (2) urrp, (3) urur, (4) uurr) (2 minute pause). The number in brackets shows the order of the town map into which the way was plotted. Ma02 (Marek draws the way (5) ruur) (4 minute pause) Ex01 “Look, I will show you the ways you found.” (The experimenter shows all the plotted ways with her finger) Ma03 (Marek plots the way (6) ruru) Ex02 (the experimenter gives Marek another sheet of paper with town maps) “Look for other ways on which you will pay no fine.” Ma04 “How many fines can I get?” Ex03 “That depends on how many times you go the same way.” (1 min. pause) Ma05 “I think I have found all ways.” Each way is described by a “word” composed of four letters (two u – walking upwards and two r – walking to the right. Analysis In Ex02, the experimenter wanted to learn from Marek whether he had found all possible ways. She did not want to ask explicitly (“Have you found all ways?”) because the possible yes and no answer would give no information about what Marek really thought. She did not expect Marek to take her intervention as a challenge to look for more ways. But she gave Marek two pieces of information – nonverbal (giving the new sheet with more maps) and verbal. Marek was in a conflict situation. He did not know whether to say: “But I have solved the problem already!” or whether to comply with the supposed teacher’s expectation that he 241 should look for more ways. He chose the second option but asked himself what other ways to look for when he had found them all already. The conflict that arouse in Marek’s mind is formulated in the question Ma04. The experimenter could have reacted quite naturally by answering: “If possible no fine.” This would allow Marek to say that there were no other possible ways to find. However, the experimenter interpreted Marek’s question Ma04 as his need to have the word “fine” explained and thus she provided this explanation in Ex03. We think the source of misunderstanding is turn Ex02. The conflict arouse from insufficient information the experimenter had about Marek’s mental activity. On the one hand it was possible that Marek was sure his activity was finished and thus could not understand the challenge to continue looking for more ways. On the other hand Marek gave no hint of having concluded the task and thus the experimenter could not know whether this idea was present in Marek’s mind. This information vacuum needed some question or challenge that would clarify whether Marek was fully aware of completeness of his solution. The experimenter could have asked Marek how he would convince his friend that there was no other possible way. Conclusion The paper presents two examples of misunderstanding between a pupil/pupils and an experimenter-teacher. What we find important is that while conducting the experiment the communication interference or misunderstanding went unnoticed by us as experimenters. All became apparent only when looking for the answer: Why did the pupil use this particular word? In what sense was it used? What did he mean by saying ... ? Why did it take him so long to answer? How did he understand the question? Why did he answer in a different way than was expected? How did I interpret his question, his reaction to my words? etc. We came to realize the difficulties of a teacher who tries to provide space for meaningful class discussions as often as possible. If the teacher wants to direct the discussion well, to allow pupils to construct new knowledge in the discussion in a way that is most convenient for each of them, not to force on them their own ideas and ways of discovery, they must understand the discussion, must respect differences among individual pupils, must give pupils equal opportunities in the discussion and not select only those pupils whose perception of things is similar to the teacher and whom they understand most naturally both in cognitive and social aspects. This means the teacher should a) pay attention to the content of their statements, b) get feedback on how pupils interpret these statements, c) listen to pupils and interpret their statements, d) evaluate cognitive and communication phenomena, including non-verbal aspects (Kratochvílová and Swoboda 2002), e) choose suitable means of communication adequate to the pupils’ level, f) detect strategies of speakers. 242 The misunderstandings revealed in the first two illustrations came to us as a surprise but then made us conduct more experiments and analyses in which we were trying to learn to understand our pupil/pupils better, to look for the context in which the pupil is thinking and interpret their statements. The meaning of statements not only about mathematical concepts is connected to the idea of these concepts. This may result in a situation that what may seem meaningful to one person (albeit a teacher or a pupil) may seem to be nonsense to somebody else. A teacher learns about a pupil’s knowledge structure when the pupil interprets situations and selects strategies for solving a problem. In other words from the pupil’s mathematical and social behaviour. At this point we are in intensive cooperation with several primary teachers in two EU projects (EU project Teacher community support, development project Innovation), where the key activities are reflection of teachers’ work and analyses of video recordings from lessons. In our analyses we focus not only on getting insight into pupils’ thinking but primarily on teachers’ communication and its impact on pupils. We try to engage our teachers in developing sensitivity to their intervention own in communication, to occurrence of mistakes, in becoming aware of the fact that the only way to understand their pupils and to work with them efficiently is to give them space. We encourage our cooperating teachers to video record their lessons so that they keep a record of interesting episodes. 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STUDENT TEACHERS’ DIDACTICAL COMPETENCES IN MATHEMATICS Darina Jirotková and Jana Slezáková Abstract Didactical content knowledge is a concept discussed frequently in recent didactical literature, especially in the context of improvement of teacher education. From the wide research area the here presented study focuses on one of the twenty parameters that characterise a teacher’s educational style. The perspective on teacher’s didactical competences is narrowed to an important area of the teacher’s work with their pupil’s mistake. Four didactical mistakes are described and illustrated on examples from written tests of five student teachers. The study tries to uncover the causes and the consequences of the mistakes and to find such re-educational tool that will enable the teacher to overcome their mistakes. Keywords: Diagnostic tool, teacher’s educational style, teacher’s didactical competences, diagnosis of a pupil, pupil’s mistake, teacher’s didactic mistake, re- educational tool Introduction The issue of what a primary teacher should master from mathematics has been paid much attention in the community of mathematics educators and researchers in mathematics education. The issue of mathematical knowledge needed for teaching mathematics at primary school level was the topic of the international conference SEMT’11 held at Faculty of Education, Charles University in Prague. The plenary lecture of S. Vinner, published in the proceedings of the conference (Vinner, 2011), expresses the belief that primary teacher training all over the world should pay due attention both to mathematical content knowledge and pedagogical content knowledge. Vinner recommends that the level of mathematical content should not surpass the students’ or teachers’ intellectual Charles University, Czech Republic: [email protected], [email protected] 245 abilities, i.e. the topics taught should be in their zone of proximal development (ZPD, Vygotsky, 1986). In full compliance with these ideas the team at the Department of Mathematics and Didactics of Mathematics at Faculty of Education, Charles University in Prague lead by M. Hejný created a concept of courses of mathematics and didactics of mathematics for student-teachers both in full-time and combined form of studies as well as in in-service teacher training. These courses are designed on the concept of scheme-oriented education (Hejný, 2012a; Kvasz, 2016). After long time of experiments, the concept of scheme- oriented education was fully developed even in the set of textbooks, workbooks and teachers’ books for 1st to 5th grades of primary school by the authors Hejný, Jirotková, Slezáková et al. published by Fraus. The here presented paper focuses on a narrower area of a teacher’s didactical competence, namely on the teacher’s work with a pupil’s mistake. To this topic we paid attention also in (Jirotková, 2012b). Further for the purpose of this paper, the word teacher refers to a student teacher finishing the course of Didactics of Mathematics, thus concluding their education in mathematics and didactics of mathematics. Having completed this course, student teachers meet mathematics only within their teaching practice at schools. This teaching practice, however, is in most cases supervised by mentors from the Department of Primary Education. Tool for diagnosing a teacher’s educational style In his study, M. Hejný (2012a) proposes the conception of creating a diagnostic tool allowing us to characterize a teacher’s educational style in mathematics. The diagnostic tool was further developed by Jirotková (2012a). She also illustrates the use of the tool on three case studies. The tool consists of twenty parameters divided into four areas: A. A teacher’s beliefs 1. Relationship to mathematics 2. Goals of teaching of mathematics 3. Educational style (transmission versus constructivism1) 4. Interactional style towards the pupil, to the class (from attitudinal to dialogical) and possibly their projection into the integration of the pupil with people in his/her environment (classmates, parents, grandparents, etc.) 5. Interactional style towards colleagues, school government, inspection 6. A need to develop his/her competence 1 In the sense of Noddings (1990). 246 B. Life experience as a springboard for the teacher’s pedagogical beliefs 1. where it comes from 2. what it concerns 3. what is missing 4. how it is reflected (analysed) 5. which experience resulted in the shift of the teacher’s beliefs C. Personality 1. Self-confidence in the area of pedagogy 2. Self-confidence in the area of didactic 3. Self-confidence in the area of mathematics 4. Self-confidence in the social area (towards colleagues, school management, inspection, parents) 5. Assessment of one’s educational style (does it correspond to reality?) D. Abilities / competence 1. Pedagogical: management of class discussion, organisation of work, individualisation, creating and maintaining the climate, work with mistakes, patience, etc. 2. Didactic: pupils’ motivation, conception of the ontogeny of concepts, relationships, processes, languages, problem, diagnosis of pupils (understanding their ideas), evaluation of pupils, re-education, etc. 3. Mathematical: knowledge of solving strategies of various types of problems, ability to experiment, to effectively use the trial error method, ability to create generic models both procedural and conceptual, posing problem with required characteristics, etc. 4. Social: interaction with colleagues, management of the school, inspection and parents. This paper focuses only on one of the twenty above presented parameters, namely parameter D2 – the teacher’s didactical abilities/competences. Methodology The form of the exam sat by students attending the course of Didactics of Mathematics at Faculty of Education, Charles University in Prague corresponds to the way students work in the course. One part of the exam is a test in which students prove their level of pedagogical content knowledge (Shulman, 1986), or more precisely didactical content knowing (Cochran, DeRuiter and King, 1993 247 according to Janík, 2007). The test requires from the student some of the following activities: - solution of a described didactical situation, - teacher’s reaction to a pupil’s mistake, - determination of the most likely cause of a pupil’s mistake, - plan of a re-educational process related to the given mistake, - design of a scenario for teaching a selected concept, relation, or procedure, - plan of a differentiated approach to describe types of pupils in a given situation, - description of their own experience related to teaching mathematics, - comments on and analysis of this experience. The written solutions of these tests make an important research database. We started archiving these tests more than ten years ago and now the archive features more than 200 items. Other materials that form the basis for a student’s evaluation and that are part of the database are students’ seminar works. In these seminar works students are asked e.g. to describe and analyse a conducted experiment with pupils, or to describe a teaching episode in which something important with respect to learning mathematics happened, or to describe and analyse video recordings of many lessons (these video recordings are either made by the student teachers or made during their teaching practice), episodes from their diploma thesis or other texts. This study presents four teachers’ didactical mistakes. The teacher - reads the pupil’s solution incorrectly, - is not able to diagnose the pupil’s mistake effectively, - formulates the problem unclearly, - modifies a good problem inappropriately. All these four items are illustrated on a solution of one problem by five student teachers. The analysis was conducted by comparing a number of student solutions. These solutions were classified into groups of solutions that are characterized by the same phenomenon. The method was based on grounded theory (Strauss and Corbinová, 1999). The analysis of students’ solutions begins by describing the phenomenon “the teacher’s didactical mistake”, then sources of this mistake are sought, possible consequences of this mistake described and finally possible re- educational tools discussed, i.e. the possible ways the teacher can overcome these mistakes. Illustration In the final test student teachers were solving the following problem: 248 A 5th grader solved the following problem incorrectly: The house originally cost 2.17 million crowns. Then the price was reduced by 1/7, but after its roof was repaired, the price put up by 140 000 crowns. What is the price of the house now? The pupil’s solution: 140 000 CZK + 2 170 000 CZK = 2 310 000 CZK. The reduction is 2 310 000 ÷ 7 = 330 000. Current price is 2 170 000 – 330 000 = 1.84 million crowns. a) Show the shortcomings in the pupil’s solution and write the correct solution. b) Find the possible source of the pupil’s incorrect procedure and pose a re- educational problem. c) Write a cascade of 3 re-educational problems with a comment. The teacher reads the pupil’s solution incorrectly The cause of incorrect reading of a pupil’s solution is either teacher’s lax reading but more often a graphically disorganized pupil’s written record. The consequences of this phenomenon can be serious. This depends on the way the pupil’s solution is corrected. If pupils get the chance to learn what the teacher considers to be a mistake, they are given the chance to discuss and the teacher’s mistake is clarified. If pupils do not see the corrected and marked tests, the unfortunate consequence tends to be pupils’ worsening of their attitude to mathematics. They may feel wronged and moreover do not know what mistake they made, which blocks any chance for remedy. A teacher’s wrong reading of a pupil’s solution may also cause social problems. Moreover, a teacher’s failure to grasp a pupil’s solution may result in an unsuitable re-educational teacher’s intervention. The re-educational process in case of this teacher’s mistake largely depends on the didactical contract between the class and the teacher. If the teacher-class interaction is dialogical, the pupil is encouraged to discuss all possible misunderstandings with the teacher and thus they are cleared away. In case of an authoritarian teacher re-education is problematic. Illustration Iva Iva solved the problem herself correctly but misread the pupil’s solution. Iva writes: “ … the pupils calculated the reduction from the assigned 2.17 mil. CZK, subtracted it from the original price and did not add the increase of the price after reduction.” However, that is not true. The pupil did not forget to add the increase in price, but both operations were carried out in the reverse order. The source of this misunderstanding is Iva’s lax reading. The consequence is a triplet of “re- educational” problems that fail to have the re-educational function for this particular pupil. 249 The teacher is not able to diagnose the pupil’s mistake effectively In everyday teaching we rarely came across teachers’ need to diagnose their pupils’ mistakes or shortcomings. Teachers are more likely to focus on occurrence of these mistakes and on getting rid of them, e.g. by giving advice (or command): “You must study harder, you must practice more.” However, this does not help the erring pupil find out why they have made the mistake. Then pupils do not know how they should learn and practice the particular content in order to avoid a similar mistake in the future. The teacher fails to take into consideration the variety in pupils’ thinking, variety in their individual needs. They approach all pupils as if they were the same. Let us ask here why a teacher should not feel the need to diagnose their pupils’ mistakes. We believe that teachers do not even realize this diagnosis is possible. Their belief is they cannot see into their pupils’ minds and if they were to analyse solutions of all thirty pupils in their class they would be doing nothing else. Moreover, teachers do not think this kind of work is effective. Our experience shows that this teachers’ attitude can be changed if a similar diagnostic intervention is done with biological children of the teacher. However, we also come across teachers who believe in usefulness of diagnosis of pupils’ mistakes but lacks knowledge of diagnostics and does lacks the needed diagnostic tools. A teacher is able to diagnose better if they have diagnostic tools available. For example if they know the concept of antisignal in word problems2 and its diagnostic manifestation, they are able to diagnose and name the mistake the pupil makes and to propose efficient re-education. In this case it is dramatization of the word problem. If the teacher does not know appropriate diagnostic tools, they must rely on their mathematical knowledge perceived in the polarity correct – incorrect. They diagnose prosthetically, i.e. they interpret their pupils’ mistakes through the prism of their own personal experience with the given or similar mistakes. Based on this prosthetic diagnostics the teacher poses problems for their pupils that can hardly be effective in re-education. A creative teacher often targets their creativity not on the core of the pupil’s mathematical shortcomings (they have failed to diagnose it) but on the context of the problem or its semantics. However, it is not impossible that the teacher, intuitively, not deliberately, uses some elements that prove to be truly re-educational. We know a number of 2 By an antisignal we understand such a word in a word problem that suggests a particular operation despite the fact that the correct solution of the problem requires the inverse operation. For example: There were cakes on a plate. I ate three and now there are five. How many cakes were there on the plate at the beginning? The word “ate” signals subtraction but this problems is solved by addition 3 + 5 = 8. 250 teachers who are successful in re-education and are able to pose efficient problems without actually being able to describe the pupil’s diagnosis. In consequence to wrong diagnostics of the teacher the pupil is given problems that are not re-educational, since they have no relation to the mistake. The teacher is likely to realize their mistake as soon as they watch the pupil’s solution of the assigned problems. Then the teacher sees the pupil does not have this shortcoming. Illustration Hanka Hanka solves the problem correctly. And she writes the following about the shortcomings in the pupil’s solution: “…. he failed to realize that the value 1/7 (or any other fraction) differs with respect to the size of the whole; also he forgot that he calculated 1/7 from the price of the house with the repaired roof and subtracted that from the original price of the house thus here he realized that 1/7 should be subtracted from 2.17 million crowns.” Her not really clear explanation of the pupil’s mistake points, though not explicitly, at the pupil’s rash addition of the sum 140 000 CZK. Then Hanka proposes a series of re-educational problems. These include problems of the type: Colour 1/2 of the given bar of chocolate. This implies that Hanka’s diagnosis of the pupil’s thinking is inaccurate. She fails to see the pupil’s mistake in confusing the operations. Instead she attributes it to the fact that the pupil did not realize one seventh depends on the whole from which it is taken. The cause for this inaccurate diagnostics seems to be in the phenomenon of wrong attribution: it seems she perceives the pupil’s solution through the prism of her own mathematical experience or of her didactical experience with a pupil who failed to grasp the relation between the whole and its part. Hanka thus interprets the cause of the pupil’s mistake based on wrong attribution. This is often the case with teachers who used to have problems with fractions or even still have them. Let us illustrate this on two episodes: 1. Jana, one of the solvers, presents the following re-educational problem for our hypothetical pupil: The original price of the house was 2 170 000 CZK. How many sevenths is this? The question seems to be nonsensical. Jana explains her didactical intention in the follow-up interview: “If the pupil realizes that 2 170 000 is seven sevenths, they will get a clearer idea what one seventh is.” Also in this case the pupil’s mistake is diagnosed wrongly because the pupil works with the fraction 1/7 with understanding. The proposed re-educational problem comes out of the solver’s personal experience. She used to have a lot of difficulties in grasping the fraction one seventh and it helped her considerably when she realized that a whole is seven sevenths. 251 2. Věra, another solver, posed a series of three graded re-educational problems as follows: - In the first problem the pupils should calculate one half of the price of a book (50 CZK). But this is not the particular pupil’s problem. On the contrary he is able to calculate 1/7 of a large number. Thus the problem is not re-educational. - The second problem is flooded with a lot of irrelevant information and is set in a flowery story. Intuitively, Věra sets each operation into a different temporal story, which is good, as it makes it harder to confuse the order of operations. In other words the situation cannot be perceived as a concept. It helps the pupil to grasp the temporal sequence of two events. - The third problem has again a very complex plot and has no relationship to the pupil’s problem, as it fails to work with importance of order of the carried out operations. The author’s creativity is channeled into the context of the story, not the mathematical core. The teacher formulates the problem unclearly Inaccurate formulation sometimes does not become the source of misunderstanding. In that case there is nothing to worry about. What we have to pay attention to is such unclarity of formulation that causes misunderstanding. In that case the contract teacher-pupil becomes crucially important. This contract determines how this misunderstanding is handled. An authoritarian approach is the least adequate one from didactical point of view: “A formulation must be understood as it was intended by the teacher.” What happens to pupils in this case is that instead of trying to make each other understand, they try to comply with the teacher. This slows down not only communicative but also cognitive development of the pupil. The approach that seems to be most appropriate is a dialogical approach of the teacher to the phenomenon of misunderstanding. What happens is that first the cause of misunderstanding is pinpointed in the dialogue and then misunderstanding clarified. This accelerates communicative and cognitive development of the pupil. What really supports the pupil’s development is if the teacher speaks highly of the pupil’s discovery of an alternative interpretation of the text. Formulation of one owns ideas is a very demanding activity. A teacher who is deficient in this aspect will have to work on improvement systematically and for a long time. First of all they have to read the same text several times and even after some period of time. They should also ask colleagues or a friend to read the text critically. Finally, and that is most important, they test the text on pupils. Any misunderstanding they come across should be analysed with the goal of making the text clearer and more comprehensible. 252 All of us sometimes make the mistake of formulating ambiguously. In a dialogue these mistakes are cleared away spontaneously. It is much harder in case of written communication. When posing problems, especially those used for assessment of pupils, we must try to be as precise and unambiguous as possible. Experience shows this is very demanding. Even so carefully posed problems as problems for mathematical competitions or TIMSS studies are sometimes unclear or ambiguous. Illustration Bára Bára also solved the problem correctly. She was even able to describe the pupil’s mistake correctly. She then presented a cascade of well graded re-educational problems. However, she concluded this by one extra problem: “A fully furnished house costs 2 million crowns. Without furniture it costs 140 000 less. If it is not decorated, it costs 1/7 less. How much does the house cost without furniture and without being decorated?” In this problems we do not know if the price reduction of 1/7 is taken from the sum 2 millions, or the sum 1 860 000. This mistake may have two causes. The first might be communicative. When posing the problem the student knows exactly what she means and forgets to express relevant data in the formulation. To be more precise the idea of the communicated phenomenon or situation in the communicator’s mind is much richer and more complex than the product of its articulation. The other might be unclarity of the idea that she is trying to convey. Sometimes we fail to realize that our idea is unclear, we are convinced we see things clearly and unambiguously. The teacher modifies a good problem inappropriately When an inappropriately formulated problem is solved in a classroom during frontal teaching, all ambiguities are cleared away fast. If the problem is solved individually, pupils who are trying to solve it waste time and energy. If the teacher-class interaction is dialogical, this loss is not significant. If the teacher solves the problem before the lesson, this mistake is avoided. The solution of Bára’s problem from previous section is in neither of the cases a whole number (if rounded to whole crowns it is 285 714 and 12 286 CZK). The solution is thus semantically questionable and inaccurate for calculating). The author did not realize this as she did not try to solve the problem. Conclusion In the paper we focused on teachers’ didactical competences particularly on the competence to effectively work with pupil’s mistake. We illustrated four kind of teacher´s didactical mistakes in their reaction to pupil’s mistake. Analysis of other problems from the test showed other teacher’s didactical mistakes: for example 253 the teacher poses a problem that is not associated with the given mistake, the teacher selects for their problem a context unknown to the pupils. However, the scope of this paper does not allow us to discuss these. Diagnostics of a pupil’s mistake is an important and difficult part of teachers’ work. To what extent should teachers be educated in this area and how can they be trained to develop their experience into good intuition that will help them considerably in this area? These are questions that we focus on both in teacher training and in the current research project ´Inovace´. Acknowledgement The research was supported by the research project PROGRES Q17 Teacher preparation and teaching profession in the context of science and research. References Cochran, K. F., De Ruiter, D. A., King, R. A. (1993). Pedagogical Content Knowing: An integrative model for teacher preparation. Journal of Teacher Education, 44(4), 263– 272. Hejný, M. (2012a). Exploring the cognitive dimension of teaching mathematics through scheme-oriented approach to education. Orbis Scholae, 6(2), 41– 55. Hejný, M. (2012b). Pedagogické schopnosti učitele v matematice – příběh. In J. Kohnová et al. (Eds.), Profesní rozvoj učitelů a cíle školního vzdělávání (pp. 245– 252). Prague: Charles University, Faculty of Education. Janík, T. et al. (2007). Pedagogical content knowledge nebo didaktická znalost obsahu? Brno: Paido. Jirotková, D. (2012a). Tool for diagnosing the teacher’s educational style in mathematics. Orbis scholae, 6(2), 69– 83. Jirotková, D. (2012b). Didaktické schopnosti učitele v matematice. In J. Kohnová et al. (Eds.), Profesní rozvoj učitelů a cíle školního vzdělávání (pp. 253–260). Prague: Charles University, Faculty of Education. Kvasz, L. (2016). Princípy genetického konštruktivizmu, Orbis scholae, 10(2), 15– 45. Noddings, N. (1990). Constructivism i mathematics education. Journal for Research in Mathematics Education, 4, 7–18. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 2(15), 4–14. Strauss, A., Corbinová, J. (1999). Základy kvalitativního výzkumu: postupy a techniky metody zakotvené teorie (1st Ed). Boskovice: Albert. Vinner, S. (2011). What Should We Expect from Somebody Who Teaches Mathematics in Elementary Schools? Scientia in Educatione, 2(2), 3–21. Vygotskij, L. S. (1970). Myšlení a řeč. (Transl. J. Průcha.). Praha: SPN. 254 DIVERSITY OF RESULTS IN RESEARCH IN THE DOMAIN OF PRE-SCHOOL MATHEMATICS AT KINDERGARTEN Michaela Kaslová Abstract Research guides the training of teachers and the development of teaching aids. How do changes in our society influence the interpretation of these research results? Are the results of the research independent of these changes or are there results that remain constant? Are the diversity of results and the differences between obtained data constant or not? In order to be able to answer those questions, we made a retro-analysis of data obtained in long term research and in repeated research. For this contribution we chose three groups of results which focused on the diversity in the child´s strategies in the solution process and on the diversity in the results of the child´s activity. We used data obtained in different years encompassing 300 – 960 children. In formulating the conclusion we added a new dimension, that of time. Keywords: Labyrinth, configuration, model of natural number, retro-analysis Background research During the process of pre-school teacher training in the didactics of pre- mathematics at Charles University we use the research results from different domains such as pre-mathematics, cognitive and evolutional psychology, neurology, native language etc. The system of grants policy supports short term research, usually three to five years. Long term research offering a description of changing or constant phenomena in pre-school education is rare and in the area of pre-mathematics does not exist. For this reason, I decided to repeat three completed research studies, using the same methodology, to obtain more data. In the third case I compared the results with a part of a national test (the author of this test – Kaslová) where I modified the task in one detail (material) but used the same methodology. Each part of this research, ranging over twenty years, was analysed and published separately. Now we have returned to different groups of results with the aim of comparing the results and of observing the measure of their stability. In several research contributions we can register a similarity of theme and a similarity in data (kindergarten context: in the domain of natural number, orientation in a space, structure of whole etc.), but the point of view changes to give us a new information. The facts can be outside of the scientific world and could be in the socio-cultural and political domains (for example Ernest, 2004; Valero, 2004; Lerman and Zevenbergen, 2004). The research is usually financially supported by the state, by the university, but indirectly by the actual philosophy of education and educational goals and aims. Analysis of the official materials (not only curricula) shows the metamorphoses which are stimulated by research or they Charles University, Czech Republic: [email protected] 255 self-stimulate our research. This situation offers for the researchers a large spectrum of methods which correspond with new demands, but on other hand it limits researchers, for example, in the realisation of broad-spectrum research which is regularly repeated in certain periods. If we are not satisfied with methods used by teachers in practice, we observe the school practice, we modify teacher training, (Chapmen, 2014; Fleming, 1983 et al.), we complete relatively limited research projects, we exchange our results, we publish them, we apply the conclusions in didactical recommendations, we organise post-university courses for practice etc. The majority of actual researches have a horizontal character – this means that we are focused on several aims in a limited time (day/months/year). In the school domain there are some researches in mathematics education (international PISA, TIMS, in the Czech Republic CERMAT, 2006), which have a vertical character and they can detect a certain stability or special changes in pupils´ abilities. This type of research doesn’t exist in pre-mathematics yet. By comparing a number of researches in the domain of school and pre-school didactics of pre-/mathematics and by analysing the official materials in the pre-school domain, we can recognise the development of aims, of curricula, instruments, forms and methods of teaching in kindergartens. They have a more general character. Do we know the changes in school practice exactly in different generations/groups of preschool children? We cannot do this without the characteristics of didactical phenomena, which help us to develop new parts of pre-mathematical literacy. Systematic research work in pre-school education creates not only a sound basis for pre-school teacher training as well as for primary school teacher training, but they can help us to look for eventual reasons of occurrence of certain phenomena and finally to predict some tendencies. Methodology, aims The situation mentioned above represents a starting-point for a description of our aims. The main aim is to find out whether the diversity of the child´s reactions is stable, or whether it changes for the children of the same age as the years go by. Research models in the field of education are described either as a series of steps or as planar models (2D) or spatial models (3D). Figure 1: The Scheme of dynamic research model For example, see Carr and Kemmis, 1986; Crawford and Adler, 1996; O’Brien, 1998; Fitzsimons, Cohen and O’Donoghue, 2003; Brown, 2013. Most of the models are not universal enough. Planar models are not suitable for my analysis, which takes place over the period of time that I monitor. I had to reject spatial 256 models completely as they neither allow the situation to be examined from different angles nor to vary the results depending on the chosen constants and variables at any given time. In my opinion, the ETM model (Kuzniak and Richard, 2014) was the best approximation but even this model had, according to my view, limiting factors which reduced its effectiveness. Therefore, I formulated my own model for this article, based on the research theory by Chrástka (2007). This dynamic model (scheme No 1) can change the proportions. The principal focus in our research is represented by the blue axis which passes through several levels representing different years. I approached this from the possibilities offered by axial coding (for example Strauss & Corbin, 1998). The axes of other colors represent possible points of view to the special problem, a circle (its diameter could be change in relation with the observed “area”) also contains socio-cultural aspects. The principal method used is a retro-analysis of three series of completed researches or of their parts, which were repeated in different periods and used the same methodology. All mentioned researches were analyzed separately and concurrently, their analysis was complete and combined quantitative and qualitative approaches. All this research observed the child´s reactions to the special tasks during the last school year at kindergarten; i.e. children aged from five to six and a half years. To realize the retro-analysis it was necessary to reduce and select data received in separate “floors” of the model. The character of the applied analysis is vertical. I have concentrated on a greater number of themes in the course of my repeated experiments. The three topics that are presented here are those which allowed me to formulate conclusions based on experiments which could be repeated over a longer period of time and where I had access to larger samples. The principal Task is the use of new approach “retro-analysis” based on a new research model. This model permits to discover the constant and the variability of the chosen phenomena in the flow of time. Researches Research No 1 named Labyrinth (Kaslová, 2003) was focused on the independent strategiesof solution of the shape 2D labyrinth. It was a drawing in black colour on white paper (A4). Each child worked individually at their kindergarten and could choose one colour for the following work. The work time was not limited. If a child asked for a different colour pastel /marker pen, or another sheet of paper I gave it to him/her. Each child solved 5 different labyrinths. At the end of this activity the child could swap roles with me and draw a labyrinth for me to solve. This activity was realised three times in kindergartens in Prague, in four other town sand in four villages (1994; 1999; 2006). We worked with 960 children aged from 5 to 6 (see Table 1). This research was supported by any grants. The results obtained in the first phase were compared with results obtained by L. Černá in her thesis (2016). There are no important differences. 257 1) In the first phase (1994: 320 children) from a total of 1600 of which 1% of labyrinths were not solved, 8% of labyrinths were solved erroneously and 50% (800) with good correction. I identified three principal types of correction: a) a new way (a majority of the children used a new sheet of paper; a minority retained the original sheet but used a new colour); b) turning/kink; c) T- correction – the child “jumps” back to the previous crossing. Figure 2 and 3: The Corrections types: b), c) © Kaslova 2) In the second phase (1999: 320 children; 1600 sheets) we received 1546 completed sheets. The good correction a) new way: a majority of the children used a new sheet of paper; a minority retained the original sheet but used a new colour. The children only used a colour felt pen. In addition, we analysed the number of “stops” during the solution process and the distance between them. This method allowed us to describe the perception of the labyrinth as a structured whole. 3) In the third phase (2006: 320 children; 1600 sheets) we received 1552 completed sheets. The correction a) new way: a minority used a new paper; a majority retained the original sheet but used a new colour; sometime accompanied by scratch-outs. The children only used colour felt pens. We analysed in addition the number of “stops” during the solution process and the distance between them. We did not register a significant change in the perception of a structured whole. finished with with with good a) New b) Kik/ c) T- error success correction way Turning correction Phase 1 99 % 8% 41% 50% 9% 31% 10% Phase 2 96 % 9% 40% 47% 8% 29% 10% Phase 3 97 % 10% 38% 52% 12% 31% 9% Table 1: Labyrinth solution and corrections strategies Comparing this three test analysis we can suggest: a) that the success level is relatively constant; b) that the number of variability in the strategies of correction is fundamentally unchanged, and c) that the proportions using these strategies varies. Slide augmentation of the number of unfinished labyrinths could be interpreted as a product of an easing in requirements to the child´s reactions; children have no fear of bugs; children are more open. If a child in the first phase asked for a new piece of paper, the reason could be because their teacher refused 258 work with mistakes. After the year 2000 a new period starts; supporting new strategies in experimentation. The exploratory procedure is stressed as a new approach in kindergarten after the year 2000. The possibility of using cut-outs changed the style of teaching and probably positively influenced the number of finished and successfully corrected labyrinths. Research No 2 named Configuration (Kaslová, 2003, 2010; 2015; Nováková, 2009). This activity was a part of a complex of diagnostic activities. In the section named “Configuration” we observed the child´s ability to modify/change the representation of the number six (⁝⁝). Children worked individually drawing each new configuration on new paper. Figure 4: New configuration; ©Kaslová 1) In the pilot research (1994; 20 children) we assumed that child would use the strategy of modification (change of position of only one point), but children used this method less frequently than we had assumed. We assumed that the child would be able to create usually 4 or 5 “new pictures”. The results: The minimum was 6 “new pictures” and the maximum 14 “new pictures” (1 child). In the first phase (1995-1998; 280 children) we detected 6 different strategies: a) Mb – basic modification changing the position of one point when comparing the new configuration with the original configuration; this strategy was used rarely and not systematically in 4% of the children; b) Si– similar configuration (smaller or bigger than “initial configuration”) 75% of the children at least one; c) Cn – new configuration (as a new drawing totally different from the original or initial configuration) used in the majority of their “pictures”; d) Rk – reproduction of a precedent configuration in the same position e) Rt – reproduction of an original configuration combined with translation in the plane of the square; f) Sc - similar configuration (smaller or bigger than an original configuration). Cn - represents the dominant strategy. At least 4 and a maximum of 28 “pictures” (one child). Ri is not consider as a strategy, it is the copying of the “initial configuration”. The process of correction using the stroke of the pen was not detected, but correction using the addition of point(s) was detected in 5 cases. 2) The second phase (2000-2010; 300 children); The situation changed and we detected 3 new strategies: g) Mp – partial modification born in the process of decomposition of the original configuration in two parts, child changes a position of one part; h) Rr – reproduction of an original or initial configuration but the child rotates drawing on the square; i) Cf – new configuration arises by an alteration of the original configuration. Strategy Cn is the most used. In the second place is strategy Mp; the proportion between other strategies is not as important as in the first phase. We noticed new phenomena: 12% of the children spontaneously used self-correction when they used more than 6 points. The 259 number of “pictures” made by one child oscillated between a minimum 5 different configurations and maximum 72. Distribution of “product” number corresponds with Gauss´s curve. The majority of children produced between 12 and 24 “pictures”. The group of children which drew more than 25 “pictures” increased. 3) The third phase (2011 – 2014; 300 children) showed (see table 2) similar results though the proportion between strategies varied a little bit. Three children were unable to create any new configuration (children with specific needs in the process of learning). We discovered new “drawings”. In Cf type there was a new element: 6 disarranged points. Now we distinguish Cfs with evident system of points and Cfd drawings without any system. It is possible to distinguish new differences in the character of configurations: A) the configuration is based on the rectangular or triangular system; B) the configuration is based on the use of curves (usually all points are near one another); C) the configuration change to a disarranged set of points. This difference in phases 1) and 2) was not so visible, because the characteristic B) and C) did not occur so frequently. The number in the minimum group is 4 (in 3 cases) and 6 – 8 (in 15 cases), with a maximum 54. The difference between results (see Table 2) in the phase 1) and 2), 3) corresponds to changes in the teaching style at kindergartens. Comparing phase 1 and phase 3 we can argue that over twenty years the diversity and the variability of child´s reactions changed and increased. One of the children used in his/her work at least 2-3 strategies and in maximum 5 to 8 strategies (see 12th column). Strategy Rr Rt Rk Cn Cf Cfs Cfd Mb Mp Si Sc min /max Phase 1 30 52 100 4 75 21 3/5 Phase 2 21 25 40 100 14 3 74 72 20 2 /6 Phase 3 18 28 41 98 20 8 3 90 60 27 2/8 Table 2: Using of strategies Research No3 named Basket (Chvál et al. 2015, Kaslová, 2010, Chrástka, 2007). This activity was a part of a series of complex diagnostic activities (author of the national test Kaslová). It was focused on a child´s ability to choose six or seven objects from a basket. In the preparation phase I tested this activity with different materials 1994 - 1996 (among others, Kropáčková’s thesis). Finally we put together a selection of small objects (suitable for manipulation): 2 small blue and white cars, 1 white pink-pong ball, 1 plastic giraffe, 1 plastic cow,1 small plastic cup, 1 plastic dinosaur, 1 pencil, 1 rubber, 2 marbles(yellow and red),2 red wood cubes, 2 small pictures on a square paper, 3 colour felt pens, 260 3 pegs, 3 dies, 4 beads (red, blue, white, yellow).The first observation showed that 6 children from 20 (of the same kindergarten) had a problem based on a manipulative method to select 6 or 7 objects from this basket. This means that the choice (in my research – MK) was not independent of the character of the chosen objects. For this reason we did not change the set of objects. The next observation was realised in three steps: 1) Years 1994-1996 (360 children); 2) Years 2000- 2010 (300 children); 4) Years 2014–2017 (300 children). Step 3 represents the National Test (NT – 800 children); data: see Table 3. 1) In the first phase 72% of children (259 from 360) correctly and quickly chose 6 or 7 objects from our basket; 8% of children (29) asked me if it is possible or not to use other objects, but only 8 from 29 children succeeded; this means that in the end 74% of the children mastered this task even though everybody counted with a success of at least from 1 to 10 on minimum. More than 50% of unsuccessful children argued that it was not possible to select 6 or 7 objects because of the differences between offered objects (it is not possible). 2) In second phase 81% of children (243 from 300) reacted well immediately. If we summarise the success, we can see that 83% of children answered well, but 16% had important difficulties and asked for help or finished this activity without a satisfactory result. 3) I prepared the national test (NT) with colleague M. Chvál from the Institute of Research and Development of Education at Charles University – Faculty of Education. The national test was conducted in different parts of the country, on 800 children (2014). The national test contained one task (similar in part with activity Basket) and researchers used a collection of pieces of paper – various colours and shapes and a maximum of 4 pieces n the collection were identical. 4) In third phase (300 children) Comparing results of the national test in this task with results (see table 3) obtained in the first and in the second phase (problem Basket) I decided to carry out the third phase to be able to compare actual results (Basket) with results obtained in the national test (set of papers). The difference between results of my research (MK) and the national research is important. 24% of children in their first reaction claimed that it was not possible to select 7 objects from basket (in the NT only10 % in the first step). 19% of children had no success in the second step (after a short dialogue), the arguments were similar as in phase 1 and 2. If we offered them the set of papers (collection used in the NT) half of them chose 7 objects correctly. In the final semi-directive dialogue, the children stressed that there were only pieces of paper (set of papers) but the objects in the basket differ. The colour and the shape of paper pieces did not play an important role for the children; it was the material and dimension which for them represented the common characteristic. 261 Not possible Correct Correct Not possible - immediately reaction reaction - after - after a immediately a discussion discussion Phase 1 MK 20% 72% 2% 6% Phase 2 MK 13% 81% 3% 3% Phase 3 NT 2% 90% 8% 0% Phase 4 MK 19% 74% 2% 5% Table 3: Selection of six/seven objects The failure in the second and the third part was not distributed equally. This fact offers one simple interpretation that the success in “basket” was not only conditioned by the child’s development but also by the teacher´s work as well. Conclusions If we speak about diversity, we are focused usually in a horizontal dimension: (one direction – rational- is focused on the child’s solution process or in results of activities, second - direction in the same plane is socio-emotional – the teacher, other children, parents, cooperation, atmosphere, communication), vertical dimension is connected with time and the development process. The analysis of separate parts offered to using the diversity in horizontal meaning and in its complexity, makes it possible for us to use a profound qualitative approach, more so than the vertical analysis, which goes through three parts as a whole and includes in the majority cases quantitative data in combination with chosen qualitative phenomena. The retro-analysis presents the diversity and metamorphosis over a relatively long period. This obtained selected data cannot be used for the profound qualitative analysis in this moment. Partial conclusions offer the material for a new research, for eventual changes of our curricula or of didactical recommendation. The part Labyrinth in this research detects the constant character of correctional strategy. The proportion between three principal strategies oscillates, but there are new tendencies in the level of sub-strategies. The using of T-strategy is constant too in all kindergartens, but the proportion between other strategies depends on the local conditions in kindergartens. This means that first two strategies used by the children are influenced to more by their environment and/or by their teachers. 262 The Configuration part in this research presents the metamorphosis over time in the domain of “mistake-correction” in kindergarten and a change from the direct transmissive style to constructivist teacher strategies. In addition, the variability of a child work increases. The Basket part of this research presents an interesting observation: the minor variation of used materiel used in the research can cover important phenomena in practice and can influence the pre-school teacher´s focus on the process of stimulating the development of the concept of number. We can discuss other interpretations but in this moment only on the level of hypothesis. In this contribution I used a reduced horizontal approach by the selection of data and I stressed the vertical approach. The use of retro-analysis also revealed several interesting facts. I believe that this research can give new insights to aid teacher training as well as teaching practice in kindergartens. References Brown, J. P. (2013). Inducting Year 6 Students into „A Culture of Mathematising as a Practice“. In G. A. Stillman, G. Kaiser (Eds.), Teaching Mathematical Modelling: Connecting to Research and Practice (pp. 295–305). Dordrecht: Springer. Carr, W., Kemmis, S. (1986). 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In Journal Research in Science Teaching, 20(5), 419–435. 264 EQUITABLE LEARNING OPPORTUNITIES: TEXTBOOK LANGUAGE ACCESSIBILITY IN ENGLISH MEDIUM CLASSES Lisa Kasmer,Danielle Harrison and Esther Billings Abstract In this paper we present preliminary findings regarding the accessibility of mathematics textbooks written in English in order to consider how textbooks might impact the mathematical learning experience of multi-lingual learners in Tanzania where the medium of instruction in elementary schools is English. Since mathematics textbooks are viewed as an authority by teachers and students within the classroom, it is important to investigate textbooks use in mathematics classrooms to ensure these materials are accurate, clear, and accessible. Instances of improper grammar, mathematical inaccuracies, or unclear instructions are prevalent in our examination of elementary mathematics textbooks. Implications for learning and equity are discussed. Keywords: Textbooks, multi-lingual learners, equity INTRODUCTION AND RELATED LITERATURE In mathematics classrooms across the globe, multilingualism is widespread, as many students attempt to move between the language spoken at home and the language spoken in school. Learning mathematics in a multi-lingual classroom increases the demands on students as they simultaneously navigate mathematical content while negotiating the nuances of language different from their mother tongue. We claim that mathematically accurate and high quality textbooks are essential for mathematical learning and equity in the multi-lingual classroom; research establishes that the textbooks teachers use in their classrooms impact the learning outcomes of their students (Grouws, Smith and Sztajn, 2004). In many countries, teachers rely heavily on the use of mathematics textbooks for instruction (Hiebert et al., 2003). In addition, most mathematics teachers use textbooks as the authority in their classroom, meaning the textbook becomes a script rather than a resource (Roth McDuffie and Mather, 2009). Therefore, it is important to carefully analyze the textbooks employed in multi-lingual mathematics classrooms to ensure these materials are accurate and accessible to English Language Learners (ELLs) in order to provide equitable learning opportunities for all students. With this is mind, this preliminary study analyzes various mathematics textbooks used in elementary schools in Tanzania, and considers which key design features of a textbook identified by O’Keeffe and Donoghue (2015) are present in these textbooks. There is an erroneous acceptance that ELLs do not have difficulty learning mathematics, since mathematics is numerical. Mathematics can be perceived as a subject that does not require a strong command of language since symbolic and numeric representations are often used; however language is closely linked with Grand Valley State University, USA; e-mail: [email protected] 265 mathematical reasoning and problem-solving and both rely on understanding math vocabulary (Roberts, 2009). Mathematics is especially language dependent; there is content-specific language that students must learn in order to become successful in mathematics (Langeness, 2011). Consequently, ELLs face a triple challenge in learning mathematics as they simultaneously acquire (1) “everyday” English, (2) the English of mathematics, and (3) new mathematics content (Lager, 2004 p. 2). If ELLs (or their teachers) do not have a firm grasp of English, understanding the mathematical content can be problematic. ELLs struggle with acquisition of mathematical content because they are both trying to learn the nuances of English as well as the content-specific language. ELLs often are unable to understand or utilize rudimentary terminology and do not have the English skills to pose clarifying questions (Ewing and Huguelet, 2009). Furthermore, reading comprehension in English is challenging for ELLs. Textbooks often utilize word problems in an attempt to situate mathematics in context; the greater the complexity of the language presented in mathematics word problems, the greater the difficulty for ELLs becomes (Martineillo, 2008). Moreover, in order to solve word problems, students must possess adequate reading skills as well as have analytical competence (Ewing and Huguelet, 2009). Research also confirms that in order for text comprehension to occur, between 90 to 95 percent of the words in the text must be known to the student (Nagy and Scott, 2000). Barrett (2014) found that textbooks in ELL classrooms are often difficult to read and do not support students to learn English. There is widespread documentation that ELLs in sub-Saharan Africa cannot speak English well enough to use English as a medium of learning (Alidou and Brock- Unte, 2006) and is often correlated with low school achievement (Clegg and Afitska, 2011; Rubagumya, 2003). “[L[ack of proficiency in the language of instruction, in our case English, results in poor performance in subjects taught in English” (Qorro, 2006, p. 5). This weak performance in mathematics is affirmed in the recent 2015 results of the National Examinations Council of Tanzania (NECTA). “Over 80 percent of candidates who sat for last year’s Form Four national examinations failed in the subject of mathematics”(Mbago, 2016). With this in mind, it is important to consider aspects of mathematics teaching and learning in order to improve all students’ success in mathematics, including the textbooks used for mathematics instruction. Ensuring that all students and teachers have access to quality textbooks that are accurate, clear, and accessible, which increases opportunities to learn, is one essential aspect when considering the effectiveness and equity of mathematics teaching and learning for ELLs. METHODOLOGY AND RESULTS In order to determine the accessibility of mathematics to ELLs, a sample of four elementary mathematics textbooks were analyzed using grounded theory (Strauss 266 and Corbin, 1990). The researchers started with an overarching question of in what ways might the textbooks exhibit or impede accessibility of mathematics for learners. The researchers focused the analysis on explanations and problems in the text with associated words, (as opposed to computational problems with only numbers). As the researchers examined and analyzed the text and problems, repeated elements of textual features were noticed and categories were eventually established. Categories included: (1) whether the problem stated is grammatically correct, (2) the mathematics is accurate, (3) the instructions to students are clear, and (4) the amount of written text on a page. Then, using a modified version of O’Keeffe and Donoghue (2015) key design features of mathematics textbooks, further analysis was undertaken to determine how the texts align with these elements (see Table 1 for key characteristics). Two researchers worked independently and then compared and reconciled their results. Ideational Avoid passive sentences Use active verbs to encourage participation Focus on relating mathematics Interpersonal Textbooks should be in the present tense Consistent use of symbols Plan the introduction of new words/symbols; Vocabulary use is simple Include informal sentences Words with dual meaning are carefully considered Textual Establish textbook intent Be consistent with structure Be narrative (encourage reading) Include a glossary or dictionary Encourage oral and written use of vocabulary Table 1: Modified Framework (based on O’Keefe and Donoghue, 2015) Instances of improper grammar and unclear instructions were prevalent in our examination of mathematics textbooks. For example, when an explanation of vocabulary or terminology was offered, the sentence structure was typically formal and cumbersome (see figure 1) as highlighted below in an explanation about “finding values of letters that stand for digits in numbers”. In this particular example found in Steps in Primary Mathematics 2, (Adamson, 2010, p.27) the instructions indicate that the problem is written vertically, but the example is horizontal (this is a reoccurring situation in many of the textbooks). Furthermore, the actual instructions are wordy, and difficult for an ELL to understand. And, a key textual element was missing from all examined textbooks: no glossary or dictionary was provided to provide further support or explanation of terms. These grammatical errors and lack of consistent use of symbols and clarity could also be interpreted as mathematical inaccuracies which diminish students’ opportunities to both make sense of and learn mathematics. 267 Figure 1: Finding values of letters that stand for digits in numbers Furthermore, the textbooks typically did not provide adequate support for students in terms of modeling or guided practice. Nor were the contexts, when provided, relevant or meaningful to students, hence the ideational feature of focusing on relating mathematics was often lacking in these textbooks. Here we present an example to elucidate our findings. Problem 4.5 (see figure 2) asks students to convert a given time into the 12-hour clock notation and the 24-hour clock notation. The instructions are unclear. To ELLs who do not have a firm command of the English language, this question can cause confusion. The confusion is in the “twelve in the morning”. In English we often refer to twelve as either noon or midnight, depending on what time we are referring to. This question is ambiguous because it could be asking students to write the time for 11:44 am or 11:44 pm, depending on how “twelve in the morning” is interpreted. Figure 2: Problem involving time Here is an example of mathematics that is incorrect. In Figure 3, students are given a table that instructs them how to read a whole number with a decimal. The convention for reading whole numbers with decimals (in the United States) is to use the word “and” to indicate the decimal point then the subsequent numbers as decimals (e.g. tenths, hundredths etc.). In this example, the students are instructed to read 15.2 as fifteen point two, instead of 15 and 2 tenths. 268 Figure 3: Reading whole numbers with decimals However, in Figure 4 students are instructed to use “and” to indicate the decimal point, (in Figure 3 they use point) but the decimal numbers are presented as whole numbers (e.g., three thousand and ninety-three, rather than three thousand and ninety three hundredths). Figure 4: Writing numbers with decimals Again, for all students, but especially for ELLs this conflicting presentation of can add to their confusion of learning how to read decimal numbers. In Figure 5, students are presented with a problem that uses simple vocabulary, focuses on relating mathematics, however the grammar is not correct. The problem should be written as: 423,100 road accidents were reported last year. This year 212, 034 accidents were reported . . . While this may not seem like a critical error, these errors can negatively influence ELLs written and spoken English. Figure 5: Improper grammar 269 DISCUSSION For ELLs, learning mathematics is often a difficult challenge that is compounded when the instructional materials are not accurate (grammatically or mathematically) or includes unclear instructions. Since non-native English speakers often write curricular materials in Tanzania, the problems in these mathematics textbooks are riddled with syntactic and grammatical mistakes, which can be misinterpreted or misunderstood by ELLs. It is given that teachers in English Medium schools, who are often ELLs themselves, rely on these textbooks to assist in the delivery of instruction. When these textbooks are deemed inadequate, teachers cannot deliver quality instruction. All students are negatively impacted, especially ELLs. These textbooks can inhibit a student’s learning mathematics. If ELLs see a new word that they have never seen before, they need to have some sort of definition or example to make sense of what this new word means. Just like when they are learning the English language, they also need these definitions and examples to help them learn new mathematical words. The lack of explanation of new terms as well as a lack of a glossary or dictionary can greatly hinder ELL students’ learning of mathematics and set them back greatly in their education. Another common problem with Tanzanian mathematics materials is that they introduce math-specific words without meanings or examples to explain what the words mean. For ELLs, it is hard enough not knowing what certain words mean in English without some meaning or an example given to them, but when they are given an entirely new mathematical word, it can cause even more trouble. By providing new math-specific words without relatable meaning or examples, ELLs will struggle to understand the math vocabulary as well as associate them with some mathematical concept, which can decrease their opportunity to learn. This could force the ELL student to just look at numbers or diagrams and not make sense of the problem. All of these preliminary findings related to Tanzanian mathematics curricular material must be addressed to create greater opportunities to learn for ELLs in Tanzania. This preliminary study focuses on 5 mathematics textbooks used in Tanzanian classrooms. However, inadequate textbooks may be found in other classrooms across the globe. In order to provide equitable learning opportunities for all students who are learning mathematics in a language other than their native tongue, attention must be given to the accessibility of the content. As developing countries like Tanzania continue to find ways to provide their students with a high quality education, they need to consider that ELLs often struggle with both the content and language of instruction. Curricular materials and instructional techniques need to be improved upon so that these students can have adequate and appropriate opportunities to learn. Students need to see acquiring a second language as an advancement of knowledge and societal standing, not just another roadblock to learning. 270 While this analysis is based on a limited number of textbooks, future research will include additional textbooks both at the primary and secondary levels. In addition, a modified framework that is more “user friendly” will be developed in order to equip teachers and administrators with a textbook analysis tool, enabling educators to critically review textbooks for their own schools prior to selection. References Alidou, H., Brock-Utne, B. (2006). Teaching practices: Teaching in a familiar language. In Alidou et al (Eds.), Optimizing Learning and Education in Africa – the Language Factor (pp. 84–100). Paris: ADEA. Adamson, K. K. (2010). Steps in primary mathematics pupil book 2. Arusha, Tanzania: Adamson Educational Publishers. Adamson, K. K. (2010). Steps in primary mathematics pupil book 5. Arusha, Tanzania: Adamson Educational Publishers. Barrett, A. M. (2014). Language, learning and textbooks. Language Supportive Teaching and Textbooks Research Brief no. 1. Bristol: University of Bristol. February 2014. Clegg J., Afitska O. (2011). Teaching and learning in two languages in African classrooms. Comparative Education, 47(1), 61–77. Ewing, K., Huguelet, B. (2009). The English of math – It’s not just numbers! In S. Rilling, M. Dantas-Whitney (Eds.), Authenticity in the Language Classroom and Beyond: Adult Learners (pp. 71–83). Alexandria, VA: TESOL. Flanders, J. R. (1994). Textbooks, teachers, and the SIMS test. Journal for Research in Mathematics Education, 25(3), 260–278. Grouws, D., Smith, M., Sztajn, P. (2004). NAEP findings on the preparation and practices of mathematics teachers. In P. Kloosterman, F. K. Lester, Jr. (Eds.). Results and in interpretations of the 1990 to 2000 mathematics assessments of the National Assessment of Educational Progress (pp. 221–267). Reston, VA: NCTM. Halliday, M. A. K. (1978). Language as social semiotic. London: Edward Arnold. Hiebert, J., Gallimore, R. Garnier, H., Givvin, K., Hollingsworth, H., Jacobs, J., Chui, M., Wearne, D., Smith, M., Kersting, N., Manaster, A., Tseng, E., Etterbeek, W., Manaster, C., Gonzales, P., Stigler, J. (2003). Results from the TIMMS video study. Washington DC: National Center for Education Statistics, March. Lager, C. A. (2004). Unlocking the language of mathematics to ensure our English learners acquire algebra. No. PB-2006-1004. Los Angeles: University of California. Langeness, J. (2011). Methods to improve student ability in solving math word problems. St. Paul, MN: Hamline University. Martiniello, M. (2008). Language of the performance of English-language learners in math word problems. Harvard Educational Review, 78, 333–368. Morgan, C. (1995). An analysis of the discourse of written reports of investigative work in GCSE mathematics. Unpublished, University of London. 271 Mbago, G. (2016, February 19). Form Four results: Over 80% fail in maths. Retrieved from: http://www.ippmedia.com/frontend/index.php?l=89064. Nagy, W., Scott, J. (2000). Vocabulary processes. In M. Kamil, P. Mosenthal, P.D. Pearson, R. Barr (Eds.), Handbook of reading research (pp. 269–284). Mahwah, NJ: Lawrence Erlbaum. O’Keeffe, L., O’Donoghue, D. (2015). A role for language analysis in mathematics textbook analysis. International Journal of Science and Mathematics Education, 13, 605–630. Porter, A. C. (2002). Measuring the content of instruction: Uses in research and practice. Educational Researcher, 31(7), 3–14. Qorro, M. (2006). Does language of instruction affect quality of education? HakiElimu Working Papers. Dar es Salaam, Tanzania. Roberts, K. (2009). Math instruction for English Language Learners. Retrieved September 10, 2015, from http://www.colorincolorado.org/article/math-instruction- english-language-learners. Rubagumya, C. M. (2003). English medium primary schools in Tanzania: a new linguistic market in education? In Brock-Utne, B., M. Qorro, Z. Desai (Eds.) Language of Instruction in Tanzania and South Africa (LOITASA). Dar es Salaam: E&D Limited. Roth McDuffie, A., Mather, M. (2009). Middle school teachers use of curricular reasoning in a collaborative professional development project. In J. Remillard, G. Lloyd, B. Herbel-Eisenmann (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 302–320). Oxford, UK: Routledge. Strauss, A., Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage. PYRAMID OR TRIANGLE - ISN’T IT ONE AND THE SAME? (A CASE STUDY) Janka Kopáčová and Katarína Žilková Abstract The examination of preschool children’s ideas about geometrical shapes became the focal point of our research activities. The project “Geometric conceptions and misconceptions of pre-school and school age children” (VEGA no. 1/0440/15) also includes the research of pre-school children’s concepts related to identification, sorting and properties of geometrical shapes. Pre-school children often identify or categorize geometrical shapes by prototypes or real- world objects which the shapes resemble. To describe their understanding of the various prototypes they often use various phrases, labels and nicknames. While examining pre- Catholic University in Ružomberok, Slovakia; e-mail: [email protected] Comenius University in Bratislava, Slovakia; e-mail: [email protected] 272 school children’s understanding, we found a girl with unique misconceptions. Examining the girl, Ivanka, one year later, as a first-grader, we found surprising progress. Keywords: Conceptions, misconceptions, preschool children, geometric shapes, case study INTRODUCTION A term or concept is a mental representation of a certain group of objects with characteristic attributes (Čáp and Mareš, 2001). Thus, speaking about a geometric shape, we include all mental representations tied to a particular term in geometry. For instance, if we think about a triangle - by imagining a triangle many will visualize a certain type. To build a correct idea of a term it is paramount to employ such mental schemes that syncretize not only visualizations of a given type of a triangle, but mainly significant common characteristics and attributes of different types of triangles that are later important for the term’s exact determination. Because of this it is important to examine children’s ideas about geometric shapes and to find out what visualizations are the most common for a given term, or what vocabulary they use to describe them. THEORETICAL BASIS The van Hieles were the first to describe the levels of geometric thinking characteristic for hierarchical stages of geometric cognition (1986). Outcomes of their research in this field are fundamental and form the core of the theory of geometric thinking often called the van Hiele theory of geometric thinking. Our research is based on the van Hiele theory, whereas we look for connections and parallels in pedagogical-psychological theories and mathematics education theories. At the same time we are looking for potential influences of the socio- cultural background in context of the child’s natural language on the description of the geometric term as well as the influence of the differences in inducing the terminology in higher stages of education on exactment of the ideas in geometric terms. The brief overview of other theories that underlie geometrical knowledge is described in Swoboda and Vighi (2016) who say that building the geometrical knowledge of students has many different aspects. One of them is the language and communication. It is important to survey the words used by preschool children to describe differential specifics of a given term. Communication plays the biggest role in this process. Murínová and Mastišová (2011, p. 41) substantiate the significance of the child’s language and speech that “it helps a child understand basic and fundamental attributes of objects and phenomena, uncover their relationships and connections and form generalizations”. Hejný and Littler (in Stehlíková, 2006) point out that in mathematical education we use three vocabularies that change depending on age and level of mathematical knowledge: 273 ● everyday vocabulary for communication; ● mathematical vocabulary; ● everyday/mathematical vocabulary (see in Hejný and Littler, in Stehlíková, 2006). Vighi (2015) in his research study follows two aspects in context of the significance of language development for mathematic education: “the use of natural language in mathematical communication and the specificity of mathematical language”. He observes that “the language employed to describe geometrical situations plays a fundamental role: it is important to take care of the language and of its precision” (Vighi, 2015, p. 341). It is natural that with preschool children there is a big tolerance for inaccuracies in terms of exact mathematical terminology. It is often apparent that children can, in their natural language, communicate even abstract geometric terms. At first they think about shape and colour in this context. According to Brian and Goodenough (1929, in Košč, 1972) shape is the most often used criterion of sorting geometric figures with children up to 3 years of age. In the next stage (3-6 years of age) children prefer sorting the figures by colour. Shape as the more important sorting criterion reappears with children aged 6 and more. In this process children perceive geometric figures holistically, which concurs with the van Hiele level of visualization. In the shape identification process the child compares the figure to a respective figure prototype. She decides if the figure is or is not ‘similar’ to the prototype. There are basic prototypes of planar geometric shapes (e.g. the prototype for triangle is an equilateral triangle with a vertical-horizontal side position). Children have their own names, commonly emerging from previous experience or communication in their common setting, to mark these prototypes. A common name for a triangle is ‘roof’. In our research we observed that children coming from different backgrounds use different names (words) for prototypes, but in general they decide according to the mental, visual imagination of the prototype. According to the research of Clements and Sarama (2000) conducted which 128 children aged 3-6, children have noticeably lower success rates in identifying triangles compared to other basic geometric shapes (circle, square, rectangle). At this age children usually accept triangle-like objects with rounded sides, but reject triangles that are elongated or not in the horizontal-vertical position. The mentioned indicators are typical for the lowest level of geometric thinking that is connected to visual imaginings tied to a certain term. Visualization and language are clearly important factors determining the development of mathematical thinking in preschool children and are crucial to pay attention to. 274 RESEARCH A. Objectives of the research One of the objectives of the research (VEGA no. 1/0440/15) is to ascertain the ideas and imaginings of preschool and younger school children in Slovakia about geometric figures, ways of their perception and description. We assume that the target group (of preschool children) will accord with the van Hiele level of visualization. We aim to find out what prototypes the children use in the geometric figure identification process, what vocabulary they use and which of these are the most common. We are equally interested in finding out the difficulty of naming a planar geometric figure based on its pictured model, and also the reverse process of identifying the model based on its name. B. Case study research design Within the existing analysis of about 100 video recordings of preschool and young schoolchildren we encountered a special case - a girl that we will call Ivanka in this paper. Ivanka’s reactions were unique within all recorded reactions. According to Shuttleworth (2008) “the advantage of the case study research design is that you can focus on specific and interesting cases“. Ivanka was indeed an interesting case and given the nature of the research method and course of the study we decided to use illustrative case study, that is the study will have a descriptive character and will offer two separated records of Ivanka taken after a lapse of over one year. We shall describe her thinking and comments on triangles as recorded on two occasions. The research was conducted via a “face to face” semi-structured interview. The researcher used models of planar geometric figures and pictorial models (Fig. 1a, 1b, 1c). Figure 1a Figure 1b Figure 1c A set of questions for the child came with the models and the researcher was allowed to slightly rephrase the questions, change their order or react interactively to better suit the flow of the interview. Video recordings and transcriptions of the duration of the research allowed us to conduct a thorough analysis. The aim of the interviews with the first set of models was finding out if the child knows the names of planar geometric figures, as with the second set of models we aimed to 275 determine if the child is able to correctly identify the shape based on the name. The tasks are seemingly equivalent, but the difficulty of each task is different. It is easier for the child to pick a model by name than otherwise. Illustrative case study – Ivanka Ivanka was, during the first research (2015), 5 years and 10 months old and was attending a kindergarten. In account of her interview that differed radically from those of other observed children, we decided to return a year (exactly 16 months) later with the same research tools and revisit the research. At the time of the second research (2017) Ivanka was 7 years and 2 months old and had been attending 1st grade for 5 months. Ivanka comes from a complete family of university educated and employed parents and has two older siblings, a 13 year old sister and a brother aged 15. Ivanka, even being a kindergartener, had an above average vocabulary, cooperated well and answered when asked a question. Figure 2 The first interview (2015) Within the first activity Ivanka was given models of geometric figures and was asked to build something using the models. She took to work immediately and constructed a facsimile of a person (Fig. 2), commenting on her work. She participated in a dialogue with the researcher eagerly. When asked ‘What shape is the head, body, arms and hat?’ she answered: sphere, cuboid and pyramid, respectively. Even in the following communication she named the model of a square as a cube, a rectangle as a cuboid, a triangle as a pyramid and a circle as a sphere. When the researcher repeatedly asked for triangle, she pointed out the vertices and emphasized: “ihlan, lebo má také ihly” (a wordplay: Slovak word for pyramid - ihlan meaning ‘consisting of needles’, ihly meaning needles or pricks). We are aware that the models used were not 2-dimensional and that this fact could lead to misunderstandings between the researcher and Ivanka. Ivanka, however, 276 without swerving, identified the planar geometric figures on paper models with the names of their spatial counterparts as well. During the first interview, on the first, planar, model (Fig. 1a) Ivanka without hesitation pointed out the shapes and named them: a rhombus and a square in standard position as cubes; a slightly turned square as a diamond. Circles she named as spheres and rectangles as cuboids and neither position nor size unsettled her. It is of note that from the triangles she chose the obtuse-angled one before the equilateral , whereas she declared “pyramid, pyramid“. She was not familiar with the trapezoid : “I don’t know, it’s like a house, if it was refilled, then it’s a pyramid“. By the second model (Fig. 1b) she surprised us further. On the researcher’s (M) questions Ivanka (I) answered like this: I: ...this is a pyramid (pointing at triangle) and this is a cuboid (pointing at both rectangles). M: And do you see a triangle? I: Triangle? No, you just say that… There is no triangle, but these are pyramids (tracing the bottom triangle with her finger and pointing out the other triangle), or a triangular… triangle. M: So what do you think this is? (M pointing at the triangle in the lower part of the model) I: I think this is some kind of, some kind of a pyramid; and this is also s kind of pyramid. (Pointing at both triangles) It was clear from the communication that Ivanka encountered the term triangle and had seen its’ various models. Within the identification she was certain and decided almost without hesitation. From reasons unknown she refused to name the triangle models as triangles and used the term pyramid. By the third model (Fig. 1c), Ivanka, given the instruction to point at the triangles, said: “I don’t know what triangles are.“ She picked the triangles one by one and named them “hat, tent“, , “tent, roof“ and rotated the paper. Again, she assigned her own names: “I call this a cube ( ), and I call this a pyramid ( ).“ As the term triangle resonated with her, she decided to use it for such non- models of the triangle ( , , ) that resembled it holistically, commenting: “Because this doesn’t have those pointy… it doesn’t have needles.“ As long as the shape had three vertices, she named it a pyramid no matter the position or proportions. The communication between Ivanka and the researcher confirmed that Ivanka assigns the names of spatial solids to planar figures unambiguously and 277 decisively. The girl’s decisions were based on the holistic perception (visualization level), although she supported her statements by counting vertices or other arguments (elements of description level). The second interview (2017) After a year Ivanka stated that she does not remember neither the activities nor the models. The identification process looked a lot different now. The results of the repeated interview was that Ivanka correctly named the square and diamond as quadrilaterals; a slightly turned square as a ‘diamond - quadrilateral’. She did not refuse the term square - she identified the square models correctly. This means that the assignment of term -> model is learnt correctly; only Ivanka has a little trouble otherwise. Rectangles and triangles were named correctly. The interesting fact was, that she still chose the obtuse- angled triangle first; she didn’t spot the equilateral one until a little later. She identified the triangle models without hesitation (Fig. 1c). She also identified commenting that these are two triangles. She called a non-model of a triangle a “softened triangle“, which shows that she started to perceive the details of the figure. She clearly understood that the figure has a triangular shape, but that it is not really a triangle. As she used the word triangle, sensing a disproportion, she presented her reservation by adding the adjective “softened”. With other children the statements were often “It is a triangle, but not quite...” and “It is a triangle, but...”. Ivanka identified the figure as a triangle with a reservation saying “this is a different triangle, because it’s fat”. After comparing Ivanka’s statements from the two interviews 16 months apart we find that her perception of triangle models was more precise and that she focused considerably more on the details of given models by the time of the second interview. She included only the rounded non-models of triangular shape. Particularizing their name (by adding adjectives like “softened” or “fat”) indicates that Ivanka appreciates this handicap of the figures in question. After a year, we gave Ivanka a cube and a cuboid for manipulation and identification. She did not use the names of the solids (cube or cuboid), but pointing out a face of the cuboid, identified it as a rectangle. We did not examine her ideas about spatial solids more closely. C. Results and discussion Ivanka, being two months short of 6 years old, differed from other children in the same age group by confidently using names of spatial solids, assigning them to planar geometric figures (Tab. 1). 278 Table 1: Comparison of figure names according to Ivanka First hypothesis about the problem origin Ivanka’s names for the planar geometric figures are specific within the research sample. This specific is interesting mainly because usually children assign the names of planar figures to spatial objects. The reason may be simplified representations (drawings) of 3-dimensional objects as 2-dimensional shapes. Further studies confirmed this fact too and “pointed to the use of plane geometry terminology when young children describe three-dimensional figures“ (Levenson, Tirosh and Tsamir, 2011, p. 17). For instance, a real object in the shape of a cube is often drawn as a square, or a pyramid’s simplified schematic drawing is a triangle. Hansen et al. (2005) describes some of these misconceptions and points out, that the problem of representation, or interpretation of 3D vs 2D objects may be present in adults as well. Second hypothesis about the problem origin Ivanka’s ideas about triangles were first tied to the acute (sharp) angles that she called “ihly “ (needles), which enabled her to identify even the obtuse-angled triangle as well as rotated or resized triangles of all types. She not only was familiar with the word triangle, she refused to use it and used pyramid instead. The reason may well be a linguistic one, because in Slovak word for pyramid – “ihlan“ is similar to the word “ihla“, meaning needle. 279 We inquired into Ivanka’s learning the names of spatial solids, but even her mother could not explain. She (the mother) herself tried to rectify but was not successful. Ivanka differentiated the planar figures correctly by type, but assigned them with wrong names. We assume the reason for this, in addition to the two aforementioned hypotheses, may be also a consequence of the influence of Ivanka’s two older siblings or her kindergarten teacher. After a year, there was noticeable progress in Ivanka’s identification of triangles. She exchanged the use of the van Hiele visualisation level nomenclature (hat, roof, tent) for using the correct terminology (triangle). She was also able to express her reservations about the non-models by adding descriptive commentary in the form of adjectives. Ivanka’s ideas of triangles thus started to correspond with the second van Hiele descriptive level. CONCLUSIONS The study of Ivanka’s process of describing triangles as pyramids indicates that language specifics may have an influence on identification of geometric figures by children and, subsequently, on forming a child’s primary conceptions of said figures. This influence may result in conflict between natural language and correct mathematical terminology. The illustrative case study of Ivanka proves that this conflict is not unresolvable and doesn’t hinder the development of geometric thinking in a child. It’s important to provide an educational environment providing opportunities for children to correct and further specify their primary conceptions and ideas. In accordance with the results of Clements and Sarama (2014) we are convinced that the child needs “varied examples and nonexamples, discussions about shapes and their attributes, a wider variety of shape classes, and a broad array of geometric tasks” (Clements and Sarama, 2014, p. 153) to build her concepts about shapes. Simultaneously it’s essential to support the child in communication about geometric figures or shapes. Inaccuracies in terminology may be tolerated in this age group. Within the communication, it’s possible to focus more closely on detailed description, development of the ability to determine and reason whether a certain shape belongs into a certain geometric figure category or not given the shape’s attributes. This illustrative case study about Ivanka’s pyramids and triangles being the same, or, that her mental figurative idea of a triangle evoked the term pyramid, facilitated several alternative interpretations. The first interpretation is based on the problem of transformation between real 3D objects and drawn 2D shapes. This is a problem inspiring us to further research. The second interpretation is based on the problem of the specifics of Slovak language influencing Ivanka’s terminology. The third interpretation is a more prosaic one, taking into account the influence of Ivanka’s immediate surroundings (her siblings or kindergarten 280 teachers). In this case, we cannot determine the influential prevalence of any of the three interpretations. Acknowledgement This study was supported by VEGA 1/0440/15 “Geometric conceptions and misconceptions of pre-school and school age children“. References Clements, D. H., Sarama, J. (2014). Learning and Teaching Early Math: The Learning Trajectories Approach. Oxford: Routledge, 2014. 394 p. ISBN 978-0-415-82850-5 Clements, D. H., Sarama, J. (2000). Young Children's Ideas About Shape. [Online]. The National Council of Teachers Of Mathematics, 2000. Retrieved from http://Gse.Buffalo.Edu/Org/Buildingblocks/Writings/Yc_Ideas_Shapes.pdf. Čáp, J., Mareš, J. (2001). Psychologie pro učitele. Praha: Portál, 2001. 656p. ISBN 80- 7178-463-X. Hansen, A., Drews, D., Dudgeon, J., Lawton, F., Surtees, L. (2014). Children’s Errors in Mathematics. Understanding common misconceptions in primary schools. Learning Matters, 2014. 256p. ISBN 978 1 84445 032 9. Levenson, E., Tirosh D., Tsamir, P. (2011). “Preschool Geometry. Theory, Research, and Practical Perspectives”. Rotterdam: Sense Publishers. 2011. ISBN 978-94- 6091-600-7 (e-book). Košč, L. (1972). Psychológia matematických schopností. Bratislava: SPN, 1972. 276p. Murínová, B., Mastišová, J. (2011). Rozvíjanie komunikatívnych kompetencií v predprimárnom a primárnom vzdelávaní. Ružomberok: VERBUM, 2011. ISBN 978-80-8084-804-0. Shuttleworth, M. (2008). Case Study Research Design. Retrieved Mar 18, 2015 from Explorable.com: https://explorable.com/case-study-research-design. Stehlíková, N. (Ed.) (2006). Creative Teaching in Mathematics. Prague: Charles University, 2006. ISBN 80-7290-280-6. Swoboda, E., Vighi, P. (2016). Early Geometrical Thinking in the Environment of Patterns, Mosaics and Isometries. In: Part of the series ICME-13 Topical Surveys (pp. 1–50). Springer. ISBN 978-3-319-44271-6. van Hiele, P. (1986). Structures and Insight. A Theory of Mathematics Education. London: Academic Press. Vighi, P. (2015). Language for Learning: Spontaneous vs specific geometrical language. In J. Novotná, H. Moraová (Eds.), Proceedings of SEMT ´15 – Developing mathematical language and reasoning (pp. 341–349). Prague: Charles University, Faculty of Education. ISBN 978-80-7290-833-2. 281 THE INVESTIGATION OF CO-OPERATIVE-INTERACTIVE LEARNING SITUATIONS IN AN INCLUSIVE ARITHMETIC CLASSROOM Laura Korten Abstract This paper presents parts of a study, which addresses the current topic of teaching mathematics in an inclusive classroom. How can we face the increased diversity of learners and make sure that all children work and progress on their individual level but at the same time learn with and from each other? Based on this question the aims of this research project are the development of, and the research on a teaching-learning arrangement for the inclusive mathematics classroom. The approach is a Design Research approach to face research interest on the level of design (to improve) and on the level of research (to understand). This contribution focuses on the level of research: Interpretative data analyses showed that co-operative-interactive learning situations of elementary school children with and without cognitive learning disabilities can be fruitful for all participants. Children have progressed according to their individual abilities triggered by ‘impulses’, which occurred in the interaction. The project and selected insides about the investigation of co-operative-interactive learning situations in an inclusive classroom will be presented. Keywords: Inclusive education, diversity, co-operative learning, flexible mental calculation Introduction Learning takes place in a context of social construction processes on the basis of individual interpretation processes (Krummheuer and Voigt, 1991; Miller, 1986, 2006; Schwarzkopf, 2003). Consequently, learning mathematics requires inter- action and co-operation. In social processes, mathematical learning develops through contrasting, contradictions and re-interpretation (Steinbring, 2005). Also for children with cognitive learning disabilities the construction of mathematical knowledge is an active, explorative and social process (Scherer, 1995). Further on, the communication about mathematics (e.g. about task characteristics, relations, strategic tools) is especially important for children with cognitive learning disabilities (Schröder, 2007). Thus, learning mathematics in an inclusive classroom equally requires interaction and co-operation to support learning. Supporting everyone’s learning process and at the same time encourage co-operative learning with and from each other are the two central matters of inclusive education in Germany. The UN-Convention on the ‘Rights of Persons with Disabilities’ intends to protect the rights equality, participation and dignity of everybody. The debate as its consequences for school systems is about inclusive education with the aim of Technical University of Dortmund, Germany; e-mail: [email protected] 282 Mutual Learning. The expression of Mutual Learning as it is used here combines both individualizing and co-operating. It means to consciously induce learning situations as often as possible in which all children work and learn at a common content, in co-operation with each other, on their individual level, by use of their current individual skills, and in orientation on their ‘next zone of development’ (Amongst other based on Feuser, 1997). This definition is based on a wide sense of inclusion, acknowledging the diversity of all children and counteracting all forms of special needs (Prengel, 1993). Nonetheless this research project focuses on learning and co-operation processes of children with and without cognitive learning disabilities.1 In the German discussion wide consensus prevails on the idea that no “new didactics” is needed, we “just” have to enrich what we do, to support students with special needs (Lütje-Klose and Miller, 2015). However, from a mathematical didactic perspective, little is known about teaching and learning processes in inclusive classrooms (ibid.), which implies both matters: individualizing and co-operating. Korff (2015) found out that teachers in school practice especially emphasize the difficulty of Mutual Leaning in mathematics. Interviewed teachers explained that this is due to the arithmetical content with its focus on the symbolic level and the fewer opportunities for hands-on work (ibid.). In order to contribute to the closure of this research gap, this article describes empirical findings about Mutual Learning in arithmetic classrooms. Therefore, learning processes and interaction processes, which take place during co-operative-interactive learning situations of children with and without cognitive learning disabilities, will be investigated, to find out, when fruitful Mutual Learning takes place and which support means can be reconstructed. Theoretical Background In consideration of the two central matters of individualized learning and at the same time learning with and from each other, first theoretical supportive principles for successful Mutual Learning can be derived: ‘content variability’, ‘goal-differentiated learning process-‘ and ‘interaction orientation’. Having regard to these, the theoretical background gets presented. In this process a link is made between ‘mathematics education’ and ‘special needs education’. Cooperation in mathematics classrooms (interaction orientation) As pointed out before, the construction of mathematical knowledge is an active and explorative process; also for children with cognitive learning disabilities as several studies show (e.g. Scherer, 1995). Gaidoschik (2009) points out, those children with problems in mathematics need more time and more support to learn arithmetic but they don't need something different from others. The exploration and use of arithmetical patterns as well as the communication about those should 1 This distinction is not used to label deficits, but rather with regard to make research and communication possible. 283 be in the focus to support students with difficulties (ibid.). In social- communicative processes, individual mathematical learning develops through ‘productive irritations’ (Nührenbörger and Schwarzkopf, 2015), ‘contrasting’, ‘contradictions’ and ‘re-interpretations’ on the basis of individual interpretation processes (Steinbring, 2005). “The subject matter of mathematics is […] not understood as a pre-given, finished product but interpreted according to the epistemological conditions of dynamic, interactive development (ibid., 34). Therefore, this very individual processes of learning in the context of interaction processes, needs to be supported on different cognitive levels for successful Mutual Learning, aiming for different goals to encounter the rich diversity in an inclusive mathematics classroom. With regard to the structure of co-operation, the project will concentrate on the I- You-We principle (Gallin, 2010). It has been adapted to create co-operative- interactive learning situations that contribute successful Mutual Learning (Fig.1). Singular accesses and comprehensions evolve, through communicative exchange, to new comprehension and understanding. Some studies investigated the interaction between children. Naujok (2000) for differentiated between three types of co-operation: “working side by side“, “helping“ and “collaborating“. Littleton et al. (2005) focused on the content linkage during communication processes and distinguished between „dispuational talk“, „cumulative talk“ and „explorative talk“. This contribution refers to these findings in order to reconstruct interaction processes between children with and without cognitive learning disabilities to. Fostering flexible mental calculation on different cognitive levels (content variability, goal-differentiated learning process orientation) Developing flexible mental calculation competences is a ‘critical point’ in everyone’s learning process (Heinze, Star and Verschaffel, 2009). In this process error-prone counting strategies should be replaced by more beneficial strategic tools. Individuals should be able to solve mathematical tasks not only quickly and accurately, but also flexibly and adaptively (Threlfall, 2002). In most of the cases flexibility is understood as the ability to switch between different solution tools (Rathgeb-Schnierer and Green, 2013; Verschaffel et al., 2009). While varied definitions of adaptivity more emphasizing the selection of the most appropriate strategy. Referring to Rathgeb-Schnierer and Green (2013), in this project flexible mental calculation involves ‘flexibility’ as described above and ‘adaptivity’ in relation to the recognision of problem characteristics, number patterns and numerical relations. Consequently, flexible mental calculating is a situation- dependent and individual response to specific number and task characteristics and the corresponding construction of a solution process using strategic tools (Rathgeb-Schnierer and Green, 2013; Threlfall, 2002). Therefore, the fostering of flexible mental calculation needs to support children to focus on these characteristics, patterns and relations. 284 Today it is proven that also less advanced students can develop flexible mental calculation (Verschaffel et al., 2009) and that the focus on problem characteristics, number patterns and numerical relations especially supports less advanced students in developing flexibility (Rechtsteiner-Merz and Rathgeb-Schnierer, 2016). Consequently, those will be fostered on different cognitive levels as an aim of the designed teaching-learning arrangement, in which children explore sums of neighbouring numbers in substantial tasks. The communication about these arithmetic characteristics, relations and strategic tools is especially important for children with cognitive learning disabilities. Schröder (2007) points out their problems in the choice and usage of flexible strategies: Even if they know some strategies, they very often cannot adapt and use those. At the same time this mathematical content is supportive and preventive for everyone’s learning process, because generally children show little task-adequate action when solving problems (Selter, 2000). Regardless of their effectiveness, they prefer written algorithms (ibid.). Further, the content provides opportunities and richness for high-performing students to establish mathematical structures and to generalize. Consequently, it meets the requirements for a variable common content for successful Mutual Learning to encounter the diversity of cognitive abilities, skills and differentiated goals. Research Questions and the Design of the Investigation Beyond the described research findings, little is known about individual learning processes and interactive structures during co-operative-interactive phases of Mutual Learning. Appropriately, the following two research questions are addressed in the study presented in this contribution: R1: How do individual learning processes of elementary students with and without cognitive learning disabilities concerning flexible mental calculation develop during the co-operative-interactive phase of Mutual Learning? R2: Which interactive structures can be reconstructed during the co-operative- interactive phase, which support fruitful Mutual Learning for all children? Overall a Topic-specific Design Research approach is used, which is based on the FUNKEN-model by Prediger et al. (Prediger and Zwetzschler, 2013). This requires both research questions on the level of design and on the level of research. Both levels are related and influence each other. Nevertheless, this contribution only focuses on the level of research and mainly on R2. Iterative design research cycles As an approach for answering the questions, a teaching-learning arrangement was designed, tested and refined by conducting design experiments in three iterative cycles. Within each cycle the individual learning processes concerning flexible metal calculation as well as the interaction processes were reconstructed to be 285 able to evaluate weather Mutual Learning took place. To, in a next step, reconstruct support means for successful Mutual Learning. Theoretical sample: The design experiments were conducted in classes two and three (7-9 years old), at three different German primary schools. Laboratory situations with couples of learners allowed to learn more about their thinking, their individual learning and interaction processes. Each design experiment consisted of three phases (Fig.1) and took place in a pair setting with one child tested and “termed” with and one child without learning disabilities. The participants were selected with the help of the class teacher and the special needs teacher in order to find pairs of children who like each other to have a positive basis of communication. In the design experiments, the learners processed the learning activities largely by themselves. The researcher, on the one hand, acts as a teacher, in order to give the learner a stimulus or help, and on the other hand as a researcher, who wishes to learn more about the thinking processes and the ways of proceeding by means of observation and targeted inquiry (Cobb, et al., 2003). Teaching-learning arrangement - "We explore neighbouring sums": After a mutual introduction (Fig.1), the children individually explore neighbouring numbers on a 20frame (I-/individual-phase). The focus on neighbours – which are next to, under or crosswise to each other – and their sums enables them to discover number and problem characteristics and relations, as well as to develop mental calculating strategies based on individual abilities, arithmetic-, and context- characteristics. In the following, two children – one with and one without learning disabilities – work together (You-/co-operative-interactive-phase), which enables them to communicate, use, reflect, refine, and/or improve their discoveries and strategic tools. In this way, singular accesses and comprehensions can evolve, through interaction, to new comprehension and understanding. Due to the focus on neighbouring sums the arithmetical patterns stay the same even in higher number ranges. This makes communication possible, even though some children already transfer their discoveries to neighbours on the 100frame or generalize the mathematical structures. … Figure 1: Structure of a design experiments Two analytical perspectives The process of generating local theories gets content-specific theoretically and empirically justified. The data was collected in form of transcribed videos and 286 gets analysed from two perspectives: 1) An epistemological perspective, to reconstruct individual learning processes on different cognitive levels in terms of the common content of flexible mental calculation. 2) An interactionist perspective, to reconstruct interactive structures during the co-operative- interactive phase of student with and without learning disabilities, to find out how these interaction processes influence fruitful Mutual Learning. In order to address the two central matters of successful Mutual Learning, both perspectives are essential to evaluate if the children progress on their individual level and at the same time learn with and from each other. The interpretation of statements and actions, reconstructs interactive knowledge construction. Accordingly, an Interpretatively Epistemological Analysis Approach of Interactive Knowledge Construction (Krummheuer and Naujok, 1999; Steinbring, 2015) is used. At the same time this reveals information for the analysis of the teaching-learning arrangement and gives answers weather Mutual Learning in the sense of inclusion is supported or not. So generally, the findings allow elaborating and enhancing the teaching-learning arrangement, as well as local theory building about co-operative-interactive learning processes of children with and without cognitive learning disabilities in an inclusive classroom. Selected Results In this section, the described analysis approach is illustrated with a short exemplary co-operative-interactive phase. Afterwards, selected general results mainly concerning the research question R2 will be presented. In the exemplary situation, a child with learning disabilities (S1) and a child with average mathematical skills (S2) work together. They explore crosswise neighbouring numbers and their sums. Figure 2 shows an example: Figure 2: Crosswise neighbouring numbers (constancy of two sums) S1: We need 24 in between (she points between 7+16=23 and 7+18=25) S2: Wait. (she writes down the problem 8+17=25) ... S1: 23 (points on 7+16=23) #, 24 (points on the gap), 25 (points on 7+18=25) S2: # No, this is… No... Here is the same. (points on 4+13=17) Also always one. (points on 3+12=15) See, there is 16 missing. (points between 3+12=15 and 3+14=17) … here 14 is missing. (points between 287 2+11=13 and 2+13=15) Here 18. (points between 4+13=17 und 4+15=19) Oh here even (points between 5+14=19 und 6+17=23) two … no, 3… yes, 3 S1: What? Now I am confused. S2: Why? Ah! See, … Interactionist perspective - S1 ‘assumes’ that the sum 24 is missing and thus she questions the completeness of the sum. This ‘impulse’ triggers S2 to exemplify relations between the sums. Her empirical argumentation leads to the assumption that the sum 24 does not exist. Both participants communicate with each other about the common content, according to individual assets. A ‘balanced co- operation’, in which both are involved and all utterances are linked (Littleton et al., 2005) can be observed. Regarding to Naujok (2000) the children are ‘collaborating’ with the focus on the same topic. Epistemological perspective - The children respond to the same ‘incorrect assumption’ (Fig. 3 and 4, sign/symbol) in different ways by referring to number relations on the basis of their individual cognitive abilities. Figure 3 and 4 show the progress of the scene from an epistemological perspective: S1 argues with counting and refers to the number word series (ordinal). S2 uses empirical arguments to prove that the sum of 24 does not exist by referring to arithmetical patterns (relational). Due to S1´s incorrect assumption, S2 discovers, exemplifies and later even generalizes number relations between the addition problems. S1, like this situation shows, is able to see and to question number patterns. This focus of attention on number characteristics and relations only started due to the interaction with S2. Before she solved every task isolated by using counting strategies. In the following, this situation leads S2 even to explore, explain and generalize the constancy of two sums (Fig.4). From this point on, as a reaction on the interactive situation, she is not only referring to numbers characteristics and relations anymore but to problem characteristics and relations, which she is using later to solve new addition problems. 288 Figure 3 & 4: Epistemological analysis of learning processes (Steinbring, 2015) The example shows how the individual learning processes developed during the co-operative-interactive phase (for detailed analisys concerning R1 see Korten, 2017). They progressed according to their individual levels, triggered by an ‘impulse’ in the interaction, in this case ‘sharing an assumption’. These key impulses in the interaction – here called ‘productive moments’ – seem to be opportunities for fruitful Mutual Learning. As Figure 5 shows, they influence not only the learning processes but also the following interaction, mostly in a positive way. All research cycles showed regularity in the appearance of these 'productive moments'. They mainly appeared during a ‘balanced co-operation’, like defined in the example. Generally, a distinction can be made between direct-didactical, indirect-didactical and interactive productive moments: Figure 5: ‘Productive moments’ in the interaction Outlook In the future, research questions on the level of design will be addressed in order to reconstruct support means for successful Mutual Learning. It will be investigated in more detail how the developed teaching-learning-arrangement can 289 specifically foster these ‘balanced cooperation’ and the 'productive moments'. First analyses show that beneficial and meaningful interaction must be specifically encouraged by an ‘extrinsic positive dependence’. This, for example, can be a goal, which they can only reach together. It picks up the idea of the principle ‘positive dependency’ from the concept of ‘co-operative learning’ (e.g. Johnson, Johnson and Holubec, 1994) and advances it for the special conditions of an inclusive arithmetic classroom. Without this ‘extrinsic positive dependence’ a ‘balanced co-operation’ with ‘productive moments’ seems to be impossible in an inclusive setting. References Cobb, P., Confrey, J., diSessa, A., Lehrer, R., Schauble, L. (2003). Design experiments in educational research, Educational Researcher, 32(1), 9–13. Feuser, G. (1997). 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Teacher use of the Livebinder, an electronic training resource, was associated with collecting almost twice as many evidences per child. Similarly, teachers who found the progressions useful for instructional planning collected more evidences than those who did not. Keywords: Formative assessment, implementation fidelity, object counting INTRODUCTION Formative assessment is much more than a set of measurement tools; it is a process that is integrally interconnected with both high quality instructional practices and positive student learning outcomes across all domains of child growth and development (Heritage, 2010). Formative assessment has also been shown to increase student growth through much more cost effective means than many other educational reforms (National Council of Teachers of Mathematics, 2007). The National Mathematics Advisory Panel (2008) found that the use of formative assessment strategies in mathematics instruction specifically correlates with enhanced student growth. Formative assessment processes lead to these gains by providing teachers and students with timely feedback about teaching and learning, help students learn to assess themselves and monitor their own learning and growth, and impact student motivation positively by helping students engage more fully in the learning process (National Council of Teachers of Mathematics, 2013). Despite these benefits, it is difficult to implement large-scale formative assessment programs. Educators and researchers continuously redefine formative University of North Carolina at Charlotte, USA; e-mail: [email protected], [email protected] University of South Carolina, USA; e-mail: [email protected] 292 assessment to ensure its relevance and effectiveness in practice. Reflection on the process of formative assessment allows teachers to determine how interventions are influencing student learning. Formative assessment systems have been one innovation implemented in educational programs that successfully address the achievement gaps that often widen during Kindergarten through third grade (Graves, 2006). It has since been determined that optimal learning may only take place when there is both a recognition of student strengths, opportunity for student growth, and assessment of the whole child which may include the child’s culture, family, healthy and early experiences (Black and William, 1998; William and Thompson, 2007). Black and William (2009) further define the responsibility of the teacher and the learner during the process of formative assessment. They assign responsibility for designing and implementing effective learning environments to teachers and place responsibility to learn within that space to students. Their research implicitly emphasizes the teacher’s role to elicit and interpret evidences of student growth, but also recognizes that effective implementation of formative assessment programs requires intentional planning time to allow teachers to process assessment information and consider how context has influenced the data obtained. Black and William (2009) suggest that a teacher’s implementation of formative assessment and the determination of student contributions to growth and learning involve complex analyses that teacher often have little time to consider. Current research suggests that ensuring the successful application of formative assessment programs not only requires time to establish a solid framework but also effective communication between educational stakeholders (teachers, parents, policy makers, researchers and students) beginning with a clear understanding of the program’s goals and lexicon (Dunn and Mulvenon, 2009). Benner and Hatch (2004) advocate for approaches that ensure: 1) individual attention, 2) instruction customized to their development, and 3) frequent communication. These elements emphasize the relationship between and necessity of accessible and effective resources, supports and tools for teachers implementing new programs. Documenting children’s development through portfolios, which include samples of children’s activities, photographs, videos, and anecdotal records, has the potential to help teachers reflect on their teaching methods and children’s needs, which then translates into better-developed lessons and activities for their classrooms (MacDonald, 2006). Technological developments such as electronic portfolios have changed the mode of evidence collection for formative assessments from paper to electronic versions, and may be perceived either as helpful or demanding depending on teacher skill and training, and communication from school leadership. Successful implementation of electronic portfolios depends on teachers’ perceptions about technology integration in the classroom, 293 the cost of new technology, including time invested, and administrative and peer support (Meyer et al., 2011). The assessment that was the focus of this study, the North Carolina K-3 Formative Assessment Process: Kindergarten Entry Assessment (KEA), was the initial step in the development of a comprehensive electronic portfolio and formative assessment process for young children from kindergarten entry through third grade. The complete assessment process includes progressions in the areas of approaches to learning, development, emotional and social development, health and physical development, and language development and communication. Each progression outlines skills children acquire around the age of kindergarten entry and includes a developmentally sequenced list of behaviors describing a child’s progression toward skill mastery. However, for the first year of full- implementation, 2015, teachers focused only on progressions for literacy and math outcomes: Object Counting, Print Awareness, and Book Orientation to allow them time to learn the overall assessment process. Over the assessment period, teachers documented evidences of learning in the form of student work samples, photographs, videos, audio recordings, and anecdotal notes from student observations. Teachers uploaded these evidences to an electronic platform, where they interpreted the evidences and used them to assign their students a learning status along the appropriate construct progression. This study aimed at answering two questions: 1) How does the quantity of collected evidences and children’s placement in the progressions differ based on children’s demographics and teachers’ resources? 2) How do teachers’ perceptions about KEA affect their implementation of the assessment? METHODS Population The original dataset of evidences collected from the KEA electronic portfolio included data from 5,252 teachers and 86,913 children attending 1,105 schools in 113 districts. Teachers and administrators received an invitation to participate in a survey to evaluate the electronic portfolio implementation. At the end of the survey, they could volunteer to be interviewed. From the 736 survey respondents, 104 volunteered, and 44 followed through and were interviewed. This study includes only participants who were interviewed, and uploaded evidences in the electronic portfolio for at least 10 children. Therefore, our final dataset for this study includes 36 teachers, 678 students from 35 schools in 22 districts. The mean age of the children was 66 months, and the majority was boys (53.54%). The racial distribution was: 47.05% White, 21.39% African American, 20.01% Hispanic, and 11.55% unknown race. On average, teachers had 15 years of experience, and 8 years of experience teaching kindergarten. They had 19 children in their classrooms, on average, and 25% of them were teaching in a rural county. Only a few had an advanced graduate degree (13.9%). 294 Data and Instrument Progressions Measurement. The Object counting progression is used to analyze children’s knowledge of numbers, counting sequence, understanding that quantities remain the same when they are rearranged, and ability to recognize relationships between numbers. The Object Counting scale includes steps from 0 to 9. Children who are in the beginning of the scale, for example 1, are able to count randomly, but not in sequence. For example, Sarah has 10 toys and she is on level one, she randomly points or touches a toy and gives that toy a number. Children who at the end of the scale count sequentially, count each object only once, and state that the last number counted is the total quantity. When counting the same objects again, the child recognizes that the quantity is the same. If objects are rearranged, the child recognizes that the quantity is still the same, and if another object is added to the sequence, the child can add the object to the sequence without counting all the objects again. For example, Emely has 10 toys, her teacher rearranges the toys and add two other toys, Emely recognizes and states that now there are 12 toys without counting all of them again. Book Orientation is used to analyze children’s basic understanding of what is a book, for example recognizing that books have pages, and how to use books. Book Orientation has a scale from 0 to 5. Children in the beginning of the scale hold a book and flip the pages in a random way without looking at the pages. Children in the end of the scale can hold a book upright, open it from the cover, and turn pages one by one. Print Awareness is used to analyze children’s understanding that there are words and pictures in books. Print Awareness has a scale from 0 to 8. Children in the beginning of the scale only pay attention to pictures, ignoring words completely. Children in the end of the scale differentiate words from pictures, pretend to read, show an understanding that the reading is from top to bottom and left to right, indicate correctly where reading starts, differentiate between letters and complete words, and point to a word when the teacher reads it. For example, David’s teacher is going to read a book for him, and she asks him to point to each word as she reads. If David is in level 8, he will match the word that he hears from his teacher to the word on the book. Electronic Portfolio Procedure. Teachers uploaded evidences such as notes, videos and photos that were related to the three progressions to the KEA platform. They had the option of using a website or the KEA app, through which a teacher could, for example, take a photo using an iPad and upload directly to the platform. After analyzing all the evidences, teachers placed children on each progression. The quantitative data for this study does not include the actual evidences, but it includes the number of evidences collected, which is the main measurement for implementation, and the teachers’ final placement of children on each progression, i.e., the status summary for children. 295 Survey and Interviews. The questions in the survey and interviews included themes such as teacher’s perceptions about effectiveness of their training, easiness of using KEA website and the app, relevance and usefulness of KEA instructions and collected evidences, and demands related to assessment tasks and the KEA process. The survey items took the form of Likert scales or “yes” or “no” answers. All questions in the survey were recoded as dummy variables. The qualitative data for this study also includes statements of 36 teachers who were interviewed. Data Analysis Hierarchical linear models (HLM) with robust standard errors that examined the associations between child demographics, classroom features, and teacher perceptions as independent variables, and evidences collected and children’s placements in each construct progression as dependent variables. Specifically, the dependent variables were overall evidences per child, evidences for Book Orientation per child, evidences for Print Awareness per child, evidences for Object Counting per child, and final child placements on each of the progressions. Level 1 is the student level, and has the following variables: new student (student started school after September 2015), age (in months), repeated preschool or kindergarten (age more than 72 months is the proxy), girl, African American, Hispanic, and free or reduced price lunch eligibility (proxy for poverty). Level 2 is the teacher level, and the variables for this level varied for different dependent variables. The level 2 models included the following variables: rural county, computer or laptop in the classroom, iPad or another tablet in the classroom, teacher had a coach or mentor, teacher had peer support, teacher used the Livebinder, teacher had administrative support, teacher had a fulltime teaching assistant, and teacher perceptions that the trainers were unprepared. Additional variables are from two survey questions: Where you able to make instructional decisions for your students based on the data generated from the Book Orientation and/ or Print Awareness progressions? (variable is called BP Decision); and 2) Where you able to make instructional decisions for your students based on the data generated from the Object Counting progression? (variable is called OC Decision). BP Decision was included in the regressions where the dependent variables were overall evidences, evidences for Book Orientation, and evidences for Print Awareness. OC Decision was included for overall evidences and Object Counting evidences. Because there is no reason to believe that the answer to the BP Decision and OC Decision questions affected the placement of children on each progression, these variables were excluded from the models examining children’s placements on the progressions. Other variables such as number of students in the classroom and teaching experience were also tested and found not to be associated with the outcomes in single-level regression models with heteroscedasticity consistent standard errors. They are not included in the final models because this study only 296 includes variables that were relevant in the single-level models and/ or the dual- level models. Although the state where the electronic portfolio was implemented provided materials such as webinar, the Livebinder, and a manual, individual districts were in charge of planning and implementing their own training. Almost all North Carolina school districts reflect the demographic characteristics of the counties they serve, with only a few counties having more than one school district. Therefore, variation in the implementation is more likely among districts than school buildings, although there was some between school variance within districts. RESULTS On average, teachers collected 4.78 evidences, 1.51 for Object Counting, 1.43 for Book Orientation, and 1.33 for Print Awareness. The means for children’s placements for Object Counting (scale from 0 to 9), Book Orientation (scale from 0 to 5), and Print Awareness (scale from 0 to 8) were 4.71, 4.06, and 5.63 points respectively. As the scales for the progressions are different, even though the means for Object Counting and Book Orientation appear to be close, they are actually very different. While the average score for Book Orientation was close to the maximum, 5 points, the average score for Object Counting was about 50% of the maximum, which is 9 points. The average for Print Awareness was also low, since the maximum is 8 points. Most of the variance in the evidence counts was between classrooms: 79% (overall evidences), 76% (Object Counting evidences), 59% (Book Orientation evidences), and 67% (Print Awareness evidences). Close to half of the variance for children’s placement in the progressions was between classrooms: 44% (Object Counting), 49% (Book Orientation), and 46% (Print Awareness). This section only reports results from coefficients that were statistically significant at the 0.05 level. As the reported numbers are regression coefficients, they are expected values across all teachers in the sample, and can be interpreted as averages. For overall evidences, at level 1, gender was the only statistically significant predictor. Teachers collected 0.31 more evidences for girls than boys. For level 2, the following variables were statistically significantly associated with evidences collected: BP Decision, coach or mentor, Livebinder, and teaching assistant. Teachers who answered the survey question BP Decision positively collected 2.36 more evidences. Teachers who had a coach or a mentor collected 1.89 less evidences. Teachers who accessed and used the Livebinder collected 3.19 more evidences. Teachers who had administrative support collected 2.05 more evidences. Teachers who had a teaching assistant collected 4.55 less evidences. Only the variable new student was statistically significant at level 1 for Object Counting evidences. Teachers collected 0.344 less evidences per child for new students. At level 2, the following variables were statistically significant for 297 Object Counting evidences: rural, computer or laptop in the classroom, iPad or tablet in the classroom, and the Livebinder. Teachers whose school was in a rural county collected 0.68 more evidences. Teachers who had a computer or laptop in the classroom collected 1.20 less evidences. Teachers who had an iPad or computer in the classroom collected 1.90 less evidences. Teachers who accessed and used the Livebinder collected 0.81 more evidences. At level 1, gender is the only statistically significant predictor for Book Orientation evidences. Teachers collected 0.09 more evidences for girls than boys. At level 2, the following variables were statistically significant for Book Orientation evidences: BP Decision, computer or laptop in the classroom, iPad or tablet in the classroom, coach or mentor, Livebinder, administrative support, and teaching assistant. Teachers who answered positively to BP Decision collected 0.63 more evidences. Teachers who had a computer or laptop in the classroom collected 0.97 less evidences. Teachers who had a coach or a mentor collected 0.55 less evidences. Teachers who had an iPad or computer collected 0.58 more evidences. Teachers who accessed and used the Livebinder collected 0.53 more evidences. Teachers who had administrative support collected 1.09 more evidences. Teachers who had a teaching assistant collected 0.96 less evidences. At level 1, none of the variables were statistically significant for Print Awareness evidences. At level 2, the following variables were statistically significant for Book Orientation evidences: BP Decision, computer or laptop in the classroom, and coach or mentor. Teachers who answered positively to BP Decision collected 0.51 more evidences. Teachers who had a computer or laptop in the classroom collected 1.14 less evidences. Teachers who had a coach or a mentor collected 0.61 less evidences. Age and Hispanic were statistically significantly associated with the placements on the Object Counting progression. For each month increase in age, children’s placement increased by 0.09 points. Hispanic children placements were 0.51 points lower than White children. At level 2, the only variable that was statistically significant for Object Counting was rural county. Teachers in rural counties placed children 1.37 points lower than in other counties. At level 1, age and gender were statistically significant for Book Orientation. For each month increase in age, children’s placement increased by 0.03 points. Girls’ placements were 0.16 points higher than boys. At level 2, coach or mentor was statistically significant. Teachers who had a coach or a mentor placed children 0.45 points lower. At level 1, age, gender, and Hispanic were statistically significant for Print Awareness. For each month increase in age, children’s placement increased by 0.06 points. Girls’ placements were 0.37 points higher than boys. Hispanic child placements were 0.49 points lower than for White children. At level 2, rural was statistically significant. Teachers in rural counties placed children 1.08 points lower than those in other counties. 298 DISCUSSION AND CONCLUSION Teachers collected only 4.78 evidences, on average, per child during the first semester of the academic year. According to interviews with the teachers, this relatively low level of implementation was related in part to technical problems with the electronic platform. The qualitative results also suggest that lack of clarity about the purpose of the assessment and inadequate training also contributed. Even though most teachers experienced problems with the KEA electronic platform, interviews with teachers also indicated that those who felt supported and had a clear understanding of KEA’s purpose had a better experience implementing the new tool. One of the most effective resources was the Livebinder, which contributed to almost doubling overall evidences collected. Teachers who used the Livebinder collected on average 3.19 more evidences than teachers who did not. The Livebinder is a web-based resource accessible to all teachers. It contains many very helpful resources for teachers including training materials, how-to videos, and updated materials and announcements. Perhaps most importantly, it contains very thorough and clear explanations of the purpose of the FAP. Other key variables were the ability to use the Book Orientation and Print Awareness progressions to make instructional decisions and administrative support. Teacher who perceived the progressions as useful for instructional purposes collected more evidences. Another relevant resource was having a coach or mentor. Contrary to what was expected, having a coach or mentor was negatively related to all dependent variables. Yet the qualitative data explained the negative repercussion from having a coach or a mentor. Coaches, except for peers, were the closest supports that teachers had to assist them with KEA. However, the interviews suggested that coaches and mentors did not have the support and training that they needed to have a positive effect on the implementation. As many coaches did not understand the purpose of KEA, the usefulness of the portfolio, and how to implement the new tool in an effective manner, teachers therefore lacked the same clarity and information about KEA. Consequently, their perception of KEA was negative, which likely affected their implementation. A comprehensive training for coaches and mentors is central for an effective implementation. Likely, if coaches and mentors have a clear understanding of KEA, they will be able to assist teachers not only with strategies for collecting evidences, but with developing their own clear understanding of the purpose of KEA. Another important finding from this study is the influence of geographical location, which indicates differences in training. Teachers in rural locations collected more evidences, but placed children lower in the progressions. Although one of the limitations of this study is the small number of teachers, if this finding holds for larger samples, more investigation is necessary to understand what are 299 the differences in resources and training between rural and other counties. The extension of this study for 2017 will include interviews with principals and administration, which may elucidate how training was organized and implemented in different schools. The variation in training among different counties in the first year of the program is not necessarily harmful, as long as there is a clear understanding of what was effective. For the subsequent years of the program, because the electronic portfolio is being implemented statewide, a consistent training strategy may be essential to obtain an effective implementation throughout the state. With respect to issues of student diversity, the findings of this study are mixed. There were no differences between African American and White children in their average placements on the progressions. Similarly, there were no differences between children from economically disadvantaged families and all other children in terms of mean placements. However, Hispanic children were placed, on average, lower than other children across all three progressions. These findings raise questions about teacher perceptions of children who may be dual language learners and may be going through a process of acculturation. Teacher preparation programs may need to consider how well they are preparing teachers for the multicultural realities of the American classroom. Furthermore, future research may need to focus on whether there is any evidence of cultural or linguistic biases in the use of the progressions. This study only includes a small fraction of teachers who implemented the portfolio, which is a limitation. However, the survey and interviews provided valuable information about teachers’ experience and perceptions of KEA. The findings indicate that coaches and mentors had a strong influence on teachers’ perceptions and actual implementation. Therefore, an important area of improvement is training for coaches and teachers that assists them with strategies to implement the portfolio and give them a clear understanding about KEA’s purpose and usefulness. As the interviews with teachers reveal, KEA can potentially assist teachers, but to become a useful tool, the first step is to improve teachers’ perceptions of the tool. Also, they need to understand more clearly the mechanics of using the electronic platform, so teachers can upload, analyze, and use evidences for instructional purposes. It is important to notice that this study only included data from KEA’s first year of full-implementation, and as in any beginning, setbacks can be commonplace. References Black, P. J., William, D. (1998). Inside the black box: Raising standards through classroom assessment. Phi Delta Kappan, 80, 139–48. Black, P.J., William, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation and Accountability (formerly: Journal of Personnel Evaluation in Education), 21(1), 5–31. 300 Benner, S. M., Hatch, J. A. (2004). A talent development approach to assessing pre- service performance in early childhood teacher education. Journal of Early Childhood Teacher Education, 25(1), 29–37. Dunn, K. E., Mulvenon, S. W. (2009). A critical review of research on formative assessment: The limited scientific evidence of the impact of formative assessment in education. Practical Assessment, Research & Evaluation, 14(7), 1–11. Graves, B. (2006). PK-3: What is it and how do we know it works? FCD Policy Brief: Advancing PK-3, (4). Retrieved from http://fcd-us.org/resources/pk-3-what-it-and- how-do-we-know-it-works. Heritage, M. (2010). Formative Assessment and Next-Generation Assessment Systems: Are We Losing an Opportunity? Washington, DC: Council of Chief State School Officers. MacDonald, M. (2007). Toward formative assessment: The use of pedagogical documentation in early elementary classrooms. Early Childhood Research Quarterly, 22(2), 232–242. Meyer, E. J., Abrami, P. C., Wade, A., Scherzer, R. (2011). Electronic portfolios in the classroom: Factors impacting teachers’ integration of new technologies and new pedagogies. Technology, Pedagogy and Education, 20(2), 191–207. National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington. DC: U.S. Department of Education. National Council of Teachers of Mathematics. (2007). What is formative assessment? A research clip of the National Council of Teachers of Mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2013). Formative Assessment: A position of the National Council of Teachers of Mathematics. Reston, VA: Author. Wiliam, D., Thompson, M. (2007). Integrating assessment with instruction: What will it take to make it work? In C. A. Dwyer (Ed.), The Future of Assessment: Shaping Teaching and Learning (pp. 53–84). Mahwah, NJ: Lawrence Erlbaum Associates. USING MULTIPLE REPRESENTATIONS OF FRACTIONS TO ENHANCE PROBLEM SOLVING Bojan Lazićand Jasmina Milinković Abstract In this paper we report on a study of the effects of propaedeutic approach related to development of understanding of multiple representations of fractions. Grounded in the literature review on the children’s understanding of representations on one hand and problem posing on the other, we analyze students’ achievements in solving problems, University of Novi Sad, Serbia; e-mail: [email protected] 301 particularly examining transfer of knowledge in solving non-realistic problems and problems in real context involving different representations of fractions. A sample of fourth grade school children were enrolled in the experimental program in regular school setting. The results obtained in the experimental study show that the integration of learning content in propaedeutic learning has significant impact on the performance of students in non-realistic problems as well as in problems presented in a realistic context. Our discussion of results point for the need for further examination of the effects of using multiple representations of math concepts on children’s adeptness in solving particular types of problems. Keywords: Fractions, representations, non-realistic context, real context INTRODUCTION The paper is one of the outcomes of the research project focused on investigation of a propaedeutic approach to teaching fractions in primary grades (Lazić and Maričić, 2015, Lazić et al., 2012). Considerable challenge in developing teaching lessons is linked to the choice of representations of concepts. On the other side problem design has been on the spotlight in last years (Singer et al., 2015, Margolinas, 2013). In this study, we focused attention to students’ achievements in solving problems including various representations of fractions. Representations, as a ways for presenting concepts are considered to be important means for understanding. The importance of mathematical models that accurately reflects the characteristics of the abstract concepts has been pointed by researchers (Doerr and English, 2003). It is believed that using multiple representations in problems may contribute to higher achievement (Cai, 2013). Contemporary trends in teaching mathematics put emphasis on learning by solving mathematical problems through models from real life, where mathematics becomes a tool for predicting possible outcomes of real life situations. For example, the concept of fractions appears naturally to children since early childhood. The process of learning about fractions starts in family setting, occurring in sharing (e.g. apples, oranges, bread, cakes, chocolate). Notice that number of parts depends on the number of parts participating in the division, and that these parts are equal. Propaedeutic approach has characteristics similar to realistic mathematics education. Real situations are taken as a starting point in teaching and learning of mathematics, because they stimulate students to develop models of abstract mathematical ideas through visualization. Precondition for the organized, proper, semi-formal learning is timely use of carefully crafted realistic visually appealing models (pictures and manipulatives) and realistic problem situations. At the beginning, children observe real, concrete object as a whole, and then parts of the whole, equal split in two-division of the whole into two equal parts, as well as explicit and visual presentation of the two parts of this whole. 302 According to research findings (Charalambos and Pitta-Pantazi, 2007; Pita- Pantazi et al., 2004) one of the factors that contributes to a marked complexity of learning and knowledge transfer is introduction of multiple representations of fractions. It is widely agreed that representations play an important role in learning fractions, and should be treated as essential elements in supporting the understanding of the mathematical concept of fractions (NCTM, 2000). Only after the concepts are visualized, named and explained, mathematization begins with the introduction of symbols (reading and writing the symbolic notation). In the interactive dialogue, teachers are directing students' attention towards models that present a fraction. Researchers emphasize convenience of particular representations. While Fosnot and Dolk (2003) discuss convenience of symmetrical figures, Cramer and others (2008) point to suitability of circle representation, but many others emphasize functionality of number line (Ball, 1993; Behr et al., 1992; Lesh et al., 1987; Bright et al., 1988). Ball (1993) notes that the presentation of the fractions on the number line helps students to focus on the basic characteristic of fractions as parts of the given whole, by visually stunning division of the given segment as a whole to a certain number of equal parts as the visual display of certain fraction. Presented geometric model of visualization of fractions is especially useful in measurement, as illustrative presentation of measurement units and their relationship in a fraction form. Using symbolic representations is final goal but some of teachers rely on various representations as models of abstract math ideas in process of learning while others quickly move to symbolic ones. Although using different representations require some additional effort, the use of different forms of representations of mathematical concepts in teaching has been strongly advocated (Abrams, 2001; Cuoco and Curcio, 2001). Given the above, we can conclude that, especially in the early stages of learning and teaching fractions, multiple representations of fractions play essential role as visual models. Beside importance for the initial introduction of fractions, visual representations are extremely useful at other levels of learning. Constructing geometric models suitable for displaying fractions, allows to connect the abstract concept of fractions with concrete presentations. This content makes it easier and more understandable, which further affects learning in a positive and functional knowledge transfer. METHODOLOGY Overall goal of our research was to study primary children initial learning of fractions. An experimental program based on propaedeutic teaching method, consisting of 28–lessons for 22 teaching units in fractions was designed, implemented and observed in a 4th-grade primary school classes. Each learning unit consists of a sequence of integrated activities designed to explore concept of fractions with multiple opportunities for using different representations of 303 fractions. This approach is unlike conceptual structure of the mathematics curriculum in the Republic of Serbia, which is dominantly characterized by atomization of content matter and teaching practice focused on mastering bits and pieces of knowledge. The control group had this approach in place. The goal was to investigate impact of this alternative approach on students’ problem solving. The current study is based on data collected before the experimental program started when initial test was distributed and at the end, during the final class when teacher evaluated children’s progress in solving problems. The study was designed as experiment with parallel groups. We formed two even groups – experimental (E) and control (C) with the total sample of 140students aged 10. The students in experimental group were selected from four classes of a primary school, and the control group consisted of students from four classes of another primary school. The experimental and the control group were balanced by the number of students, gender structure, average grade and average score in mathematics at the end of the 3rd grade, as well as by the equal arithmetic mean of points scored on the initial test. The aforementioned equalization was possible because we separated 70 students for the control and 70 students for the experimental group from the total sample of four classes from both schools. The research was longitudinal and realized in the course of two academic years, through several stages. In first year, after the initial testing of students, we introduced an experimental program during regular mathematics classes in the experimental group. At the same time, mathematics classes were conducted in a usual way, in accordance with the mathematics curriculum for the 4th grade. General description of the experimental program based on the propaedeutic approach as well as exemplary activities are reported elsewhere (Lazić and Maričić, 2015, Lazic et al., 2012). The priority was in the integration of elements of math curriculum related to fractions with division and measurement. Models of fractions used in the experimental program are described in the introductory part of the paper. Here we report on the results of the first final testing, which measured the short term effects of propaedeutic learning about fractions. The test consisted of problems with various representations of fractions. Both, the initial and the final test consisted of 12 items. Examples of problems are presented in Figure 1. These items are exemplary for the types of representations used in problems: pictorial, number line, in words and symbolic ones. Some of the problems are in non-realistic context while others are in real context. For example, in the first problem, students are asked to express in fractions part of the shaded picture (non-realistic context), while in the last item they answers are based on survey data presented in the pie chart (real context). 304 Figure 1: Examples of final test items We ensured test objectivity by placing each student in approximately the same test situation, by ensuring that independent examiners acted on uniform instructions, and that task assessment was conducted on the basis of the key. We determined the logical and content validation of tests by defining the tests' correspondence with the requirements of the curriculum and the contents they refer to. Instrument accuracy was determined by calculating the Cronbach’s alpha coefficient (α = .88) which indicates a high instrument reliability. Discrimination of the test was determined by item analysis. The coefficient of discriminative value of each task in the test varies from .14 to .25. The data obtained in this research were processed by using the software package IBM Statistics SPSS20, where a single factor analysis of variance (ANOVA) was used to longitudinally monitor the effects of the experimental program in the experimental group. RESULTS AND DISCUSSION Applying knowledge in solving problems in a non-realistic context First, we examined whether propaedeutic learning fractions based on the vertical integration of mathematical content has positive effects on students’ ability to solve problems. In the initial testing both groups of students achieved approximately similar uniform results: experimental (M = 4.67, SD = 2,394), a control group (M = 4.61, SD = 2.896) (Table 1). Calculated variance (F (1,138) = 0.016; p = 0.899) indicated that there was no statistically significant difference in the performance of students in the separate tasks of resolving the problem in the initial check (Table 2). 95% Confidence Interval for Mean Тest Group N Std. Lower Upper Min Max Mean Std.Er. Bound Bound Dev Initial test Experimental 70 4.67 2.394 2.394 4.10 4.10 0 12 Control 70 4.61 2.896 2.896 3.92 3.92 0 12 Total 140 4.64 2.647 2.647 4.20 4.20 0 12 305 Final Test 1 Experimental 70 11.47 3.068 3.068 10.74 10.74 3 15 Control 70 3.99 2.966 2.966 3.28 3.28 0 10 Total 140 7.73 4.811 4.811 6.92 6.92 0 15 Table 1: Descriptive indicators of performance of experimental and control group on nonrealistic tasks After the initial measurements in the experimental group we carried out an experimental program with propaedeutic learning methodical approach to the introduction of fractions, based on the vertical integration of content, while the control group pursued working in a customary manner. After the realization of the experimental program followed by final measurement. Looking at Table 1 we can see a considerable improvement in the average of the number of winning points at the final measuring the pupils of the experimental group (M = 11.47, SD = 3,068), while the control group of the students has shown a decrease compared to the initial measurement (M = 3.99; SD = 2.966) (Figure 2). Figure 2: Box plot diagrams of achievement on problems in non-realistic context The variance at the final test (F (1,138) = 215.415; p <0.001) indicates that there are no statistically significant differences between the experimental and control groups at the final measuring ability of the student to solve mathematical problems (Table 2), which confirms our assumption about the influence of the applied methodical approach. Sum of Mean Square df F Sig. Squares Between Groups .114 1 .114 .016 .899 Initial test Within Groups 974.029 138 7.058 Total 974.143 139 Between Groups 1961.257 1 1961.257 215.415 .000 Final test Within Groups 1256.429 138 9.105 Total 3217.686 139 Table 2: ANOVA analysis 306 In order to remove the suspicion that the result is the result of imbalances experimental and control groups, we calculated the covariance (ANCOVA). As covariate was taken at the initial measurement result. The covariance between the groups (F 1,137) = 365.449; p <0.001) negate the suspicion that the differences are the result of unevenness in the experimental and control groups, but that the result of the action of the applied methodology point (Table 3). Confirmation of the effects size of the experimental program comes from calculated the resulting partial eta squared (0.727) which indicates a high impact of methodical approach (Cohen, 1988, Under: Pallant, 2011). In addition, we established a strong link between the results of investigation of the effect of the applied methodical approach to solving its own problem tasks before and after the experimental approach (partial eta squared is 0.849).The results indicate that the propaedeutic learning fractions, based on the vertical integration of the content, in teaching mathematics brings a considerable effect on the ability of children to the solve tasks in nonrealistic context. Type III Sum Mean Partial Eta Source Df F Sig. of Squares Square Squared Corrected Model 2875.102a 2 1437.551 574.881 .000 .894 Intercept 356.708 1 356.708 142.649 .000 .510 Initial test 1932.135 1 1932.135 772.666 .000 .849 Group 913.845 1 913.845 365.449 .000 .727 Error 342.583 137 2.501 Total 11580.000 140 Corrected Total 3217.686 139 Table 3: Analysis of covariance The results and their analysis indicate that the propaedeutic learning fractions, based on the vertical integration of the content, in teaching mathematics leads to a considerable effect on the ability of children to solve problems in non-realistic context. Applying knowledge in solving problems in real context We wanted to examine whether the applied methodical approach had effects on the application of acquired knowledge in solving problems in realistic context. Looking at Table 4 we see that in the initial testing of both groups reached a students’ average an equal number of points (the experiment (M = 4.81, SD = 1.739), a control group (M = 4.81, SD = 3.281)), wherein among students of the control group were no major fluctuations in the average number of points Calculated variance (F (1,138) = 0.000; p = 1.000) indicated that there was no statistically significant difference in the pupils success in solving the tasks given in the context of realistic (Table 5). 307 95% Confidence Interval for Mean Test Group N Mean Std.Dev Std.Er. Lower Upper Min Max Bound Bound Initial Experimental 70 4.81 1.739 .208 .306 5.23 0 10 Control 70 4.81 3.281 .392 .405 5.60 0 12 Total 140 4.81 2.616 .221 .408 5.25 0 12 Final Experimental 70 13.06 2.559 .306 12.45 13.67 5 15 Control 70 5.50 3.391 .405 4.69 6.31 0 12 Total 140 9.28 4.831 .408 8.47 10.09 0 15 Table 4: Descriptive indicators of performance of experimental and control group on realistic tasks At the final measure students of the experimental group were significantly more successful in solving the real context tasks compared to the initial measurement (M = 13.06, SD = 2.559), but also in relation to the students in the control group (M = 5.50; SD = 3,391) (Table 4, Figure 3). Figure 3: Box plot diagrams of achievement on problems in real context The variance in the final measurement between the experimental and control groups (F (1,138) = 221.513; p <0.001) suggests that the differences between the groups was statistically significant, indicating that the experimental program a substantial effect on the performance of students in performing tasks in a realistic context. Sum of Mean Square df F Sig. Squares Between Groups .000 1 .000 .000 1.000 Initial test Within Groups 951.171 138 6.893 Total 951.171 139 Between Groups 1998.864 1 1998.864 221.513 .000 Final test Within Groups 1245.271 138 9.024 Total 3244.136 139 Table 5: ANOVA table 308 The calculated covariance (F (1,137) = 845.922; p <0.001), confirming results of the overseas obtained about the progress of the students in the experimental group under the influence of the applied methodical approach and rejects the suspicion that differences occur as a result of unevenness in the group (Table 6). Partial eta squared (0.861) indicates that the impact of the methodical approach is large (Cohen, 1988, Under: Pallant, 2011, p. 212), and that there is a strong link between the results of the methodical approach to the application of knowledge in solving problems in a realistic context before and after its application (0,740) (Table 6). Type III Sum Mean Partial Eta Source df F Sig. of Squares Square Squared Corrected Model 2920.413a 2 1460.206 617.962 .000 .900 Intercept 654.084 1 654.084 276.809 .000 .669 Initial test 921.549 1 921.549 390.001 .000 .740 Group 1998.864 1 1998.864 845.922 .000 .861 Error 323.723 137 2.363 Total 15297.000 140 Corrected Total 3244.136 139 Table 6: ANCOVA Table Knowledge in solving problems involving different representations Finally, we attended to the achievement of students in solving problems involving different types of representations discussed earlier: figural form (symmetrical 2D shapes), number line, pie chart diagram, word form and symbolic form. Based on their performance on the tasks, students appear to find problems with certain representations more challenging than with others. The best results were obtained with problems involving fractions represented on a pie chart. Similarly high percent of correct answers was obtained in tasks involving number line. More than half correct answers were given in problems involving other graphical representations. In contrast, problems with symbolic representation seem to be relatively difficult (Table 7). Representation used in problem Group Control Experimental Fractions in symbolic form 29,42% 45,86% (2/5) Fractions as parts of figures 42,37% 60,15% (stars, polygons) Fractions on pie charts 47,00% 72,65% Fractions on number line 45,20% 70,25% Fractions in words 28,70% 52,41% (two fifths) Table 7: Percentages of correctly solved problems 309 CONCLUSION Our findings show that propaedeutic learning content of fractions in elementary grades, based on the vertical integration of mathematics content, significantly affect the performance of students in independent problem solving tasks and apply the acquired knowledge in solving problems presented in a realistic context. The findings also point to importance to attending to features of the problems that make problems easier or difficult. 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THE USE OF THE REFERENCE POINT STRATEGY FOR MEASUREMENT ESTIMATION Alenka Lipovec and Jasmina Ferme Abstract Measurement estimation is an important skill in daily life and it is unavoidable during formal education. In this paper, we discuss the use of the reference point strategy, which can help us estimate a measure of each quantity and it also has many advantages in comparison to other measurement estimation strategies. Based on a survey of students aged 11 to 12 (N=214) we found out that more than 40 % of study participants do not use spontaneously the reference point strategy. Estimation skill of those participants could be improved, if they are explicitly taught that strategy. Based on results of study we list some recommendations for school work. These recommendations include use of concrete illustrations and class discussions. Keywords: Measurement, estimation, reference points, students’ strategies University of Maribor, Slovenia; e-mail: [email protected], [email protected] 311 Theoretical framework Measurement estimation or estimation in measurement can be described as making a measurement without using measurement tools (Bright, 1976 in Gooya et al., 2011). In the process of measurement estimation, we make subjective judgements or express opinions about measures of quantities (Adams and Harrell, 2003 in Gooya et al., 2011). Measurement estimation is used consciously or subconsciously, with or without a purpose every day. Because of its often use in daily life, it is also unavoidable during formal education. According to Chang et al. (2011) a measurement estimation has a key role for learning measurement of quantities. Activities which comprise measurement estimation also enable transmission of knowledge to other mathematical areas, mainly to the area of numbers and counting (Bright, 1976, in Joram et al., 2005) and to fractions and rations (Coburn and Shulte, 1986 in Joram et al., 2005). They help students develop problem solving skills and encourage them to explore the link between abstract mathematical ideas and real-word situations (Hodgson et al., 2003 in Gooya et al., 2011). When solving problems, which demand measurement estimation, we can use several measurement estimation strategies. It is about methods and procedures which help us determine the estimate of measure of some quantity. In this paper, we focus on the reference point strategy. Reference points are also called benchmarks, mental rulers (in case of estimating length) or individual frames of reference. It is about measures of objects, which are known to estimators and help them estimate measure of another object (Hartono et al., 2015) in a way that estimator compares the object to be estimated with the chosen reference point (Joram et al., 2005). Reference points typically pertain to objects, which are well- known to estimators and have great importance for them. Maybe they are even more meaningful than the corresponding standard units (Carter, 1986 in Joram et al., 2005). For example, a student who knows the length of a new pencil, object, with which he is often in touch, compares this with the height of a milk package. Then the student finds out that the milk package’s height is »a little« larger as the length of a pencil and based on the length of a pencil gives an estimate of the milk package’s height. Estimation, where we use reference points, has at least two advantages in comparison to estimation where we do not use them and only use standard measurement units. First is that reference points make standard measurement units more meaningful for students, they deepen knowledge of them and show their sense (Joram et al., 1998). In addition, measurement estimation with reference points may lead in executing less mental steps, thereby the probability of errors made in a process of measurement estimation decreases (Joram et al., 2005). Joram et al. (2005) have found out in their research that use of the reference point strategy positively affects estimation accuracy of lengths, heights and widths of objects, and 312 also the accuracy of representations of standard units of mentioned quantities. Mentioned authors suggest an additional plus of use of the reference point strategy, important especially for young estimators. The process of measurement estimation is namely more explicit as estimation where we use only standard measurement units. Mainly the transmission of reference object to the object to be estimated, which is first executed physically and then mentally, is much better illustrated. This makes it more accessible to students and consequently students can follow correct execution of measurement estimation with ease. Measurement estimation with reference points is included in study process as a part of realistic teaching of mathematics (Buys and De Moore, 2004). In the Netherlands for example, as the reference point for one litre they introduce a volume of a milk bottle, for one meter a length of a long footstep or for one kilogram a mass of sugar package (Buys and De Moore, 2004). Purpose of research The main aim of this study is to research use of the reference point strategy for measurement estimation. We have been mainly interested if knowledge of some object’s measure influences on the estimate of measure of the same quantity of another object. With that we have been actually interested in frequency of students’ (aged 11 to 12) use of the reference point strategy. Studies (Gooya et al., 2011 and Joram et al., 2005) indicate that students very often do not spontaneously and arbitrarily use the reference point strategy for measurement estimation. According to Clements's (1999) or Bright’s (1976 in Hartono et al., 2015) opinion, the primary goal of (length) estimation instruction should be exactly developing students’ points of reference. This is namely among other things related to developing students’ sense of measures and their understanding of measurement (Bright, 1976 in Hartono, 2015). Methodology Methods of quantitative empirical pedagogical research which we used are descriptive and causal non-experimental method. This study has been conducted based on fulfilled surveys of 214 students. Survey was executed in November 2016 in Slovenia. The sample has been conventional. Data gained from surveys has been analysed with a program named IBM SPSS Statistics 23. Here we list only part of results gained from described study. In statistical sample were included 214 of primary school students, aged 11 to 12, with average age 11,689 years and standard deviation 0,4626. The percentage of male participants was 43 %, 57 % participants were females. Anonymous survey was, besides questions about age, grade and sex, comprised of 8 open ended questions, pertained to measurement estimation. Questions were divided in two sections. Questions of first part required measurement estimation of objects, measures of which were supposed to be 313 known to participants of the study and were supposed to represent their reference points. In this section of questions students had to estimate length of a new pencil, area of A4 format paper, volume of a can of Coca Cola drink and mass of a »large« Milka chocolate. Measures of these object were supposed to be students’ reference points. Similarly, the second section contained four questions. They were pertained to the same quantities as questions of first part, but required to estimate measures of objects, which are in our opinion not exactly known to students. They had to estimate height of a carton package (cardboard container, shape of rectangular solid) of one litre of milk, area of the biggest surface of »large« Milka chocolate, amount of water in a toilet tank and mass of a paperback notebook with 52 sheets. Validity of survey was provided by a pilot experiment and by discussing with experts from the area of early didactics in mathematics. Reliability of the survey was provided by thoroughly formulated questions. According to the aim of study, we asked ourselves the following two study questions. 1. Are length of a new pencil, area of A4 format paper, volume of a can of Coca Cola drink and mass of a »large« Milka chocolate reference points for students, aged 11 to 12? 2. Do students use the reference point strategy for estimation of lengths, areas, volumes and masses of objects? Results Reference points If given estimate of measure belongs to an interval, which comprises values, from accurate value differ 10 (for length) or 30 % (for other quantities) and if boundaries are rounded to integers (lower bound downwards, upper bound upwards), we will name it appropriate estimate. The range of appropriateness of estimates is determined by Van de Walle (2014). Estimated quantity The number of students, who have given an appropriate estimate f (f %) Length of a new pencil 55 (25,7%) Area of A4 format paper 38 (17,8%) Volume of can of a Coca Cola drink 65 (30,4%) Mass of a “large” Milka chocolate 61 (28,5%) Table 1: The number of students, who have given an appropriate measure estimate 314 Table 1 shows the number (and proportion) of interviewees who have given appropriate estimates of measures of object from first section of questions. We can see that for each quantity less than a third of participants have given an appropriate estimate of a measure of an object, therefore we conclude that these measures do not represent reference points for students, aged 11 to 12. The use of the reference point strategy In continuation, we will present results which demonstrate the relation between appropriateness of estimate of supposed reference point and appropriateness of given estimate for length, area, volume and mass of entitled object (i.e. object, of which the same quantity the participants had to estimate). Only 15 (7 %) participants wrote appropriate estimates of the length of a new pencil and the height of a carton package of milk, however, 94 (43.9 %) participants did not give an appropriate estimate of any of the two measures of mentioned objects. The percentage of people who gave an appropriate estimate of the length of a pencil and an inappropriate estimate of the height of a carton package of milk, is 18.7 %. Based on results of χ2 test (χ2 =3,233, P=0,072), we cannot conclude that the appropriateness of estimate of a new pencil’s length and appropriateness of estimate of the height of a carton package of milk are dependent. We found out that 15 (7 %) participants gave appropriate estimates of areas of a paper and a surface of a chocolate, however, 161 (75.2 %) participants did not give an appropriate estimate of areas of any of mentioned objects. The percentage of interviewees who wrote an appropriate estimate of the paper’s area and inappropriate estimate of area of chocolate surface, is 10.7 %. Based on results of χ2 test (χ2=24,838, P=0,000), we found out that variables, appropriateness of area estimate of A4 format paper and appropriateness of area estimate of the biggest surface of chocolate, are dependent. The percentage of study participants who gave appropriate volume estimate of a can of Coca Cola drink and appropriate estimate of the amount of water in a toilet tank is 7.9 %. The percentage of participants who gave inappropriate estimates of both mentioned objects is 50.5 %. The percentage of students who estimated appropriately the volume of a first, but inappropriately the volume of a second object, is 22.4 %. Based on results of χ2 test (χ2 =0,043, P=0,837) we cannot conclude that the appropriateness of volume estimates of a can and of water in a toilet tank, are depended. We found out that 6.5 % participants wrote appropriate estimate of a mass of “large” Milka chocolate and appropriate estimate of a paperback notebook with 52 sheets, however, 57.9 % of study participants wrote inappropriate estimates of both object. The percentage of people, who gave appropriate estimate of a mass of chocolate and inappropriate estimate of a notebook’s mass is 22 %. Based on results of χ2 test (χ2=0,434, P=0,510) we cannot confirm the dependence of appropriateness of estimates of chocolate’s and notebook’s mass. 315 Discussion Results of our study show that dependence of estimates of measures of two entitled objects occurs only for area estimation. This indicates the use of the reference point strategy for area estimation, where the reference point is area of an object from the first section of questions (area of A4 format paper). However, it is also possible that participants used some other measurement estimation strategy (for example comparing areas of two objects). For estimation of length, height, volume and mass results show two possibilities: a) students don’t use reference point strategy or b) quantities of objects used in our study are not reference points for participants. Several studies confirming low level of intuitive use of the reference point strategy are consistent with the first case (Joram et al., 2005 and Gooya et al., 2011). On the other hand, there are some authors who point out the importance of choosing the appropriate object as reference point (Joram et al., 2005, Van de Walle, 2014, Buys and De Moore, 2004). Since the results of estimates for reference point objects are quite low (30, 4 % or less), we cannot confirm which of cases, a) or b), describes the results of our study in the optimal way. We believe that results are simultaneously shaped by both cases. Students, who did not provide appropriate estimate of quantity of any of two related objects (e.g. can of Coca Cola drink and toilet tank), don’t use the reference point strategy since they have not developed reference point yet. Results of our study show that the average share of such students is above 42 %, what is in accordance with results of other studies. When estimating area more than 75 % of students don’t comprehend area of A4 format paper as reference point object since they are not familiar with the quantity of it. The results confirm the hypothesis of persistence of illusion of linearity in human thinking (De Bock et al., 2007). Students who did not provide appropriate estimate of the second quantity, whether use reference point strategy unsuitably or they don’t use it at all. Maybe these students haven’t developed suitable reference points for measurement estimation yet. The fact is that the development of students’ reference points sometimes cannot be achieved. This is because of student’s own internal process, which cannot be forced, because it relates to a mental perception depending what students perceive and experience (Hartono et al., 2015). Maybe students, aged 11 or 12, are not old enough that they would develop enough wide conscription of suitable reference points. Percentage of participants who didn’t use reference point strategy when estimating is quite high (more than 40 %, for area estimation even more than 75 %). This is congruent with a study, conducted by Joram et al. (2005). They note that only 9 % of students in school spontaneously use the mentioned strategy. As a solution of a relatively poor quantity estimation, the reference point strategy is encouraged to be used to improve estimation. 316 The research has some limitations. The sample includes students, who were not chosen randomly or objectively. This sample is consequently not representative. The questions of the survey were not precise enough, description of certain objects did not ensure that all participants imagined themselves the same object as we. For example, the description of the notebook at mass estimation (paperback notebook with 52 sheets) allows more options, because such notebooks may vary in format, weight of the cover, thickness of sheets … all of which affect its mass. We assumed that the students know length of a new pencil, area of the sheet A4 size, volume of a can of Coca Cola drink and mass of the "large" chocolate Milka, and therefore that measures of the quantities of those items correspond to the students’ reference points. Further research could determine if students, aged 11 to 12, are too young to use the reference point strategy or to develop a collection of reference points. Further research would also need to examine reference points of students. We believe that on the basis of the obtained findings it would be reasonable to create a basic set of reference points, development of which would be encouraged during formal education. Conclusion Based on our findings that children aged 11 to 12 do not use the reference point strategy to a large extent and quantities are relatively poorly estimated, we suggest building up the use of this strategy and thereby encouraging the development of students’ reference points. Bright (1976 in Joram et al., 2005) has stated that a good estimator has effective and a wide range of reference points, which he is able to apply flexibly. Above mentioned author and others, for example (Clements, 1999 and Hildreth, 1983), believe that the development of reference points should be a significant goal in teaching measurement estimation in schools. Hartono and others (2015) see a solution in discussion of students which refers to their reference points. In their opinion, a person can on the basis of discussion about the perception of quantities develop effective personal reference points and improve them. Hogan and Brezinski (2003 in Hartono, 2015) think that when measuring and estimating quantities students’ interaction with physical objects is important. Application of concrete objects should reinforce and improve students’ knowledge about measures of individual objects (Hartono et al., 2015), that is why when estimating and developing reference points use of concrete objects should be encouraged. In addition, we propose developing of reference points that are thoughtfully selected and adapted to the entire class, and yet general enough to allow estimation of different quantities of diverse measures. When choosing reference objects, we have to bear in mind that reference points may be in standard and nonstandard units. It may be useful to point out that despite the positive impact of reference 317 points, one should not exaggerate with their range because it can lead in confusion and short duration of their memorization. It is necessary to be aware of the fact that the development of students’ reference points, as we have written yet, sometimes cannot be achieved (Hartono et al., 2015). Estimation can be improved through the implementation thereof (Gooya et al., 2011). We should provide specific instructions for estimation in tasks which require estimates of measures. Chang and others (2011) argue that the poor estimation (of lengths) of objects is caused by the excessive openness of tasks, which is due to the nature of the content expected and has certain advantages (possibility of exploration). A lack of precise instructions causes confusion and problems (with understanding and misdirection of attention). When preparing tasks, state precisely how accurate estimation has to be, how it should be obtained and provide explicit description of the object being estimated. References Buys K., De Moore E. (2004). Domain description measurement. M. van den Heuvel- Panhuizen in K. Buys (ur.). Young children learn measurement and geometry: A learning-teaching trajectory with intermediate attainment targets for the lower grades in primary school (pp. 15–36). Rotterdam, Taipei: Sense. Chang, K. L., Males, L. M., Mosier, A., Gonulates, F. (2011). Exploring US textbooks’ treatment of the estimation of linear measurements. ZDM, 43(5), 697–708. Clements, D. H. (1999). Teaching length measurement: Research challenges. School Science and Mathematics, 99(1), 5–11. De Bock, D., Van Dooren, W., Janssens, D., Verschaffel, L. (2007). The illusion of linearity: From analysis to improvement (Vol. 41). Springer Science & Business Media. Gooya, Z., Khosroshahi, L. G., Teppo, A. R. (2011). Iranian students’ measurement estimation performance involving linear and area attributes of real-world objects. ZDM, 43(5), 709–722. Hartono, R., Ilma, R., Hartono, Y. (2015). Supporting the development of students’ reference points for length estimation. Retrieved 03.03.2017 from http://eprints.unsri.ac.id/5783/1/ hartono.pdf. Hildreth, D. J. (1983). The use of strategies in estimating measurements. The Arithmetic Teacher, 50–54. Joram, E., Subrahmanyam, K., Gelman, R. (1998). Measurement estimation: Learning to map the route from number to quantity and back. Review of Educational Research, 68(4), 413–449. Joram, E., Gabriele, A. J., Bertheau, M., Gelman, R., Subrahmanyam, K. (2005). Children's use of the reference point strategy for measurement estimation. Journal for Research in Mathematics Education, 4–23. Van de Walle, J. A., Karp, K. S., Lovin, L. A. H., Bay-Williams, J. M. (2014). Teaching Student-centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (Vol. 2). Pearson Higher Ed. 318 STUDENTS’ VISUAL REPRESENTATION OF FRACTIONS AND EXPONENTIATION Alenka Lipovec and Manja Podgorošek Abstract Visual representations are an important part of learning mathematics and also a good indicator of mathematical knowledge of students. This paper presents the results of research conducted with 1272 students, aged between 10 and 12 years. Students were asked to provide a drawing depicting numerical expressions 15 and 2 . The data were analysed through three aspects: (a) adequacy of the drawing to embody the required concept, (b) conceptual or procedural orientation of the drawing, and (c) correctness of the result of numerical expression. It turns out that approximately 60 % of students represented the required concept adequately. Procedurally oriented drawings prevailed for both concepts, and predominantly for exponentiation. Additionally, procedurally oriented drawings were more prone to arithmetic mistakes than conceptual ones. Keywords: Representations, drawings, conceptual and procedural types of knowledge, drawings, fractions, exponentiation Theoretical Framework The mathematical representations allow us to think and communicate mathematically, to explore and interpret the meanings of mathematical concepts, relationships and processes (NCTM, 2000), therefore they play an important role in mathematics education. The term visual representation is used in various contexts, including static graphic representations (pictures, schemes, diagrams, etc.), dynamic graphic representations (video, applet, etc.), and even concrete representations that are viewed, but not changed, by the students. According to Duval (2014), visual representations are all kinds of representations used in mathematics and in the teaching of it; on the other hand, visualisation is the recognition, more or less spontaneous and rapid, of what is mathematically relevant in any visual representation. ‘For a mathematician and a teacher, there is no real difference between visual representations and visualisation. But for students, there is a considerable gap that most are not always able to overcome even throughout their mathematics education. They do not see what the teacher sees or believes that they will see (Duval, 2014, p. 160). Presmeg (2014) states several questions that need to be addressed in this research domain, and points out that an overarching theory of visualisation in mathematics education has not yet been established. She urges researchers to explicitly state their definition of visual representation in their research. In this paper, the term graphic representation will be used to describe static pictures. It is known that University of Maribor, Slovenia; e-mail: [email protected], [email protected] 319 graphic representations can help students to visualise mathematics concepts, foster their understandings, and help them communicate these understandings (Ryve, Nilsson and Pettersson, 2013). Graphic representations are used regularly in schools in the form of drawings, diagrams and tables and are the leading communication tool in some areas of mathematics, such as geometry or graph theory. In some cases, graphic representation seems to be a proficient strategy in mathematics education for all students (Arcavi, 2003). On the other hand, an older meta-analysis found no significant benefits of pictorial representations when compared with concrete or symbolic representations (Sowell, 1989). Some researchers even reported harmful effects of pictorial representation oriented instructions (de Bock e al., 2007; Siegler and Thompson, 2014). We have chosen two basic mathematical concepts: exponentiation and fractions. Exponentiation is an important mathematical concept that is useful in understanding the phenomena of life i.e. surface, volume … Even though some research show existence of an early intuitive knowledge about the characteristics of nonlinear growth (Ebersbach et al., 2010), it was argued that most university students not progress beyond an action-level understanding of these topics (Weber, 2002). The similar conclusions were drawn for fractions, where several studies have showed that it is quite demanding concept. Siegler et al. (2012) argued that students' early understanding of fraction magnitude predicts general mathematics achievements. This hypothesis was verified also by Torbeyns et el. (2015) in a study conducted with 6th and 8th graders from USA, China and Belgium. Nevertheless, it is well documented that students have severe difficulties in learning fractions (McMullen et al., 2015). It has been stated before that measuring mathematical knowledge is a demanding task for researchers. In 1976, Skemp proposed categorising relational and instrumental knowledge. This led to two types of knowledge: conceptual and procedural (Hiebert and LeFevre, 1986; Rittle-Johnson and Siegler, 1998). Although cognitive approaches to mathematics education equally emphasise conceptual and procedural types of knowledge, the debate in the research community about defining and measuring conceptual and procedural knowledge is ongoing (Rittle-Johnson, Schneider and Star, 2015). There is a lot of criticism of this distinction, especially regarding the definition of procedural knowledge (Star, 2005). Ainley, Pratt and Hansen (2006) also propose a third dimension of understanding, which takes into account the usefulness of the idea. Despite these shortcomings, we decided to adopt categorization on procedural and conceptual types of knowledge as defined by Rittle-Johnson and Siegler (1998) and Hiebert and LeFevre (1986). Procedural knowledge is therefore defined as knowledge of sequences of steps or actions that can be used to solve problems, and conceptual knowledge is defined as knowledge that is rich in relationships. It has to be stated 320 that our belief is that neither of these two types is better than the other, or is it more superficial or deep than the other. It is our believe that analyzing students’ knowledge as seen through their representations might reveal gaps and misconceptions in students’ understanding and help teachers to develop efficient teaching strategies. Aim The purpose of the study was to examine the 6th grade students' knowledge of exponentiation and fractions in order to develop instruction that would enhance their constructions of mathematical meaning. The main goals of the study were to find out: whether students’ representations represented the required mathematical concept, which type of knowledge (procedural or conceptual) is reflected in the representations of certain mathematical concepts, and how the correctness of the numerical expression result is connected to the type of knowledge. With that in mind, we wanted to identify targets for instruction. Methodology The sixth grade students were selected as the target population. At that age, according to the Slovenian curriculum, students have already been introduced to both mathematical concepts used in the study. This study was based on a convenience sample of 1272 sixth graders. Data were collected through an anonymous questionnaire, the results of which did not affect students’ grades in any way. Participants were provided very scant instruction, namely to: Draw pictures representing: 15 and 2 . Due to objectivity and because of arguments presented by diSessa (2004) regarding the limitations of stating the research context, additional explanations were not given, despite the possibility that mathematical representations might not prevail. In other words, we cannot be sure if this task was perceived by students as a mathematical model-eliciting task (Doerr, 2006, p. 255). Through the process of coding, precise criteria for categorising participants’ drawings into a certain code were defined. In order to strengthen the validity and reliability of the findings that emerged during data analysis, the presentation of findings will be illustrated with concrete examples of participants’ drawings (Hesse-Biber and Leavy, 2004). After coding, we used a quantitative methodology and presented data using descriptive and inferential statistics. 321 First, participants’ responses were evaluated with respect to the adequacy of the representation to represent the concept. Some drawings could not be categorised according to the adequacy of the represented concept, therefore they were not included in further analysis. A description of the adequacy criteria is as follows. Representation of the numerical expression was considered adequate, if prescriptive and descriptive notion (according to Vom Hoffe, 1998) of the concept coincided. Some representations were therefore considered as non-adequate, for instance a drawing of the exponentiation showing 3∙2. Drawings, that were not connected to the concept in any way, were categorised as other. Figure 1 illustrates the examples. Figure 1: Drawings demonstrating (a) adequate representation (exponentiation), (b) adequate representation (fractions), (c) non-adequate representation (exponentiation), (d) non-adequate representation (fractions), (e) other We included only those participants' drawings that adequately represented the required mathematical concept in the further analysis. Drawing was considered to be predominantly procedural if procedures prevailed. For instance, when calculating a part of a whole, Figure 2d show a typical example how to calculate three fifths of 15. The drawing follows calculating procedure used in Slovenian schools (word od means from), replacing number symbols with concrete objects. On Figure 2 typical examples of drawings are shown. Pupils’ drawings were contextualized (depicting apples, flowers etc. instead of circles). Figure 2: Drawings demonstrating: (a) predominantly conceptual type of knowledge (exponentiation), (b). predominantly procedural type of knowledge (exponentiation), (c) predominantly conceptual type of knowledge (fractions), (d) predominantly procedural type of knowledge (fractions) 322 Results and interpretation Our first aim was to find out if students adequately represented the mathematical concept. Results are shown in Table 1. Adequate Non- Other Total drawing adequate drawing f f % f f % f f % f f % Fraction 667 52.4 302 23.7 303 23.8 1272 100.0 Exponentiation 878 69.0 238 18.7 156 12.3 1272 100.0 Total 1545 60.7 540 21.2 459 18.0 2544 100.0 Table 1: The adequacy of the students' representations of mathematical concepts Table 1 shows that the majority of students represented the required concept adequately (60.7 %). The category other mainly consisted of drawings, which suggested that the task was not perceived as a mathematical model-eliciting task; the models presented were more art oriented and did not show mathematical aspects. The differences among the concepts were statistically significant (χ = 83.480, P = 0.000) The lower degree of adequately representing the mathematical concept was observed for calculating part of a whole (52.4 %). Our result showed that students are less successful with fractions then with exponentiations. Several studies have showed that teachers themselves often have severe difficulties with fractions (Dixon et al., 2014; Isik and Kar, 2012) and exponentiation (Confrey and Smith, 1995). These difficulties occurred even on very simple tasks, for instance ‘shade 1/4 of objects’ (Depaepe et al., 2015; Van Steenbrugge et al., 2014). Therefore, the answer to why pupils are less successful with these concepts, as reflected in our study, could lie in their teachers' content pedagogical knowledge (Ball, Thames and Phelps, 2008), since students’ achievements are related to teachers' mathematical knowledge (Hill et al., 2005). Simultaneously, these effects accumulate and there is little evidence that more effective teachers in later grades can be compensatory (Sanders and Rivers, 1996). The other explanation of our results lies in well documented difficulties in understanding the various meanings of rational numbers (McMullen et al., 2015). Results in Table 2 show that the proportion of drawings that reflected conceptual drawings occurred in smaller share than procedural ones. 323 Procedural Conceptual Type not Total recognisable f f % f f % f f % f f % Fraction 393 58.9 117 17.5 157 23.5 667 100.0 Exponentiation 737 83.9 28 3.2 113 12.9 878 100.0 Total 1130 73.1 145 9.4 270 17.5 1545 100.0 Table 2: Number of drawings for each category according to predominating type of knowledge Only 9.4 % of pupils clearly demonstrated conceptual orientation, whereas 73.1 % of drawings reflected a procedural aspect. The proportion of pupils who represented a mathematical concept with conceptual knowledge was higher for the calculation of part of a whole (17.5 %) than for exponentiation (3.2 %). The differences were statistically significant (χ = 140.321, P = 0.000). Among other things, students' representations also reflect students’ numerical answers to the mathematical expressions. Therefore, we were interested to investigate the connection between the correctness of the result and the type of the representation. Correct Incorrect No answer Total f f % f f % f f % f f % Procedural 572 50.6 325 28.8 233 20.6 1130 100.0 Conceptual 134 92.4 9 6.2 2 1.4 145 100.0 Type not 1 0.4 2 0.7 267 98.9 270 100.0 recognisable Total 707 45.8 336 21.7 502 32.5 1545 100.0 Table 3: Correctness of the result according to type of representation In most cases (45.8 %) the results of the numerical expressions were correct, regardless of the type of representation. However, findings indicate that a higher rate of correct result was found among students whose representations were conceptual (92.4 %). The rate of correct result was lower for students who used procedural representations (50.6 %). For the not recognisable type, correctness or incorrectness could not be determined since most of those representations do not show a numerical result (98.9 %). The difference between types of representation 324 in terms of the correctness of the result is statistically significant (χ = 751.366, P = 0.000). One of the major limitations of this study is that children were not asked to explain their representations; hence the true meaning behind them could be quite different to that inferred by the authors. Verbal representations, in addition to the pictorial, could have been of a big importance. Conclusion Our study showed that an inspection of drawings can be a good indicator of students’ mathematical understandings. Our participants were young, but in spite of that they provided a rich pool of different representations, which suggests that fostering visual representations in classrooms may not be as demanding as it would seem. The same idea can be found in constructivist literature (diSessa, 2004). Lipovec and Antolin (2015) conducted a study on future primary teachers and future teachers of mathematics (N = 336) in three different countries (Slovenia, Spain and Slovakia). The participants were asked to draw a picture that represents 2³. Only one fifth of representations could be categorised as conceptually oriented drawings of exponentiation. We believe that the use of drawings should be encouraged in teacher training programs as a means of developing preservice teachers’ content pedagogical knowledge. Acknowledgement The authors acknowledge the project Development and realization of innovative learning environments and flexible learning approaches for general competencies, no. 5442-138/2016 was co-funded by the European Social Fund and the Ministry of Education, Science and Sport. References Ainley, J., Pratt, D., Hansen, A. (2006). Connecting engagement and focus in pedagogic task design. British Educational Research Journal, 23(1), 23–38. Arcavi, A. (2003). 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In Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Athens, GA: University of Georgia. 327 SUPPORTING MATHEMATICS LEARNING THROUGH PROJECT BASED LEARNING: A FIFTH GRADE CASE STUDY Christie Martin,Richard G. Lambert, Chuang Wangand Drew Polly Abstract This paper discusses the results of a four month long pilot study in a fifth grade classroom in the Southeastern United States. The study introduced students to using project based learning as a way to infuse learning activities that are student-centered, long-term, and interdisciplinary with connections to real world issues (Holbrook, 2007). The project consisted of two parts and within each part were several components. The project was created around a school fundraiser that had participation across grade levels and the components covered a multitude of mathematical standards. The findings indicated that students engaged multiple strategies to solve problems, students remained engaged and on task, and their partners became sounding boards and they were able to scaffold each other’s learning. The writing components within the project revealed students’ reflections on their own learning. From the analysis of student work, observations and writing the researchers identified areas for further research and practitioner implications. Keywords: project based learning, reflection, engagement, student-centered, collaboration Introduction In the United States, the National Council of Teachers of Mathematics (NCTM) articulates in the Principles to Action a principle for teaching and learning “An excellent mathematics program requires effective teaching that engages students in meaningful learning through individual and collaborative experiences that promote their ability to make sense of mathematical ideas and reason mathematically.” This fundamental principle put forth by the NCTM aligns with the theoretical underpinning of project based learning. Mergendaller (2006) defines project based learning (PBL) as teaching that fosters students’ learning through an extended, student-influenced inquiry that is created around authentic questions and carefully designed tasks. Steffe and Gale (1995) suggest project- based learning is a constructivist pedagogy that focuses on projects that are both individual and collaborative. The literature on PBL varies across elementary, secondary, and higher education. Most studies examine PBL in STEM with a primary focus on science. The research on PBL in the content area of mathematics is limited. The researchers were particularly interested in the discourse and reflection that occurred in the context of students collaborating on a project rooted in the tenets of PBL. NCTM suggests meaningful learning experiences promote the ability to make sense of mathematical ideas and reasoning. Our research University of South Carolina, USA; e-mail: [email protected] University of North Carolina at Charlotte, USA; e-mail: [email protected], [email protected] 328 questions sought to uncover students’ reasoning and sense making in their engagement and collaboration. Marks (2000) conceptualizes student engagement as attention, interest, effort, and investment of students toward classroom instruction. Engagement defined in this way is in agreement with prior definitions of engagement that include motivation (Steinberg, 1996), and involvement in school (Fin, 1989). Several studies indicate student engagement has a positive effect on mathematics achievement (Steinberg, 1996; Yair, 2000), and authentic instructional work contributes strongly to the engagement of students (Marks, 2000). Student engagement is a factor in students’ achievement in mathematics and is positively influenced by authentic tasks that engage students in multiple strategies and reflection. This study intended to examine how students engaged with project based learning in mathematics and how they collaborated with their peers. The research questions that guided the study were: 1) How did project based learning impact students’ engagement in mathematics? 2) How did students collaborate with their partners throughout the project? Literature Review Project based learning (PBL) is a student-centered practice that occurs over an extended amount of time that is responsive to real-world questions; students produce a final product as an outcome of the project (Holm, 2011). Moursand (1999) include authentic content, authentic assessment and teacher as facilitator not director as important components of project based learning. This method is considered superior to other more traditional methods in improving problem solving and thinking skills for students (Berends, Boersma and Weggemann, 2003). Project based learning also include cooperative learning, collaboration, and reflection (Diehl et al., 1999). This environment creates an opportunity for students to engage in complex tasks. Thomas (2000) notes that there is not a universally accepted model for PBL, which is evident in the variety of research and activities labeled PBL. In order to begin to define a model Thomas (2000) offers five criteria for PBL; centrality, driving question, constructive investigations, autonomy, and realism. Centrality which means the project is the curriculum and the central teaching strategy. The second criterion is a driving question which makes the connection between the activities and conceptual knowledge. Third criterion is constructive investigations which are goal-directed process that include design, decision-making, problem- finding, or problem-solving. Fourth criterion is autonomy, meaning that the projects are student driven to some significant degree and that the students get unsupervised work time. Lastly, the project should be realistic and present a sense of authenticity to the students. Blumenfeld et al. (1991) found the fit between activities within the project and the subject matter concepts were not always complimentary. Blumenfeld et al., 329 (1991) and Barron et al. (1998) suggest that finding ways for projects to center on learning goals and maintaining those goals through explicit design requirements would improve the connection between activities and concepts to be learned. Polman and Pea, (1997) highlight a method of coaching for teachers to be able to address the tendency of students to go of course but without intervening to the point of directing the inquiry. Method Setting and Participants This study took place in one fifth grade classroom in a parochial elementary school in the southeastern United States. The study began in the middle of the school year due to the fifth grade teacher leaving midway through the year and one of the researchers volunteering to teach this one mathematics class. The study includes 21 fifth grade students that were tracked into this higher level mathematics class. At the start of the class the researcher was informed by the principal that although this was the higher class there was a lack of rigor in the instruction. The researcher observed teacher led direct instruction in this class and the other classes the students attended. The context of the classroom is a factor to be considered as project based learning is in opposition to a teacher directed class. The researcher conducted action research as the teacher and implementer of PBL. Research Design A case study design (Creswell, 2009; Stake, 2000) was used to explore the impact of project based learning on student engagement and collaboration. Yin (1994) suggests an appropriate design for a beginning investigation is a case study, and this study focused on students who were not accustomed to project based learning. The primary data sources in this study are students work and writing on the project and observations. The researchers examined the data, coded, and triangulated it to answer our research questions. The researchers used the literature on engagement and PBL to identify codes that were used to organize the data. Some example codes were motivation, decision-making, attention, and interest. The codes were condensed into themes to address our questions: 1) How did PBL impact students’ engagement in mathematics? 2) How did students collaborate with their partners throughout the project? The analysis of the student data was kept in a spreadsheet. Data Collection The project (Figure 1.) included several parts and students’ submitted their work for each component. The project included authentic content (Moursand, 1999) related to a school wide event. It was a student-centered, long-term, connected to real world issues and included components that were interdisciplinary (Holbrook, 2007). The duration of the project, the centrality of the work, the autonomy students were given to work aligned with Thomas (2000). The explicit design and learning goals of the project addressed Blumenfeld et al., (1991) and Barron et al., 330 (1998) findings related to connecting activities with concepts to be learned. This context provided observations, student work, writing, and classroom discussions to be analyzed. Findings Our research questions guided the reporting of our findings. Within the findings we provided the project parts. Research Question 1) How did project based learning impact students’ engagement in mathematics? There were a few days before the project was distributed when the students were told about the project. The Run with the Saints 5k was a school wide fundraiser that many students and parents participated. The event included a shorter fun run which increased the participation of younger students. The school calculates participation rates to award the grade level with the most student participants and the grade level with the most parent participants. The students knew that this year they would be the ones figuring this out, and then they received the project. Welcome to the Run with the Saints Project This project has several components and will have multiple due dates. You will have one partner that you will have time to work with, but everyone will turn in their packet for individual grades. All work must be organized and included for full credit. Class time will be provided; however, you will need to use time management skills to ensure you will turn in each part in a timely manner. Assignment Description Due Date Winners! This includes: Feb. 27, 2017 Solving for which grade level had the most participation in the 5k. In each grade level you will also calculate which class had the most participation. Identify the grade level winner and the class that had the most participation. Solve for which grade level had the most parent participation. In each grade level you will also calculate which class had the most participation. Identify the grade level winner and the class that had the most participation. Organize your data and show all your work from solving these problems. 331 Money, Money, Solve for how much money the 5K generates by Due March Money $ participants. 3, 2017 Calculate the expenses (shirts, food, timers) Determine the profit made by the 5K Getting to the goal- a goal number will be provided and you will need to figure out how to reach it. Data Now that we have the data from solving our Due March Representation! problems we will use bar graphs, pie charts, line 10, 2017 plots, to share our data in multiple forms. These representations will be drawn and created on the computer. Discussion Use a calculator to check your work. Due March 17, 2017 Share a few paragraphs about the skills needed to work on this project. Discuss the experience of checking your work with the calculator and partner. Figure 1: Run with the Saints Project Figure 1 shows the table of contents of the project, each part had its own page with more details and the students received the data to analyze. Once they were able to look over the project one student asked repeatedly “Are we the only ones solving this for the school? Are we really in charge of this?” These questions were asked with a slight grin, which seemed to portray a sense of empowerment that they were tasked with this project. The students began making fractions for the amount of participants for the grade level (example: 10/35 first graders participated). The teacher researcher worked with groups and offered some prompting questions to help students get from fraction to percent in their division. The students were grouped in pairs, one group had a student that simplified the fraction before dividing and the other was not quite ready for that step. They both solved and got the same answers and shared their work. The different ways of solving and arriving at the same answer seemed to reinforce a sense of autonomy within the students. The project seemed to foster students’ ability to use number sense and judge whether their work was reasonable. One example was when several groups were solving a fraction of 1/22 and found it was equal to 45%; they had done the division and did not recognize .045 would be 5%. Many of the groups knew something was not right, they looked at their previous work, talked with each other, and some came to the realization on their own. The groups that were not sure were asked to consider place value; this prompt was enough for them to recognize their error. 332 One of the parts of the project asked students to reflect and write about their experience using a calculator after they had solved, to think about the skills needed to complete the project, and to identify areas to work on. Below is two excerpts from Ivan and one excerpt below are from John. The first paragraph discusses checking with a calculator and the other describes the skills needed for the project. “My partner and I first made an estimate for each problem. Then we each solved the problem and the one that got the answer closest to our original estimate we decided was correct. Most times my partner and I did get the same answer. We actually did not use the calculator during the project. I checked my calculations with a calculator for Mrs. C’s class. Mrs. C had 2 out of her 12 students participate. My class work calculations for her participation was 17%, when I solved on the calculator I got 6. My first reaction was that the calculator was wrong. I tried again and got 6, I realized that I should switch from 12 divided by 2 to 2 divided by 12. When I made that change the calculator and my answer were the same.”- Ivan “The mathematical skills needed for this project are estimating, dividing, and multiplying. The other skills are adding, subtracting, rounding, fractions and percent. We counted each grade and class participation. We made these into fractions, one example would be 4/24 of the 4k students participated. My partner and I estimated the answer. Then we used long division to get the decimal, which we had to round and convert to a percent.” - Ivan Although Ivan and his partner were able to calculate by hand all of the problems issues arose when using the calculator. The immediate reaction was that the calculator was wrong, but Ivan does not share how he knows it must be wrong. It takes a few tries before realizing that he is not entering the numbers in properly. Since Ivan does not elaborate on how he knows the calculator is wrong, besides it not matching his result, it is difficult to tell if the error would have been caught if calculators were used first. The next paragraph from Ivan is only a part of the whole paragraph. He clearly identifies the many skills needed to complete the project and explains an example for each skill. The reflection was intended to reinforce the interconnectedness of mathematics skills and real life problem solving. In this project we used many calculations; we used multiplying, such as finding out percentages for the class and parent levels. We solved a lot of problems. My hand was broken after. We also used division. We used it to found out the percentages too. It was a lot of work, but it was worth it. - John John identifies the rigor of the project and the extensive strategy and computation needed to solve. His response shows a definite recognition of the challenge, but reaffirms the experience as worth it. There appears to be an understanding of the benefit of this type of work for building mathematical understanding. In summary the project based learning experience created opportunity for students to be connected to authentic tasks, engage in multiple strategies, share ideas with 333 partners, develop number sense, and be reflective about their work. The project being connected to actual grade level winners for the school seemed to inspire greater investment in accuracy. There were decisions to be made about the data and the percentages were very close. Students recognized that careful calculating and rounding made a difference. Research Question 2) How did students collaborate with their partners throughout the project? Students were assigned one partner for the project and each student was responsible for each task. Students were encouraged to discuss methods of solving, work through problems together, and share their work for each problem. Once they shared their work they would then move on to the next step if they were in agreement, otherwise they would communicate with one another to decide on a correct answer. Observations indicated that students engaged with their partners and valued their input. They made plans and strategies together. This excerpt from a written reflection shows how partners interacted. “My partner and I first made an estimate for each problem. Then we each solved the problem and the one that got the answer closest to our original estimate we decided was correct. Most times my partner and I did get the same answer. When my partner and I got different answers we would both do the problem again and we would see who was closest to our estimate. My partner and I had a few disagreements within the math problems, but we worked through them by trying the problems again and talking about the math. We would ask questions of each other like “Does this look right, is this close to our estimate?” This is how this project was hard and easy.” In this response the student indicated that they engaged each other in probing questions about their disagreements. The students questioned the reasonableness of their answers in order to come to an agreement. This response includes the phrase “talking about the math” as a strategy for working together. Observations supported that students collaborated which increased the mathematical discourse. Another student’s response offered in-depth discussion of the social aspect of effective mathematical discourse. Below is Leon’s response: We had to be cooperative by listening to each other’s reasoning even if we thought it was wrong. We also had to be patient with each other if one of us was behind on our work. We had to be kind of one of us was sad or mad or frustrated. We had to be kind if one of us got a little off track, and just politely tell them to focus. Last but not least, I feel we had to be able to argue our answers and be able to support the data. - Leon This response highlights how Leon navigated discourse with his partner to ensure that both parties felt valued and could contribute. The last line seems to show that although politeness and good listening were important the need to argue and support answers was just as important. 334 Conclusion The goal of using project based learning was to provide students with the opportunity to engage in authentic real world tasks, communicate their thinking in a collaborative setting, to justify their work, and reflect on mathematical concepts. In this study students exhibited varied strategies and discussed their thinking with their partner. Students were immersed in traditional learning before the change in mathematics class and continued to be immersed in traditional learning in other classes throughout the day. This was a factor in how well students adapted to PBL and it also appeared to have an influence on their perception of their learning. This research is continuing with a part two of the project, researchers will continue to observe, collect student, work, and interview students to better understand the impact of PBL. The next project is wrapping up at this point and the students’ familiarity with PBL and group work as advanced the progress. Teachers that embark on PBL would be best served by making a commitment at the beginning to foster the environment needed to thrive. References Creswell, J. W. (2009). Research design: Qualitative, quantitative, and mixed methods approaches. Third Edition. Thousand Oaks, CA: Sage Publications. Barron, B. J. S., Schwartz, D. L., Vye, N. J., Moore, A., Petrosino, A., Zech, L., Bransford, J. D., The Cognition and Technology Group at Vanderbilt. (1998). Doing with understanding: Lessons from research on problem- and project-based learning. The Journal of the Learning Sciences, 7, 271–311. Berends, H., Boersma, K., Weggeman, M. (2003). The structuration of organizational learning. Human relations, 56(9), 1035–1056. Blumenfeld, P., Soloway, E., Marx, R., Krajcik, J., Guzdial, M., Palincsar, A. (1991). Motivating project-based learning: Sustaining the doing, supporting the learning. Educational Psychologist, 26(3&4), 369–398. Diehl, W., Grobe, T., Lopez, H., Cabral, C. (1999). Project-based learning: A strategy for teaching and learning. Boston, MA: Center for Youth Development and Education, Corporation for Business, Work, and Learning. Finn, J. D. (1989). Withdrawing from school. Review of Educational Research, 59, 117–142. Holbrook, J. (2007). Project-based learning with multimedia. Retrieved from http://pblmm.k12.ca.us/PBGuide/WhyPBL.html. Holm, M. (2011). Project-based instruction: A review of the literature on effectiveness in prekindergarten. River Academic Journal, 7(2), 1–13. Marks, H. (2000). Student engagement in instructional activity: Patterns in the elementary, middle, and high school years. American Educational Research Journal, 37(1), 153-184. Retrieved from http://www.jstor.org/stable/1163475. 335 Mergendoller, J. R., Markham, T., Ravitz, J., Larmer, J. (2006). Pervasive management of project based learning: Teachers as guides and facilitators. In C. M. Evertson, C. S. Weinstein (Eds.), Handbook of Classroom Management: Research, Practice, and Contemporary Issues. Mahwah, NJ: Lawrence Erlbaum, Inc. Moursund, D. (1999). Project-based learning using information technology. Eugene, OR: International Society for Technology in Education. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author. Stake, R. E. (2000). The art of case study research. Thousand Oaks, CA: Sage Publications. Steinberg, L. (1996). Beyond the classroom: Why school reform has failed and what parents need to do. New York: Simon and Schuster. Thomas, J. W. (2000). A review of research on project-based learning. Online Retrieved June 24, 2011, from http://www.bobpearlman.org/BestPractices/PBL_Research.pdf. Yair, G. (2000). Educational battlefields in America: The tug-of-war over students’ engagement with instruction. Sociology of Education, 73, 155–174. HIGHER ORDER THINKING SKILLS IN CLIL LESSON PLANS OF PRE-SERVICE TEACHERS Hana Moraová and Jarmila Novotná Abstract The paper focuses on skills of future teachers of mathematics when planning CLIL lessons (Content and Language Integrated Learning, i.e. partly in a non-mother tongue) to pupils up to the age of 12. The paper presents results of a detailed analysis of pre- service teachers’ mathematics lesson plans developed within the frame of an undergraduate course at the Faculty of Education, Charles University. In the research study reported in this paper the authors focus on how higher order thinking skills and their development are addressed in the lesson plans and whether the pre-service teachers are aware of the need to target more than just lower order thinking skills. Development of higher order thinking skills might be more intricate when teaching in a non-mother tongue but must never be given up entirely. The findings from this analysis are of interest to teacher educators in general, to researchers interested in CLIL as well as to practicing teachers. Keywords: CLIL, lesson planning, language and content goals, higher order thinking skills Charles University, Czech Republic; e-mail: [email protected], [email protected] 336 Introduction Support of language education is one of the priorities of the EU to make its citizens understand each other, to be able to communicate across borders, to be functionally bilingual, i. e. to be able to switch between languages according to the situation easily. CLIL (Content and language integrated learning) may be seen as one of the tools of supporting pupils’ competence in a non-mother language. CLIL refers to the teaching of a non-linguistic subject such as mathematics through a foreign language. It suggests an equilibrium between content and language learning. In CLIL, the subject understanding and thinking manifested by the language of the subject are developed through a foreign language (L2). Conversely, the L2 is developed through the non-language content, such as mathematics. CLIL provides plenty of opportunities for incidental language learning, which has been shown to be effective, deep and long-lasting (Pavesi et al., 2001). A CLIL lesson has two objectives – the goal is not only learning the subject but also learning the language itself. Theories (Teaching Knowledge Test. Content and Language, 2009) distinguish between soft CLIL and hard CLIL, where in the first case 30% of communication in L2 is sufficient. The latter case asks for at least 70% of communication in L2. The advantage of CLIL thus is that it does not disqualify its learner from sitting various national exams in mother tongue. The possibility to combine mother tongue and foreign language also allows development of mathematical reasoning in case of A1 or A2 learners of the foreign language according to European Framework of Reference for Languages. Moraová and Novotná (2015) show that already very young learners with very basic knowledge of L2 can cope in a CLIL lesson as long as the demands on their language production are limited. However, mathematical reasoning and its development need a very elaborate level of language and thus can only be achieved in L1. In (Moraová and Novotná, 2017) the authors focused on the general ability of students to plan lessons, to deal with the obstacle of teaching a lesson in another language, on how the lesson plans are structured, in what language they are written and whether both language and subject goals are clearly defined. In this paper the authors focus on a narrower area – the area of the potential of CLIL lessons with young learners with respect to development of thinking skills. 4 Cs in CLIL As presented in (Moraová, 2015), there are certain underlying principles in a CLIL lesson that must be paid due attention to. When planning CLIL lessons, the teacher should always bear in mind that there are some areas the lessons must not fail to address. One of the possible underlying structure to bear in mind when planning CLIL lesson are Coyle’s (Coyle, 1999) 4 Cs, i.e. Content - Progression in knowledge, skills and understanding related to specific elements of a defined 337 curriculum, Communication - Using language to learn whilst learning to use language, Cognition - Developing thinking skills which link concept formation (abstract and concrete), understanding and language, Culture (or Citizenship) - Exposure to alternative perspectives and shared understandings, which deepen awareness of otherness and self are based on the basic principles described by 21st century skills and key competences (FEP, 2010). Bloom’s taxonomy of cognitive skills Bloom’s Taxonomy is a multi-tiered model of classifying thinking according to six cognitive levels of complexity. Throughout the years, the levels have often been depicted as a stairway, leading many teachers to encourage their students to “climb to a higher (level of) thought”. The lowest three levels are: knowledge, comprehension, and application. The highest three levels are: analysis, synthesis, and evaluation (Forehand, 2010). During the 1990’s, a former student of Bloom’s, Lorin Anderson, led a new assembly which met for the purpose of updating the taxonomy, hoping to add relevance for 21st century students and teachers (Anderson and Krathwohl, 2001). The revision includes several seemingly minor yet actually quite significant changes. The changes occur in three broad categories: terminology, structure, and emphasis. Changes in terminology between the two versions transformation of the categories from noun to verb forms (knowledge – remembering, comprehension – understanding, application – applying, analysis – analyzing, synthesis – creating, evaluation – evaluating). The lowest level of the original, knowledge was renamed and became remembering. Finally, comprehension and synthesis were retitled to understanding and creating. The new terms are defined as: Remembering: Retrieving, recognizing, and recalling relevant knowledge from long-term memory. Understanding: Constructing meaning from oral, written, and graphic messages through interpreting, exemplifying, classifying, summarizing, inferring, comparing, and explaining. Applying: Carrying out or using a procedure through executing, or implementing. Analyzing: Breaking material into constituent parts, determining how the parts relate to one another and to an overall structure or purpose through differentiating, organizing, and attributing. Evaluating: Making judgments based on criteria and standards through checking and critiquing. 338 Creating: Putting elements together to form a coherent or functional whole; reorganizing elements into a new pattern or structure through generating, planning, or producing. (Anderson and Krathwohl, 2001, pp. 67-68) Structural changes of the revised taxonomy are the following. Bloom's original cognitive taxonomy was a one-dimensional form. With the addition of products, the Revised Bloom's Taxonomy takes the form of a two-dimensional table. One of the dimensions identifies The Knowledge Dimension (or the kind of knowledge to be learned) while the second identifies The Cognitive Process Dimension. The Knowledge Dimension is composed of four levels that are defined as Factual, Conceptual, Procedural, and Meta-Cognitive. The Cognitive Process Dimension of six levels that are defined as Remember, Understand, Apply, Analyze, Evaluate, and Create. (Forehand, 2010) Emphasis is the third and final category of changes. The revised version of the taxonomy is intended for a much broader audience. Emphasis is placed upon its use as a more authentic tool for curriculum planning, instructional delivery and assessment (Forehand, 2010) Both the original Bloom’s taxonomy (Bloom et al., 1956) and its revised version help to define the teaching content with respect to the given educational aims, the method of meeting them (in other words the educational activity and the instructions that the teacher gives for their evocation), and the level of their achievement. Evaluation at the same time helps to answer the question of whether there is any coherence between the aims and instruction. Table 1 is a taxonomic table adapted for the needs of a mathematics lesson. The Cognitive Process Dimension Create = Analyse = put Apply = Evaluate = Remember break material elements carry out or make = retrieve Understand = into parts and together to relevant use a determine judgments form a The Knowledge construct knowledge procedure how parts based on whole; Dimension meaning. from long- in a given relate to one criteria and reorganize term situation. another and to standards. elements into a new memory. an overall structure. pattern or structure. Classify Order = to List = to Summarize according place Combine Factual Rank = to identify the given important e.g. Knowledge (the e. g. what was arrange the concepts relations or events in the quantities basic elements). give, said, facts into and ideas. principles, order in e.g. or represented, structures. in own words. e.g. colour, which they measures. size, … happened. 339 Conceptual Experiment Describe Knowledge (the with the Plan e.g. interrelationships Interpret the numbers or situation Assess if a the among the basic obtained geometrical Explain what and solution is procedure elements within results in shapes to was done and conditions right or how to a larger structure relation to the find the why. for the use wrong. situation. solution of solve the that enable them of the the problem. to function concept. together). problem. Procedural Conclude – Knowledge (how to draw to do something, Predict the conclusions methods of future steps Differentiate Compose a Tabulate based on inquiry, and of a solution between items problem in data, results Calculate. their criteria for using and/or and given etc. knowledge skills, proximate part/whole. situation. of how a algorithms, results. system techniques, and works. methods). Meta-Cognitive Appropriate Knowledge Action = use of Actualize a (knowledge of apply concepts Execute Construct a concept cognition in Achieve a concepts and independently procedure structure general as well level of learned in procedures the solution to solve a according as awareness and understanding. class in real in given of a problem. problem. to knowledge of life situation. experience. one's own situation. cognition). Table 1: Taxonomic table – adapted for mathematics education All this implies that a teacher of mathematics teaching through CLIL should plan their lessons carefully with attention to development of higher order thinking skills. This is of course more difficult with young learners whose proficiency in the language of instruction is very limited as one can expect higher order thinking skills to ask for elaborate language and good ability to express oneself. It might be difficult to be evaluating or passing judgements, to be reasoning at A1 or A2 level of L2 according to European Framework of Reference for Languages. Thus the research question asked in this paper concerns the 3rd C of Coyle’s structure. With respect to the age and language competence of learners up to the age of 12, what are the possibilities in a CLIL lesson for development of cognition, of thinking skills, especially higher order thinking skills? And how ready are pre- service teachers to pay attention to this C when planning their lesson? Materials and Methods The design of this study is qualitative. The authors of the paper analyse 16 lesson plans written by pre-service teachers within the frame of their work in the undergraduate course Integrovaná výuka matematiky a angličtiny (Integrated Teaching of Mathematics and English) at Faculty of Education, Charles 340 University. The course is compulsory for pre-service teachers of mathematics only and optional for pre-service teachers studying mathematics in combination with another subject or for pre-service primary teachers. In Czech system, the course participants will also be teaching primary children even if they study mathematics for upper secondary school as 11 and 12 year old children are taught by them. The course is run at the Department of Mathematics and Didactics of Mathematics in the cycle of master studies, i.e. its participants have also other courses of didactics of mathematics. Within the course the students are introduced to CLIL, its principles and obstacles a CLIL teacher faces. In the introductory seminars they get introduced to the concept of 4 Cs (Coyle, 1999) and taxonomy of thinking skills (both Bloom and Krathwohl), the concepts of soft CLIL and hard CLIL. They know that a CLIL lesson has two objectives – mathematics and language. Their task is to choose the mathematical content and the target group and plan a lesson for this specific target group with respect to the pupils’ language skills. While planning the lesson they should try to target all four Cs and pay attention to development of higher order thinking skills. To answer the research questions formulated for this study, the authors of the paper focused on how students work with the issue higher order thinking skills. Are thinking skills described explicitly in the lesson plan? If not, does the lesson plan include activities that target both the lower and the higher order thinking skills? If higher order thinking skills are not developed in the lesson, what advice could the lecturers in the course give the students to realize the weaknesses of their lesson plans? The findings from this analysis are presented in the following section results. Results and discussion The content of the plans Out of the 16 pre-service teachers who were attending the seminar, 13 created their lesson plans for a lesson conducted in English, one in Russian, one in German and one in Italian. Out of these 16 lessons plans 2 were excluded for the here presented study because they were developed for older pupils. The remaining lesson plans were developed for 3rd to 6th grades, i.e. 8 – 12 year old pupils. It was quite surprising that the students preferred geometry to arithmetic. 10 of the papers present a lesson focusing on geometry, only four focus on arithmetic. The most appealing topic for the students was line symmetry – 5 of the geometry lesson plans. 2 focused on properties of angles, 1 on planar geometrical figures, 1 on properties of triangles (medians and centroid) and one on area and perimeter. The lesson plans were handed it at the end of semester and thus the authors could not ask why students preferred geometry so much. It would be interesting to find 341 out whether their preference of geometry is anyhow language conditioned or whether the selection was based on their preferences in mathematics as such. Only one of the students defined the level of pupils’ knowledge of English according to the European Framework of Reference for Languages (6th grade, A1 level, which corresponds to objectives for Foreign Language where A2 level is expected in the 9th grade). Two of the students clearly defined the cognitive skills developed in the planned lessons, one of the students defined the key competences developed (with respect to Framework Education Programme for Elementary Education). The rest of the lesson plans do not have cognitive skills described explicitly, however, in some of the lesson plans development of higher order thinking skills is implicit as will be shown in the following section. Attention to development of 4 Cs and higher order thinking skills As stated above, a CLIL teacher should always bear in mind the 4 Cs’ of CLIL, i.e. content, cognition, culture and communication. The analysed lesson plans clearly show that students focus most on the Content of the plans. They carefully select the topic to be covered in the lesson and define mathematics and (and some also language) goals of the lesson. Mathematics goals are not defined in the sense of what competences and skills pupils will acquire but in the sense of what topic they will have been working on, discovering and practicing. As far as development of Cognition is concerned, many of the lesson plans have activities that allow individual discovery, require analysis and synthesis. Only lower order thinking skills (knowledge, comprehension and application of a learned procedure) are in 5 out of the 14 lesson plans, which means majority of the students were able to include development of higher order thinking skills in their lesson plans (even though implicitly). 2 of the lesson plans describe explicitly which cognitive skills each of the activities targets. The lesson plans that do not develop higher order thinking skills are the following – three of the lesson plans in arithmetic. One of the lesson plans (3rd grade, conversions) builds on the constructivist Hejny method of teaching mathematics (Hejný, 2008). However, in the particular lesson planned the highest order thinking skill developed is application (the children get a conversion principle and use it in solving word problems). Similar can be said about some of the geometry lesson plans. In one of the lesson plans the student plans to show formulas for area and perimeter to pupils and asks them to use them (apply). The other lesson plan focusing on planar geometrical figures is more elaborate and the development of cognitive skills is on the border between applying and analysing. The pupils are first taught names of planar geometrical figures (remember) but at the end of the lesson pupils are asked to put cards with planar shapes into groups according to their properties – this is the activity that is somewhere between apply the learned 342 knowledge to classify the figures and analyse (analyse the properties of figures on the cards and use the findings for their classification). The lesson plans focusing on line geometry (implicitly) address higher order thinking skills, especially create. In the lesson plans, pupils are asked not only to analyse the images and state whether they are symmetrical, but also to create symmetrical images on their own. This is an example of activity in which A1 or A2 level of L2 does not block development of higher order thinking skills as the learners develop them by doing something, not speaking about it. An interesting example of development of higher order thinking skills is to be come across in the lesson plan focusing of properties of a triangle, namely on medians and centroid. The pupils are first shown how to construct one median. Then their task is to construct the other two (apply). Having done that, they analyse the picture and try to discover the properties of the centroid. In the end of the lesson, a whole class discussion is planned on the importance of centroid in real life. One might ask whether 12 year old learners will be able to have this discussion in L2 but soft CLIL does not disqualify use of mother tongue at this stage of the lesson. The activity is on the level of evaluate since the pupils try to pass judgements on the importance of a concept based on the properties and qualities of the concept. Analyse is also implicitly present in the lesson plan on properties of angles. Fans are used to look for and analyse the properties of angles. Definition of key properties of angles is based on this analysis. Two of the lesson plans by the students have cognitive goals defined in the lesson plan – one defines the cognitive skills developed in the whole lesson, the other defines the cognitive skills developed in each of the activities included in the lesson plan. The first of these two lesson plans is from the area of arithmetic, the topic is fractions (How did ancient Egyptians divide loaves of bread?). The student defines goals in each of the four dimensions: Remember – recall knowledge from previous lessons Understand – explain their procedure in their own words, i.e. show they understand the procedure and can reformulate Apply – apply the rule for naming any fraction when presenting their own results, divide loaves of breads following the conditions from the assignment Analyse – analyse solutions of classmates, look for possible mistakes, look for whether the solutions meet conditions in the assignment Evaluate – evaluate one’s own work and progress in the lesson Create – create original solutions of problems, formulate general rules for work with fractions 343 The other lesson plan in which cognitive skills developed were mentioned explicitly focused on angles and their properties. All 6 levels are included. Conclusions The analysis of lesson plans shows that thinking skills are described explicitly by minority of students. However, implicitly a vast majority of lesson plans includes activities that support development of higher order thinking skills. Many of the properties are discovered by doing and by analysis, the pupils are expected to achieve much more than just remember and understand or apply. In general it can be stated that the students are able to plan lessons in CLIL that develop the 3rd C of Coyle even in case of young learners with limited command of language of instruction. The analysis shows that students should be asked to match their activities to the different cognitive objectives and state it in their lesson plans explicitly. Thus the objectives will be met clearly and the students will be fully aware of having included them. In the next academic year lesson plans developed with the condition that thinking sills must be defined for each activity will be analysed again. This will show how well student understand the concept and how they can use it in their teaching deliberately. Acknowledgement This research was supported by the project PROGRES Q16 – Environmental research – UK 2017-2020. References Anderson, L.W., Krathwohl, D.R. (Eds.) (2001). A Taxonomy for Learning, Teaching, and Assessing: A Revision of Bloom’s Taxonomy of Educational Objectives. New York: Addison Wesley Longman. Bloom, B.S., Englenhart, M.D., Fursdt, E.J., Hill, W.H., Krathwohl, D.R. (1956). The taxonomy of educational objectives. The classifications of Educational goals. New York: David Mc Key Company. Coyle, D. (1999). Theory and planning for effective classrooms: Supporting students in content and language integrated learning contexts. In Learning through a foreign language: Models, methods and outcomes, 46–62. Forehand, M. (2010). Bloom’s taxonomy. Emerging perspectives on learning, teaching, and technology, 41, 47. Framework Education Programme for Elementary Education (2010). Prague: MSMT. Hejný, M. (2008). Scheme-oriented educational strategy in mathematics. In B. Maj, M. Pytlak, E. Swoboda (Eds.), Supporting Independent Thinking Through Mathematical Education (pp. 40–48). Wy. Univ. Rzeszow. Moraová, H. (2015). Going interactive and multicultural in CLIL. In I. Krejčí, M. Flégl, M. Houška (Eds.), Proceedings of the 12th International Conference Efficiency and 344 Responsibility in Education 2015 (pp. 377–384). Prague: Czech University of Life Science. Moraová, H., Novotná, J. (2015). Teaching maths in English at primary school level – utopia, nightmare or reality. In J. Novotná, H. Moraová (Eds.), International Symposium Elementary Maths Teaching SEMT ‘15, Proceedings. Developing mathematical language and reasoning (pp. 249–-257). Prague: Charles University, Faculty of Education. Moraová H., Novotná, J. (2017). How do pre-service teachers of mathematics plan their CLIL mathematics lessons? In M. Houška, I. Krejčí, M. Flégl, M. Fejfarová, H. Urbancová, J. Husák (Eds.), Proceedings of the 14th International Conference Efficiency and Responsibility in Education 2017 (pp. 262–269). Prague: Czech University of Life Science. Pavesi, M., Bertocchi, D., Hofmannová, M., Kazianka, M. (2001). Teaching through a foreign language. Milan: M. I. U. R. Teaching Knowledge Test. Content and Language. Integrated Learning. Glossary (2009). Cambridge: University of Cambridge. CHILDREN’S PROGRESSIVE CONSTRUCTION OF NUMBER AND NUMERALS BY MEANS OF NUMERICAL DIAGRAMS Adalira Sáenz-Ludlow Abstract The paper focuses on the mathematical activity of two fourth graders to understand their conceptualization of numerals and number. These children were in the fourth-grade class participating in a year-long constructivist teaching-experiment. The tasks analyzed here reflect not only the progress of these two children but also the progress of the class as a whole. At the beginning of the school year, the fourth graders verbalized the memorized addition algorithm unaware of the meaning of the position of digits in the numerals. Four months later, children constructed their own numerical diagrams (ND) to mediate not only their de-composition of numbers into constituent units and the composition of units into abstract composite unites but also their own ways of expressing their numerical strategies verbally. Children’s ND supported their progressive conceptualization of number as a manifold of units as well as their re- constructed meaning of numerals (i.e., the place-value representation of numbers). Keywords: Number, number representations, numerical diagrams DIAGRAMS AS SIGN-VEHICLES MEDIATING POSSIBLE RELATIONS The notion of ‘sign’, in the Peircean semiotics, is a system of three elements and the three dyadic relations among them. The three elements are the sign-vehicle, University of North Carolina at Charlotte, USA; e-mail: [email protected] 345 the interpretant, and the Object. The three inter-dependent dyadic relations are between: (1) the sign-vehicle and the Object it purports to represent (R1); (2) the sign-vehicle and the interpretant it engenders in the mind of the Interpreter (R2); and (3) the emerging interpretants and the ‘dynamic objects’ the Interpreter generates (R3). In the mind of the Interpreter, the ‘dynamic objects’ are in continuous state of refinement, and they will eventually approximate the Object that the sign-vehicle purports to represent. It is important to note that the interpretant is not the Interpreter. The interpretant is the effect of the sign-vehicle on the mind of the Interpreter. In contrast, the Interpreter is the agent who takes part in and presumably exerts control over the process of interpretation. Although in this paper I am only explicit about R1, the other two relations remain in the background supporting and sustaining its existence. In other words, R1 cannot exist isolated from the other two. Furthermore, R1 could be of iconic, indexical, or symbolic nature. When is R1 of iconic nature? The iconic sign-vehicle partakes certain characteristics of the Object; it bears some sort of resemblance or similarity with the Object. Peirce subdivides the icons into three types: diagrams, images, and metaphors. The diagram is characterized by some kind of similarity with its Object in the sense that it displays, in a skeleton-like manner, the existing relations between the parts of the Object. In contrast, the image represents the Object through simple qualities, while the metaphor represents the Object through a similarity found in something else. For example, a numeral is numerical diagram hiding, in the position of the digits, the relations among the powers of ten. Children also generate collective metaphors to interpret numerals (Sáenz-Ludlow, 2004) When is R1 of indexical nature? The index is a sign-vehicle determined by its Object by being in its individual existence and connected with it. The index has a cause-effect connection to its Object, and it directs the attention to that Object by blind compulsion that hinges on association by contiguity (CP 1.558). For example, numerals (in any base) become indexical-numerical-diagrams when it denotes a particular quantity as a function of powers of that base. How? It uses the digits (in that base), in relative positions, to indicate a quantity constituted by the addition of powers of that particular base while hiding the addition and the actual power. When a numeral (i.e., numerical diagram) is interpreted as being of indexical nature, the mind has transformed the iconic-numerical-diagram into an indexical-numerical- diagram that indicates particular powers of the base in increasing or decreasing order (depending on the direction in which the numeral is read). When is R1 of symbolic nature? The symbol is a sign-vehicle that denotes the Object as consequence of habit that hinges not only on intellectual operations but also on cultural conventions (CP 3.419). For example, numerals become symbolic- numerical diagrams, for the Interpreter, when and only when they are conceptualized as manifolds of powers of a particular base. When this happens, the 346 Interpreter is in the process of developing both the concept of number along with number-sense. In fact, as argued in the three preceding paragraphs, the three subtypes of R1 constitute a nested triadic relation in which the more complex involves specimens of simpler ones. In other words, symbols typically involve indices which, in turn, involve icons. This also means that icons are incomplete indices, which in turn, are incomplete symbols. With respect to our numeral example they can be interpreted by children as icons (diagrams devoid of meaning to them), as indices (diagrams with some meaning of units), or as symbols (diagrams with intimately connected with mental operations). The fact that a child recites the number word corresponding to the numeral does not mean that he/she interprets it at a symbolic level. So we have to ask ourselves the level at which the child is interpreting the numeral and the meaning of it for that child. In general, Relation (1) between the sign-vehicle and its (mathematical) Object clearly depends on what the Interpreter ‘makes of it’ at his/her particular level of evolving understanding and conceptualization at a particular time. On the one hand, the same sign-vehicle can be interpreted by the same Interpreter as iconic, indexical, and symbolic given that his/her interpretants are evolving and producing, each time, more advanced ‘dynamic objects’ that eventually will approximate the (mathematical) Object that the sign-vehicle purports to represent. On the other, the same sign-vehicle could simultaneously have different meanings for different Interpreters (iconic, indexical, or symbolic) depending on their level of conceptualization. Numerical diagrams (ND), as icons, could implicitly represent the unit-structure a number (i.e., the mathematical Object or concept) through some kind of skeletal structural relations. Thought-experimentation with ND and on ND facilitates the progressive conceptualization of units in the mind of children (i.e., Interpreters). The children, who participated in the constructivist teaching-experiment, initially interpreted their own ND’s as icons to convey their emerging sense of number as constituted by units. Then, they interpreted the relations among the parts of their ND’s as indicating the possibility of operating with those units. Throughout the school year, their ND’s were also interpreted as symbolic in the sense that relations among the parts of these diagrams became the product not only of what they could visually perceive but also of what they could mentally conceive and operate with. This is to say that children recursively played with the perceptual-elements in their thinking and the thought-elements in their perception to conceptualize number. Peirce argues that the skeletal structure of a diagram (i.e., an iconic sign-vehicle) convey the potential structure of the abstract Object that it purports to represent and that such a similarity enables and instantiates thought-experimentation in order to infer the actual existence of the structural relations among the parts of the abstract 347 Object. Peirce calls this type of thinking process diagrammatic reasoning. By diagrammatic reasoning, I mean reasoning which constructs a diagram according to a precept expressed in general terms, performs experiments upon this diagram, notes their results, assures itself that similar experiments performed upon any diagram constructed according to the same precept would have the same results, and expresses it in a general form. (CP 2.96, italics added) METHODOLOGY Teaching-Experiment. The teaching-experiment methodology predominantly focuses on students’ conceptual constructions (Steffe, 1983) as the product of their own interpretations during teacher-student and student-student interactions with the purpose to initiate and sustain their mathematical activity. This methodology harmonizes the co-existence of personal constructions of mathematical meanings, of the meanings intentionally encoded in socio-cultural mathematical definitions and algorithms, and of the inter-personal influence on one another. This teaching- experiment lasted one academic school-year (September-June) and the goal was twofold. The first was to explore children’s interpretation of numerals and their understanding of the place-value as well as their conceptualization of number as manifold of units of ten. The second was to influence, indirectly, their ways operating with whole numbers and their ways of interacting with others during classroom mathematical discussions. The teaching-experiment consisted in teaching-episodes (five per week) and in small-group interviews (one per week) during the school-year. The teaching- episodes were conducted by the researcher with the collaboration of the teacher. When tasks were posed, children were expected to work on their own first, then to work with others and, finally, to participate in classroom discussions presenting not only their answers but also their strategies to arrive to their answers. Each teaching-episode was marked by continuous teacher-student and student- student interactions as children solved arithmetical tasks in different ways. In each teaching-episode, the teacher attempted to view any posed task both from her own perspective and from that of the child. As Vygotsky (1986/1934) argues, both the child and the adult may refer to the same object, but they will think of it from a fundamentally different frame of reference: “The child’s framework is purely situational, with the word tied to something concrete, whereas the adult’s framework is conceptual” (p. 133). Thus, the teacher has to de-center from his or her own mental actions to be able to generate hypotheses about the interpretations of the children so that he/she will be able to indirectly support and guide their progressive constructions of number as manifold of units. Subjects. The elementary school in which the teaching-experiment took place had a population of less than 400 students who came from low income parents with a minimal number of years of schooling. The fourth-grade class that participated in the study was constituted by six girls and ten boys. 348 Data Collection. To analyze the evolving activity and changes in children's meaning-making, the daily teaching-episodes and weekly-interviews were videotaped, and field notes of children's solutions were kept. The task pages and children's scrap papers were also collected. The researcher and the teacher continuously shared and exchanged perspectives about children's interpretations and progress. Starting from the beginning of the school year, there was an explicit and mutual agreement between the teacher/researcher and the students about their responsibility to listen carefully to the solution of others and to express their agreement or disagreement by giving justifications. This socio-cognitive strategy was consolidated throughout the teaching-experiment; such a strategy had also been useful in other studies (Lampert, 1990; Yackel, 1995). Arithmetical Tasks Posed to the Children. The instructional tasks for this teaching- experiment were prepared beforehand; nonetheless, they were subject to change according to the mathematical and cognitive needs of the children. The objective of the tasks was to promote children’s conceptualizations of units and abstract composite units (von Glasersfeld, 1981; Steffe, von Glasersfeld, Richards and Cobb, 1983) and their construction of ‘meaningful’ arithmetic instead of ‘meaningless arithmetic’ (Brownell, 1947). Emphasis was put on mental ways of operating with numbers, number sequences, counting strategies, composition and de-composition of numbers in terms of different units of ten without excluding other units that were conceptualized by the children. In the following section we present the analysis of the solutions of three tasks given by two of the children. The tasks chosen for this papers were posed at different times during the first fourth months of the teaching-experiment. They were chosen because they give a clear indication were children were at conceptually at the beginning of the teaching-experiment and how they started to make progress. The solutions to the first task indicate their initial conceptualization of numerals; the solutions to the second indicate their emerging notion of units of ten represented in a numeral; and the solutions of the third indicate their conceptualization of units other than units of ten using numerical diagrams (ND). The two children were chosen because they fairly illustrate how all the children in the class were operating with ‘numbers’: memorized number facts, memorized addition algorithm, and meaningless numerals. The first teaching-episodes during the teaching-experiment indicated that children understood “counting” as memorized patterns of number-words without any quantitative meaning. At this point, they have not developed a conceptualization of number as a manifold of units (Dewey and MacLelland, 1895; Steffe et al., 1983). ANALYSIS Task 1. At the beginning of the school-year (September), the fourth graders were given the task of adding, mentally, two two-digit numbers. Numbers were chosen 349 in such a way that the addition of digits representing units of one and ten were larger than ten in order to inquire about their understanding of place-value. Teacher: Add 85 and 37 in your head. Randy: That’s hard without paper. Richard: Uh-huh! Teacher: Try it. Randy: One twenty-two. Teacher: How did you get the answer? Randy: In my mind I put 85 on top of 37. 5 and 7 is 12. Put down the 2 and carry the 1. 1 and 8 is 9; 9 and 5 is 12. Teacher: When you added 5 and 7 and carry the 1, you carry 1 what? Randy: You carry 1. Teacher: When you added 1 and 8 and 9 you got 12 what? Randy: 12 ones. Teacher: OK. Richard, how did you do it? Richard: I put 37 on top of 85. 7 and 5 is…8-9-10-11-12 [showing five fingers and verbalizing a number sequence that was increasing by 1]. Write down the 2 and carry the 1. 1 and 3 is 4. 4 and 8 is…5-6- 7-8-9-10-11-12 [showing eight fingers and again verbalizing a number sequence that was increasing by 1]. One two two. Teacher: When you added 7and 5 you wrote down 2 and carry 1 what? Richard: You carry 1. Teacher: When you added 1, and 3, and 8 you got 12. This 12 is 12 what. Richard: 12 ones. The above dialogues are only an illustration of how all fourth graders were adding ‘numerals’ by means of the memorized addition algorithm. These dialogues indicate that: (a) neither of the two children had any awareness of the value of a digit (in terms of units of ten) in a particular place of the numeral; (b) Richard’s description left no doubt that when he added 7 and 5, 5 was a unit of five 1’s (composite unit) but not ‘one’ unit of 5 (abstract composite unit), reason for which he counted on five 1’s starting at 8; (c) Randy’s description left implicit what type of unit 7 was for him—‘one unit’ of seven 1’s or ‘one unit’ of 7; and (d) neither child was aware of what type of units of ten he was operating with when adding. When Randy adds and ‘carries 1’ he is unaware of carrying 1 unit of 10 as well as 1 unit of 100. The case of Richard is not unlike the case of Randy. This simply indicates that their memorized algorithm was independent of the symbolic place- value hidden in numerals even though, in the absence of paper and pencil, they were able to perform computations in their minds. 350 The Peircean semiotics shades light into this memorization: children were using numerals as iconic sign-vehicles (i.e., pure tokens) with no indexical or symbolic meanings indicating different units of 10. Let’s note here that the same numeral could be seen by the teacher as a symbolic sign-vehicle representing different units of ten and by the children as an iconic form with no quantitative meaning. Children’s solutions to similar tasks made it clear that our instructional priority should be directed to helping children develop their conceptualization of different units of 10 and their understanding of the role of these units in the numerals (i.e., the place-value notation). Money was used as an epistemological tool and several money related tasks were generated using fake-bills of denomination 1, 10, 100, and 1000 dollars. Classroom banks were prepared. They were simple boxes with fake-bills organized by denominations that were only powers of ten (five- and twenty-dollar bills were avoided while non-existing denominations like thousand- and ten-thousand-dollar bills were invented). A bank was available for each group of three children. By using money, children could visually perceive and mentally conceive 1 ten-dollar bill as one abstract composite unit of ten or as 10 one-dollar bills (i.e., a composite of 10 units of 1) (Steffe et al., 1983). Task 2. Three months later (November), children were given $374 from the classroom banks represented as 3 one-hundred-dollar bills, 7 ten-dollar-bills, and 4 one-dollar bills to parallel the representation in the number-word (three hundred seventy four). They were asked to mentally exchange this money into bills of different denominations. As customary, children were asked to think first on their own, then to discuss their solutions within the group and arrive to a consensus, and finally to share their solutions with the whole-class. Collectively during the whole-group discussion, the class came up with the following ways to talk about $374. Children volunteered to write their solutions on the board: 3 hundreds, 7 tens, and 4 ones 3 hundreds and 74 ones 30 tens and 74 ones 37 tens and 4 ones 2 hundreds, 15 tens, and 24 ones 1 hundred, 25 tens, and 24 ones 1 hundred, 27 tens, and 4 ones In addition, several children were willing to discuss the numerical pattern they saw when the units of ten were increasing while the units of a hundred were decreasing. This decomposition indicates that the fake-money banks were powerful epistemological tools to help students to construct mental images of different units of ten as well as their de-composition into other units of ten. All in all, the above representation constitutes in itself a ND that indicates that children went back and forth between abstract composite units and composite units. Seven 351 ten-dollar bills were de-composed into 70 ones (i.e., the composite unit of 70 ones); three hundred-dollar bills and seven ten-dollar bills were also de-composed into 37 tens (i.e., the composite unit of 37 tens). At this point in time, these children, collectively, were not only able to interpret the meaning of the digits in the numeral 374 and make sense of its corresponding number-word (three hundred seventy four) but they were also able to see the same numeral as a manifold of different units of 10. This numeral as an iconic sign- vehicle started to acquire, in the mind of the children, indexical and symbolic meanings. This was an indication of conceptual progress. This progress came to expedite their re-conceptualization of the addition and subtraction algorithms as well as the conceptualization of the multiplication and division algorithms. It is important to note that the algorithms for addition, subtraction, and multiplication consider the units of ten in increasing order while the division algorithm considers them in decreasing order. Task 3. With the construction of abstract composite units and their de- composition into composite units, children began to generate different strategies to operate with numbers without algorithms. Among those strategies were numerical diagrams (ND) (i.e., iconic sign-vehicles) that came to acquired indexical and symbolic qualities in the mind of the children. They use their ND to represent their conceptualizations of number as a manifold of units. Four months later (December), the teacher/researcher posed the following task to the students: “Mrs. Walgamuth wants to share $84 dollars among 6 students, how many dollars will each student get?” Children were allowed to use paper and pencil to solve the task and as customary they solved it first on their own. Then, during the whole-group discussion, Richard and Randy presented the following solutions to the class. Here the same children were chosen to illustrate not only their progress but also the progress of the whole class. All students generated different own numerical strategies and ND’s. Richard: I split 84 in 42 and 42. Then I split 42 into 30 and 12. I know 30 is 3 tens, and 12 is 3 fours. Then I put together 10 and 4. 14 dollars each. Teacher: Would you like to show us what you have on your paper? Richard: [goes to the board and reproduces the ND he has on the paper]. 1-2-3-4-5-6 [counting the lower lines of the diagram]. 14 each. 352 Randy: I did it some other way. Look [he makes this diagram on the board and says] 84 is 42 and 42. I know that 14, 14, and 14 is 42. 1-2-3- 4-5-6 [counting the lower lines of the diagram]. Each will get 14 dollars. The above ND’s indicate that Richard and Randy have started to conceptualize number as a manifold of units other than units of ten. They did not de-compose 84 only into 8 tens and 4 ones as they have done in the past. Instead, Richard de- composed 84 into two abstract composite units of 42 and each unit of 42 into three abstract composite units of 10 and three abstract composite units of 4. Finally, he added each unit of 10 with a unit of 4. Randy was able to ‘see’ abstract composite units of 14 almost at once. He de-composed 84 into two abstract composite units of 42 and each 42 into three abstract composite units of 14. Their solutions and ND’s indicate they self-scripted their own goals—to have six equal amounts of money. CONCLUSION Being able to ‘see’ mentally different ‘units of ten’ in a numeral is, in Peirce’s sense, a process of objectification. What is objectified is the quantity (i.e., the mathematical Object) which the numeral (i.e., sign-vehicle) purports to represent. The numeral (in base ten) is intended to be an iconic-indexical-symbolic sign- vehicle that implicitly represents a quantity organized into units of powers of ten by making specific only the maximum number of these units while hiding the actual powers of ten. However, the child, as an Interpreter, has to de-code from the position of the digits in the numeral (a diagram in itself) the different powers of ten and their inter-relations. This numeral, which is initially for the child an icon hiding numerical relations, has to be transform, in his/her mind, into a symbolic-diagram that conceptually and explicitly decodes different ‘units of powers of ten’ organized in decreasing or increasing order. In other words, the numeral/iconic-diagram has to acquire, in the mind of the child, indexical and symbolic characteristics to re- present a particular number (i.e., quantity). When this happens, the child starts to construct number and number-sense in meaningful ways. Children’s ND’s mediated both the composition and de-composition of units and their ways of expressing them verbally. ND’s transformed these children’s interpretations of numerals as meaningless iconic sign-vehicles into meaningful indexical-symbolic sign-vehicles in which the position of digits was indicating so many powers of ten and so many units of one. At this point, children were at a conceptual level to re-produce and re-conceptualize the arithmetic algorithms. 353 References Brownell, W. A. (1947). The place of meaning in the teaching of arithmetic. Elementary School Journal, 47, 256–265. Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27, 29–63. McLelland, J., Dewey, J. (1895). The psychology of number. New York, NY: D. Appleton & Co. Peirce, C. S. (1931-1966). Collected papers of Charles Sanders Peirce (CP). Edited by Ch. Hartshorne, P. Weiss, A. W. Burks. Cambridge, MA: Harvard University Press. Sáenz-Ludlow, A. (2004). Metaphor and numerical diagrams in the arithmetical activity of a fourth-grade class. Journal for Research in Mathematics Education (JRME), 1(35), 34–56. Steffe, L. P. (1983). The teaching experiment methodology in a constructivist research program. In M. Zweng, T. Green, J. Kilpatrick, H. Pollack, M. Suydam (Eds.), Proceedings of the Fourth International Congress on Mathematical Education at Berkeley (pp. 469–471). Boston, MA: Birkhäuser. Steffe, L. P., von Glasersfeld, E., Richards, J., Cobb, P. (1983). Children’s counting types: Phylosophy, theory, and application. New York, NY: Praeger Scientific. von Glasersfeld, E. (1981). An attentional model for the conceptual construction of units and number. Journal for Research in Mathematics Education, 12, 83-94 Vygotsky, L. (1986/1934). Thought and language. Translation newly revised and edited by A. Kozulin. Cambridge, MA: The MIT Press. Yackel, E. (1995). Children’s talk in inquiry mathematics classroom. In P. Cobb, H. Bauersfeld (Eds.), The emergence of mathematical meanings: Interaction in classroom cultures (pp. 131–162). Hillsdale, NJ: Lawrence Erlbaum. PLANNING AND CONDUCTING INQUIRY BASED MATHEMATICS COURSE FOR FUTURE PRIMARY SCHOOL TEACHERS Libuše Samková Abstract In this contribution I would like to present some methodological issues related to a teaching experiment conducted under an educational research aiming at implementation of inquiry based mathematics education into university courses for future primary school teachers. The text introduces rudiments of inquiry based mathematics education, describes a one-year inquiry based teaching experiment which I prepared and conducted within a compulsory university course on arithmetic, and focuses on questions related to planning and performing such a course. The final part of the paper briefly mentions results of the education research associated with the experiment. University of South Bohemia, Czech Republic; e-mail: [email protected] 354 Keywords: Future primary school teachers, future teacher education, inquiry based mathematics education, teaching experiment Introduction This paper is devoted to relation between inquiry based mathematics education (IBME) and university preparation of primary school teachers, a topic that was already reported at previous SEMT conference where we discussed it from the perspective of pre-service and in-service primary school teachers who had attended standard university courses on mathematics accompanied by a short instruction and self-study on IBME (Hošpesová et al., 2015). This time I will approach the topic from a different perspective that includes future primary school teachers attending a one-year experimental course on arithmetic that was entirely conducted in inquiry based manner, and the main focus of this paper will deal with questions related to planning and performing the course. Issues reported here are a part of a larger educational research project supported by Czech Science Foundation named Enhancing mathematics content knowledge of future primary teachers via inquiry based education. The goal of the project is to implement inquiry based education into university courses on mathematics and didactics of mathematics for future primary school teachers, and observe how active participation in the courses can influence professional competences of project participants. The impact that IBME might have on students was investigated by many educational researchers up to now. An extensive overview of the research was provided e.g. by Bruder and Prescott (2013) but none of the studies mentioned there focused on the impact that IBME might have on future teachers when implemented systematically during their university preparation. Inquiry based (mathematics) education Recently a lot has been said and written about inquiry based education as one of the important means of gaining new knowledge (Bruder and Prescott, 2013; Jiang and McComas, 2015; Savelsbergh et al., 2016). But the concept of inquiry is not recent in pedagogy: the term can be traced long way back to the work of Dewey (1938) who characterized it by means of transformations of indeterminate situations. Nowadays, inquiry based pedagogy is usually characterized as a way of teaching in which students are invited to work in ways similar to how scientists work (Artigue and Blomhøj, 2013) and to use procedures known from scientific inquiries: to observe, pose questions, reason, think, search for relevant information, collaborate, collect data and interpret them, solve and discuss problems that come out from real life or can be applied in everyday life contexts (Dorrier and Maaß, 2014). These procedures are naturally adapted to school context, so that during inquiry based mathematics lessons students do not discover new scientific issues but rediscover issues from school mathematics or solve 355 simple problems of everyday application character. The role of the teacher in the inquiry based mathematics lesson consists mainly in creating a suitable environment, building upon students' reasoning, giving students support, and in connecting to students' experience (Dorrier and Maaß, 2014). In mathematics education we can find various frameworks that employ approaches similar to IBME, many of them did it long before the term IBME has appeared: problem solving (Pólya, 1945), theory of didactical situations (Brousseau, 1997), realistic mathematics education and guided rediscovery (Freudenthal, 1973), mathematical modelling (Kaiser-Meßmer, 1986), substantial learning environments (Wittmann, 2001), etc. Also in the Czech educational past we can find issues related to the concept of inquiry, e.g. the concept of strengthening contacts of mathematics education with everyday reality and with other school subjects, and the concept of the process of grasping situations that were discussed by Koman and Tichá (1998), or the concept of experiments in the mathematical classroom (Hruša and Vyšín, 1964). IBME in the classroom In the mathematics classroom, the starting point for inquiry activities of students consists in creating an appropriate learning environment, usually in the form of a task or a problem that the students have to solve. To stimulate inquiry activities of students, the task should contain something unknown for the solver what is perceived by the solver as thought-provoking or interesting. But this unknown part of the task should not be too distant from student's actual knowledge, because inquiry is possible only when the unknown part can be approached through something known – only known facts and their relations might lead to conjectures and judgements that allow the solver to seek the solution. Tasks that may lead to inquiry activities of students are called inquiry tasks. The employment of an inquiry task within the lesson does not automatically guarantee that inquiry activities of students really occur in the classroom – this can be achieved only when the difficulty of the task is suitably chosen with respect to the actual students' knowledge, and when the characteristics of IBME are met by the teacher as well as by the students. From the perspective of the task design, inquiry tasks are usually open in the sense of open-approach to mathematics, i.e. at least one of the following parameters of the task is not exactly given: starting situation, process, end products, ways to develop (Nohda, 1995, 2000). So that, solving an open task may consist of various ways of formulating the task mathematically, of investigating various approaches to the formulated task, of various interpretations of the found results, and/or of posing various advanced tasks. Nohda (2000) offers four dimensions that can be used to evaluate students' responses to open tasks: 356 fluency – how many solutions the student produced? flexibility – how many mathematical ideas the student discovered? originality – to what degree is the student's idea original? elegance – to what degree is the student's explanation simple and clear? Inquiry in the classroom may be classified in various manners; one of the most common is the classification according to the kind of information that the teacher provides the students (Banchi and Bell, 2012; Bruder and Prescott, 2013): confirmation inquiry – the teacher assigns the students a question, and an appropriate method, results are known, the students are asked to confirm the results and explain them to others; structured inquiry – the teacher assigns the students a question and an appropriate method, the students are asked to find results and explain them to others; guided inquiry – the teacher assigns the students a question, the students are asked to find an appropriate method and results, and explain them to others; open inquiry – students themselves pose questions, search for an appropriate method and results, and explain them to others. As Bruder and Prescott (2013) present in their survey study, the greatest positive knowledge gains for the students in both content and process are documented in studies that monitor effects of guided inquiry. This is the reason why I choose the guided inquiry level for the experiment. The experiment The referred experimental course was intended for 33 future primary school teachers, university students of the second year of the five-year master degree study at the Faculty of Education. The teaching experiment was conducted within a compulsory course on arithmetic which acquaints the students with introduction to logic, set theory, and number systems. The course lasts 28 weeks; each week of schooling consists of a one- hour lecture and a two-hour seminar. In the referred year the students were divided into two groups for the seminars, so that the two-hour seminar on arithmetic was conducted twice a week, once for each group. For the purpose of the experiment, the course on arithmetic was entirely rearranged into inquiry-based manner on guided inquiry level. Being the course compulsory, the overall mathematical content of the course had to be preserved, and standard ways of assessment had to be used during the course (four standard 357 written tests, and two oral examinations). These requirements complicated the rearrangement, and made it really challenging. The following text will address some of the aspects related to the rearrangement. The known and the unknown The first step in the process of planning an inquiry based mathematics course of a long-term character consisted in overviewing the syllabus of the course, in determining which subject matter would have to play the role of the known part, and which subject matter would become the object of inquiry activities. The known part mainly included necessary definitions, axioms, etc. For instance, for the topic "introduction to set theory" I had to establish the meaning of the following words and the convention for their notation: "set", "element", "subset", "empty set", "union", "intersection", "complement", "difference", "Cartesian product", "Venn diagram", "cardinality". Such matter was presented at lectures. The inquiry activities based on the known part then took place at seminars. For instance, at one of the seminars to the topic "introduction to set theory" the students themselves undertook inquiries on whether and how it was possible to utilize Venn diagrams during solving various types of word problems. Time constraints Along with the process of determining what will play the role of the known, I also had to pay attention to time constraints. I had to check whether all the inquiry activities planned for the course would get enough time to enable inquiry of decent quality. Some types of the inquiry activities seem to be too time-consuming to be successfully implemented into the course but when we suitably choose the object of such an inquiry, the time devoted to the inquiry activities may be compensated by saving the time elsewhere. For example, when the inquiry results in discovering new subject matter, then no instruction on this subject matter may be needed any more. As an illustration I will describe a part of one of the seminars devoted to the topic "introduction to set theory", namely to Venn diagrams. In the first part of the seminar the students were solving many tasks focusing on how to draw and read Venn diagrams to various sets and subsets and to various set operations. In the second part of the seminar, without any previous instruction on using Venn diagrams in solving word problems, the following word problem was assigned to the students as individual work: Some children from our class went on holiday trip and visited Praha, Brno or Olomouc. Three boys went on the trip, one to each town. Jitka travelled to Brno 358 and Olomouc, Vlasta travelled to Brno and Praha, Eva with Sylva travelled to Praha and Olomouc. Dana and Alena visited all three towns. How many children visited Brno? How many children visited Praha or Olomouc? How many children visited Brno but not Prague? 1 The students worked on the task quite a long time, more than 20 minutes, but this activity successfully replaced the original "non-inquiry" part of the seminar that had lasted 45 minutes the year before. The last-year "non-inquiry" activities had consisted in an instruction on using Venn diagrams in solving word problems, and in a graded series of word problems solved with help of Venn diagrams on the blackboard; the task about the children on holiday trip being the last in the order. Comparing the non-inquiry and inquiry approaches, we can see that the inquiry approach saved us 25 minutes. This saved time was devoted to whole-class discussion on the holiday task and on methods of solving the students had used, as well as to individual consultations with weaker students. Some of the inquiry tasks do not allow compensating the time similarly as in the previous case, so that I had to find the needed time otherwise. I removed from seminars several passages devoted to training of calculations (e.g. calculation tasks on addition and subtraction of decimal numbers), and assigned them to students as homework. As the standard assessment written tests that are compulsory for the course comprise all the related subject matter (including word problems solved with Venn diagrams, and calculation tasks), the tests gave me the feedback on how the students mastered the subject matter mediated through inquiry activities as well as through homework. There were no significant differences between the test results from years when the inquiry was not implemented and the test results from the year when the inquiry was implemented. Types of inquiry tasks At seminars I assigned various types of inquiry tasks to students. From the perspective of general content targets the tasks fell into four groups: 1) Tasks that employed the most recently acquired knowledge, and applied it in new contexts, e.g. when discovering new solution methods. The above mentioned holiday task belongs here. 2) Tasks that employed knowledge that had been acquired by students some time before, and offered a new view on it: by linking topics with their practical applications, by combining various topics in one multiple-topic task, etc. For instance, to gain a new perspective on divisibility, on properties of natural numbers operations, on the concept of decimal numeral system, and on notation in non-decimal numeral systems, I invited the students to work on a multiple-topic task consisting in discovering a 1 Jitka, Vlasta, Eva, Sylva, Dana and Alena are girly names. 359 rule specifying how to recognize even numbers in non-decimal numeral systems (for more details see Samková and Tichá, 2016a). 3) Tasks that prepared the students for gaining knowledge on a quite new topic (in that case, the seminar with the inquiry task preceded the lecture assigned to the topic). For instance, the lecture on the topic "equivalence of sets" was preceded by a seminar with a series of inquiry tasks that employed manipulative activities focusing on comparisons by matching, and on rudiments of combinatorics (for one of the tasks see Fig. 1) Figure 1: Take 3 cubes and 3 hearts of different colours, and use all of them to make pairs consisting of one cube and one heart. How many pairs do you get? How many different ways of pairing exist? Show all of them! Repeat with more cubes and hearts, and generalize 4) Tasks that actually responded to a difficulty with which the students were not able to deal. For instance, as a reaction on difficulties that the students faced on one of the seminars while solving tasks on properties of canonical decompositions of the second and the third powers of natural numbers, I added into the following seminar an inquiry task that first invited the students to search for all the possible decompositions of various given natural numbers into a product of two natural numbers, and to search for all divisors of the given numbers. Then the task invited the students to investigate relations between the decompositions and the lists of divisors for various given numbers provided the numbers are (or are not) powers. Such a task successfully helped to overcome the initial difficulties. From the perspective of the amount of information given in the assignment, the two extremal cases were of special interest during inquiry based lessons: 1) Tasks with sparse starting information. These tasks are much indeterminate in their assignment, thus offer a lot of space for inquiry, a lot of possibilities that can be taken into account. For example: The sum of two unknown numbers is 10. How can these numbers look like? (Note that no particular numeral system is assigned to the addends.) 2) Tasks with dense starting information. These tasks contain a lot of facts or a lot of terminological terms in their assignment, and the solver must orientate in them. Such tasks prepare students for handling amounts of data 360 and for reading dense (scientific) texts. For example: What is the greatest product that can be obtained by decomposing number 10 into a sum of natural numbers and multiplying the addends of the sum? Particular inquiry tasks could be also put together to form a composite task. From the perspective of the inner structure, the composite tasks may be divided into three groups (for illustrative schemes see Fig. 2): 1) Tasks of hierarchical structure. These tasks are composites of two or more sub-tasks, where the end products of some of the sub-tasks can be utilized as a part of a starting situation of the other sub-tasks. For example: a) The sum of two unknown natural numbers is 10. How can these numbers look like? b) Decompose number 10 into a sum of two natural numbers. What is the greatest product that you can obtain by multiplying these numbers? 2) Tasks with dynamic starting. These tasks are composites of two or more sub-tasks with the same question, where every sub-task adds new starting information or new conditions to the task. For example: a) Decompose number 10 into a sum of two natural numbers. What is the greatest product that you can obtain by multiplying these numbers? b) Decompose number 10 into a sum of three natural numbers. What is the greatest product that you can obtain by multiplying these numbers? c) Decompose number 10 into a sum of natural numbers. What is the greatest product that you can obtain by multiplying these numbers? d) How does the number of addends influence the greatest product? e) What changes if decomposing numbers 7, 8, 9, or 11? f) What changes if decomposing into a sum of rational or real numbers? (The last sub-task is here just for illustration, since being too difficult for future primary school teachers.) 3) Tasks with dynamic ending. These tasks are composites of two or more sub-tasks with the same starting situation, where every sub-task adds a new question to the task. For example: a) Decompose number 10 into a sum of two natural numbers. What is the greatest product that you can obtain by multiplying these numbers? b) Decompose number 10 into a sum of two natural numbers. What is the smallest product that you can obtain by multiplying these numbers? The decomposition tasks were inspired by Artigue and Baptist (2012, p. 7). 361 The above mentioned groups of tasks are not disjunctive; one inquiry task may belong to more of them. More about suitable inquiry tasks and their typology can be found in (Samková et al., 2015). Figure 2: Schemes of composite tasks – hierarchical task (left), task with dynamic starting (right top), task with dynamic ending (right bottom); schemes taken from (Samková et al., 2015), and translated Instead of conclusion ... some partial results As mentioned in the Introduction, the here referred course was realized as a part of a larger educational research project. During the whole course we continuously collected various data from course students: completed worksheets, standard written tests, written solutions to problems or tasks, reflections, etc. In our contributions related to the project we already discussed some aspects that we revealed in collected data. We presented: - positive changes in students' approach to argumentation (a shift towards more efficient use of counter-examples, and a shift from using empirical arguments to attempts of using deductive arguments), for details see (Samková and Tichá, 2016a); - positive changes in students’ open approach to mathematics (a shift towards seeking more or all solutions, and a shift towards accepting more forms of notations), for details see (Samková and Tichá, 2016b); - changes in students' beliefs about mathematics and mathematics education (e.g. the newly emerged belief that discovering a thing by oneself helps remembering the thing, and also an ongoing change of the belief that 362 mathematics is about memorizing formulas and procedures), for details see (Samková and Tichá, 2016c). These results indicate that inquiry based mathematics education might play an important role in university preparation of future primary school teachers. From my personal perspective, I would recommend this approach to other educators of future primary school teachers. Acknowledgement: This research was supported by the Czech Science Foundation, project No. 14-01417S. References Artigue, M., Baptist, P. (2012). Inquiry in mathematics education (resources for implementing inquiry in science and mathematics at school). Available online from http://www.fibonacci-project.eu. Artigue, M., Blomhøj, M. (2013). Conceptualizing inquiry-based education in mathematics. ZDM Mathematics Education, 45(6), 797–810. Banchi, H., Bell, R. (2012). The many levels of inquiry. NSTA Learning Center. Science & Children. Available online from http://static.nsta.org. Brousseau, G. (1997). Theory of didactical situations in mathematics. New York: Kluwer Academic Publishers. Bruder, R., Prescott, A. (2013). Research evidence on the benefits of IBL. ZDM Mathematics Education, 45(6), 811–822. Dewey, J. (1938). Logic: The theory of inquiry. New York: Holt. Dorier, J.-L., Maaß, K. (2014). Inquiry-based mathematics education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (pp. 300-304). Dordrecht: Springer. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel Publishing Company. Hošpesová, A., Samková, L., Tichá, M., Roubíček, F. (2015). How primary school teachers posed problems for inquiry based mathematics education. In J. Novotná, H. Moraová (Eds.), Proceedings of SEMT ʻ15 (pp. 156-165). Prague: Charles University, Faculty of Education. Hruša, K., Vyšín, J. (1964). Vybrané kapitoly z metodiky vyučování matematice na základní devítileté škole. Praha: SPN. Jiang, F., McComas, W. F. (2015). The effect of inquiry teaching on student science achievement and attitudes: evidence from propensity score analysis of PISA data. International Journal of Science Education, 37(3), 554–576. Kaiser-Meßmer, G. (1986). Anwendungen in Mathematikunterricht [Applicati-ons in mathematics education]. Bad Salzdetfurth: Franzbecker. Koman, M., Tichá, M. (1998). On travelling together and sharing expenses. Teaching Mathematics and Its Applications, 17(3), 117–122. Nohda, N. (1995). Teaching and evaluating using "open-ended problems" in the classroom. ZDM Mathematics Education, 27(2), 57–61. Nohda, N. (2000). Teaching by open-approach method in Japanese mathematics classroom. In T. Nakahara, M. Koyama (Eds.), Proceedings of PME 24 (Vol. 1, pp. 39–53). Hiroshima: Hiroshima University. 363 Pólya, G. (1945). How to solve it. New Jersey: Princeton University Press. Samková, L., Hošpesová, A., Roubíček, F., Tichá, M. (2015). Badatelsky orientované vyučování matematice. Scientia in educatione, 6(1), 91–122. Samková, L., Tichá, M. (2016a). Developing views of proof of future primary school teachers. In L. Bal'ko, D. Szarková, D. Richtáriková (Eds.), Proceedings of Aplimat 2016 (pp. 987-998). Bratislava: STU. Samková, L., Tichá, M. (2016b). On the way to develop open approach to mathematics in future primary school teachers. ERIES Journal, 9(2), 37–44. Samková, L., Tichá, M. (2016c). On the way to enhance future primary school teachers' beliefs about mathematics via inquiry based university courses. Research report presented at ICME-13 Conference, July 2016, Hamburg, Germany. Savelsbergh, E. R., Prins, G. T., Rietbergen, C., Fechner, S., Vaessen, B. E., Draijer, J. M., Bakker, A. (2016). Effects of innovative science and mathematics teaching on student attitudes and achievement: A meta-analytic study. Educational Research Review, 19, pp. 158–172. Wittmann, E. Ch. (2001). Developing mathematics education in a systemic process. Educational Studies in Mathematics, 48(1), 1–20. PREPARING PRE-SERVICE TEACHERS FOR INCLUSIVE MATHEMATICS CLASSROOMS – CONCEPTS FOR PRIMARY EDUCATION Petra Scherer Abstract In Germany inclusive education becomes more and more common for students with special needs. Teacher education programs preparing teachers for an inclusive school system are in the state of development at the moment. The project ProViel (“Professionalisation for Diversity”) at the University of Duisburg-Essen aims at developing concepts and modules for teacher education for mathematics as well as for other subjects. The paper presents the project’s aims and objectives, followed by data concerning primary teacher students’ existing experiences and beliefs with respect to inclusive mathematics. Moreover, the design and try-out of a first teacher education course will be shown. Keywords: Inclusive education, special education, teacher education programs, clinical interviews, substantial learning environments Introduction In Germany students with special needs either visit special schools for handicapped children or visit regular schools in inclusive settings. Both settings show extremely heterogeneous groups in classrooms and the teacher is confronted with individual handicaps, for example deficits in language or visual perception, University of Duisburg-Essen, Germany; e-mail: [email protected] 364 failure of concentration or reduced memory and transfer so that a high degree of differentiation is needed. The proportion of students with special needs in inclusive settings increases continuously. In Germany, for the school year 2013/14 about 50 % of the students with special needs on the primary level visited regular schools (Klemm, 2015). Teacher Education Programs Until now teacher education programs for special education on the one hand, and for the regular school system on the other hand, follow different concepts and not all universities offer both programs. The University of Duisburg-Essen only focuses on teacher education for regular school systems including inclusive education. Teacher education programs preparing teachers for an inclusive school system are in the state of development at the moment. For example, by law the future teacher education programs have to provide for each subject matter – like mathematics or German language – 5 Credits for relevant aspects of inclusive education (LABG, 2016). Looking more detailed at the concepts of teaching and learning mathematics it can be stated that special education followed a more traditional view for a long time. As a consequence, the concrete situation in classroom for students with special needs was quite different from teaching practice in regular schools: Whereas constructivism and investigative learning could be seen as a guiding principle for mathematics education, in special education it was mostly disregarded. Instead, behaviourism remained the central principle. This practice can still be observed in teacher education and school. The Project “Mathematics Inclusive” within the Project ProViel The project ProViel “Professionalisation for Diversity” (https://www.uni- due.de/proviel/) is funded by the Federal Ministery of Education within the frame of a program for teacher education (2016–2019). The project at the University of Duisburg-Essen with 22 sub-projects follows three fields of action: “Diversity & Inclusion”, “Skills Labs – New Learn Labs” and “Quality Assurance/Quality Development”. Numerous university departments are involved to ensure the development of a coherent conceptual program for teacher education. The sub-project “Mathematics Inclusive” aims at implementing subject-specific concepts and modules for inclusive mathematics education. The project has started with the BA program for primary mathematics; lower and higher secondary mathematics will follow later on. Teacher students can choose “Inclusive mathematics” as a focal point and – if pursued consequently – achieve a specific profile. Courses for didactics of mathematics in semester 1 to 4 will touch the topic of inclusion casually before in semester 5 and 6 two practice- oriented courses “Learning Mathematics with Substantial Learning Environments (SLE)” and “Diagnosis and Support” are offered. Moreover, for the MA program further courses are planned. 365 Concept and Objectives of the Course “Learning Mathematics with SLEs” The developmental work for the course “Learning Mathematics with SLEs” (3rd year, BA-program for primary mathematics) started in 2016 and the first course has been running during the winter semester 2016/17. The course concept is as follows: The course contains a weekly 2 hours lecture combined with a weekly 2 hours seminar. The lecture should be attended by the whole cohort and it covers the theoretical background of SLEs and the concept of natural differentiation, examples for planning and designing concrete learning arrangements as well the analyses of concrete interview or classroom situations for various SLEs and various mathematical contents (cf. Krauthausen and Scherer, 2013, 2014; Wittmann, 2001; KMK, 2005). For the corresponding seminars the cohort is distributed in groups of about 15 persons with different focal points like differentiation, difficulties in language or inclusive mathematics. The latter one is part of the sub-project “Mathematics Inclusive” and in the next sections first results and experiences with three of these seminar groups will be reported. In the seminars the teacher students have to work in small groups up to four persons over the whole semester. Each small group has to choose a SLE from a given catalogue (see section First Results of the Seminar). The students have to design and carry out clinical interviews (cf. Hunting, 1997) with pupils from primary school. Each teacher student has to interview two ore more children with and without special needs. They should offer one and the same substantial learning environment and tasks to the different students and videotape the interviews. Within their small group they have to analyze and reflect the interviews in general, the concrete learning processes and existing competences as well as existing difficulties. During the semester each small group has to give two presentations within the seminar: The first presentation contains the concept and design of the planned interview. In the second presentation, later on, the teacher students have to present their data (video episodes, students’ written documents etc.) with the results of their small group analysis. Initial Questionnaire To gather information with respect to the teacher students’ pre-experiences in the field of inclusive mathematics a questionnaire was given to the whole cohort (N = 135) in the first lecture at the beginning of the semester. The questionnaire includes items concerning attitudes and beliefs with respect to inclusion and inclusive mathematics (cf. Meyer, 2011). The evaluation of these items will be completed in the next time. Moreover, an open item was integrated asking for individual experiences with inclusive mathematics. 366 Table 1 shows that less than 50 % of the students have made school-related- experiences whereas the others had no experiences or made experiences out of school or in other fields. These out-of-school-experiences cover private contexts (for example reports from friends) or more general experiences (for example theoretical university lectures in pedagogy). school-related- out-of-school-experiences no N = 135 experiences (unspecific/private/university) experiences 64 12 59 Table 1: Teacher students’ experiences with regard to inclusive mathematics Looking more detailed on the school-related-experiences, a first step is the categorization of statements. For generating the categories the combination of theory-based analyses and empirically found data was used (inductive and deductive, cf. Mayring, 2000). For some categories one can find statements only named by one or two persons referring for example to their own experiences as a child in inclusive settings at school, or referring to the meaning and effects of inclusion for the children or referring to questions of specific performance appraisals. For other categories you can find numerous statements, so that they are called main categories. Those main categories touch different fields: Some categories are specific for mathematics, others are more general and relevant also for other subjects or inclusive settings at school in general. To the main general categories belong structural or organisational conditions for inclusion in school (technical equipment or support for specific special needs, extra rooms for differentiation etc.), personal conditions at school (primary teachers and special education teachers working together, teaching assistants, further professions involved in the process etc.) or the naming of specific impairments and special needs (children with visual impairments, autism, children with language deficits etc.). Moreover, one could identify main categories for mathematics that are of great importance for the course concept “Learning Mathematics with SLE” (see section First Results of the Seminar), namely differentiated learning offers and forms of inner differentiation and outer differentiation. The category differentiated learning offers covers a wide range of aspects: offering more time, more or less number of tasks, different worksheets or tasks on different levels of difficulty, different textbooks, different mathematical topics, additional materials and manipulatives, additional help, learning step-by-step, more repetitions. Although a questionnaire does not allow in-depth analyses of the underlying concepts of teaching and learning and especially the concept of differentiation, 367 one might assume that the classroom situations the teacher students have experienced did not follow the concept of a natural differentiation and the children did not work on common problems and tasks. Some of the statements might lead to the conclusion that the teaching and learning setting more or less represents an exclusive setting with separate learning situations than inclusive education (see also Scherer et al., 2016, p. 640 ff.). Moreover, it remains as an open question if those underlying concepts for inclusive mathematics represent the teacher students’ own attitudes and beliefs or only represent the situation they had observed at school. In contrast, the course “Learning Mathematics with SLE” focuses on common learning situations for all students, enabling individual as well as cooperative learning situations, for example by realizing the concept of a natural differentiation (cf. Krauthausen and Scherer, 2014). It became clear that certain attitudes might have to be revised. First Results of the Seminar For the teacher students a catalogue of SLEs with topics related to arithmetic, geometry or selected contexts were offered. All topics and examples should show the central characteristics of SLEs like a necessary complexity, the opportunity of open tasks or problems with a common mathematical structure (cf. Krauthausen and Scherer, 2013, 2014). The concrete topics offered to the teacher students were the following: - Open problems within a substantial arithmetical learning environment (number walls, number triangles or number chains; cf. Krauthausen and Scherer, 2014, 2013) should enable the children to work on their individual level (for example size of numbers, level of representation) and include the opportunity for the children to choose easy, difficult or special tasks by themselves and to reflect on the level of difficulty or numbers and patterns (fig. 1; see also Grossman, 1975; van den Heuvel-Panhuizen, 1996, 144 p.). Figure 1: “My easy number walls” (student with special needs, 3rd grade) - For the SLE activities with digit cards problems with similar structures (cf. Nührenbörger and Pust, 2006; Scherer, 2015) should be offered. The concrete tasks should focus on numbers with two or more digits and reflect on place value. - Doubling with the mirror (cf. Scherer, 2005) is an activity from grade 1 on and focuses on the important competence of doubling, here in the geometrical 368 context of reflection (fig. 2). Herewith, arithmetical and geometrical objectives are connected in a natural way. Figure 2: Doubling five counters with the mirror - Activities with tetrominos (cf. Hirt and Wälti, 2008) should be realized material- based starting with twins and triominos and offer a variety of substantial mathematical problems. - The learning environment Shopping refers to the handling of money and is related to real-life contexts (for example buying a self-chosen number of different objects for a fixed amount of money, cf. Hirt and Wälti, 2008). - Context stories should offer authentic context situations with relevant mathematical contents (cf. Erichson, 2003), and a specific topic can be designed in different forms with individual levels for problem solving (example animals; cf. Scherer, 2016). All teacher students groups had to develop an interview concept and design concrete problems and material for primary students including a didactical analysis of the mathematical content and offering at least two strategies for solving the designed problems. All concepts were discussed in seminar sessions before the field test followed and the teacher students carried out their interviews. For the second seminar presentation the small groups had to present their data. Summarizing their results and the observations and discussions of the presentations the following aspects can be noted: - Coping with the organisational requirements of such a field test was a great challenge for the teacher students. But many of them expressed that it was an important experience with relevance for their future work at school. - It became also obvious that some students did not use the whole potential of the SLEs. For example, not being aware of the variety of strategies for finding all tetrominos. As a consequence they were not flexible enough to accept different ways and interpretations. 369 - With occurring difficulties during the interviews there was sometimes a tendency of helping too quickly and helping too much. For example the group with the SLE context stories had chosen the concrete topic schedule of the emperor penguins (Erichson, 2003, p. 87 ff.). The children had to understand and order different events in the life of the penguins. As a possible additional help the students had prepared in advance a table representing the months of a calendar. In one case the teacher student offered the table as a scaffolding very soon, and one cannot be sure if the child would have been able to develop a diagram or visualization by herself. - Some of the teacher students did not consider all requirements of language, technical terms or specific mathematical terms in advance and had difficulties how to react during the interviews when children did not understand the given task. Sometimes a teacher student did not identify that understanding the task or the used technical terms caused the problem. - Further reflections about the situations made clear that in some cases the teacher students were not aware in advance about the possible errors and problems the children might have although the didactical analysis was an obligatory part of the first presentation. - It became obvious that in some cases it would have been more substantial asking for and fostering more clarification or explanations. In contrast, students were satisfied with a correct answer and this was especially true when interviewing students with special needs (see also Beswick, 2007/2008). - It could also be worked out that some of the children’s problems were caused by the teacher students’ imprecise or too advanced language or unsuitable materials or arrangements of the material: In the frame of the SLE shopping the teacher students wrote down for the same product different terms on the shopping list and on the price tags (“water” and “bottle of water”) that irritated the children. Difficulties also could occur due to false decisions with respect to the curriculum: One group asked second graders for multiplication tasks, using this technical term, although multiplication was not dealt with in class. To be understood correctly: It might be meaningful to ask children about mathematical topics that have not been dealt with in classroom but one has to be aware that these are questions about pre-existing knowledge and possible pre-concepts and not about topics learned already. Conclusions The first results of the project show that the concept of SLEs and natural differentiation are suitable for inclusive classrooms. But setting SLEs into practice 370 of this more or less new field of inclusive mathematics is a great challenge for teacher students. However, the value of SLEs became obvious. On the one hand, teacher students have to master the mathematical content and be flexible in their reactions to different students with their variety of strategies and ways of thinking. On the other hand, they have to be aware of a variety of difficulties, like general impairments in language comprehension or reading, or specific difficulties in technical language in mathematics. Children might have problems with arithmetic skills or problem solving. Moreover, a lack of motivation or mathematics anxiety could occur. When being confronted with those difficulties a tendency of reproducing some of the patterns they experienced at school could be observed, for example more traditional forms of differentiation that low achievers need different learning offers, different tasks and materials or a prescribed program. These concrete experiences have to be made a subject of discussion to widen the repertoire of teacher students (cf. Scherer and Steinbring, 2006). Analyses and reflections on videos, materials and examples given in the lecture have a high value. Extended by the teacher students’ own experiences and common reflections in a seminar can increase their knowledge and teaching repertoire for the future. The above mentioned aspects are important for all kind of classroom situations but seem to be more challenging in inclusive settings. For the project in addition to the initial questionnaire a second questionnaire was presented at the end of the semester (pre-post-design): This questionnaire includes the items with respect to the teacher students’ attitudes and beliefs as well as some content-related items asking for a retrospective self-assessment of their own competence development. Moreover, in-depth interviews with selected teacher students are planned. These data will give more insight in teacher students’ knowledge and beliefs and will lead to further consequences for course concepts in the BA- and MA-program. Realizing the UN Conventions (see UN, 2006) in school requires adequate teacher education programs and integrating this topic in various pre-service as well as in- service courses will be necessary (see also Scherer, 2015). 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Mathematics in Designing Substantial Learning Environments. In M. Van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 193– 197). Utrecht: Freudenthal Institute. Wittmann, E. C., Müller, G. N. (1990). Handbuch produktiver Rechenübungen. Band 1: Vom Einspluseins zum Einmaleins. Stuttgart: Klett. STUDENTS CONSTRUCTING MEANING FOR THE DECIMAL SYSTEM IN DYADIC DISCUSSIONS: EPISTEMOLOGICAL AND INTERACTIONIST ANALYSES OF NEGOTIATION PROCESSES IN AN INCLUSIVE SETTING Christian Schöttler and Uta Häsel-Weide Abstract Participation in mathematical interaction is widely accepted to be important for students’ meaningful learning of mathematics. But how can interaction be initiated when students show different levels of understanding as it is common in inclusive classrooms? In the study “Decimal” learning environments focusing on decimal system understanding are developed, in which students can cooperate on different levels. The paper discusses the video recorded negotiation processes of pairs in cooperative settings. Objective of the observation is to analyze how the students interact and to what extent mathematical learning processes take place. Keywords: Cooperative learning, discourse, inclusion, decimal system understanding Introduction As a result of the United Nations Convention on the Rights of Persons with Disabilities (United Nations, 2006) the German school system is in a process of change. Especially in secondary schools, where a separate education of students with and without special needs as well as low achievers was common for many Universität Paderborn, Germany; e-mail: [email protected] 373 years, this change can be observed. Students with learning disabilities and children of risk are still being taught by their own individual curriculum, customized to their special needs and abilities, but nowadays have the option to visit the regular secondary school as well. Therefore, learning environments are needed which allow students to work individually on their level of understanding but also facilitate participation, collaboration and mathematical discourse. To enable this learning environments could be designed according to a substantial mathematical idea, for example the decimal system. This idea is fundamental for mathematical understanding (Wittmann, 1995) and can be edited at different levels and in various ways. If a fundamental idea is the core of the learning environment, it must be possible for students to learn on their individual level on the one hand and in interaction with others on the other hand. Thus, the pedagogical idea of learning together in inclusive classes on a common subject and at different levels (Feuser, 1995) can be put into practice in this way. Theoretical framework Decimal system understanding Doing mathematics, we generally use the decimal system, which is marked by its concise, effort and economy. Thus, in the decimal system, each number can be written down with a small number of digits. The decimal system contains various characteristics and principles which must be connected to each other. Treffers (2001) distinguishes between structuring and positioning: Structuring is based on specific structural characteristics and includes positional, base-ten, multiplicative and additive property (Ross, 1989, p. 47). The combination of all properties is needed to interpret a numerical sequence as a number and to deal with it adequately. Positioning focusses on the number row and the idea that the numbers form an ordered series in which all numbers take a firm position. For a comprehensive understanding of the decimal systems both ideas with all aspects have to be integrated. The aspects refer to both natural numbers and decimal fractions. However, the decimal system is also an extremely abstract and complex number system and student’s decimal system understanding develops gradually and over several years (Ross, 1989). A full understanding is not easy and without problems. Moser Opitz (2007) shows that decimal system understanding is very important for learning mathematics, but it is difficult for some students to develop a profound understanding. Difficulties in learning mathematics often become apparent while appropriating different aspects of the place value system such as grouping, degrouping and understanding place value (Scherer et al., 2016). In the first school years, understanding of the decimal system is first built up in smaller number spaces. Then, the understanding will be extended and deepened according to the concept of the spiral curriculum and related to larger numbers or 374 number ranges. In Germany numbers are introduced in steeps. Beginning with numbers up to 20 (first grade, 6-year-old students), with each grade the number space increases (up to 100, to 1000 and to one million in grade four). In fifth and sixth grade, numbers bigger than one million and decimal fractions are taught. Due to the analogues and hierarchical structure, decimal system understanding is particularly suitable for inclusive education of mathematics. This allows all students to work on a common idea at different levels. Decimal understanding of natural numbers is the core for all activities and it is connected with the understanding of place values on decimal fractions. For the work on this core different aspects, which are important for the understanding of the place values, can be distinguished. These aspects can be elaborated and clarified by a networked facilitation and a comparison between different numbers and number ranges: Analogies between relations of natural numbers and decimal fractions Structure of numbers (e.g. meaning and order of place values) Decimal subdivision as a constitutive principle (Endless Base 10 Chain) Spatial relationship of numbers as part of the basic orientation within the series of numbers (e.g. neighboring units, (approximate) position of numbers) Participation in mathematical interaction Students’ Collaboration is one central aim of inclusive teaching and learning. But it is not only central from an inclusive-pedagogical point of view, learning together is important from an epistemological perspective, too. Mathematical knowledge is not a ready-made curricular product which can be introduced into the learning process, it can only develop during interactive negotiation of meaning (Steinbring, 2005). Indeed, every student has to construct and develop knowledge independently and actively and therefore learning processes are individual, but the emergence of new knowledge is integrated in social-interactive processes. Consequently, the individual acquisition of mathematical knowledge is closely linked to social learning processes. The student’s interaction and communication play a decisive role in constructing new mathematical knowledge (Steinbring, 2005). Therefore, it is important to initiate interaction and discourse between students. Discourse is especially important, because if the students are in agreement about their results, strategies or interpretations, they won’t feel any necessity for a mathematical dispute. Instead, they need tasks which cannot be solved by one person him or herself. Hence, specific activities and tasks as well as limited materials or methods which cause positive interdependence serve as so-called formal occasion for collaboration. Further, informal occasion like mistakes, gaps or questions can be followed by interaction, too (Häsel-Weide, 2017). In inclusive classrooms, students with different learning competences study together. This may increase the variety of interpretations of the students to a wide 375 range of mathematical ideas and approaches. However, it must also be considered that the range of interpretations may not be too great. The students should still have the chance to exchange ideas and agree on a common solution. In other words, if the differences were too great, they might also have an adverse effect on the collaborations success (Webb, 1989). Taking this into account, the design of an interesting and suitable task seems to be an important criterion for productive communication. The tasks should be designed in such way that collaboration brings substantive profit, the students cannot work the task alone, and the students achieve more together, e.g. by solving a problem that could not have been solved without collaboration. However, the construction of tasks is a particular challenge and its effect is not guaranteed. Hence, the same task can lead to a productive communication for one pair, whereas another pair does not benefit the potential for conversation. Method and proceeding Methodological considerations: In the tradition of design science, developing appropriate teaching examples is a central task of research in mathematical education (Nührenbörger et al., 2016). According to content-related theoretical concepts learning environments are constructed with the aim of investigating learning processes. In order to analyze the student’s mathematical ways of thinking and their mathematical activity, a descriptive and analytic point of view is taken and methods of interpretative education research are used. The interpretative mathematics education is concerned with “ideas of mathematical concepts or problem solving, with viewpoints of mathematics [...] as well as, in particular, with the teaching and learning processes themselves, so with subject- related teaching interactions, participation structures and collective themes” (Jungwirth 2003, p. 190, translated by the authors). For the data analyses two perspectives are taken into consideration: the epistemological and the interactionist. The epistemological perspective deals with the meanings of mathematical signs and symbols constructed interactively by the students (Steinbring, 2005). The interactionist perspective investigates how the each student individually is involved in the negotiation processes and his or her roles as he or she participates in this process (Krummheuer, 2010). Project “Decimal”: In the project Decimal two learning environments have been developed for an inclusive mathematical classroom, in which students of fifth and sixth grade should extend their place value understanding. One learning environment focusses structuring, the other focusses positioning. Various collaborative settings are used to initiate social phases where students are asked to interact and collaborate, respective to their different levels of understanding. Six classes took part in the study. The teachers were asked to choose pairs with different competences in order to work together. Two pairs were video recorded 376 in each of classes. Chosen for the video were pairs of students where a student with learning disabilities collaborates with a partner without difficulties in learning mathematics. The dyadic discussions are carefully transcribed and the corresponding transcripts are interpreted according to the following questions: ‐ What is the student’s mathematical understanding of the structure of the decimal system? ‐ Which interaction and interpretive processes are observed during a dyadic discussion in an inclusive setting and to what extent are they productive? For analyzing the students’ interactive negotiation processes, the ideas involved in the interaction are first being identified and worked out. Meanings for mathematical ideas are constructed by constituting relations between signs/ symbols and suitable reference concepts (Steinbring, 2005). The students’ interpretations of the decimal system are reconstructed and analyzed whether the students recognize underlying decimal structures or remain on an empirical- concrete level (cf. Nührenbörger, 2009) by working with visible given numbers. Furthermore, the students’ participation in the negotiation process of the mathematical ideas is to be reconstructed. There are different opportunities for participation in a negotiation process. One can give a contribution, which has an organizational, documentary or content-related relevance. These different functions of contribution and according roles are called “organizer”, “documentator” and “contentor”. In the case of a content contribution, the responsibility of the utterance is also examined. According to Krummheuer (2010) two components of an utterance for which one can assume responsibility are distinguished: once for “the syntactic structure with a specific word choice and form […]. [As well for] the content-related (semantic) contribution” (Krummheuer, 2010, p. 85). Additionally, Krummheuer differentiates four kinds of speaker statuses: An “author” assumes responsibility for both content and formulation. A “relayer”, on the other hand, assumes no responsibility for the content and the formulation. A “spokesman” takes the idea of another and expresses it in own words. Finally, a “ghostee” uses the identical formulation of the parts of another’s utterance to express his own idea (Krummheuer, 2010). In the present study both interpretive approaches are connected and extended as it is taken into consideration what happens with the idea in the negotiation process. Here, different possibilities are distinguished: Another’s idea is picked up and developed further, so it can be regarded as a common idea. Or the idea is directly accepted by the partner (monologue idea). As an alternative, the idea is taken up and concretized more precisely or is rejected or ignored by the partner. Another possibility is that the idea is picked up and related to another task (application of the idea). 377 All aspects come together in an analysis instrument (see p. 8), making it possible to precisely grasp and analyze the joint negotiation process of the mathematical ideas as well as to reconstruct the active individual participation in this process. Negotiation processes in an inclusive setting: an exemplary analysis1 Learning environment: “Zooming on the number line” The learning environment “Zooming on the number line” is designed according to the idea of positioning. It consists of three parts: in the first two parts, relationships between different number spaces are discussed and in the third part, the relationships between natural numbers and decimal fractions are considered. Overall aim of the environment is to promote the positioning by using the “zooming-function” of the number line. By refining certain sections of numbers, the students have to determine neighboring units of the searched numbers and the appropriate position on the different number lines (spatial relationship of numbers). In addition, they have to subdivide sections into ten new parts when they refine (decimal subdivision). By comparing the series of number lines the students should recognize decimal analogues between different number ranges. Thus, they can see that both numbers are of analogous structure and that the basic principle of decimal subdivision also applies in the range of the decimal fractions. The following episode is from the third part. Each student received a series of number lines, either with natural numbers or with decimal fractions, and should first by him or herself, beginning with a partially labeled number line, refine sections of the number lines (indicated by lines and magnifying glass). In doing so, they label respective number lines, determine the space between the units on the different number lines and find at least one searched number on the number lines. After this, the students are asked to compare their series of number lines. Here, they should recognize similarities and differences and discover same and different relationships between the number ranges. The use of analogous tasks enables the students to work in different number ranges but on the same mathematical structure. Also, a subject-related communication about their findings and discoveries is possible. The joint activity is to compare analogues tasks. Ina and Akim compare the number lines The following sequence shows a section from a partner work, where the two students Ina and Akim work together. Before the sequence starts they have worked on their series of number lines individually and labeled them completely (see Fig. 1). 1 The data originate on the PhD-project of Christian Schöttler. 378 Series of number lines in the range of natural numbers, edited by Ina Series of number lines in the range of decimal fractions, edited by Akim Figure 1: Decimal relations between natural numbers and decimal fractions when zooming on the number line Now they are asked to compare the series of number lines regarding to similarities and differences. Prior to the start they have already mentioned that they worked with different number ranges. 1 Ina So. In my case was zoomed between the numbers seven hundred and eight hundred. Akim, what did you have? Where was zoomed in your case? Between what numbers? 2 Akim Between seven and eight. 3 Ina Curious, in my case was between seven hundred and eight hundred. There are nevertheless quite different numbers. 4 Akim Yes. You have hundred, I have one and ten. 5 Ina Yes. 6 Akim And you have thousand. [...] 7 Ina So, actually, there were exactly the same numbers, just that were several of tens. 379 8 Akim Yes. In your case were two zeros and in my case, were no zeros. 9 Ina And by the fact that there are several tens, so higher numbers than in his case, other numbers came out in the end. Because it was getting smaller and smaller. That is our result. After both students had mentioned between which numbers they refined at the beginning Ina discovers something interesting (“curious”). Potentially she compares the two refined number line sections and the numbers mentioned together (700 and 800 resp. 7 and 8). Although both have refined between the seventh and eighth scales and thus have similar numbers, the numbers are quite different. Her expression does not show explicitly which kind of differences she has noticed. Akim agrees and seems to describe the “curiosity” more precisely: He mentions that Ina worked in the range up to 1000 and he himself up to 10. Presumably he refers to the respective second-highest number lines. Here, Ina has labelled her number line from zero to thousand, with units of hundred (“You have hundred” and “thousand”), while Akim has labelled his number line from zero to ten, with units of one (“I have one and ten”). In doing so, he probably concretizes the “different” numbers but does not see the multiplicative structure between the parallel number lines. In this few turns it seems that Akim and Ina have recognized analogies between the number lines, but without verbalizing precisely the underlying, invisible structural and decimal contexts. The differences are only explained by the various number spaces and not by explaining in which way the compared numbers are different, why they differ or by relating the numbers to one another. They use the given number lines and units in a concrete way and thus remain in the interpretations on the empirical-concrete level (cf. Nührenbörger, 2009). The conceivable features of the number lines represent an “empirical justification context” (Steinbring, 2005, p. 191) because the concrete given number lines and units are used for explaining the differences. In the next turn Ina seems to bring in a new idea. She compares the both second- highest number lines and recognize that their structure is similar: “there were exactly the same numbers, just that were several of tens”. It is possible that Ina wants to express that the digits are the same, but she lacks the word “digit” and thus says same numbers. Despite the similarities recognized, the numbers differ. Viewed optimistically it could be that Ina sees a hundredfold between the units of her and Akim’s number line, which she expresses with “several of tens”. Akim seizes up the idea and gives concretes examples. He expresses the visible difference between the units which differ by two zeros. In the following, Ina seems to try to verbalize the decimal structure “several tens, so higher numbers”. She may use “tens” as a term for place values. In this case, she seems to recognize the importance of the 10 in the place value system. Her numbers are greater than Akim’s by a hundredfold, here a central idea of the decimal system can be identified. 380 The analyses of the discourse clarify that the students recognize structural relationships by comparing the number lines. It also shows the difficulty in conceptualizing the structures of the decimal system. The perceptible patterns (same digits, hundredfold, two zeros, no zeros) of the series of number lines represent an “algorithmic justification context” (Steinbring, 2005, p. 191) when comparing the number lines: they compare the units of the different number lines without seeing structural relations. With regard to the interaction processes it can be supposed that the comparison of analogous tasks leads to a mathematically focused conversation, which is characterized by related contributions (reciprocity). Despite of their different learning competences, both learners are actively involved in the negotiation process and show own, task-oriented contributions which are negotiated jointly. As shown in the table of production design (Fig. 2), both ideas are introduced by Ina and are jointly concretized and further developed. Especially the collective (further) development and concretization of ideas is important in the scene and also shows a symmetry between the two learners on the content level. Speaker Utterance (line) Idea Dealing with Organizational and role the idea function (Status) Ina: Akim, what did you have? Reconciliation organizer Where was zoomed in your case? to the Between what numbers? (l. 1) comparison of the refinement Ina: Curious, in my case was between recognize from contentor seven hundred and eight Ina’s idea is analogies (author) hundred. There are nevertheless taken up and between quite different numbers. (l. 3) further numbers up to developed by Akim: You have a hundred, I have one ten and numbers Akim contentor and ten. (l. 4) up to thousand (author) And you have thousands (l. 6) Ina: there were exactly the same contentor numbers, just that were several recognize from (author) of tens (l. 7) decimal Ina’s idea is Akim: In your case were two zeros and in analogies first contentor my case, were no zeros. (l. 8) between concretized (spokes numbers up to by Akim and man) ten and numbers further Ina: And by the fact that there are several up to thousand developed by contentor tens, so higher numbers than in his and structure of Ina (author) case, other numbers came out in the numbers end. (l. 9) Figure 2: Table of the production design 381 Final remarks Aim of the study was to design learning environments which allow students with different level of mathematical understanding to extent their decimal system understanding in collaborating and discussing. The selected scene gives a short insight into how the students communicate with each other in a stimulating way about the mathematical task. Here as well as in other episodes, the learners express their interpretations with their capabilities of communication (everyday language, more detailed explanations). This episode can give an insight into how hard it is to describe the abstract concept of place value and describe the decimal structures precisely, but at the same time it is obvious that this is a challenge of all students. In the scene presented, the students remain on a superficial and empirical- concrete level. They refer to visible and concrete characteristics and numbers of the series of number lines (e.g. units, digits). It is also striking that the two students remain exclusively within the range of the natural numbers during the scene and do not compare natural numbers and decimal fractions as it was intended. Further analyzes will investigate the extent to which the joint negotiation of meaning leads to an extension of the understanding of the decimal system among the students as well as whether the students develop real mathematical knowledge. From an interactionist point of view, both students actively participate on a content level of the negotiation process, they are equal partners. Further research will show, if this is an individual finding. Therefore, in further analyses, the episode will be compared with others. Among other sections of the learning environment, the second is also considered. Criteria for successful cooperation are to be worked out through a comparison of different scenes as well as the participation possibilities of pupils in an inclusive setting. References Feuser, G. (1995). Behinderte Kinder und Jugendliche. Zwischen Integration und Aussonderung. Darmstadt: Wissenschaftliche Buchgesellschaft. Häsel-Weide, U. (2017). Mistakes as occasions for productive interactions in inclusive mathematics classrooms. Proceedings of CERME 10. Dublin, Irland. Jungwirth, H. (2003). Interpretative Forschung in der Mathematikdidaktik – ein Überblick über Irrgäste, Teilzieher und Standvögel. ZDM, 35(5), 189–200. Krummheuer, G. (2010). Representation of the notion „learning-as-participation” in everyday situations of mathematics classes. ZDM, 43(1), 81–90. Moser Opitz, E. (2013). Rechenschwäche/ Dyskalkulie. Theoretische Klärungen und empirische Studien an betroffenen Schülerinnen und Schülern. Bern: Haupt. Nührenbörger, M. (2009). Interaktive Konstruktionen mathematischen Wissens – Epistemologische Analysen zum Diskurs von Kindern im jahrgangsgemischten Anfangsunterricht. Journal für Mathematik-Didaktik, 30(2), 147–172. 382 Nührenbörger, M., Röseken-Winter, B., Fung, C. I., Schwarzkopf, R., Wittmann, E. C., Akinwumni, K., Lensing, F., Schacht, F. (2016). Design Science and Its Importance in the German Mathematics Educational Discussion. Springer. Ross, S. (1989). Parts, Wholes and Place Value: A Developmental View. The Arithmetic Teacher, 36(6), 47–51. Scherer, P., Beswick, K., DeBlois, L., Healy, L. Moser Opitz, E. (2016). Assistance of students with mathematical learning difficulties: how can research support practice? ZDM, 48, 633–649. Steinbring, H. (2005). The Construction of New Mathematical Knowledge in Classroom Interaction – An Epistemological Perspective. Berlin: Springer. Treffers, A. (2001). Numbers and number relationships. In M. van den Heuvel- Panhuizen (Ed.), Children Learn Mathematics: A Learning-Teaching Trajectory with Intermediate Attainment Targets for Calculation with Whole Numbers in Primary School (pp. 101–120). Utrecht: Freudenthal Institute. UN-United Nations (2006). Convention of the Rights of Persons with Disabilities. New York: United Nations. Webb, N. M. (1989). Peer interaction and learning in small groups. International Journal of Educational Research, 13(1), 21–39. Wittmann, E. C. (1995). Aktiv-entdeckendes und soziales Lernen im Arithmetikunterricht. In G. N. Müller, E. C. Wittmann (Eds.), Mit Kindern rechnen (pp. 10–41). Frankfurt a. M.: Arbeitskreis Grundschule. STUDENT PORTFOLIO AS A TOOL FOR DEVELOPMENT OF PRE- SERVICE PRIMARY TEACHERS’ COMPETENCES IN TEACHING MATHEMATICS Jana Slezáková, Darina Jirotková and Jaroslava Kloboučková Abstract The paper reports on a research focusing on one of the possible ways of leading pre service primary student teachers to constructivist educational style effectively. The research base for the study is formed by student portfolios. The goal of analysing these portfolios is to look for the turning points in a student’s development and for impulses that brought about any changes in students’ attitudes. Analyses of several portfolios showed that creation of a portfolio itself and the following work with it has a significant and positive impact on formation of pre-service teachers’ pedagogical beliefs. Keywords: Student portfolio, roles of portfolios, changes in attitudes to teaching mathematics, pedagogical beliefs Charles University, Czech Republic; e-mail: [email protected], [email protected], [email protected] 383 Introduction The unfavourable results of Czech pupils in the international comparative tests TIMSS 2007 and PISA 2009 stirred a society-wide discussion on the need of improving the quality of mathematics education at primary schools and of bringing about a change to Czech schools that would have positive influence especially on Czech pupils’ attitude to mathematics and their attitude to learning in general. The community of mathematics agree that the decisive element in the didactical triangle pupil – subject – teacher is the teacher, their personality and approach to teaching. McKinsey study (2010) clearly points at the need of bringing about a change in attitudes of Czech teachers: “If there is to be an improvement in the quality of education, it is essential to change people’s attitudes and behavior – in the Czech Republic attitudes and behavior of more than 100 000 teachers – and this is an extremely difficult task for any organization.” (p. 4). Any changes in systems of education are often very difficult and long-term. Moreover, about one half of our teachers are happy with the current state and feel no need of a change, which is shown in research, e.g. in (Straková et al., 2013), (Voda, 2010). Our experience from dozens of in-service teacher training seminars and summer schools for in-service teachers only confirms this. Experience also shows that changes in pedagogical beliefs (Žalská, 2012) of every single teacher who is trying hard to change their educational style from instructional to constructivist does not happen at a snap of their finger but takes many years, is very painstaking and the teacher must overcome many obstacles. This process is described e.g. by G. Hlavatá in her diploma thesis (2014). The difficulties spring out (apart from other) from the fact that a change in beliefs intervenes with the person’s hierarchy of values where any changes are difficult or even impossible. The paper focuses on pre-service primary teachers and the changes in their pedagogical beliefs during their studies at the Faculty of Education. At the outset of their studies their beliefs are formed mostly by their experience as pupils, experiences with teachers, by their relationship to mathematics built while at primary and secondary schools, by certain personal traits such as self-confidence and autonomy in the area of mathematics as well as in social area, the skill of self- reflection and self-assessment. An efficient tool for observing the incentives that result in changes in students’ pedagogical beliefs are, apart from on-going reflections on teaching and numerous discussions with students, also student portfolios. Students develop their portfolios over a long period of time. Some of them start already in the first year of their undergraduate studies at the Faculty of Education. Analyses of the final form of these portfolios give us an idea about the changes the students underwent, about the causes of them and about the great variety of students’ personalities. It could seem easy to direct a pre-service teacher’s style to constructivism during their studies. We exert influence on them in courses of mathematics and didactics of mathematics (DM) all five years of their studies. We build all courses of 384 mathematics in the spirit of constructivism. Our objective is not only future teachers’ thorough content knowledge and pedagogical content knowledge but also their sufficient experience with learning in the spirit of constructivism. We strive to build in our students a model-for creative, reflexive, constructivist and inquiry based education (Gravemeijer, 1999). Our long-term experience says that it is not easy to change a student’s beliefs about goals of mathematics education and about effective educational style. The fact that students experience constructivist education in the role of learners, that they get the needed theoretical background knowledge does not necessarily imply that they will be able to use all this in the role of a teacher. Recent research confirms this. For example, Samková et al. (2016) describe a two year long project within whose frame pre-service primary teachers experienced Inquiry Based Mathematics Education in their undergraduate courses. Although their subject content knowledge improved and their attitude to this model of teaching was positive, when actually teaching they tended to seek the “secure” environment of transmissive teaching. Students often start their undergraduate studies with decided attitudes about what mathematics education should be like. Their opinions usually copy the educational style of their former favourite teacher or contradict the educational style of a teacher who made their life in mathematics very difficult. The goal of research studies in (Hejný, 2012) and (Jirotková, 2012) was to create a tool that would allow characterization of a teacher’s educational style but mainly to find the way of influencing a student or a teacher. For these ends 20 parameters characterizing a teacher’s educational style were created. These are presented also in (Jirotková and Slezáková, 2017). We chose those parameters that we believe can be influenced during undergraduate studies. These parameters are attitude to mathematics, self-confidence in the area of mathematics and didactics1, mathematical, didactical and pedagogical competences and experience with the role of a teacher. The last two listed parameters can hardly be influenced in compulsory courses of DM as they are organized at this point. Students lack the opportunity to gain experience in the role of a teacher. Teaching practice has not had sufficient time allocation at the Faculty of Education for years. The efficiency of these two subjects despite our effort to develop reflective practice (Korthagen, 2011) is not sufficient for more substantial changes. Students get further experience with teaching as late as in the 5th year during their teaching continuous practice at schools. This does not allow students to try out what they have come across in theoretical preparation. It does not help them avoid their fear of didactical or 1 Analysis of the latest TALIS 2013 survey from 2015 shows that Czech teachers were one but last in the area of self-confidence.[http://www.oecd.org/edu/school/talis-excel-figures-and- tables.htm] 385 mathematical mistakes, get deeper insight into their pupils’ minds, get hands-on experience with developing pupils’ personality. We look for ways that would give students the opportunity to get the needed experience in the role of a teacher that might affect their pedagogical beliefs. We are convinced direct reflected students’ experience is the most meaningful way of achieving this. That is why in the 4th (and also the 5th year) – parallelly with the courses DM with teaching practice II and III students are offered an elective course Extension module from DM. From the point of view of students’ beliefs and their future educational style, this course seems to be most effective. Methodology Collection of materials into student portfolios – research database The basis of a student portfolio is formed by materials from the following activities assigned during pre-service teachers’ studies: a) At the beginning of the 1st year students write an essay about the development of their attitude to mathematics from their early school years until their university studies. b) After the 1st year they write an essay on changes in their attitude to mathematics. c) In the 2nd year they work on a selected mathematical topic. In the selected area they pose several problems for a 6-12 year old child, set this problem to one child and observe the child while solving them. This seminar work is peer-assessed in a study group. By this, students develop their mathematical competences, ability to formulate an age appropriate problem for a selected pupil, conduct and record an experiment with one pupil and assess work of their colleagues. d) In the 3rd year they pose graded problems, i.e. a series of problems of growing difficulty for weak to talented pupils in their seminar work. Students develop their ability to poses graded problems. e) In 1st to 3rd years they write various types of seminar works (their own collections of problems from given areas of mathematics with comments, first reflections from lesson observations) and they start building their portfolio. The portfolio also includes their own solutions of problems, especially those in which they face some difficulty or in which they changed their solving strategy or made a mistake. f) In the courses DM with teaching practice II and III in the 4th year students make detailed lesson plans for their teaching experiments which include didactical analyses of problems, objectives of the lesson and mathematical problems. All this is supplemented by the student’s own reflection and a record from a joint reflection in which their peers and mathematics educator, sometimes also the class teacher participate. 386 g) In the 5th year students have two continuous teaching practices – 2 week and 4 week long. Only a small part of the students can select to be supervised by an educator from the Department of Mathematics and Didactics of Mathematics. h) In the 4th and 5th years students can add into their portfolio more materials from the optional Selective seminar from DM I and II. In this seminar attention is paid to analyses of video recordings from lessons, of stories, of pupils’ solutions of problems, problem posing including posing of graded problems, lesson observations on “alternative” schools, e.g. Scio school, which deconstructs all contemporary prejudice about how to conceive education, Jedlička’s institute, where children with serious learning difficulties are educated, e.g. children after cerebral palsy. i) The richest source of material for their portfolio is the selective Extension module from DM. In this module students gain a lot of experience with the role of a teacher in which they are placed once a week for two semesters. In each of the lessons they have a discussion with the mathematics educator and their peers. There is no compulsory curriculum to be covered. This selective Extension module will be discussed in more detail in the following text, as the time when students attend this module is the time in which we can observe striking changes in pre-service primary teachers, especially in the area of goals of mathematics education and effective educational style. The results from this module, which is attended by fewer than one half of students from one year, confirm that what students lack if they are to be able to introduce innovative methods into practice, is experience with teaching pupils in laboratory conditions. Extension module – the source of experience allowing a shift in a student’s pedagogical beliefs The Extension module was conceived in a way to make it linked to the school practice as much as possible and to provide space for linking theory from courses of DM to teaching practice. The module consists of three two-semester courses, each of them with time allocation 0/2. This means that in academic year in which students select this module they spend 6 hours a week in the environment of a primary school. The school runs a mathematical club for primary pupils. In recent years the cooperating school was primary school Barrandov II, a large school whose pupils are eager to participate in the club – in 2014-2015 the club was attended by 80 pupils, in 2015-2016 by 110 pupils and the interest is the same in 2016-2017. The pupils are divided into groups of about 15 pupils according to their age. Three to four students work with the same group for the whole year. In the first part of six hour block students share their lesson plans for the club activity and discuss the anticipated didactical phenomena and mathematical obstacles. It is important for students to have the chance to choose the mathematical content and its methodological handling. Students often chose those areas and environments where they have most doubts about how to approach them 387 methodologically, how to introduce them to pupils, how pupils accept them and what mathematical knowledge is developed in them. The students’ task is to ask two questions for each lesson plan: one of the questions must concern mathematics or didactic of mathematics, the other pupils. The first question is discussed immediately, series of graded problems are solved and posed and parameters of gradation are discussed. The answer to the second question is sought while running the club. Teaching in the club with pupils and reflection is the following stage. Each group of pupils is video recorded. Students pay attention especially to whether they are able to create safe environment, appropriate motivation and suitable gradation of problems. Reflection is conducted in the form of joint reflection (Hošpesová and Tichá 2007) and is done on two levels: a) in groups, b) as a whole group with the mathematics educator. Joint reflection is followed by planning the activities of the clubs to come. In the last block the mathematics educator guides the students to focus on the mathematical depth of the different activities. This is achieved by solving more demanding problems. Students discover the schemes of mathematical concepts, phenomena and processes built through the given problems (Hejný, 2012), they look for the didactical and mathematical goal of each activity and also discuss their potential for personal and social developments both of the pupils in the club and the students who teach there. Let us point out here that the relationship between students in the group and between the students and their teacher is based on trust and security, which gives the students the chance to try out new didactical approaches that they would never dare to try in everyday teaching or that they would be dissuaded to use by their mathematics educator. Students can make use of their own mistakes, learn more about their pupils’ mistakes and in discussions look for their causes and possible consequences. This triggers or accelerates the students’ shift to constructivism. Finalization of portfolio and self-assessment The attendees in the Extension module create a partial portfolio as the basis for being assessed in the module. This portfolio is made in at least two stages. In the first semester it is a collection, an on-going work portfolio – students collect all available materials. At the end of the term the portfolio is presented and objectives they want to focus on in the next term are formulated. The students try to formulate the specifics of the portfolio they will be creating. They might focus on a long-term observation of one pupil, on an analysis of pupils’ solutions and pupils’ as well as their own mistakes, on analysis of selected video recordings etc. They discover that for example more thorough thinking about pupils’ motivation has direct impact on their pupils or that analysis of mistakes will help them understand pupils’ thinking processes, which in consequence improves their ability to create a suitable re-educational tool. Students experience 388 meaningfulness of work on a portfolio and their processual, formative, work portfolio (Spilková, 2007) gradually changes to developmental portfolio. The change in students’ thinking is clearly visible e.g. in their reflections. In the beginning they write more about feelings and emotions (“I worked well with the pupils”, “the pupils enjoyed it”), at the end of the semester their reflections become more descriptive, they write about the course of their teaching in the club in detail, they think about individual pupil’s reaction when solving problems, about sources of their mistakes. They also pay more attention to the atmosphere and student-pupil as well as pupil-pupil communication. Processing research material When concluding their studies students may choose an alternative form of the final state exam, which is defending their portfolio during the state exam from didactic of mathematics. Their portfolio becomes the tool for documenting and assessment of learning in a complex form (Košťálová, Miková and Stang, 2008). This option is selected predominantly by students who attended the Extension module because in the module they gained the experience of how to build a portfolio on two levels, which gives them the courage to prepare their portfolio for the state exam. However, the possibility to defend their portfolio is selected also by students who did not attend the extending module. These are likely to be autonomous students who perceive the portfolio as a tool for their further development and who know how to make use of it in their teaching practice. It is these portfolios finalized for defense at the state exam that form the most substantial part of our research database. These portfolios have to be produced in electronic form as well. At this point we have 15 portfolios available for our research. These portfolios are subject to qualitative and comparative analyses. In various stages of our research we focus on various phenomena (motivation impulses, getting to know the pupils, building of mathematical knowledge, …). Now we want to identify those phenomena that describe the turning points in formation of pedagogical beliefs. Results Let us present here excerpts from portfolios of three students: Klára, Sylva, Zita. Let us illustrate the change they show having finished compulsory as well as optional mathematics education at Faculty of Education, Charles University in Prague as well as the variety of their focus. Klára after 1st year on Hejny’s method In the beginning of the first year I was writing in my essay ´Me and mathematics´ about my positive relationship to mathematics. To be honest this second essay will be quite negative. In the beginning I liked teaching at Faculty of Education. I saw similar method at the initial one week practice. The teacher had 20 pupils in her class and everything worked really well. The second time it was different. I didn’t like it so much. There were about thirty children. The active pupils were really involved, 389 the rest of the class quite passive. My initial enthusiasm for this way of teaching is weakening now. Klára after the extending module There are differences between pupils in all classes. There will always be active, extrovert children as well as shy or slower ... The thing I want to react to is “passivity” of some children. The observer might think they are doing nothing but we do not see into a child’s head. Even if the child was only listening to a discussion, next time they might solve the same problem on their own thanks to it. I have often come across a situation that a pupil looked disinterested or was even doing something else but at the same time they were in mathematics and knew what was being solved and sometimes even reacted unexpectedly and to the point. Today I know that discussion is important for all pupils, even for those who are not involved actively. Klára shows in her reflection how important in her case linking of theoretical preparation to teaching practice was. Only when she sees in practice what is discussed in theory, theory becomes meaningful to her. On the basis of repeated reflection of her own experience she becomes aware of the importance of discussion in a class and the need of a teacher’s thorough knowledge of their pupils. Sylva after 1st year I feel I gained nothing in the second semester. I miss “real” mathematics, like a set of inequalities on an A4 sheet. On the other hand I understand that this is easier for our classmates who have some difficulties in mathematics and their reflections are much more positive. Sylva after the extending module What does it tell about a person that they are able to solve a set of inequalities on an A4 sheet? Does it mean they really understand the inequalities? I think I used to feel that I am good at mathematics but now I see that my knowledge was merely formal. I managed to get A’s from tests but did not really understand what I was calculating. I was thinking about the turning point in my perception of the goals of mathematics education …. The first was in my 2nd year of studies when I saw during a teaching practicum from pedagogy how a 1st grade teacher was putting a lot of emphasis on individualization and it was clear that mathematics was her pupils’ favourite subject. What was most important for me was the 4th year of studies when I was attending the Extension mathematical module whose part was teaching in a mathematics club and also my teaching practice. The teaching practice in the 5th year inspired me in the sense that I got direct experience with how children can be developed if taught mathematics well. Both during my teaching practice and in the mathematics club we were making video recordings. I would definitely like to continue because I know I will learn a lot from this. 390 Sylva talks about her relationship to mathematics as such but also her personality – thoughtfulness, perception of great diversity of a group of pupils/students as far as their mathematical competences are concerned. The re-thinking of her experience allows her to see an important phenomenon – mechanical (parrot-like) knowledge. A stimulus for the change in her pedagogical beliefs is her good experience with teaching mathematics, but in the role of an observer. However, what played the biggest role was her rich experience with the role of a teacher and her pleasure from developing her pupils. Sylva also shows she got familiar with a strong tool for development of a teacher’s competence – video recordings of one’s own lessons and their repeated analyses. Her portfolio shows that Sylva is aware of the progress she has made in the ability to individualize teaching in a heterogeneous group of pupils. She describes the places in which she felt the most marked progress in this skill: observing good individualization in lessons (2nd year) – posing graded problems (3rd year) – posing graded problems for particular pupils and individualization (4th year) – repeated reflection, if needed of video recordings when finalizing the portfolio (5th year). An excerpt from Zita’s portfolio “The ability to create a portfolio which is further developed: I started developing this ability by creating a mathematical portfolio and now I apply it in other subjects (e.g. creation of an art portfolio). What I could not imagine before now becomes clearer and even by writing this I am becoming more professionally self-confident. When I wanted to start an art portfolio, it took much shorter than starting the mathematical portfolio. Thanks to the mathematical portfolio I already had a system that was comprehensible to me. The ability of further development: …. I learned to set goals! … I tried to make the goals realistic even though I have a lot of visions. Sometimes I do not have time enough to make my vision come true but it is important for me at least to write it down. Then I know what I want to achieve in the future.” Zita uses skills acquired in one subject at university in other subjects and while doing it she reflects on her increasing competence in work with a portfolio. This is very promising for her future career as a teacher. The need to have a portfolio will become the tool of her professional development. Zita shows that her work with the portfolio as such, her effort to formulate goals for the nearest future leave imprints on her system of values and form her pedagogical beliefs. Conclusion We could be citing much more from students’ portfolios about their experience while creating them. We are aware that the purpose of some of the students’ communication may be their attempt to influence the educator’s opinion of the student. However, so far we have come across balanced, both critical and self- critical students’ opinions and conscientious work. Creation of a portfolio in a way that makes it useful for the student in their future, in their teaching career, in 391 a way that all components required for the state exam are reflected especially with respect to individual growth and development is demanding. In fact it is more time demanding than studying all the areas tested during the state exam. This implies it would make no sense for the student to try to present a distorted view of the situation. Thus students’ statements do not need to be questioned. The students are aware they do the work for their own benefit and that they will be able to use their portfolios e.g. in the new career system valid from 2018. Comparison of our current and former experience allows us to state that students undergo a more substantial change when they start working with portfolios. Thanks to portfolios students get a base for reflection (even repeated) of their own professional development. Getting aware of the turning points which are different for different students significantly contributes to their development. Getting to know these turning points is very important for us as mathematics educators. We immediately project it into our teaching and work with students. At the same time we are fully aware of the fact the newly coming students are changing fast. This means we have to work hard on updating our experience. The above described turning points of crucial importance in development of our students may be marginal in the generation of students to come. Thanks to portfolios we are able to keep in touch with reality and respond to changes accordingly. 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INTERACTIVE GESTURING: DEEPENING UNDERSTANDING OF MATHEMATICAL ACTIVITY WITH YOUNG LEARNERS IN A SOUTH AFRICAN CONTEXT Debbie Stott Abstract A multimodal approach was used to analyse video data collected from paired learner task-based-interviews with learners aged between 9 and 10 years participating in after school maths clubs in South Africa. I explore the idea that a multimodal analysis with a specific focus on gesture can help us deepen our understating of what unfolds during a mathematical learning activity. In the context of a mathematical learning activity, particularly with young learners where the language of learning and teaching (LOLT) is a second or third language and where there is traditional reliance on verbal mediation, I propose that spontaneous, in-the-moment, interactive gestures could serve as an important mediation strategy for out-of-school time practitioners as well as pre- and in- service teachers. Keywords: Multimodal analysis, gesture, gesture as mediation, mediation, spontaneous gesture, primary mathematics, out of school time Rhodes University, South Africa; e-mail: [email protected] 393 INTRODUCTION, RESEARCH CONTEXT AND AIMS As part of my doctoral study, I recently carried out a multimodal analysis of video data of three task-based interviews with 9 and 10-year-old South African learners. This analysis explored the ways that participants caught each other’s attention during mathematical learning activity. During the analysis, certain thought- provoking insights emerged, particularly regarding spontaneous gesturing used by the adults who facilitated the interviews. In this paper, I share some of these insights and suggest what they may point to concerning gesturing and mediation in primary mathematics education, especially where English, as the predominant language of teaching and learning (LOLT), is not a mother tongue. The field of gestures in mathematics education has recently expanded. Various avenues of research are evident, for example the role of gestures in learning, cognitive development and embodied thinking (e.g. Goldin-Meadow and Alibali, 2013), gestures related to young children learning mathematics (e.g. Murphy, 2014) and using gestures to construct meaning in mathematical activity (e.g. Maschietto and Bartolini Bussi, 2009) to mention a few. As the insights that emerged from the aforementioned analysis concentrated on the gestures used by the facilitators in the task-based interviews, I narrow my focus to look at research on gestures used by teachers, particularly spontaneous in-the-moment gestures. Research context Since 2012 I have facilitated many after school mathematics clubs (as examples of out-of-school time programmes (OST)) with children aged between 9 and 12 years old. Two such after-school mathematics clubs formed the empirical field for this research. These were run within the South African Numeracy Chair (SANC) Project at Rhodes University. The SANC project, to which Mellony Graven is the incumbent Chair focuses on a dialectical relationship between research and development in numeracy education in South Africa. Clubs are an important part of the project’s ongoing interventions both locally and around South Africa and are conceptualised as places where learners make sense of mathematics and where they are encouraged to develop positive learning dispositions towards mathematics (see Graven, 2016; Graven, Hewana and Stott, 2013). Many South African schools choose English as their LOLT as English is seen as a language of power and access, but this means that many learners are learning in a language that is not their mother-tongue. Stein and Newfield (2006) contend that in such classrooms, a multimodal pedagogy could inform a social justice and equity agenda and has the potential to make classrooms “more democratic, inclusive spaces in which marginalised students’ histories, identities, cultures, languages and discourses can be made visible” (p. 11). Indeed, over the last decade, multimodal approaches to research and pedagogy have increased in South Africa, potentially providing a beneficial alternative to “monolingual and logocentric approaches to meaning-making” (Archer and Newfield, 2014, p. 3). 394 Within this specific field of mathematics education research and the early foundational years in a South African context, this paper focuses on how facilitators gesture in-the-moment and explore how these gestures might guide learner attention and sustained engagement with a mathematical activity. Despite the growth of multimodal research here in South Africa, I note a lack of research in mathematics education, particularly in primary mathematics and early numeracy. Thus, research into how gesture may be used as an additional meditational practice could be useful in this context. MEDIATION AND GESTURING USED BY TEACHERS Taking a broad socio-cultural perspective of learning of Vygotskian origin, I work on the premise that learning is both social and individual. Learning often begins when children are no longer able to continue on their own and require some kind of mediation (from a person or object) in order to continue. Mediation was Vygotsky’s key contribution to understanding child development. To say that learning is mediated means that something acts as a facilitator to learning, something moves between the learner and the concept to be learned. In this paper, the something is facilitator gesture. From a mathematical perspective, Anghileri’s (2006) work provided a range of useful mediational (scaffolding) practices observed in mathematics teaching. In the context of my broader research, this framework was employed as an analytic tool to explore the nature of mediation in the two research clubs. Space prevents me from describing the framework in detail, but the mediational practices are presented as a 3-level hierarchical model of observed patterns of interaction: 1) environmental provisions, 2) explaining, reviewing and restructuring and 3) development of conceptual thinking (Anghileri, 2006). An examination of the descriptors for these practices suggests that most of them describe mediational practices based on verbal communication. There are no explicit references to the use of multimodal (including gesture) resources as mediational practices. Anghileri does allude to other possible multimodal forms, but they are by no means predominant. Examples include: learners develop their own meanings if they are encouraged to look and touch what they see as well as verbalising, to bring different senses to bear on a problem; drawing explicit attention to learners’ strategies and actions to focus “joint attention on a critical point not yet understood” (p. 36) and finally emotive feedback which includes interjection of remarks and actions to gain attention, encourage, and approve learner activities. Anghileri however argues that a close analysis of observed interactions between learners and teachers highlights those practices that experienced teachers “often implement subconsciously” (p. 38) but may be difficult for inexperienced practitioners to identify. Her claim echoes the findings I discuss in this paper regarding the use of spontaneous in-the-moment gesturing. 395 At the time Anghileri put this model together, gesture research was in its infancy and as she pointed out, her model included practices that were observed in research contexts up to 2006. Taking into account findings from my broader study and considering Stein and Newfield’s (2006) argument that a multimodal pedagogy would be useful, I hypothesise that the model may benefit from incorporating some of the recent findings on gesture research. I note with interest in 2012 (six years on), Alibali and Nathan were still drawing attention to the lack of attention given to gesture in pre-service teacher education method courses and argue that there is “an overwhelming emphasis on the verbal channel” (p. 276). This point is well observed and supports my argument regarding Anghileri’s framework. There is some evidence of multimodal pedagogies being explored in South Africa over the last decade (see Archer and Newfield, 2014 for example), but this is by no means widespread. Gestures I use the term gesture broadly to mean “movements of the arms and hands... closely synchronized with the flow of speech” (McNeill, 1992, p. 11). McNeill’s well-known typology of gestures included pointing gestures, representational (both iconic and metaphoric) gestures and beat gestures. I expand on pointing gestures briefly as at first glance the gestures used by the facilitator in the data extracts below, seemingly fall in the pointing category. Pointing gestures are often used along with speech and are generally used to “index physically present objects or inscriptions and to evoke nonpresent objects or inscriptions” (Alibali and Nathan, 2012, p. 253). However, I will argue that the gestures used by the facilitator are not pointing gestures. Rather, I surmise that the gestures she uses are more generic in nature and perhaps could be traced back to what Bavelas, Chovil, Lawrie et al. (1992) term “interactive”, conversational gestures that accompany and illustrate talk, are improvised and synchronized to speech and are made by the person who is currently speaking. They are generally directed at an addressee and do not provide information about the topic under discussion (Bavelas, 1994). Although Bavelas provides a breakdown of these gestures based on their function, I did not work at this level of granularity for this analysis. Gestures made by teachers in learning contexts Goldin-Meadow and Alibali (2013) report that there is growing evidence to support the idea that the gestures teachers use, are important for learning and that a better understanding of gesture is crucial for a deeper understanding of the communication that takes place during teaching. Alibali et al. (2013) argue that classroom interventions are often conceptualised at a ‘macro’ level such as a series of lessons extending over days, weeks or years. Their research focused on the ‘micro interventions’ that teachers use as a lesson unfolds. They maintained that teachers dynamically adjust their instruction and use gestures adaptively in these 396 in-the-moment responses to learners’ varied contributions to establish and maintain shared understanding (what could be called intersubjectivity) in the mathematics classroom. The facilitator in the data presented here, responded in- the-moment to the learners’ multimodal contributions as we will see shortly. METHODOLOGICAL APPROACH The broader study as part of my doctoral research (Stott, 2014), was designed as a longitudinal, interpretive multi-site case study, with a focus on two after school maths clubs. The data presented here were drawn from three video-recorded task- based interviews with pairs of club learners who attended South African state schools and who spoke English as a second or third language. The interviews aimed to elicit and encourage mathematical learner talk and interaction. As co- club facilitators, Mellony Graven (the Chair of the SANC project) and myself additionally facilitated the three interviews between pairs of learners, four females and two males. Each interview was about an hour long and consisted of a range of activities. The puzzle in Figure 1 below was one of three activities used in the task-based interviews. In the excerpts below, learners worked in pairs, to find the values that related to each shape to ensure that the row and column numbers would stand true and work for all instances in the puzzle. Figure 1: Activity used for analysis The video recordings of the interviews were systematically transcribed and coded in several passes. In the first pass, all multimodal forms were transcribed for each participant, both verbal and non-verbal. As my broader research aim was to establish how participants caught each other’s attention, I was not concerned with gestures specifically, thus gestures were not classified into the kind of categories described earlier. Rather my concern was with the entire range of multimodal forms evident in the data and which of those acted as instances of attention catching. Whilst acknowledging that there are many ways to classify and analyse multimodal communication, I employed Bourne and Jewitt’s (2003) four multimodal codes of speech, gaze, posture and gesture as the initial codes. Two additional modes visible in my video data, namely expression (e.g. laughter, frowning) and mathematical action (e.g. mathematical writing, workings on paper, calculating using fingers) were added for analysis purposes. These six codes were inserted into the transcript at the places they were observed. In the 397 second pass, I looked for and coded instances where the attention of the participants was caught in some way that involved a modification of what they subsequently did or said. Finally, a spreadsheet function was used to count the instances for each participant in the interview, which allowed for analysis of the forms and range of interactions for each participant (including the facilitators) in the task-based interviews video recordings. During this analysis, I noted how the use of gestures, particularly by the facilitators stood out as possible mediating responses, examples of which form the basis for this paper. DATA AND ANALYSIS The data are presented and discussed below in two parts. Firstly, a quantifiable summary of the number of coded instances of multimodal forms for the facilitators across the three interviews appears in Table 1 revealing that a range of multimodal forms were used to supplement speech for communicating during the mathematical activity. This data frames gestures in the broader analysis along with other semiotic resources. Instances counted Multimodal Task-based Task-based Task-based Combined forms interview 1 interview 2 interview 3 Totals Debbie Mellony Debbie 2 facilitators Speech 83 64 94 241 Gesture 28 20 38 86 Gaze 2 12 1 15 Posture 3 - - 3 Expression - 2 1 3 Mathematical 6 4 2 12 action TOTALS 122 102 136 360 Table 1: Instances of multimodal forms for facilitators (across 3 video episodes) Predictably, speech-based interaction was the predominant form for both facilitators across the three interviews (with 241 instances in total). Of note for this paper was the gesture-based interactions of which there were 86 combined instances. Examples of gestures used by both facilitators were: pointing to an aspect of the activity or learner workings on scrap paper; gesturing towards a learner and touching learners on the shoulder, hand or back. These findings cohere with results from the literature that teachers (in this case facilitators) use gestures more frequently than other non-verbal communication (see Alibali, Nathan, Wolfgram et al., 2014). 398 Spontaneous gesturing to maintain the involvement of the participants This section presents two transcript extracts (Table 2 and Table 3) which are taken from the second interview facilitated by Mellony, with two boys, Zac and Nate. These highlight the spontaneous facilitator gestures as in-the-moment responses. The boys had been working on the puzzle activity (Figure 1) for about four minutes and had found the value of the circle to be five using the clues in the 3rd row of the grid. In the seconds before this extract, they focused their attention on finding the value of the triangles in the fourth column of the puzzle. We pick up the transcript at line 192. Participant Multimodal form What is said [what is done] 192 SPEECH Hey! NATE [whispers to Zac and pokes him with his pencil] 193 SPEECH / GESTURE 8 [Mel touches Nate's shoulder with her finger] 194 MELLONY GESTURE / SPEECH Ok, come [Then he says more loudly] 195 NATE SPEECH it’s 8! 196 NATE POSTURE Nate leans back in his chair Ja. Ja [rising intonation] 197 MELLONY SPEECH / INTONATION How did you get that? 198 POSTURE Nate leans back in to the table and looks at Zac NATE 8 and 8 [with Zac] is 16 199 SPEECH [joins in] ZAC is 16 Table 2: Transcript extract 1 (lines 192 to 199 from task-based interview 2) Zac did not respond to being poked with a pencil by Nate and appears not to have heard Nate’s exclamation, “Hey!” (line 192) and whispered number (line 193). Mellony however did hear what Nate said. She touched him on the shoulder and encouraged him to continue by saying “ok come” (line 194). Mellony synchronised her verbal utterance with a pointing gesture that incorporated touch to gain Nate’s attention and draw him into the conversation. This is the first example of a conversational interactive gesture (Bavelas, 1992; 1994). Nate repeated what he said more loudly (“It’s 8!”) and leant back in his chair (line 196) signifying a distancing from the group or activity. Mellony encouraged him verbally (“Ja. Ja. How did you get that?”). Nate made eye contact with Zac and leant forward into the table (line 198) indicating a change in his posture. Nate said “8 and 8 is 16” and Zac joined in to say, “is 16” at the same time (line 199). From lines 200 to 207 (not shown here), Mellony made an encouraging celebratory utterance “woo hoo”, the boys shifted in their seats and looked at each other. Mellony asked Nate again how he got 16. The next interaction (lines 208 to 215) follows. 399 Participant Multimodal form What is said [what is done] Well, I was counting, so what equals… so 10, 18 plus what equals 208 NATE SPEECH 26? And I just counted on … in 8s … 209 NATE GESTURE Points to the shapes in column 4 whilst explaining ACTION Zac listens and appears to be checking the numbers himself as he 210 ZAC GESTURE points to the shapes in column 4 211 NATE GAZE Nate looks at Zac 212 MELLONY SPEECH Because 8 and 8 is …? 213 MELLONY GESTURE Mellony points to Nate 214 NATE GAZE Looks at Mellony making eye contact [pauses whilst thinking] 215 NATE SPEECH 16 Table 3: Transcript extract 2 (lines 208 to 215 from task-based interview 2) Nate explained his thinking (line 208), using his own pointing gesture as he did so. Zac listened as he appears to check Nate’s reasoning (line 210), also using pointing gestures. Nate looked at Zac again. Mellony prompted Nate to clarify his thinking “because 8 and 8 is …?” (line 212). She used a pointing gesture to specifically get Nate’s attention, which caused him to make eye contact with her (lines 213/4). He paused before giving his answer of “16” (line 215). In the remainder of this episode, Mellony continued to clarify how Nate arrived at his answer and checked that Zac concurred with Nate’s thinking. Mellony’s pointing gesture towards Nate’s was generic. She was not pointing to a mathematical object, inscription or referring to anything mathematical. Rather, she was pointing to Nate to gain his attention (which she does as he makes eye contact with her). In this instance, the gesture followed her speech. Although the gesture was given meaning by the context, the gesture carried no mathematical meaning. Thus, I describe the gesture as ‘interactive’. A physical characteristic of interactive gestures is that they include some kind of iconic reference to the addressee, in this case Nate. The gesture also facilitated the interaction and kept Nate involved in the interaction rather than conveying mathematical meaning. Bavelas et al. (1992) state that the function of an interactive gesture in social interaction such as this is to maintain involvement of the other participants without interrupting the verbal flow conversation. Alibali et al. (2013) also talk of gestures as “scaffolding student participation” (p. 426) by drawing them into the activity which could be the case here. DISCUSSION AND CONCLUDING REMARKS These excerpts illuminate that learners’ multimodal communication conveys a wealth of information about their current level of confidence or state of mind pertaining to the activity at hand. This can deepen our understanding of “what may otherwise remain at the level of intuitive response (Van Leeuwen and Jewitt, 400 2001, cited in Archer and Newfield, 2014, p. 1). More to the point, within the actual mathematical context, the multimodal communication also provides the facilitator with much to respond to. As these learners were second language English speakers, their verbal contributions were not elaborate or succinct. This multimodal analysis illuminated the way that Nate was drawn back into the mathematical conversation through subtle, interactive gesturing, thus foregrounding his dis-engagement. The multimodal analysis, particularly with an emphasis on the gestures, gives a richer picture of how the interaction played out and provides a deeper understanding of this. Mellony’s speech was directed at both boys but it seems that her in-the- moment interactive gestures were targeted specifically at Nate with the intention of encouraging him to verbalise what he was thinking. I surmise that her gestures also served the important function of ensuring that he did not disconnect from the activity when he leant back in his chair after getting no response from Zac (lines 193, 199 and 211 for example). I have noted that the gestures highlighted here are not mathematical in that they do not relate to explaining a mathematical concept, nor do they refer to a mathematical object or symbol: they are non-domain specific and in fact could be used in any learning context. Nonetheless, when used within this specific mathematical context and as a response to what unfolded, they have a useful function which is to extend our understanding of the combined mathematical activity. Together, these gestures encouraged Nate to maintain focus on the activity, which in turn helped him and Zac solve the puzzle. These short exemplifying extracts supplemented by analysis of instances in other episodes highlighted some thought-provoking insights and questions for further research. For example, I wondered if the gestures could serve a role in sustaining attention during the activity (see Stott, in press). Here, however, I suggest that when the facilitator is mindful of the full multimodal range of learner contributions, this brings an understanding of the learners and their needs and allows the facilitator to respond in a way that keeps them engaged in the activity. Past research has shown that if teachers receive guidance on the importance of gesture in the mathematics classroom, they can adjust the way they use gestures (Alibali, Nathan, Church et al., 2013). Thus, the gestures highlighted here could be a worthwhile extension to Anghileri’s’ (2006) mathematical mediational toolkit, expanding the practices to include other multimodal forms. Drawing on the extensive research on gesture, an expanded set of mathematical practices based on Anghileri’s work would perhaps be a useful tool for pre-service teacher education method courses in mathematics. I propose that these emergent findings point to the value of undertaking a systematic multimodal analysis on video data as it provides rich, nuanced insight into the ‘micro’ interactions and mediational practices within mathematical 401 activity. Such an analysis may help facilitators in the OST space, and perhaps pre- and in-service teachers to consider the ways in which they currently use conversational, interactive gestures in mathematical activity and how they could use and adjust these as mediational practices in the future. Furthermore, Ball’s plenary “Uncovering the special mathematical work of teaching” at ICME13 (2016) urged the mathematics education community to look at how mathematical listening, speaking, interacting, acting, fluency, and doing are part of the work of teaching, and to hear, see, and read students, in ‘real time.’ This echoes Anghileri’s argument highlighted at the beginning of this paper. The emergent insights I have shared may contribute to understanding the special mathematical work of teaching both in classroom and OST contexts. Acknowledgments The SANC project at Rhodes University is supported by the FRF (with the RMB), Anglo American Chairman’s fund, the DST and the NRF. I would like to thank Mellony Graven for her supervision and participation in this research. References Archer, A., Newfield, D. (2014). Multimodal approaches to research and pedagogy: Recognition, resources and access. New York: Routledge. Alibali, M. W., Nathan, M. J. (2012). Embodiment in mathematics teaching and learning: Evidence from learners' and teachers' gestures. Journal of the Learning Sciences, 21(2), 247–286. Alibali, M. W., Nathan, M. J., Church, R. B., Wolfgram, M. S., Kim, S., Knuth, E. J. (2013). Teachers’ gestures and speech in mathematics lessons: Forging common ground by resolving trouble spots. ZDM - International Journal on Mathematics Education, 45(3), 425–440. Alibali, M. W., Nathan, M. J., Wolfgram, M. S., Church, R. B., Jacobs, S. a., Johnson Martinez, C., Knuth, E. J. (2014). How teachers link ideas in mathematics instruction using speech and gesture: A corpus analysis. Cognition and Instruction, 32(1), 65–100. Anghileri, J. (2006). Scaffolding practices that enhance mathematics learning. Journal of Mathematics Teacher Education, 9, 33–52. Ball, D. L. (2016). Uncovering the special mathematical work of teaching. In 13th International Congress on Mathematical Education (ICME). Hamburg, Germany. Retrieved from https://deborahloewenbergball.com/presentations-intro/. Bavelas, J. B., Chovil, N., Lawrie, D. A., Wade, A. (1992). Interactive gestures. Discourse Processes, 15(4), 469–489. Bavelas, J. B. (1994). Gestures as part of speech: Methodological implications. Research on Language and Social Interaction, 27(3), 201–221. Bourne, J., Jewitt, C. (2003). Orchestrating debate: a multimodal analysis of classroom interaction. Literacy (formerly Reading), 37(2), 64–72. Graven, M. (2016). Strengthening maths learning dispositions through “math clubs.” South African Journal of Childhood Education. 402 Graven, M., Hewana, D., Stott, D. (2013). The evolution of an instrument for researching young mathematical dispositions. African Journal of Research in Mathematics, Science and Technology Education, 17(1–2), 26–37. Goldin-Meadow, S., Alibali, M. W. (2013). Gesture’s role in speaking, learning, and creating language. Annual Review of Psychology, 64(1), 257–283. Maschietto, M., Bartolini Bussi, M. G. (2009). Working with artefacts: Gestures, drawings and speech in the construction of the mathematical meaning of the visual pyramid. Educational Studies in Mathematics, 70(2), 143–157. McNeill, D. (1992). Hand and mind. Chicago: University of Chicago Press. Murphy, C. (2014). Collaborative group work pointing to ’that’: Deixis and shared intentionality in young children’s collaborative group work. For the Learning of Mathematics, 34(3), 25–30. Stein, P., Newfield, D. (2006). Multiliteracies and multimodality in English in education in Africa: Mapping the terrain. English Studies in Africa, 49(1), 1–21. Stott, D. (in press). Attention catching: Connecting the space of joint action and togethering. In N. Presmeg, L. Radford, W.-M. Roth, G. Kadunz, L. Puig (Eds.), Signs of signification in mathematics education research: Semiotics in mathematics education. Springer. Stott, D. (2014). Learners’ numeracy progression and the role of mediation in the context of two after school mathematics clubs. Grahamstown, South Africa: Rhodes University. TEACHER REGULATION OF INSTRUCTIONAL PRACTICES: A CASE STUDY WITH A NOVICE TEACHER Eva Thanheiser, Randolph Philipp Abstract Effective teachers must simultaneously attend to many facets of the classroom, including the mathematics topic to be learned, the way students think about that topic, and the cultural contexts in which the topic may be situated. In this study, we focus on that to which prospective teachers attend in a teaching scenario involving a culturally- based nonstandard algorithm. In this scenario, a teacher dismisses a multi-digit subtraction algorithm because it is different from the standard algorithm. We surveyed 23 prospective elementary teachers, and using open coding we found that while most of the prospective teachers disagreed with the teacher’s dismissal of the algorithm, their disagreement focused on mathematical or student thinking arguments, but not on culturally based arguments. We discuss the implications of these findings. Keywords: Prospective elementary teachers, cultural responsiveness, teacher education, number and operation Portland State University, USA; e-mail: [email protected] San Diego State University, USA; e-mail: [email protected] 403 Setting Eva Thanheiser (first author) is a mathematics teacher educator (MTE) in the United States, where she teaches mathematics content courses to prospective elementary school teachers (PSTs). In the United States, PSTs generally take content courses focusing on the mathematics of elementary school isolated from and long before they enroll in methods courses focusing on teaching pedagogy (Tatto and Senk, 2011). For an example of the courses focused on mathematics in teacher preparation see Figure 1. Two challenges associated with the separation of teaching mathematics content from teaching pedagogy are that often, elementary PSTs think they already know the mathematics that they will one day be responsible for teaching (Thanheiser, 2017), and second, elementary PSTs’ interests lie more with wanting to be with and support children than with teaching mathematics to children (Philipp et al., 2007). To address these issues, Eva has incorporated a variety of experiences designed for her students to notice and work with children’s mathematical thinking, thereby increasing the likelihood that her students will see a need to learn mathematics at a deeper conceptual level so as to become mathematically proficient (National Research Council, 2001). Figure 1: Timeline of content vs pedagogy courses in United States teacher education. Recently a renewed focus has emerged in the United States focusing on issues of equity and social justice in the educational arena. The Association of Mathematics Teacher Educators (AMTE), the National Council of Teachers of Mathematics (NCTM), the National Council of Supervisors of Mathematics (NCSM), and TODOS: Mathematics for ALL have all established statements highlighting their commitment to equity and social justice. For example, NCTM includes in their position statement, “Creating, supporting, and sustaining a culture of access and equity require being responsive to students' backgrounds, experiences, cultural perspectives, traditions, and knowledge when designing and implementing a mathematics program and assessing its effectiveness” [bold added by the authors] and a joint NCSM/TODOS position statement includes, “A 404 social justice approach to mathematics education assumes students bring knowledge and experiences from their homes and communities that can be leveraged as resources for mathematics teaching and learning.” AMTE published a book Cases for Teacher Educators: Facilitating Conversations about Inequities in Mathematics Classrooms (White, Crespo and Civil, 2016). Several of the organizations are also facilitating year long conversations among mathematics educators, see for example AMTE’s Collective Actions for AMTE to Develop Awareness of Equity and Social Justice in Mathematics Education (AMTE, 2016). In this context of a national focus on equity and social justice Eva was rethinking her course and wanted to incorporate more focus on equity and social justice. So she bought and read the Cases for Teacher Educators (White et al., 2016). Literature In the Cases for Teacher Educators, Marta Civil and Mathew D. Felton-Koestler discuss Conversations About Inequities in Mathematics Content Courses (2016, pp. 215 -218). They discuss how PSTs may expect the mathematics content course to be about mathematics only. However they agree with Bishop (Bishop, 1988) that “it is important that students see that mathematics is not neutral, is not culture free, and is not value free” (p. 216). Marta Civil takes up the cultural aspect and the valorization of knowledge (de Abreu, 1995), which refers to the notion that the knowledge that immigrant children bring is not as valuable as the knowledge that they would acquire in the U.S. schools. She argues for an integration of cultural aspects into the mathematics classroom and a de-valorization of the culturally dominant algorithm. PSTs entering courses in their teacher education programs as well as practicing teachers are often unaware that various algorithms exist for the four basic operations (addition, subtraction, multiplication, and division). They may only be familiar with the algorithm they learned in school and may be under the assumption that that is the only algorithm that exists. Thus, this algorithm is viewed as the only way to solve a problem and may be the only way teachers look for when working with children. To counteract this notion, many MTEs introduce various algorithms in their content courses for teachers and use articles written to illustrate algorithms such as Philipp’s (1996) article titled Multicultural Mathematics and Alternative Algorithms. In this article, Philipp lays out various algorithms used around the world. However, even after being exposed to various alternative algorithms, PSTs often find that while those algorithms are interesting, the one they themselves learned is the easiest and children should learn that way (Strand and Thanheiser, 2017). Eva had experienced PSTs valorizing the US algorithms and had worked hard at combating that notion in her content courses. She exposed PSTs to various algorithms and discussed the mathematics behind various algorithms with them. 405 Now she was wondering how her PSTs would react to a situation in which the US standard algorithm was valorized. To this end she collected data to establish a foundation for understanding the extent to which PSTs consider cultural factors in mathematics. Research Questions 1. How will prospective elementary school teachers (in the United States) react to a case of a teacher dismissing a student’s nonstandard algorithm? 2. Do prospective elementary school teachers (in the United States) notice cultural aspects in a mathematics teaching scenario? Methods To answer the research questions, data were collected at the end of a mathematics course for prospective elementary school teachers (taught by the first author). At the time this tasks were presented to students in the course, they had previously examined the four operations in detail, had examined how children think about the four operations, and had been exposed to algorithms from various cultures, including reading Philipp’s (1996) paper. This paper helped students recognize that the algorithms people use are culturally-based. Following are two examples of students’ reflections after reading the paper. After reading the article I now realize that depending on which country you live in or even where you live in the US you may be taught algorithms that are unique to that area.…. As they said in the article, our traditional algorithms are only "one way" of finding solutions and are not "the way. [Jason] I also used to think that everyone had the same type of math system as we do in America. It was very interesting to see all the different ways to work through a problem. The reading stated that most people think that the way they have been taught makes more sense than other ways. I have to say I am guilty of thinking this way as well. [Katie] All 23 participants in the study were presented with the case shown in Figure 2 and responded in written form. 406 Figure 2. The case presented to the PSTs The data were analyzed using Thematic Analysis (Braun and Clarke, 2006). All responses were read and re-read to become familiar with the data. Initial codes were created to capture ideas that showed up across various responses. For example, most of the PSTs stated that they would have responded to the student differently or would have wanted the teacher in the case to respond differently. Thus, a code for each of these two categories was created The PSTs would have responded differently and The teacher should have responded differently and all instances labeled. For example, Alexa gave the following response. I think the teacher should have responded differently. She should explain why he shouldn't do it that way. She could've asked him to explain his thinking. In the problem it is not clear where Karl gets the 1 that he puts below the 3 but he does get the right answer. There isn't one right way to solve a problem. There can be several ways to solve a problem, By having Karl explain as he does the problem again the teacher can see if he is understanding it. Her response was tagged with the code The teacher should have responded differently. Once the first round of codes was created the data was coded using that set of codes. Codes were refined as the data was analyzed to ensure that the code described the category of response well. Similar codes were sometimes combined into one larger theme, for example, the two above codes were consolidated into one, The PSTs would have responded differently/The teacher should have responded differently. This process was repeated until all the data was coded and all themes were captured through the coding. Results Three main themes emerged in the coding. The themes with the number of occurrences in brackets. 407 1. The PSTs would have responded differently/The teacher should have responded differently [22 of the 23 PSTs]. What the PSTs thought should have happened in this classroom or what the PSTs would have done differently if they were the teacher. 2. Focusing on the mathematics and/or children’s mathematical thinking [22 of the 23 PSTs]. a. Asking Karl to explain (why his) algorithm (works) (to the class) [15 of the 23 PSTs]. b. PSTs emphasized that a) there is more than one way, b) there are many correct ways, or c) there is not only one correct way to solve a problem [20 of the 23 PSTs]. c. Karl’s method could help other students/teach other students a new way [9 of the 23 PSTs]. d. Comparing/connecting methods (Karl’s and the standard) could lead to a better/deeper understanding of both [3 of the 23 PSTs] e. The teacher needs to figure out how the student is thinking and then build on that thinking. This allows for a discussion around student thinking [1 of the 23 PSTs]. 3. Focusing on Karl’s background/culture [3 of the 23 PSTs]. a. teacher celebrating the mom and grandma [3 PSTs], b. States that the teacher invalidating mom and grandma [3 PSTs]. Discussion In this section, we will discuss some of the themes and then reflect. As to the first theme that most elementary PSTs would have responded differently or would have wanted the teacher to respond differently, this was not surprising (yet nice to see). After 10 weeks of discussing various ways to operate on numbers including historical systems and various children’s solutions, the PSTs in this course were used to considering alternate perspectives. When discussing what they would do differently or would want the teacher to do differently, 22 PSTs focused on children’s mathematical thinking. Of those 22 PSTs, 20 PSTs emphasized that a) there is more than one way, b) there are many correct ways, or c) there is not only one correct way to solve a problem. This aligns with the focus of the class on requiring and valuing more than one way to solve a problem. And 15 of the 22 PSTs wanted (the teacher) to ask Karl to explain (why his) algorithm (works) (to the class). The focus in this response was on allowing Karl to share his thinking. This also aligns with the focus of the class on requiring and valuing student thinking/explanation and requiring justification. Of those 15 PSTs, 9 PSTs thought Karl’s method could help other students/teach other students a new way and 3 PSTs though that comparing/connecting methods (Karl’s and the standard) could lead to a better/deeper understanding of both. The PSTs, themselves, experienced learning from each other in the course and 408 compared and contrasted across number systems and algorithms as well as across each other’s’ solutions. So, these themes are also aligned with the course, however, especially the last one (comparing/contrasting) is somewhat low. Only 1 PST stated that they thought that the teacher needs to figure out how the student is thinking and then build on that thinking. This would allow for a discussion around student thinking. While 22 PSTs focused on children’s mathematical thinking, only 3 PSTs focusing on Karl’s background/culture. We would have liked more PSTs focusing in this category. This made us think about how we could be successful in having PSTs care for children’s mathematical thinking as well as elevating background and culture. Distinguishing cultural and children’s thinking perspectives In our reflections on the above results, we began to draw a distinction between a response that focused on children’s mathematical thinking and a response that highlighted cultural perspectives. Although both are important, they are different. Focus on culture (C) RC: Respect for the culture of a child. Manifest the respect for the culture of our students by proactively seeking alternative mathematical ways and bringing them into the class to unpack. Celebrate them as cultural. MC: Focus on multiple cultural methods MMC: Elevating the mathematics among different culturally based methods Focus on children’s mathematical thinking (CMT) RCMT: Respect for the children’s mathematical thinking. Manifest the respect for the children’s mathematical thinking by proactively allowing children to use their own ways to solve problems and bringing them to the class to unpack. Celebrate as mathematical thinking. MCMT: Focus on multiple methods of individual children’s mathematical thinking MMCMT: Elevating the mathematics among different child based methods Figure 3: Separating cultural and children’s mathematical thinking For example, stating that we ought to allow children to invent their own strategies highlights children’s mathematical thinking, but does not explicitly address cultural issues, whereas stating that we would like an immigrant student to show how his grandmother solved the problem in her home country explicitly addresses culture. Figure 3 presents how we distinguished between these two stances. Using this framework we found that almost all students were aligned with the lower half of the framework, but only about 13% were aligned with the upper (cultural) half. 409 Conclusions and Implications Prospective elementary school teachers enrolled in a mathematics content course for prospective mathematics teachers learned that an important goal of mathematics teaching is to focus on students’ mathematical thinking and encourage students to share their reasoning. Furthermore, this focus was valued even when students were sharing non-standard approaches that reflected innovative student thinking. However, even when faced with a case involving a culturally-based algorithm, only 13% of the students explicitly commented on the importance of focusing on the cultural aspects. Whereas the authors were pleased with the focus on students’ thinking, we are deeply concerned about the lack of focus on culture in mathematics. The results indicate a need to focus more on cultural aspects in our courses. We might accomplish this by focusing on both children’s mathematical thinking and cultural aspects, and we believe that we must explicitly distinguish between the two. We also suspect that we must look for or create opportunities to highlight the richness of algorithms used in different cultures, to emphasize the fact that these algorithms are valid, and to consider the historical context associated with such algorithms. We think we must highlight salient features so that attention is paid to the cultural components, and we would hope that these would be valued by students as they emerge into future teachers. Mathematics is often viewed as culture free. We understand why people believe that mathematics is culture free, but this is simply incorrect. Culture interacts with mathematics in many ways, and if we are to support all our students in feeling that mathematics is for them, then we believe one way to promote this attitude is by helping all our students recognize that their cultural background has a rich and valid mathematical history to be understood, honored, and shared. And for teachers to develop this stance, those of us teaching prospective teachers must create opportunities for these issues to arise in our classes. By affirming these commitments, we will promote an appreciation for differences so that instead of viewing differences as problems that teachers must face, they may view differences as opportunities to be embraced. And although this approach has always been important in the world of mathematics teaching, we believe that this is even more important in a political climate in which linguistic, cultural, and religious differences are not being recognized often enough as rich opportunities to be appreciated. References Association of Mathematics Teacher Educators. (2016). Collective Actions for AMTE to Develop Awareness of Equity and Social Justice in Mathematics Education. Retrieved from https://amte.net/2016/collective-actions-equity. 410 Bishop, A. J. (1988). Mathematics education in its cultural context Mathematics education and culture (pp. 179–191). Springer. Braun, V., Clarke, V. (2006). Using thematic analysis in psychology. Qualitative research in psychology, 3(2), 77–101. de Abreu, G. (1995). Understanding how children experience the relationship between home and school mathematics. Mind, Culture, and Activity, 2(2), 119–142. National Research Council (Ed.) (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press. Philipp, R. (1996). Multicultural mathematics and alternative algorithms. Teaching Children Mathematics, 3(3), 128–134. Philipp, R., Ambrose, R., Lamb, L. L. C., Sowder, J. T., Schappelle, B. P., Sowder, L., Chauvot, J. (2007). Effects of early field experiences on the mathematical content knowledge and beliefs of prospective elementary school teachers: An experimental study. Journal for Research in Mathematics Education, 38(5), 438–476. Strand, K., Thanheiser, E. (2017). NCTM Articles as Reading Assignments to Motivate Prospective Elementary Teachers in Mathematics Courses Paper presented at the Research in Undergraduate Mathematics Education, San Diego. Tatto, M. T., Senk, S. (2011). The mathematics education of future primary and secondary teachers: Methods and findings from the teacher education and development study in mathematics. Journal of Teacher Education, 62, 121–137. Thanheiser, E. (2017). Brief Report: The Impact of Preservice Elementary Teachers' Accurate Self-Assessments in the Context Whole Number. Journal for Research in Mathematics Education. White, D. Y., Crespo, S., Civil, M. (Eds.). (2016). Cases for Mathematics Teacher Educators: Facilitating Conversations about Inequities in Mathematics Classrooms (Vol. 1). AMTE. HOW DO PUPILS OF THE 5TH AND 6TH GRADE STRUCTURE SPACE Veronika Tůmová Abstract Measurement of area and volume seems to be one of the challenges for Czech pupils. Structuration of space into arrays of rectangular tiles or unit cubes has been found as one of the key building blocks needed to understand measurement of area and volume. In this article, I investigate strategies and misconceptions that appear when pupils solve structuration problems in 3D. In addition to a written test (1203 pupils from grades 4 to 9), interviews (57 pupils from grades 5 and 6) specifically aimed at space structuration were analysed. It appears that the structuration of space represents a problem for many 5th and 6th graders. Several types of problems have been identified including incorrect Charles University, Czech Republic; e-mail: [email protected] 411 use of formula, problems in interpretation of words “rectangle 3 cm wide and 4 cm long” and not distinguishing between spatio-graphical and theoretical worlds Keywords: Volume measurement, structuring of space, array of unit cubes, misconceptions Introduction Czech pupils scored in TIMSS well below average in most of the tasks concerning area or volume (Rendl and Vondrová, 2014). Many researchers see structuration of rectangular space (2D or 3D) into square or cubical units (further referred to as structuration) as one of the main building blocks on the way to understanding area and volume concepts (Battista, 2007; Sarama and Clements, 2009, Smith et al. 2016). Dorko and Speer (2013) hypothesize that the ability to structure space into arrays of cubes is directly related to the computational success in all the tasks used in their research (including volume calculation of triangular prism and cylinder). This hypothesis seems to be confirmed by our recent research (Tůmová, 2017) which also indicates that there is a rather strong relationship between structuration and the ability to solve area and volume problems. The aim of the article is to investigate how Czech pupils can structure the rectangular 3D space into the cubical or prism units and what strategies and misconceptions can be observed in grades 5 and 6. Theoretical Background According to Battista (2004) and Sarama and Clements (2009), structuring 3D space into arrays of cubical units seems to develop from chaotic structuration to structuration along the sides/faces with multiple counting (only visible faces are included, cubes along the edges are double counted or triple counted, because their faces are visible from 2 or even 3 sides and the cubes inside are usually ignored). Later, correct structuration follows along the sides/faces (no double counting along the edges, but cubes inside are often ignored or not calculated correctly) and after that, structuration into layers (seen as composite units, correct coordination of the 3 dimensions). The next level of structuration is represented by the skill to calculate correctly the number of units in a rectangular 3D space when the units are not cubes but rectangular prisms. The highest level of structuration consists of replacing discreet structuration with operations (i.e., multiplication) on lengths of the sides. Structuration becomes only implicit and pupils no longer need to draw or even imagine it. Battista (2007) further distinguishes pupils’ problems with space structuring of the situation and problems with connecting space structuring with appropriate numerical procedures. The latter is apparently very difficult. Pupils often use a completely different formula (e.g., for the surface or area instead of volume) or substitute wrong measures into the formula (Vondrová and Rendl, 2015; Tan Sisman and Aksu, 2016; studies cited in Battista, 2007, p. 893). As one of the 412 reasons for these problems, the authors often point to the early introduction of formula, i.e., early algebraization (Zacharos, 2006). In our previous research (Tůmová, 2017), we tested 735 pupils in grades 6 to 9. The tasks in the test were classified either as structuration ones – i.e., aimed at testing the ability to structure space into arrays of cubes/prisms (calculate the number of cubes or layers to fill certain space) or as conceptual ones – i.e., non- routine or novel tasks (both for area and volume) that require more advanced understanding of the underlying principles and concepts than a simple use of formulas; these tasks could not be solved by (mental) manipulation of discreet square or cubic units. We found a relatively high correlation (Pearson correlation of 0.66) between results of the structuration tasks and the results of the conceptual tasks. This suggests that structuration plays an important role in solving volume and calculation problems. The structuration tasks will be the focus of this paper. Following research question will be dealt with: How do pupils in grade 5 and 6 structure space (in comparison with pupils from grades 4 and 7 to 9) and what strategies and misconceptions appear in their work? Methodology The research question was investigated first via a written test which included structuration and conceptual tasks (see above, the tasks are presented in more detail in Tůmová, 2017). In view with the focus of the article, only two selected tasks will be presented here. Task H12 had three Figure 2: Task H.12 sub tasks similar to the following: H12.3: The blue cube building has 20 cubes in the first layer. How many cubes do you have to add in order to get the smallest possible completely filled prism? (Fig. 1) Task H13: You have exactly 59 cubes (with the edge of 1 unit) to build a cube building, the base of the first floor is a rectangle 4 units long and 3 units wide. You must use all the cubes but the building has to be as LOW as possible. How many layers will there be? How many cubes will there be in the top layer? [No drawing provided.] In H12, at least two strategies are possible – calculating the cubes in the building/each layer and subtracting these from the maximum number of cubes that can be in the layer. Or calculating the missing cubes directly. In this case, the cubes that are next to the other missing cubes (like the two corners) are most likely to be left out since they share no common face with the building depicted. In H13, two solving strategies were expected. The first is calculation-based: divide 59 by the product of 3 and 4 (i.e., the number of cubes in one layer). The resulting whole number means the completely filled layers and the rest is the number of cubes in the last incomplete layer: 59 : (3 ∙ 4) = 4 (rem. 11). The other approach is partially manipulative: draw cubes in the bottom layer and see how high we can continue building until all 59 cubes are used (using repeated addition or multiplication). 413 The tests were distributed in September 2015 to more than 1300 pupils (grades 4 to 9) from 8 different primary schools in Prague with no specialisation and attended by pupils from their vicinity. None of the classes were special ones for pupils with special needs. The sample was a convenience one but in all cases, the whole classes not individual pupils took part. The test was administered by appointed mathematics teachers who got instructions from the researcher. The test took 45 to 55 minutes. The test was completed by 692 pupils from grades 4 to 6 and by 511 pupils in grades 7 to 9. Pupils’ written solutions were coded by the author and two more coders. Based on the a priori analysis of tasks (not presented here due to the space limit), we had some preliminary codes. The coders first coded for them and while doing so, assigned points for each task and noted other phenomena such as what mistakes appeared, what other calculations appeared in the written solution, etc. Finally, each written solution was characterised by the presence (or not) of the identified codes. These characteristics were further elaborated in a quantitative way. To answer the second part of the research questions regarding strategies and misconceptions, a selection of pupils for clinical interviews was done. Based on the test results, six classes were selected and interviews with 6 to 12 pupils from each class were conducted (n = 57, 30 in grade 5 and 27 in grade 6). Since the interviews took place one and half years after the testing, I decided to use task H13, since in its written solution, many phenomena connected to structuration in 2D as well as in 3D and to the use of multiplication could be observed. The task was presented written on paper, only the word ‘units’ was replaced by cm after the first 6 interviews in which the absence of concrete units proved to be quite an obstacle to understanding. There were pencils, a ruler and calculator readily available for the pupils to use. In case a pupil was not able to solve the task using drawing, he/she was to be given 24 wooden cubes to manipulate. Each interview was conducted by the researcher in a separate room and video-recorded. All interviews lasted between 8 and 20 minutes. All 57 videotapes of interviews were transcribed and analysed using techniques based on grounded theory (Strauss and Corbin, 1990). In this manner, the researcher open coded the data and while doing so, looked for patterns in the data. Gradually some categories of pupils’ approaches and mistakes emerged. Examples of codes are in the Findings section. The codes were further grouped and analysed both quantitatively and qualitatively. Findings Structuration proved to be not easy for pupils. The chart in Figure 2 shows success rates in percentages for tasks H12 and H13 per grade. The success rates for task H13 are below 30% for all grades except the 9th. For grade 6, it is only 17%. This seems to be alarming (given the strong correlation with calculation results). 414 The fact that the 5th graders show better results than 6th graders might be explained by the fact that in the Czech Republic, the best pupils usually leave for the eight years secondary grammar school after grade 5. Table 1 shows the number of pupils that tried to solve H13 and the number of pupils who did not structure the first layer of the building correctly. In grade 6, 64% of pupils did not even try Figure 3: Success in tasks H12 and H13 to solve this task (this is the highest per grade percentage of non-solvers of all the grades). H13 overview per grade 4 5 6 7 8 9 No. of solvers 96 163 86 112 64 124 % did no solve 56% 32% 64% 45% 52% 29% Incorrect structuration of one layer - 29 31 15 18 4 5 No. of pupils Incorrect structuration in % out of 30% 19% 17% 16% 6% 4% solvers Total No. of pupils 218 238 236 203 133 175 Table 1: Overview of H13 results per grade The most frequent incorrect structuration strategy in grades 4 to 6 was the one in which pupils considered 7 cubes in the first layer. This is probably based on the calculation 4 + 3=7; the pupils either did not picture the first layer at all or drew the building from the side view – with 7 cubes at the base; this suggests that they took the first layer to be a 1 x 7 rectangle. In some cases, this strategy was modified into 2 (4 + 3), i.e., adding the lengths of all sides of the rectangle. Both cases represent a strategy of using some calculation with the numbers given to determine the number of cubes – in the first case it can be seen a “random operation” with the numbers given (most frequently used operation being the addition, in grades 4 and 5), the second points to the use of the wrong formula (it occurs more frequently in higher grades and comes from mixing the formula for calculating area with the one for perimeter). These phenomena were observed with 9, 17 and 7 pupils from grades 4, 5 and 6 respectively (these are only the cases that could be interpreted clearly from the written solutions). The strategies described above could also be observed with pupils in the interviews. The trust of some of them in the calculation method was so strong that 415 they claimed that the physical model was wrong and should contain another number of cubes. This type of errors can be labelled as disconnection between geometrical representation of the situation and the arithmetic representation thereof. Paul (grade 6) expects to find 14 cubes in the first layer and models it first as a rectangle 1 x 14 (Figure 3). When asked to re-read the assignment, and show where the rectangle 3 by 4 is, he builds a correct rectangle 3 x 4 but he says: Figure 4: Paul: rectangle 1x14 as the number of cubes in one layer S47: This is 4 cubes in length a 3 cubes in width (he´s pointing along the edges of the building 3 by 4 cubes) and all this together should be 14. [T nods] Well, this cannot be done with these cubes here. And again this is the second floor ... (he is finishing the second layer of 3x4 building). OK, the second one ... T48: Wait, when you said that this cannot be done with these cubes ….? S49: Well, it cannot be done with these cubes because there is not enough of them and if there was … and moreover 14 cubes is equal to what the amount of cubes that together would fill everything in width as well as in length and in this it just cannot be done. Because it´s not enough. […] T54. I understand this, but I don´t understand the part that there is not enough cubes? , S55: Well, because 14 cubes equals the sum of 4+4=8 +3=11 and +3 =14 and 14 cubes would equal the whole layer/floor, one. T56: And what have you calculated by adding those four measures? S57: How many it has, how many cubes one layer would have. Max (grade 5) on the other hand, very persistently claims that the rectangle 3 x 4 is made from 7 cubes. When asked whether he means only the cubes along the edges, he puts his hand on top of the whole rectangle and says: “these 7 cubes” (see Figure 4). Figure 5: Max: "No these seven" 416 The second most frequent incorrect structuration strategy in the written test (17 pupils) consisted of creating the first layer in the form of rectangles 4 x 4 or 5 x 3. The hypothesis was that the pupils first drew 4 cubes in a row and then added a column of 3 right next to it (not taking into account that one cube out of these 3 is already in place) (see Figure 5 for solutions of two 6- graders). We observed the same error in the interviews. For example, when Amalka made this mistake Figure 6: 3 x 4 rectangle (Fig. 6), the incorrect drawing was in structuration of two 6-graders disagreement with her assumption that it should only be 12 cubes and she corrected her error later after the manipulation. S22: This way.... (drawing 4 squares in a row) four (continues to draw other 3 squares next to each of the four in the perpendicular direction). This way? T23: OK, so this would be the … the first floor, yeah? S24: Hm. (agrees) The bottom [layer], whole. Figure 7: Amalka´s solution 4 x 4 T25: Yeah? The rectangle with length 4 and width 3? S26: Hm. (agrees) T27: Well done. So how many cubes do you have there? In this? […]S30: Sixteen? […] No twelve. No... yep... no. (she is unsure) The problem seems not to be the structuration itself, but rather connecting the description (rectangle 4 cm wide and 3 cm long) to a physical or graphical representation. This type of mistake seems to be related to the mistakes we frequently observed in the interviews. The pupils did not know how to fit the centimetre cubes onto the rectangle – they did not realize that 4 cm corresponds to 4 cubes. Sometimes they presumed that the cubes would be(come) smaller – like we saw in interview with Paul – when he tried to squeeze 14 cubes on one side of the rectangle. Another example is Aniko who trying to make the building as low as possible, put all the cubes on the same level. She used the length and width and divided the total number of cubes with those numbers, interpreting the results as the number of cubes that must be on each side. Such a rectangle would have 300 cubes. Aniko could only reproduce the structure after the researcher had put 3 cubes next to each other and drawn the line along the edges. Another major obstacle (which was only identified in the interviews) to solving H13 is connected to understanding the formulation “the base of the first floor is a 417 rectangle 4 units long and 3 units wide”. 13 out of 57 pupils presumed that the first layer (a cuboid) is the rectangle and therefore should be excluded from the calculation or that the length of the rectangle is in the vertical direction (the same direction as height). Adam is showing where the rectangle forming the base is (Figure 7) and one of the sides of the “rectangle” is vertical. S36: Wait, this is 4 cm and (he is adding 2 layers to his building 4 x 1 x 2). These are the 3 cms (pointing down – see Figure 7) and these are, yep, the 4 cms according to the task (pointing to the 4 cubes in the third layer, the top layer represents the actual building), and when this is one cube with the edge of 1cm, then the edge is 1 cm. The problem of vertical length was most frequent with the pupils who stressed the “formal correctness” of their drawing; they found it important to distinguish which direction which dimension should be drawn, used the ruler and measured the sides exactly, were concerned when the opposite sides were not exactly of the same length, etc. This might be connected to their fixation to the spatio-graphical properties rather than theoretical properties of objects (Laborde, Figure 8: Adam´s solution - 2005; Vondrová, 2015) and/or uncertainty about vertical rectangle the solution strategy, so they try to do at least what they can (drawing a rectangle is a standard task in the curricula). The reasons for this misconception of vertical length remain to be found – our hypothesis is that it might be either due to the representation of the objects on a blackboard (the length is in this case really vertical). Other possibility would be that when the pupils are taught how to construct a rectangle with tools, they are told that the length should be in the upward direction as opposed to left-right/horizontal direction of the width – this is said with respect to the orientation of the paper, but some pupils may derive from this that the length is vertical also in 3D space. This issue would require deeper investigation. The last noteworthy feature would be the tendency of many pupils to use the ruler to depict the situation – 15 out of the 57 pupils (almost 23%) preferred to use the exact measurements to draw the situation. Some of those pupils even used the ruler to measure each side of the cube representation in 2D and draw the squares one by one. For Charlota, it brought a conflict between the spatio-graphical and theoretical worlds. She correctly thinks that rectangle 3 x 4 has 12 cubes, then she draws the rectangle measuring the sides exactly, but the division into rows and columns is done by hand. She first draws the two horizontal lines but the distance between them is less than 1 cm (Figure 8), consequently, the last row would be much wider. She therefore corrects her assumption based on what she sees in the drawing. 418 I asked her why it is now 16 cubes while before she said 12. S32: Because I did not put the one line in, I put two lines instead of one. T33: OK. So, it should be…, there should be 12 and this is drawn incorrectly, or? S34: No, I probably calculated it incorrectly, it should have been 16 instead of 12 cubes. Figure 9: Charlota´s T35: And when you calculated 12, how did you go about it? solution S36: Ee, I told myself there could be 3 this way from the wall, when I was thinking 12. So [T nods] I was thinking about the width, the 3cm, that next to each cm there could be 1 cube and next to the 4, there could be 4, this way. [T nods] And then I just filled it in and it was 12. […] S48: To put it correctly, I was not thinking about this line. I moved it a little this way to make it 1 cm and then I told myself that it would not come out correctly. If I leave this [the lowest horizontal line] out, that the one cube would be more like 2 cm. … (T asks which solution is correct) S50: That one [12] was wrong and this (points to the drawing) is correct. Discussion and Conclusions We found out that over 70% of pupils even in grade 8 continue to display some problems with space structuration. The problem appears especially in the situation when they have to structure the space by themselves. According to the Czech curricula the pupils in grade 6 should be able to structure 2D rectangular shape and enclosed rectangular 3D space using square or cubic units and they should be transiting to the level, when the structuration is connected to the multiplication. According to our data this seems not to be the case (this is also confirmed by Rendl and Vondrová, 2014). It might be caused by the fact that there is not enough time dedicated to the space structuration in the elementary school and this may lead to the problems in using correct formulas later (Vondrová, 2015). The misconceptions and errors identified in our study are the evidences indicating a lack of comprehension of 6th graders in terms of space structuring. According to Duval (2006), most of them could be categorised as problems in coordination between various registers of representation. This coordination and building connections between registers is considered to be an essential condition for understanding mathematics. Namely, strategy called “7 cubes” can be seen as an inability to coordinate the register of symbolic system for arithmetic (using numbers and operations or formulas) with the representation in natural language (the text of the task) or even with the (non-iconic?) representation of the situation using physical cubes. Strategy “structure 4x4” and “vertical height” reveals a lack of coordination between natural language and a representation of the situation using physical cubes (the pupil cannot model the situation based on the 419 description). Charlota’s example confirms that it is not easy for pupils to distinguish between spatio-graphical and theoretical worlds even in 3D, coordinating physical objects, drawing and word description to create a theoretical object; the literature usually describes problems in 2D (Laborde, 2015). The fact that all prevalent incorrect strategies originate in the inconsistency of registers of representations confirms how important it is to work on building these connections. As one of the textbook series used in Czech schools specifically focuses on these connections, in my future research I will investigate whether there are any differences between pupils using this series and the others. One of the limitations of my research is the fact that the results cannot be easily generalised as the sample was not representative. However, the study pointed to some important phenomena which must be further explored (e.g., the misconception of vertical length). References Battista, M. T. (2004). Applying Cognition-Based Assessment to Elementary School Students´ Development of Understanding of Area and Volume Measurement. Mathematical Thinking and Learning, 6(2), 185–204. Battista, M. T. (2007). The development of geometric and spatial thinking. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 843–908). Charlotte, NC: Information Age Publishing Inc. Dorko, A., Speer, N. M. (2013). Calculus students’ understanding of volume. Investigations in Mathematics Learning, 6(2), 48–68. Routledge. Duval, R. (2006). A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics. Educational Studies in Mathematics, 61(1), 103–131. Laborde, C. (2005). The Hidden Role of Diagrams in Students’ Construction of Meaning in Geometry. In J. Kilpatrick, C. Hoyles, O. Skovsmose (Eds.), Meaning in mathematics education (pp. 159–179). New York: Springer. Rendl, M., Vondrová, N. (2014). Kritická místa v matematice u českých žáků na základě výsledků šetření TIMSS 2007. Pedagogická orientace, 24(1), 22–57. Strauss,A., Corbin,J.(1990). Basics of qualitative research. Newbury Park, CA: Sage. Sarama, J.A., Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge. Tan Sisman, G., Aksu, M. (2016). A Study on Sixth Grade Students’ Misconceptions and Errors in Spatial Measurement: Length, Area, and Volume. International Journal of Science and Mathematics Education [online], 14(7), 1293–1319. Vondrová, N., Rendl, M. et al. (2015). Kritická místa matematiky základní školy v řešeních žáků. Prague: Karolinum. Tůmová, V. (2017). What influences grade 6 to 9 pupils' success in solving conceptual tasks on area and volume. In Proceedings of CERME 10, Dublin, 2017. Available from: https://keynote.conference- services.net/resources/444/5118/pdf/CERME10_0588.pdf. 420 Zacharos, K. (2006). Prevailing educational practices for area measurement and students´ failure in measuring areas. The Journal of Mathematical Behaviour, 25(3), 224–239. FIRST GRADERS WRITE ABOUT MATHEMATICS: A TEACHING EXPERIMENT Nicole Venuto and Lynn C. Hart Abstract The Common Core State Standards in the United States place an emphasis on developing children’s ability to explain their mathematical reasoning when solving problems. However, explaining mathematical reasoning verbally and in writing is often difficult for young children. This teaching experiment, conducted in a first-grade classroom in a large metropolitan school district in the U.S., looked at the benefits of adding written discourse within the context of eight conceptually based lessons grounded in Cognitively Guided Instruction. Examples of the students written explanations from before and after the instructional unit are shared. They demonstrate the change in the children’s ability to explain their mathematical reasoning in writing, suggesting that adding writing to conceptually based lessons improves children’s ability to communicate their thinking. Keywords: Written discourse, conceptually based lessons The Problem Historically much of the focus of elementary mathematics has been computation. However, many children learn to compute without making sense of the procedures they use, leading to an inability to apply procedures to solve real world problems (National Research Council, 2000). Being able to reason mathematically and communicate mathematical thinking clearly have become vital skills for children to acquire as they move into the 21st century workforce (Boaler, 2016). In recognition of the need to help children develop their ability to communicate their mathematical thinking, the U.S. Common Core State Standards Mathematics (CCSSM, 2010) place an emphasis on developing mathematical reasoning and justification. Several of the content standards suggest students need to explain the reasoning used to solve problems. Further, the third Standard for Mathematical Practice specifically states that students should be able to construct viable arguments and critique the reasoning of others (CCSSM, 2010). With changes in standards come changes in assessment. If standards are addressing the need to communicate mathematical thinking, this skill must be assessed. One way for teachers to assess understanding in their classrooms is Georgia State University, USA; e-mail: [email protected], [email protected] 421 through the use of oral discourse; however, this can be particularly difficult when teachers have twenty+ children in their class. Achievement tests frequently contain multiple-choice items, making it challenging to assess a child’s ability to construct a viable argument or critique the reasoning of others. Explaining one's thinking cannot reasonably be evaluated through use of traditional multiple-choice items. As a result, constructed response items requiring written explanations are being added to some assessments such as the Partnership for Assessment of Readiness for College and Careers (PAARC) (2016) in the U.S., creating a need to develop students' ability to explain their mathematical reasoning through writing. Theoretical Perspectives While there is an immediate need to help children communicate their mathematical thinking through writing for the purpose of assessment, the benefits of writing in the mathematics classroom go beyond preparing for the test. The act of writing encourages children to reflect on the mathematics they have learned and actively engage in thinking about mathematical experiences (Burns, 2004). The process of writing helps students consolidate their thinking and develop metacognitive awareness (Pugalee, 2004). In CCSSM (2010) students are expected to begin to verbally explain their reasoning when solving mathematical problems as young as grade 1, though they are not formally assessed on written mathematical explanations until grade 3. Since the early years of elementary school serve as a foundation for future growth (Cohen et al. 2015; Moyer, 2000), it seems reasonable for children to begin experimenting with written explanations prior to grade 3. Further, young children often find communicating their mathematical thinking difficult, whether in writing or orally (Moyer, 2000), because they often do not have the conceptual understanding of the mathematics they are trying to explain. Therefore, developing conceptual understanding of mathematical concepts along with opportunities to talk and write about concepts is essential. Through oral discourse students enhance their conceptual understandings. Making written connections to oral discourse further deepens those conceptual understandings (Casa, 2015). The use of oral and written discourse also promotes equity in the mathematics classroom. Engaging in mathematics through speaking creates a space where students begin to take ownership in what they are learning by sharing with their peers. When students share their solution strategies and reasoning, the gap between students who understand the concept and those who do not understand the concept is reduced (Boaler, 2016). Additionally, giving students the opportunity to participate in discourse provides the teacher insight as to which students have a solid understanding of a mathematical concept and which students have misconceptions (Yackel and Cobb, 1996). Understanding a 422 student's thought process helps the teacher evaluate what the child knows or does not know, in turn helping the teacher improve access and equity within the classroom (Yackel, 1995). Giving children the opportunity to write about their solution strategy affords the teacher the opportunity to understand the thinking of more students than the few who are able to orally share their solution strategies during a typical lesson. The Teaching Experiment Given the importance of oral and written discourse for young children, a project was conducted in a grade 1 classroom with 20 students from a large metropolitan school district in the U.S. to look at the benefits of adding written discourse to oral discourse within a conceptually based instructional approach. Prior to the project, the first author, a grade 4 teacher in the school, was invited in the fall of 2015 by the classroom teacher in a grade 1 classroom to conduct a series of lessons modeling discourse and building conceptual understanding in mathematics lessons. That instruction was grounded in Cognitively Guided Instruction (CGI), which naturally emphasizes oral discourse and explanations as part of each lesson (Carpenter et al., 2015). However, assessment at the end of the series of lessons suggested that the students were still limited in their ability to explain their reasoning. Therefore, a second set of lessons was conducted in the spring of 2016, adding a writing component during each lesson. This paper reports on results from this second set of lessons. The teaching experiment methodology was chosen to test the research hypotheses that adding writing would enhance the children’s discourse. Once a week for 8 weeks the 20 first graders in this project participated in a mathematics lesson that focused on simple word problems that required adding and subtracting with multiples of ten (e.g., 28 + 30). An additional component was added to each lesson that required the first graders to write about their solution process after they had discussed it orally. We wanted to know if adding written discourse to conceptually based mathematics lessons that encouraged oral discourse would enhance young students’ ability to communicate their thinking, therefore increasing equity within the classroom. A pre/post design was appropriate to assess changes in the children’s written discourse before and after the 8-lesson unit. The general sequence for both the pre and post assessments and the 8 lessons in the unit was as follows: choose a problem type such as Add to: Result Unknown, engage the students by telling them a number story and checking for comprehension, and have students solve the problem using a strategy of their choosing and concrete manipulatives as needed. Once done, have students share their strategy with a partner then write an explanation of their solution process. Finally, gather as a whole group and share. For the purpose of this paper, we will share pre/post results from three students in the class as examples of change. The three students were selected because they represent typical responses from the class. 423 Pre Lesson Results To determine if the added component of writing in each lesson would make a difference in children’s ability to explain their thinking, a number story lesson was conducted with the students before and after the set of spring lessons. An Add To: Result Unknown problem was selected for these assessments to ensure that all students had access to the problem and therefore, an opportunity to explain how they solved the problem. The problem for the first lesson was Gabby had 28 seashells in her collection. Her family went to the beach, and she found 30 more seashells. How many seashells does Gabby have now? Initial analysis of the responses was attempted by adapting a grade 3 holistic rubric from the New York Department of Education (2015). It quickly became apparent that differences in the writing competence of first and third graders made the rubric unfeasible. A qualitative analysis of the pre/post student solutions was then conducted looking for (1) differences in vocabulary usage, (2) evidence of conceptual understanding, and (3) clarity in written discourse. A sample of the results from three students is reported here. While all of the pre and post assessments were analyzed, three were selected to describe in detail for the purposes of this paper. These samples were chosen because they are representative of typical responses from the class. Following are the written explanations of three students before the 8-week instructional period. Actual student work is shown in Figures 1, 2, and 3. Figure 1 Camron I put 10 2s and 10 3s and then I put 8 1s and then I put them together. 424 Figure 2 Figure 3 Jaime Kennedy I drew two ten blocks and drew 10 20 21 22 23 24 I used ten blocks and one blocks and I got my 25 26 27 28 and that is how I got my answer. answer. All three children could solve the problem and could write a complete sentence about their work. However, their written answers lacked detail and were limited to describing their drawings. Camron (figure 1) drew a clear representation of the problem showing 2 tens and 8 ones beside 3 tens. In his written work he reversed tens and ones (ten 2s and ten 3s rather than 2 tens and 3 tens), but this did not impact his ability to solve the problem. He showed some conceptual understanding of the problem by writing that he put them (the blocks) together, suggesting a basic understanding of addition or joining. The second student, Jaime's (Figure 2) drawing shows difficulty drawing the two quantities (38 and 20). In her writing, she describes the physical act of laying out the quantity 28; however, there is no explanation of how she reached the sum of 58. Kennedy (Figure 3) also drew an accurate picture, however, her words simply say she used tens blocks and one blocks to get her answer. Again, there is no explanation of how she solved the problem. None of the three children were able to effectively write about their reasoning for adding the two numbers together. While they all demonstrated some conceptual 425 understanding of base ten in their drawings, they were limited in what they were able to communicate, using narrow mathematical vocabulary, referring only to the tens and ones. Camron communicated the most awareness of the concept by suggesting addition when he said he put all the numbers together. The Experiment: Connecting Oral and Written Discourse For the next eight weeks students engaged in mathematics lessons using the format described previously. Although they had engaged in discourse in lessons in the fall, they continued to struggle with oral communication at the start of the unit. They frequently relied on standard algorithms they had learned in their regular mathematics class. For example, in the number sentence 32 + 40 = 72 most children were able to use manipulatives and draw the two quantities. However, they tended to line the two numbers up vertically and add the ones digits (0+2) and the tens digits (3+4). Even though their concrete representation showed three ten sticks with two ones and four more ten sticks, their oral explanation was typically limited to “because 0 and 2 is 2 and 3 and 4 is 7, so I got 72.” The children could not connect the rule with their drawings. They did not understand the procedure they were using or how to talk about their drawing. By learning algorithms before developing strong conceptual understanding, they were unable to reconcile the two representations. After a few lessons it was apparent that the simple adding to problems were not challenging student thinking to move away from the traditional algorithm and did not provide an opportunity to think more deeply about their strategies. So consistent with the teaching experiment methodology, a new problem type was introduced: Take From Change Unknown (CCSSM, 2010) to promote more discourse, e.g., Toby has 43 goldfish for a snack. He ate some. He has 13 goldfish left. How many goldfish did Toby eat? Many children used direct modeling strategies to solve this problem, drawing four tens and 3 ones and crossing out until they had 13 remaining. Other students began with 13 and counted up by ten until they reached 43. As hoped, this problem type created much deeper discourse as the children felt a need to explain why they chose to count up (add) or count down (subtract); and, while the change problem type was more challenging it also made children think more deeply about what they were doing. As a result their written explanations became more detailed. A variety of problem types were integrated over the 8 weeks to encourage richer discourse and written explanations. Post Lesson Results Knowing that all the children would have access to the mathematics, the same problem type was used as in the pre assessment, Add To: Result Unknown (CCSSM, 2010). The problem for the last lesson was Eric had 36 baseball cards in his collection. He went to the store and got 40 more. How many baseball cards does Eric have now? Their work is shown in Figures 4, 5 and 6. 426 Figure 4 Figure 5 Camron Jaime So first I put down 2 10s and 6 1s and then I put I wrote 3 ten blocks and put 6 one blocks and I 40 10s down and then I added, 3 + 4 = 7 and then put 3 tens blocks and one one blocks. I knew to I added 30 + 6 + 40 = 76. add because he went to the store and got more and I counted up all my numbers and my answer was 76. Again, all three children were able to solve the problem and were able to write at least one complete sentence about their work. However, the written responses all offered more detailed explanation of how the students thought about the problem. Camron’s drawing matches his written explanation with the exception of writing 40-10s instead of 4-10s and while he continued to add digits in the tens place (3+4) he quickly explained, writing 30+6+40. Camron demonstrated conceptual understanding by stating that he drew the quantities and then added. He did, however, seem caught between his procedural knowledge of adding digits (3 + 4) and his conceptual knowledge of using place value (30 + 40). Jaime’s response at the end of the unit also provides much deeper insights into her thinking than at the beginning of the unit. In her written explanation she explains by stating that she “knew to add because he went to the store and got more.” She also used more mathematical vocabulary to explain her reasoning, incorporating words such as add, count and more in her response. Although her written explanation does not completely match her drawing, it appears that at some point she realized one of the blocks in the second quantity (40) was a ten, not a one, 427 leading her to the correct solution. Figure 6 Whether or not this can be attributed to the added step of writing about her strategy, thus activating her metacognitive awareness, cannot be concluded, but it is a likely possibility. Finally, Kennedy also explained her mathematical reasoning in her writing, stating, “I know to add because it said he got more” (spelling corrected). She also incorporated vocabulary in her explanation, such as add, more, and counted. Her drawing demonstrates conceptual understanding, indicating that she counted by tens and then added her six. However, she stopped short in her written explanation by not explaining how she got from 70 to 76. Discussion Kennedy This study was conducted to explore if I know to add because it said he got more. I put 30 tens and I put 4 more tens and I put 6 ones. I adding written discourse to counted like this 10, 20, 30, 40, 50, 60, 70 and the conceptually based lessons that answer is 76. encourage oral discourse would enhance young students’ ability to communicate their thinking. While a consistent lesson format was selected for all 8 lessons, teaching decisions were made throughout the unit to elicit richer oral and written discourse. Initially all of the students in the class had access to an Add To: Result Unknown problem type, however it quickly became evident that this problem type did not provoke substantial oral and written discourse because the problem type was not cognitively demanding. Experimenting with problem types such as Add To: Change Unknown and Add/Take From: Change Unknown, all more difficult problem types (Carpenter et al., 2015), made students more aware of their solution strategies. As the problems became more challenging the students became more metacognitive. The deeper thought needed to solve these problems elicited much richer conversations and written explanations than the lessons that used the Add/Take From: Result unknown problem types. It appears, engaging in more complex discourse during the unit transferred to the students’ ability to produce more detailed written explanations on the post assessment. The work samples were analyzed qualitatively to look at the effects of adding written discourse to conceptually based lessons and determine if future research was warranted. Several patterns were found when analyzing the data. For 428 example, on the post assessment the students demonstrated an awareness that when adding two-digit numbers, the tens are added to the tens and the ones are added to the ones. They were no longer simply adding digits. The explanations became clearer over the course of the unit, with some students including additional details within their explanations beyond the steps they used to solve the problem. For example, both Kennedy and Jaime included an explanation regarding how they knew to add on their post assessment. Further, the students’ use of mathematical vocabulary, such as tens, ones, add, equals, and count, increased between the pre and post assessments. While it appears from this small sample that adding written discourse to these lessons improved students' ability to communicate their mathematical reasoning through writing, several questions remain. Limitations Clearly this teaching experiment with a small group of grade 1 students provides limited findings about the usefulness of adding written explanations to young children's ability to communicate their thinking and is not generalizable. Larger studies are clearly needed. However, analyzing large data sets of this type would be unwieldy. Analysis would benefit from rubrics/tools that allow quantification of the data. Although not available for this paper, a rubric that is appropriate for grade one students that considers each respondents attention to mathematical content, mathematical communication and mathematical vocabulary is currently being created for a larger study. A final limitation is that it is not clear whether the students attended to mathematical content, communication, and vocabulary on the post assessment because of the added writing component or because of the conceptually based lessons. Additional study would be needed to determine that. Conclusion While this limited project with one group of first graders does not provide definitive evidence of the value of having students write about their thinking, it does suggest that, for this group of children, when writing was added to conceptually based mathematics lessons in which students verbally discuss their thinking, students improved their ability to communicate their thinking. Making their written words align with their visual representations appears to help students’ make better sense of the concept they are learning. References Boaler, J. (2011). Mathematical mindsets: Unleashing Students' Potential Through Creative Math, Inspiring Messages and Innovative Teaching. San Francisco, CA: Jossey-Bass. Burns, M. (2004). Writing in math. Educational Leadership, 62(2), 30–33. Carpenter, T., Fennema, E., Franke, M. L., Levi, L., Empson, S. B. (2015). Children's Mathematics: Cognitively Guided Instruction. Postmourth, NH: Heinemann. 429 Casa, T. M. (2015). The right time to start writing. Teaching Children Mathematics, 22(5), 269–271. Cohen, J. A., Casa, T. M., Miller, H. C., Firmender, J. M. (2015). Characteristics of second grader's mathematical writing. School Science and Mathematics, 115(7), 344–355. Moyer, P. S. (2000). Communicating mathematically: Children's literature as a natural connection. The Reading Teacher, 54(3), 246–255. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Vol. 1. Reston, VA: Author. National Council of Teachers of Mathematics. (2014). Principles to Action: Ensuring Mathematical Success for All. Reston, VA: Author. National Research Council (NRC) (2001). The strands of mathematical proficiency. In J. Kilpatrick, J. Swafford, B. Findell (Eds.), Adding it up: Helping children learn mathematics (pp. 115–154). Washington D.C.: U.S. Department of Education. New York State Education Department. (2015). Educator guide to the 2015 grade 3 common core mathematics test. Retrieved from http://www.engageny.org/resource/test-guides-for-english-language-arts-and- mathematics. Pugalee, D. K. (2004). A comparison of verbal and written descriptions of students' problem solving processes. Educational Studies in Mathematics 55, 27–47. Yackel, E. (1995). Children’s talk in inquiry mathematics classrooms. In P. Cobb, H. Bauersfeld, H. (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 131–62). New York, NY: Routledge. Yackel, E., Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for research in mathematics education, 458–477. LEARNING TO SUPPORT ALL STUDENTS’ FRACTION UNDERSTANDING Jana Višňovská and José Luis Cortina Abstract We report on our research collaboration with Irene, a Mexican teacher who undertook to trial a research-based instructional sequence on Fractions as Measures, in her underperforming fifth grade classroom. In the process of the collaboration, Irene became a strong advocate of the importance of pursing all students’ understanding, and notions of equity in access to mathematical ideas came to shape her planning and classroom decision-making. We first outline significant student fraction learning that took place in Irene’s classroom. We then document Irene’s evolving teaching practice and how she came to base her instructional decisions in students’ reasoning broadly, and in the reasoning of students for whom the ideas were proving the most challenging, in The University of Queensland, Australia; e-mail: [email protected] Universidad Pedagógica Nacional, Mexico; e-mail: [email protected] 430 particular. We discuss how the instructional sequence supported Irene’s learning, and suggest that resources of this kind are particularly well suited for supporting teachers’ transition to ambitious and equitable mathematics teaching. Keywords: Fractions, teacher learning, educative materials, design experiment Introduction Elementary teachers in classrooms across the globe encounter diverse student backgrounds and struggle with supporting reasonable progress of all their students. Differences in students’ mathematical reasoning often become particularly overwhelming when the classrooms work towards complex and notoriously problematic learning ideas related to fractions and proportional reasoning (Tzur, 2007). While some students seem to develop fraction-related insights relatively seamlessly, many others never overcome their initial difficulties (e.g., Hannula, 2003). Instructional topics with well-documented effect of exacerbating the differences in students’ mathematical competence require particular attention if equitable mathematical learning is the goal. Our contribution to the equity and diversity theme of SEMT conference draws on a dual design experiment (Gravemeijer and van Eerde, 2009) in which we tested and revised teaching resources for equitable and ambitious initial fraction instruction. The instructional sequence on Fractions as Measures was developed in prior classroom design experiments, where it successfully supported the learning of all students (Cortina, Visnovska and Zuniga, 2014). Alongside with exploring the students’ learning and how it was supported by the designed resources, we explored the ongoing learning of an elementary Mexican teacher, Irene, as she worked with these resources in her 5th grade classroom. As typical for dual design experiments, our research questions addressed the learning of both the teacher and her students. In this paper, we focus on the learning that enabled Irene to incorporate the instructional sequence into her classroom teaching. We are in particular interested in understanding how the sequence itself supported Irene’s learning and equitable teaching, and what additional means of support were required. Background of the Dual Design Experiment At the time of the experiment, the collaborating teacher, Irene, was enrolled in a Master’s degree program on educational development at a local public university. She also worked as a full-time teacher in an urban elementary school that serves children living in unfavorable social and economical conditions, with parents’ irregular access to employment. Irene’s fifth-grade classroom was composed of 20 students, 12 female and 8 male, aged 10 to 11 years. The students attended the school in an afternoon shift. Prior to this collaboration, Irene’s teaching could be characterized as traditional. 431 In a series of six one-hour meetings with the second author, Irene first became acquainted with the instructional sequence, including its rationale, and how this sequence was developed and used in prior classroom design experiments. She then used the instructional sequence in her classroom during 18 dedicated weekly sessions, approximately 35 minutes each. After each session, she met with the second author to analyze the classroom events, and collaboratively plan for the upcoming session. Methodology The data collection for the dual design experiment primarily consisted of two design research logs. Irene’s planning and teaching log included elaborated lesson plans, in which she specified the learning goals for each upcoming classroom session, and the activities she planned to use. After each classroom session, Irene annotated her lesson plan. She recorded her thoughts on what actually happened during the classroom session, what modifications she made while teaching and why, and which issues remained to be addressed in the ensuing teaching sessions. Irene also video-recorded all the teaching sessions with one camera, collected copies of her students’ initial and final written assessments, and took high- resolution pictures of students’ work produced during classroom sessions. The second author produced a research log, which included design conjectures and notes related to both students’ and Irene’s learning. First, the log documented the second author and Irene’s conversations during weekly debriefing and planning meetings, in which they relied on Irene’s notes, classroom video, and copies of student work to understand students’ learning progress. Second, this log documented weekly to bi-weekly debriefing sessions between the two authors, which focused on Irene’s teaching and planning, and on the ways in which her work was supported. In the retrospective analysis of the data, we relied on an adaptation of constant comparative method described by Cobb and Whitenack (1996) that involves testing and revising tentative conjectures while working through the data chronologically. As we analyzed new teaching episodes, we compared these with conjectured themes and categories. This process resulted in a set of the theoretical assertions that remained grounded in the data. For present purposes, we focused on the key episodes, which highlighted how Irene supported the learning of diverse learners in her classroom. We construe teacher learning as a change in participation in specific instructional practices. The Instructional Sequence and Equity The instructional sequence that was trialed in the dual design experiment has as its backbone a conjectured learning trajectory (Simon, 1995), in which a rather detailed succession of learning goals is outlined, along with the means that teachers can use to accomplish these in a classroom. The designs were guided by the instructional design theory of Realistic Mathematics Education, which 432 postulates access of all students to personally meaningful mathematical activity among its central tenets (e.g., Cobb, Zhao and Visnovska, 2008). As a result, the means of support were designed with addressing the equity in mathematics classroom at the forefront of the considerations. The means of support included instructional tasks that can be characterized as low floor high ceiling (e.g., Boaler, 2016), ways of organizing classroom instructional activities that provide students with a situational rationale for engaging in the classroom (D’Amato, 1993) and with ownership of mathematical ideas, the tools to be introduced as emerging organically from students’ activity, and productive—often initially unconventional—ways of talking about and symbolizing fraction quantities (Cobb et al., 2008). They also included the establishment of particular classroom norms that regulated the ways in which the teacher and students participated in the sessions, such as promoting that students interact respectfully, listen to each other, ask questions, and always express non- understanding (Stephan, Bowers and Cobb, 2003). As we have explained elsewhere (Cortina, Visnovska and Zuniga, 2015), in operationalizing the idea of fractions as measures, for instructional purposes, we have built on Freudenthal’s (1983) insights about two ways of construing fractions. In his analysis of the concept, Freudenthal made a distinction between fraction as fracturer and fraction as comparer. This distinction is parallel to that of Kieren (1980) and others, between the meaning of fractions when used to quantify part to whole relations, and when used to quantify measures. The main focus of Freudenthal’s analysis, however, is not on the characteristics of the mental images that emerge when construing fractions in these two ways. Instead, it is on the features of realistic situations in which one or the other interpretation may emerge as a natural way of mathematizing these situations. In the Fractions as Measures sequence, we aim at helping students develop fraction understandings squarely within the instructional context of fraction as comparer. More specifically, the learning goals are aligned with the expectation that students come to construe fractions as linear measures1, mostly of lengths. Unit fractions emerge as subunits of measure in situations where they are separate from a reference unit (see Fig. 1). Other fractions are then construed as measures produced by iterating the specific subunit of measure. Hence, it is expected that a fraction such as 7/3 would be interpreted as seven iterations of a subunit that is 1/3 as long as the reference unit. Figure 1. The fraction one third as the length of a subunit, a rod, such that three iterations of the subunit cover the same length as the reference unit 1 Using length measurement as a context for fraction instructions is not, of course, our innovation. Davydov (1969/1991), Brousseau (2004) and many others have used it. 433 We first outline the student learning in Irene’s classroom to illustrate that she came to use the fraction sequence productively, and suggest that an analysis of the means that supported her endeavor can provide valuable insights. We then discuss two features of the sequence that supported Irene in organizing the learning in her classroom and specifically contributed to all students’ learning. Students’ Learning Outcomes In the course of the 18 classroom sessions, Irene was successful in significantly advancing her students’ understanding of fractions. Results from the test that the students took prior to the experiment suggest that the majority of the students had yet to develop sound quantitative understandings of fractions. Fifteen of the 20 students compared the sizes of two given fractions by relying on the whole number dominance strategy (Behr et al., 1984). For these students, the symbol that included bigger numbers represented the bigger amount (e.g., 1/5 > 1/2). Four students compared fractions correctly but two of them only did so by first converting fractions to decimal numbers. In contrast, results from the test that students took after the conclusion of the experiment suggest that all of them developed sophisticated understandings of the inverse order relation of unitary fractions. All the students correctly responded to all 9 items that compared the sizes of two unit fractions (e.g., 1/4 > 1/456). Moreover, 18 of the 20 students developed relatively sophisticated understandings of the sizes represented by common fractions, relative to the size of the reference unit. These understandings allowed them to accurately estimate the place of 3 improper fractions (16/11, 10/3, and 15/5) on a number line2. The remaining two students each placed one of the improper fractions incorrectly, based on what appeared to be a calculation error (one estimated that 15/5 would align with 4 units). The evidence from classroom interactions that took place during the design experiment corroborated and extended the findings from final test results, suggesting that all of the students could also soundly justify their fraction comparisons and based their justifications in images of relative length. These findings are consistent with the prior design experiments (Cortina et al., 2014) and portray unusual learning progress. Yet, they do not shed light at the diversity in student reasoning and engagement that Irene encountered in her classroom, and the rather extraordinary progress that some of the students made. An illustration of Leonardo, one of the lowest performing students in Irene’s classroom, provides some of this background. Initially, Leonardo shunned from participating in learning activities. In the initial test, he did not respond to any of the items. In the third session, Irene introduced a narrative where two people had measured the same window with their hands, obtaining different measures. She asked Leonardo: “What do you think happened? If they measured the same 2 Placing fractions on a number line was not part of the teacher’s agenda during design experiment. Students only encountered it once, in the last 20 minutes of the session 18. 434 window, why did they report different measures?” Leonardo at first responded, “Because of the hands, teacher,” but then remained unresponsive to additional prompts form Irene: “What happened with the hands?” Like Leonardo, several students in Irene’s classroom never volunteered any responses and were confused, or answered briefly or superficially when prompted directly. In situations like this, teachers often wonder whether the gain of engaging an additional unresponsive student is worth the potential disengagement of the more competent, and increasingly more impatient, classmates. Irene managed to sidestep this dichotomy and supported Leonardo to make more effort to respond. His contributions improved noticeably and began to include justifications. He completed the final assessment in the format of one-on-one diagnostic interview with Irene, and answered all the items correctly. Analysis of Teaching The Opportunity to Teach All Students Irene followed the narrative of the instructional sequence, which introduces students to ways in which a tribe of ancient people, Acajay, measured. Irene first engaged her 5th graders in activities that entailed measuring different lengths with their body parts. Although her students engaged enthusiastically in these experiences, she recognized that there was a complication: when questioned, none of the students came to see measuring with body parts to be problematic. Irene knew that within the conjectured learning trajectory, students were to notice limitations of measuring with body parts (i.e., inconsistent measures produced for the same length), and develop a need to introduce a standard unit of measure. When this did not happen in Irene’s classroom, she was not sure how to proceed and considered introducing a standard unit of measure anyway. When debriefing the first classroom session, the second author used Irene’s experience to contrast two ways in which one can make instructional decisions: trying to faithfully enact an instructional activity on the one hand, and progressively supporting students to reason about specific issues in particular ways, on the other hand. The major difference between the two, they discussed, was in deciding when the instructional goals for a classroom session had been accomplished. Irene then agreed to co-design and trial further instructional activities in which the limitations of measuring with body parts could become readily noticeable to some of her students. One activity involved noticing the difference in the number the different people would get when measuring the same table with their hands, and discussing when this difference would matter. After the second session, Irene commented that she had been successful in helping some students recognize the different complications that measuring with body parts might cause. Once she recognized that instructional sequence allows her the opportunity of bringing about students’ understanding, she wanted to make sure 435 that all her students benefited from this opportunity. From the sequence design point of view, developing such awareness was important so that all students would see the introduction of a standard unit of measure, a wooden stick about 24 cm long, as a meaningful innovation. Irene decided to design additional problem scenarios to use in the following teaching sessions, in which she determined to focus on the less responsive students, including Leonardo. The determination to work with all students was Irene’s own, but it was fueled by her noticing engaged participation from a broader range of students than she initially expected. Fractions as Measures sequence supported Irene’s work by providing mathematically meaningful engagement across a range of existing students’ understandings: The activities Irene co-designed for the teaching sessions allowed students repeated opportunities to explore how measuring with body parts led to problems. Yet, these very same activities provided sustained, meaningful challenges for more advanced students who started developing explanations of effects of relative sizes of people’s hands on their measurement results. Without such challenge, Irene’s opportunities to keep working with the less responsive students would likely be lost. These initial teaching sessions document an important shift in the rationale Irene employed for making instructional decisions. She no longer focused on which activities she needed to enact, when, and how. Instead, she now focused on the mathematical issues she wanted her students to discuss and understand, and viewed the problem scenarios as the means for orienting students to do so. By and large, Irene kept focusing on forms of students’ reasoning she aimed to elicit, throughout the rest of the classroom design experiment. Sustaining the Commitment to Equity when Supporting the Foundational forms of Mathematical Reasoning Supporting all students in making sense of the inverse order relation among unit fractions (Tzur, 2007) became an important goal for Irene later in the sequence. Once students developed a need to standardize the measurement unit, a traditional measurement tool, the wooden stick, was introduced. While the stick was initially a solution to an earlier problem, its use was subsequently problematized through situations where students developed the need for measuring lengths more accurately than what the stick alone allowed. At the beginning of the sixth session, Irene shared that Acajay elders solved this problem by introducing smaller length measures, smalls, where each small fulfilled a specific iterative condition with respect to the stick: Small of two had such a size that two iterations of its length exactly covered the length of the stick (i.e., its length represented 1/2 of the length of the stick; Fig. 1 is an example of small of three). The students used the remainder of the class session creating their own small of two using scissors and a plastic drinking straw that was shorter than the stick. This constraint encouraged students to estimate the length of their ‘trial’ 436 straw, measure the stick with it, and adjust the length based on the result of their measurement (see Fig. 2). Figure 2: Estimating and checking the length of a straw intended to represent small of two In the seventh session, students created the smalls of three and four. At this point, most of the students would correctly anticipate that the ensuing small to be produced would be shorter than the prior one. However, Irene noticed that when probed, students did not offer a clear explanation for why that would be the case. Because of the importance that Irene attributed to achieving the leaning goals specified in the instructional sequence, she decided to continue with the activities of producing the smalls. As she did so, she pushed the students to explain how was it that they were able to correctly anticipate, every time, the size of the new small, relative to the size of the one that they had previously made. This clearly illustrates that for Irene, the activity was not about making the smalls. It was instead about using smalls as a means to support all students’ understanding of quantitative relationships of relative length. Irene started the eighth session by asking the class to anticipate if the small of five they were to create would be longer or shorter than the small of four. Daniela responded, looking at the smalls she had made in earlier sessions: The small of two is the bigger than the small of three, and the small of three was smaller than the small of two, and the small of four was smaller than the small of three. So the small of five will be smaller than the small of four. At this point, Daniela’s explanation was based entirely on noticing a pattern among existing smalls and predicting that it would hold for the new ones. None of the students in the classroom had a clear explanation for why this should be so. Once the students created their smalls of five, the teacher asked them to examine all the smalls again. She then asked: “Would the size of the small have anything to do with the number of times it is repeated in the stick?” Students agreed it would and Rafael explained why the small of six would be shorter than the small of five: “The small of six is going to be repeated more times in the stick. Then it has to be smaller for it to fit.” His argument mirrored those he and some of his classmates started developing when they argued why someone with a bigger hand would obtain a smaller measure. Importantly, Irene now needed all students to come to reason about relative sizes in these ways. In the ninth session Irene orchestrated an activity in which a student would choose a small from their set (e.g., small of four), decide which small they created just 437 before the chosen one (small of three), and explain which one was smaller. Irene then asked which one would be repeated more times in the stick. Some students had to check this empirically. Later, Angel shared an insight: I know how to know how many times a small repeats itself on the stick without having to use the stick and the smalls. The small of two repeats itself two times on the stick. The small of three repeats itself three times on the stick. That is, the name of the smalls has to do with how many times it repeats on the stick and it also has to do with the size. For the small of three to repeat itself three times on the stick it has to be smaller than the small of two, otherwise it would not align… By the end of this session, even some of the least sophisticated students offered sound explanations consistent with the inverse order relation: Melanie referred to the issue as being “obvious,” noting “the small of eight has to be smaller than the small of seven because it repeats itself more times.” Irene decided to continue with the same type of activities during the next session to make sure that all of her 20 students had made sense of this idea. This decision could have resulted in a problematic session, given that most of the students already considered the relationship to be obvious. The instructional sequence allowed Irene to introduce new learning elements, while keeping the focus on the same form of mathematical reasoning she wanted to promote. Specifically, at the beginning of the 10th session, Irene introduced the inscription system to keep written record of smalls. She suggested notating the small of three as a digit 3 written inside a square (here we use “[3]”) to indicate that it was a small, rather than a length corresponding to 3 sticks. Students were then asked to use the conventional inequality sings to indicate which symbol represented the bigger length. They first compared 10 pairs of smalls, which only included those previously created by the students (e.g., [3] vs. [2] and [4] vs. [8]), without using the physical smalls and the stick. Students responded correctly and were mostly able to explain their answers by referring to relative sizes. When Irene asked them to compare smalls they had not created—[2] vs. [30]—she called on one of the least sophisticated students to check for her understanding. The girl gave the correct answer but did not provide justification. Discussion and Conclusions In the illustrations above, we did not intend to suggest that supporting all students’ understandings became immediately seamless for Irene or that all her students could always contribute with clarity. We instead called attention to the two ways in which the instructional sequence on Fractions as Measures and associated resources made it possible for Irene to sustain her focus on the reasoning of all— and in particular of the struggling—students in her classroom. First, students’ participation in the initial activities in the sequence suggested to Irene that they were all able contributors. Second, the sequence architecture, in which the classroom activities can be designed ‘on the go,’ in response to need to 438 elicit or solidify specific forms of students’ reasoning, provided opportunities for Irene to experience how she can proactively support students’ reasoning. Her co- participation, with the second author, in planning and targeted task design, allowed Irene to transition beyond teaching by ‘covering’ the tasks, truthfully but rigidly, as outlined in the teaching resources. Third, even when Irene needed to stay with the same topic, the designed tasks offered new forms of mathematical engagement for the students who were already confident in their reasoning. This made it possible for Irene to work more with the less responsive students, while the classroom as the whole continued to progress. The instructional practices that generate equitable, ambitious mathematical learning for all students are non-trivial and involve substantial teacher learning (e.g., Hiebert and Grouws, 2007). Teachers who transition to equitable and ambitious teaching depend on availability of instructional resources that are well aligned with their newly established, and somewhat fragile, values and goals. We maintain that the rather unprecedented progress that all students made in their fraction understanding over the duration of the design experiment makes it worthwhile to study processes and resources that made this learning possible. References Behr, M., Wachsmuth, I., Post, T., Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323–341. Boaler, J. (2016). Mathematical mindsets. San Francisco, CA: Jossey-Bass. Brousseau, G., Brousseau, N., Warfield, V. (2004). Rationals and decimals as required in the school curriculum. Part 1: Rationals as measurement. Journal of Mathematical Behavior, 23, 1–20. Cobb, P., Whitenack, J. (1996). A method for conducting longitudinal analyses of classroom videorecordings and transcripts. Educational Studies in Mathematics, 30, 213–228. Cobb, P., Zhao, Q., Visnovska, J. (2008). Learning from and adapting the theory of Realistic Mathematics Education. Éducation et Didactique, 2(1), 105–124. Cortina, J. L., Visnovska, J., Zuniga, C. (2014). Unit fractions in the context of proportionality: supporting students' reasoning about the inverse order relationship. Mathematics Education Research Journal, 26(1), 79–99. Cortina, J. L., Visnovska, J., Zuniga, C. (2015). An alternative starting point for fraction instruction. International Journal for Mathematics Teaching and Learning. D'Amato, J. (1993). Resistance and compliance in minority classrooms. In E. Jacob, C. Jordan (Eds.), Minority education (pp. 181–207). Norwood, NJ: Ablex. Davydov, V. V. (1969/1991). On the objective origin of the concept of fractions. Focus on Learning Problems in Mathematics, 13(1), 13–64. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Kluwer. 439 Gravemeijer, K., van Eerde, D. (2009). Design research as a means for building a knowledge base for teachers and teaching in mathematics education. The Elementary School Journal, 109(5), 510–524. Hannula, M. S. (2003). Locating fraction on a number line. In N. A. Pateman, B. J. Dougherty, J. Zilliox (Eds.), Proceedings of the 27th Conference of the IGPME (Vol. 3, pp. 17-24). University of Hawai’i, CRDG, College of Education. Hiebert, J., Grouws, D. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371–404). Reston, VA: NCTM. Kieren, T. E. (1980). The rational number construct - Its elements and mechanisms. In T. E. Kieren (Ed.), Recent research on number learning (pp. 125–149). Columbus, OH: ERIC. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114–145. Stephan, M., Bowers, J., Cobb, P. (Eds.). (2003). Supporting students' development of measuring conceptions: Analyzing students' learning in social context. Reston, VA: National Council of Teachers of Mathematics. Tzur, R. (2007). Fine grain assessment of students’ mathematical understanding: Participatory and anticipatory stages in learning a new mathematical conception. Educational Studies in Mathematics, 66, 273–291. THE INFLUENCE OF CONTEXT AND ORDER OF NUMERICAL DATA ON THE DIFFICULTY OF WORD PROBLEMS FOR GRADE 6 PUPILS Naďa Vondrová and Jarmila Novotná Abstract Word problems have been identified as an area of concern and have been a focus of many studies. The article investigates the influence of two parameters on the difficulty of word problems and on the frequency of mistakes grade 6 pupils make when solving them. Four variants of a two-step word problem were made which have the same mathematical structure but differ in the order of numerical data and/or context. These variants were solved by four equally abled groups of pupils and their solutions were analysed. The order of numerical data had a statistically significant effect on the success rate of pupils and on the presence of mistakes of wrong mathematization. However, the effect of the context (familiar versus unfamiliar) was rather weak. The study brought some puzzling effects such as a smaller influence of the order of data in the problem with the unfamiliar context than with the familiar one. They will be investigated in further research. Keywords: Word problems, order of numerical data, context, cue words, distractors Charles University, Czech Republic; e-mail: [email protected], [email protected] 440 INTRODUCTION The area of word problems has been identified as an area of concern in many countries because of pupils’ difficulties in solving them. Teachers report that word problems belong among those in which pupils have the poorest performance and studies with pupils confirm this assertion (e.g., Beswick, 2011; Hembree, 1992; Zohar and Gershikov, 2008; for the Czech teachers and pupils, Novotná, 2000; Rendl, Vondrová et al., 2013, etc.). What might the causes of pupils’ difficulties with word problems be? Many of them are connected to text comprehension. The formulation of word problem assignment is fundamental. In fact, as Richard (1984, p. 228) underlines: “it is completely unjustifiable to consider that the formulation is secondary, that there is a structure and content in a problem […]. The formulation is a part of the problem as essential as the expressed relationships, as forth as it has a determining role in the construction of the representation of the problem.” In this article, we focus on two possible sources of difficulty of the word problem assignment, namely the influence of the choice of the context and of the order of numerical data on the success rate of 12-year-old pupils when solving a word problem and on the types and frequency of mistakes they make. THEORETICAL BACKGROUND Word problems are defined in different ways, there is no agreed-upon meaning in research literature. For our purposes, we characterise word problems loosely as problems which include some context (real, real-like or imaginary non- mathematical) within which some numerical data are given and a question (questions) is posed for pupils to solve using their mathematical knowledge and out-of-school experience. Problems fitting this characterisation differ in the measure of “contextualisation”, in their mathematical structure, in their relevance to pupils, in the complexity of context, etc. Obviously, various characteristics of word problem influence the way it is perceived by pupils, the solving strategy used for its solution, mistakes made, etc. Therefore, word problems have been widely researched in terms of the influence of their characteristics. Two of them are the focus of our paper. The review of literature will be limited to the studies aimed at elementary pupils. It is taken as self-evident that contexts play an important role in pupils’ understanding and success in word problems, however, the evidence of their influence is rather anecdotal and speculative (Beswick, 2011). For example, Palm (2008) found an impact of authenticity on both the presence of ‘real-life’ considerations in the solving process and on the proportion of pupils’ written solutions that were really affected by these considerations. Zohar and Gershikov (2008) showed that 5–11 year old girls’ performance was affected by the context of the task while boys’ performance was not. Wiest (2001) concluded that pupils of grade 4 and 6 expressed an interest in the fantasy contexts and solved problems 441 using these context as well as or better than real-world problems. Cooper and Harries (2002) found an effect of more realistic contexts on the pupils’ willingness and ability to introduce realistic responses to a word problem. Studies on the influence of context typically use sets of word problems of the same mathematical aspects (i.e., mathematical relationships and number size) but with differing contexts. Hembree’s (1992) large-scale meta-analysis concerned 44 of such studies for pupils of the 1st grade up to students at the college level. Better performance was most strongly associated with a familiar context (as opposed to the unfamiliar one). Concrete versus abstract contexts showed borderline significance in their positive impact on problem-solving performance. It might be surprising that contexts that are based on pupils’ personal interests or preferences are not related to pupils’ problem-solving skills while the familiarity with solving the class of problem (e.g., problems on rate) is. Another parameter possibly influencing pupils’ performance in solving word problems is the order of numerical data. It will be called proper if the numerical data appear in the sequence needed to solve the problem, otherwise it will be called mixed. Searle, Lorton and Suppes (1974) showed that the order of data was a major predictive variable. Similarly, Hembree (1992) meta-analysis reported better performance for the proper order of data. On the other hand, Nesher, Hershkovitz and Novotná (2003) found that the effect of the order of presentation of the comparison relations was negligible. Finally, an important parameter of the word problem assignment is the presence of verbal cues, that is, words that help “to choose the correct mathematical operation which is required to reach the solution” (Nesher, 1976, p. 374). However, these words may work as distractors, namely, in lexically inconsistent word problems. This type of problems is more difficult especially for younger pupils (e.g., Verschaffel, De Corte and Pauwels, 1992; Lewis and Mayer, 1987). The research questions explored in this article are: What is the effect of the order of numerical data on pupils’ performance in solving a word problem and on mistakes they make? What is the effect of the context on the above? What is the combined effect of the order of data and the context on the above? METHODOLOGY Task The task consists of a two-step word problem of the type “whole divided into unequal parts” which, in its basic variant, has a familiar context and the proper order of numerical data. The cue words (‘more’, ‘less’) are not in the role of distractors. While keeping the mathematical structure of the problem stable, three variants were formulated (Table 1). Variants A and B (and C and D) differ in the order of data, while Variants A and C (and B and D) differ in the context. Thus, variants A and D (and B and D) differ in both the context and order of data. The context of a jigsaw puzzle is considered to be familiar while the context of 442 making a medicine and adding effective chemical substances as unfamiliar for pupils of the given age. Chemistry becomes part of the Czech curriculum in grade 8 only and it seems reasonable to expect that pupils do not have experience with this topic from their everyday life. Proper order of numerical data Mixed order of numerical data Familiar Variant A. Mirka got a new Variant B: Mirka got a new context jigsaw puzzle for Christmas. She jigsaw puzzle for Christmas. She solved it in three days. The first solved it in three days. The day, she correctly places 25 second day she placed by 8 pieces pieces. The second day she more than the first day, the third placed by 8 pieces more than the day by 11 pieces less than the first day and the third day by 11 second day. The first day, she pieces less than the second day. correctly places 25 pieces. How How many pieces did Mirka many pieces did Mirka place the place the third day? third day? Unfamiliar Variant C. When making a Variant D. When making a context medicine, effective chemical medicine, effective chemical substances are added to the substances are added to the solution in three steps. In the first solution in three steps. In the step, 25 substances are added, in second step, by 8 substances the second step by 8 substances more than in the first step was more than in the first step, in the added, in the third step by 11 third step by 11 substances less substances less than in the second than in the second step. How step. In the first step, 25 many substances are added in the substances are added. How many third step? substances were added in the third step? Table 1: Four variants of word problem The problem can be solved in different ways. It has a relatively straightforward structure and we can reasonably expect that pupils will use an arithmetic strategy, i.e., they will find the input value and use it in operations: 25, 25 + 8, (25 + 8) – 11. Some heuristic strategies could also be used such as (Přibyl and Eisenmann, 2014) Solution drawing, Guess – check – revise or Systematic experimentation. Pupils might make mistakes in the mathematization of relations using different operations than required. We expected that this might be the case especially for the variants with the mixed order of data (B and D) which require that pupils read the whole problem first to find the input number. In the Czech schools, pupils are taught to make a legend for a word problem, that is a recording of the data and relationships between them in an informative way, to get an insight into the situation and its mathematization (Novotná, 2000). The common way of legends for problems in which a whole is divided into uneven parts is the arrow one (Figure 1). For the mixed order variants, the legend might start with the second day/step which might cause problems to pupils. 443 1st day … 25 2nd day … by 8 more than 3rd day … by 11 less than Figure 1: The arrow legend for Variant A Sample The sample consists of grade 6 pupils (11–12 years) from four Prague primary schools purposefully sampled within GA ČR project (aimed at investigating parameters influencing the difficulty of word problems). We selected the schools according to Reports by the Czech School Inspection and websites of schools themselves in order to get schools with no specialisation, of a medium size, attended by children from their immediate surroundings, with a varied socio- economic background, with the percentage of children from abroad not exceeding the average for the whole Czech Republic, not aimed for children with special needs and placed in the area of outer Prague. An important condition for the inclusion of the school in the GAČR project was that the whole school will participate. No selection of pupils was made; all the classes from the four schools of the same grade participated. The pupils were given initial tests in mathematics and the Czech language to be split in equally abled groups. Altogether 353 Grade 6 pupils were split in four groups. The criteria for their division into groups were: their average success rate in initial testing, standard deviation from the success rate and the minimum and maximum of success rate. This division was done on the level of each class so that each class was roughly divided into four equally abled groups. The groups differed in maximally 2 % of success rate in initial testing and standard deviation was between 20% and 25%. As not all the pupils were present when the test was assigned and we had to remove some pupils to balance the above characteristics of the groups, finally we worked with four groups of 66 pupils, each solving a different variant. Procedure and data analysis The above problem was part of a test with 6 problems. For half of the pupils in each of the four equally abled groups, the problem was placed at the beginning of the test and for the other half, at the end. The test was assigned in February 2017 by trained helpers. Pupils handed in the test after 20 to 40 minutes. Pupils’ written solutions were analysed by the two authors. First, 1 point was assigned to solutions in which the pupils found the correct mathematical model of the problem (i.e., created two numerical expressions 25 + 8 and 33 – 11) and 0 point in the other cases. It was not important for us whether the pupils correctly finished the solution as it is not influenced by either of the two parameters. The chi-square test was conducted to investigate differences between versions A versus B, A versus C, A versus D, B versus C, B versus D, C versus D, AB versus CD (familiar versus unfamiliar context) and AC versus BD (proper versus mixed order of numerical data). The null hypothesis was that there is no statistically 444 significant difference in the pupils’ performance for these variants. Finally, the solutions were coded for mistakes. Some of them were expected by us (see above), others were coded anew. Mistakes were elaborated in a qualitative way. RESULTS 0 point 1 point 0 point 1 point A (familiar context, proper AB (familiar order of data) 6% 94% context) 15% 85% B (familiar context, mixed CD (unfamiliar order of data) 24% 76% context) 19% 81% C (unfamiliar context, AC (proper order proper order of data) 17% 83% of numerical data) 11% 89% D (unfamiliar context, BD (mixed order mixed order of data) 21% 79% of numerical data) 23% 77% Table 2: Success rates for the four variants and the groups of word problems Table 2 shows that better results were reached for the familiar context (jigsaw puzzle, versions A and B) as opposed to the unfamiliar one (chemical substance, versions C and D), and for the proper order of numerical data (versions A and C) as opposed to the mixed order of data (versions B and D). The best results were reached for the original version A, followed by a different context but the proper order of data (version C), then the version with a different context but the mixed order of data (version C) and finally the version with the original context but the mixed order of information (version B). When the chi-square statistic was calculated for the distribution of 0 and 1 point for the solutions, the null hypothesis was not supported (and thus the alternative hypothesis was supported) for the following groups (with the level of significance 0.05): differences between AC and BD (χ2 = 6.03, p = 0.014), A and B ( χ2 = 8.49, p = 0.004) and A and D (χ2 = 6.43, p = 0.011). For the level of significance 0.10, we also get a statistically significant difference for A and C (χ2 = 3.69, p = 0.055). Thus, the order of numerical data indeed influenced the difficulty of the word problem but the context did not have such a big influence. 25–8 × 8 +11 ÷11 11 – 8 25 + (25 + instead instead instead instead or 8) – 11 = total of 25+8 of + 8 of –11 of –11 11 + 8 47 Variant A 1 1 1 3 Variant B 5 1 3 2 1 12 Variant C 1 2 1 1 5 Variant D 5 4 2 4 15 12 8 6 5 3 1 35 Table 3: Types of mistakes in variant A, B, C, D 445 Nearly all pupils used the arithmetic strategy as expected and some of them incorrectly mathematised the relationships. Table 3 presents mistakes in all the variants. 35 pupils made such a mistake, 6 of them later corrected themselves. The most frequent mistake was that pupils subtracted rather than added 8 from 25. Similarly, 6 pupils added 11 rather than subtracted it. More problems are associated with number 8 (which was subtracted from 25 or multiplied by 25) than number 11 (which was added to 33 or used as a dividend). Would the latter be the case if the number which was received in the second solving step was not divisible by 11 or would the pupil be alerted to something being wrong in his/her mathematization if he/she saw that a remainder would originate by dividing the number by 11? It might not be the case as one pupil’s solution of variant C shows (Figure 2). The pupil made a mistake already in the first step and arrived at the number not divisible by 11. Nevertheless, he carried out the division. In the third step, 19 chemical substances are added. Figure 2: Mistake in a pupil’s solution Again, the unfamiliar context did not lead to many more mistakes (the total of 15 for the familiar context and of 20 for the unfamiliar one) while the order of numerical data did. In the variants with the mixed order of data, the pupils made 27 mistakes of the followed type as opposed to only 8 in the problems with the proper order of data. As for the legends, altogether 32% of pupils made a short legend, none made a pictorial one. 11 pupils used a variable (or variables) in the legend and some of them also in the calculation. Nearly the same number of pupils made a legend for the proper order of data (32) as for the mixed order of data (35). In the latter case, only 7 made a legend starting with the second day/step which shows that the majority of them finished reading before starting the legend. Interestingly, while 42 pupils in the familiar context variants made a legend, only 25 pupils in the unfamiliar context variants did so. We would have expected the opposite. Did the pupils have to concentrate more on reading the unfamiliar context and thus did not have enough energy/time to make a legend for these variants? It is hard to say from the written solution whether pupils made use of the legend in their solution or if they just included it because it is a part of the didactical contract in the Czech Republic that “a legend is a necessary part of the solution to a word problem”. The latter seems probable at least for some solutions in which the legend is only written below the calculation. 446 DISCUSSION OF RESULTS First, we can say that the influence of context is far smaller than the influence of the order of numerical data (Table 2). The context by itself significantly lowered pupils’ performance only for the variant with the proper order of numerical data. It may be that the influence of the context is only applicable in some circumstances. For example, the context of chemical substance was not so unfamiliar to our pupils as expected. Or it may only apply to some types of problems. López and Sullivan (1992, cited in Wiest, 2001) found significant differences in favour of personalisation on the scores for two-steps problems but not for one-step ones. We may reach different results if we investigate the change of the context for a different type of word problem (for example, the inconsistent word problem, Lewis and Mayer, 1987). Similarly to Searle, Lorton, and Suppes’ research (1974) and to studies reported in Hembree (1992), we conclude that the mixed order of data made the word problem more difficult and led to more mistakes. However, while it caused problems to more pupils in both types of context, far more of them in the familiar one (Table 2). Thus, if the two parameters worked in combination (mixed order of data and unfamiliar context, see the difference between A and D), it had smaller effect on the pupils’ performance than if the order of data was changed by itself (see the difference between A and B). This is a source of puzzlement to us. Could an unfamiliar context used in combination with the mixed order of data cause that pupils were alerted to possible sources of problems and were more attentive when reading the text? This is something which we will investigate in the future. We did not find any indication of this phenomenon in literature. Also, we will use the IR Pro programme to re-analyse our data with the help of item response theory which will further eliminate the possible interference of differences in the ability of pupils in the four groups. The division of pupils into four groups based on their initial testing in mathematics and Czech language may have been insufficient. The pupils who used the multiplication instead of addition (25 × 8) probably misread “by 8 more” as “8 times more” (similarly for the division instead of subtraction in 33 11 and for “by 11 less” and “11 times less”). This is a well- known phenomenon (e.g., Lewis and Mayer, 1987; Schumacher and Fuchs, 2012). For pupils who subtracted 8 rather than added (or added 11 rather than subtracted), the words ‘more’ (or ‘less’) in our word problem acted as distractors rather than direct cue words even though they did not originally play a role of such. In almost all cases, this happened in the solutions of the mixed order of numerical data variants. The mixed order of data is cognitively more demanding for a pupil to make an image of the situation than the proper order which probably results in pupils’ having not enough resources to concentrate on the choice of the operation. 447 CONCLUDING REMARKS, LIMITATIONS AND FUTURE WORK Our study showed that the order of numerical data had a statistically significant effect on the success rate of pupils and on the presence of mistakes of wrong mathematization. However, the effect of the context (familiar versus unfamiliar) was rather weak. The study also brought some puzzling effects, not seen in literature, such as a smaller influence of the order of data in the problem with the unfamiliar context than with the familiar one. One limitation of our study has already been mentioned – possible bias in making the four equally abled groups. Another is that we base our considerations on the written solutions only. In further research, we will investigate a possible influence of the two parameters on the difficulty of word problems and on the presence of mistakes for both younger and older pupils than in the present study. If the puzzling effect persists, some interviews with pupils might be needed to get an insight into its causes by analysing the way pupils reason about particular word problems. We may need to use a context which is far more distant from a pupil’s experience in connection with the mixed order of data to investigate the phenomenon further. Interviews with pupils may bring information regarding what pupils consider to be familiar or unfamiliar context. The knowledge of influence of different parameters is important not only for the teaching of mathematics in general and word problems in particular, but also for those who prepare word problems for achievement tests and write textbooks. Finally, this knowledge is important for mathematics teachers as they are the ones who teach pupils to understand and solve word problems and should be aware of the way various parameters influence the nature and difficulty of word problems. The knowledge of influence of different parameters is important not only for the equitable teaching of mathematics in general and word problems in particular. Acknowledgement This research was financially supported by the grant GACR 16-06134S Context problems as a key to the application and understanding of mathematical concepts. References Beswick, K. (2011). 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ON THE REPRESENTATION OF QUANTITIES WITH MULTI-TOUCH AT THE 'MATH-TABLET' Daniel Walter Abstract This paper presents a study, which investigated students’ methods of using a tablet- application called ‘Math Tablet’. Fifteen students were interviewed, with a focus on those who predominantly solve addition problems through counting strategies. The aim of the investigation was to explore if, and how, students make use of the multi-touch Technical University of Dortmund, Germany; e-mail: [email protected] 449 potential while representing quantities on the ‘Math Tablet’. Descriptive data analyses show that many learners already use multi-touch to display quantities after short introductory phases. However, it became also clear that some children need targeted impulses, so that they utilise the potential fully. The use of the fingers as a primary hands-on material seems to be more important than the mere use of multi-touch, since the software is to be classified as a secondary hands-on material. Finally, it could also be shown that the child's methods of use can be influenced negatively by some technically evoked difficulties. Accordingly, it becomes clear that the promising potential of the multi-touch technology is not automatically exhausted, but rather appropriate mathematical-oriented accompaniment is necessary. Keywords: multi-touch, digital media, representation of quantities Introduction Since the development of touch-enabled mobile devices started, the discussion about the use of digital media in learning mathematics has been further stimulated. Particularly for the initial arithmetics lessons, the possibility of controlling tablet computers by multi-touch offers new opportunities for representing quantities. However, from a mathematical didactic perspective, little is known about whether and how children access the potentials of digital media in general (e.g. Moyer- Packenham et al., 2015) and the potential of multi-touch technology when using tablet apps in particular (e.g. Ladel and Kortenkamp, 2011). In order to contribute to the closure of this research gap, this article describes empirical findings that provide insights into the use and the thoughts of children when using an app that can be controlled by multi-touch. This article is structured as follows: In Section 2, the theoretical background to the research work is presented. The focus is both on the description of basic differences between touch-control and traditional use of the mouse and keyboard, as well as the presentation of selected research findings on the use of multi-touch in mathematics teaching and learning. Section 3 contains a description of the research questions and the design of the empirical investigation. Subsequently, the corresponding findings are presented in Section 4. To conclude, the findings are summarized and discussed (Section 5). Theoretical background Software can be controlled either traditionally with mouse and keyboard or touch- enabled devices. Intervention studies suggest that when students use software on touch-enabled devices can lead to greater learning outcomes than when using structurally comparable software which is controlled by mouse and keyboard (e.g. Paek, Hoffman and Black, 2013; Segal, 2011). A possible explanation for this result is that learners can perform their actions on touch-enabled devices directly on virtual objects. An indirect transmission of the action via the mouse is no longer necessary. The mouse as mediator is absence. While tablet computers require only reliable two-dimensional hand-eye coordination, the operation of a computer is 450 associated with the ‚difficulty of the triple coordination’, since, in addition to eye and hand, the mouse pointer must also be observed on the screen: A place entirely different from the hand (e.g. Ladel, 2016). This can evoke a greater cognitive load than when working with touch apps (Segal, 2011). In addition to this basic difference between the two operating variants, the central mathematical-didactical difference is the possibility to control tablets with multiple fingers. In international literature, multi-touch technology is given great potential to support children in the acquisition of basic mathematical concepts and, in particular, in the representation of quantities (e.g. Baccaglini-Frank and Maracci, 2015; Ladel and Kortenkamp, 2011; Sinclair and Baccaglini-Frank, 2016). Objects can not only be exclusively produced individually, as it is often the case when working with traditional software, which is operated with a mouse and keyboard. Multi-touch technology also allows students to add multiple objects simultaneously using multiple fingers on the touch screen. Children are thus given the opportunity to represent quantities not only sequentially, but also in the sense of the part-whole concept (Resnick, 1983). Various research works provide information on whether and how children use this potential and which difficulties can arise. In their experiments on the use of the multi-touch-table (MTT), Ladel and Kortenkamp (2011, 2012, 2014) investigated how internalization and externalization processes can proceed on this digital medium. It has been found that the formulation of a task can have an impact on the use of the child. For example, children tended to assign individual tokens on tasks such as "Please put x tokens on the table". However, if the children were additionally encouraged to display the tokens "all at once," many learners changed their approach by representing quantities quasi-simultaneously with fingers. In addition, the authors refer to different methods of use: „Some children first counted their fingers one by one and then put them all at once on the table. Other children did the opposite, showing fingers all at once when asked for a certain number, and working one-by-one on the MTT“ (Ladel and Kortenkamp, 2014, p. 250). Accordingly, it must always be taken into account that both fingers and digital media (here: MTT) are representative media. The quasi-simultaneous representation of quantities at the MTT does not necessarily indicate a cardinal number concept when a finger set was previously derived sequentially. At the same time, the sequential representation of quantities on the MTT can not be automatically characterized as a sequential conception if the finger set was previously determined quasi-simultaneously. Thus, the existence of multi-touch in software does not guarantee its adequate use by children. It is also possible to display objects one at a time via single-touch, which is structurally consistent with successive single mouse clicks (e.g. Ladel and Kortenkamp, 2014). Previous research has also shown that children can also have difficulties with handling touch-enabled software. In the study of Sinclair and Heyd-Metzuyanim (2014), 451 some children touched the screen surface while representing quantities on the app TouchCounts unintentionally with more fingers than wanted. So more counters appeared on the screen as required. This difficulty can particularly be seen as an obstacle if the child can not locate faulty touch inputs itself and considers incompatible representation as belonging to one another. The development of erroneous numerical representations can be a consequence of this. Some difficulties with the MTT could be spotted in the studies of Ladel and Kortenkamp as well. In some cases, the children did not exercise enough pressure with their fingers to produce tokens, so they often switched to the one-by-one method to sequentially display tokens with the index finger (Ladel and Kortenkamp, 2014). Furthermore, in the experiments by Barendregt and colleagues (2012), some children had motoric difficulties while using the app Fingu for representing quantities, too (Barendregt et al., 2012). Overall, it is important to note that the possibilities of multi-touch and the chance of Direct Manipulation (e.g. Sarama and Clements, 2016) of objects with regard to virtual-enactive representations are becoming increasingly important (e.g. Krauthausen, 2012). This is especially evident in the ever-growing range of tablet apps that allow multi-touch operations. On the theoretical level, multi-touch seems to be a promising design element that can support the advancement of children in mathematics. However, the consequences and implications of this technical innovation for mathematics teaching and learning have not yet been sufficiently explored. In addition, possible further difficulties and hurdles in the handling of other soft- and hardware must be identified in order to elaborate appropriate strategies for dealing with them. Research questions and the design of the investigation Research questions In Section 2, theoretical considerations and empirical findings on the usage of the multi-touch potential in mathematics teaching and learning were presented. Beyond the described research findings, little is known about methods of use while dealing with multi-touch software – especially tablet-apps – for representing quantities. Appropriately, the following two research questions form the starting point for the empirical investigation: To what extent is the potential of multi-touch technology used by students for representing quantities? What are the particularities in children's representation processes when using multi-touch-capable software? The tablet-app ‘Math Tablet’ As an example for multi-touch capable tablet apps, which have not yet been fully evaluated, the tablet app 'Math Tablet' (Urff, n. y.) was used in the study described. 452 Counters can be displayed in two fields, separated by a line, by touch control. There are as many counters as fingers touching the screen (see Figure 1). The corresponding numeric symbols for each field are displayed in accordance with the iconic representation. In addition, the number of all placed counters is also represented in numerical form. Figure 1: Multi-touch interactions at the ‘Math Tablet’ Process of the investigation and interview tasks In order to investigate whether and how children use the multi-touch technology at the 'Math Tablet' in the course of the representation of quantities, this process was investigated by means of exemplarily selected quantities in a qualitative interview. In the present study, quantities consisting of 9 and 15 elements were used. For both, it seems to be expedient to add multiple counters simultaneously. For the representation of quantities the children were at first asked to present counters only in the right one of the two fields. This purpose of this approach is to investigate whether the dimensions of a field with standard hardware (9.7 x 12 cm for a 4th generation iPad) are sufficiently large to use the multi-touch technology adequately. In the event that a child represented a number in the first attempt purely sequentially with only individual counters, the child was asked to display the corresponding number of counters again, but this time in the left field "in as few steps as possible". It was also emphasized that they can add multiple counters, too. In this way the children were explicitly encouraged to use the multi- touch technology. At the beginning of the interview, a ten- to fifteen-minute introduction to working with the app took place, which was not in the form of an instruction, but a joint preparation. In this phase, the children have discovered that multiple counters can be added simultaneously. It has also been discussed, among other things, why it can be helpful to use multi-touch. The interviews were videotaped from two perspectives. While one camera provides a front view of the child and the interviewer, the second camera is positioned next to the child and 453 directed at the 'Math Tablet'. Thus, both the mimic and gesture as well as the actions on the tablet could be observed. Information on the sample and data assessment A total of 15 children ages 7 to 8 took part in the interview described. The children have learning difficulties in mathematics, at the beginning of their second school year and solved simple addition tasks preferably with counting strategies. None of the examined children knew the 'Math Tablet' before the interview, whereby the fluency of the functions and usage of the app could and should be trained only by the introductory phase. Accordingly, the observed methods of use can be regarded as the first intuitive approaches of the children. The assessment methods used were qualitative content analysis (Mayring, 2015) and comparative analysis (Glaser and Strauss, 2005). Based on the data material, categories were developed for the different methods of students’ use of the application ‘Math Tablet’. Accordingly, inductive category formation was undertaken which developed into a structured analysis of the contents as the assessment process continued. Results This section presents selected results of the investigation along the described research questions.1 Section 4.1 deals with the question of whether the children used the multi-touch technology when representing quantities. Subsequently, in Section 4.2 the particularities while using multi-touch are described. Representation of quantities on the ‘Math Tablet’ Table 1 shows the absolute number of students who used multi-touch or only single-touch to represent quantities consisting of nine or fifteen elements. The data show that a total of nine children used only individual counters to represent a set consisting of nine elements. Six children used the multi-touch potential and added multiple counters simultaneously. In representing a set consisting of fifteen elements, there were six children who initially added only individual counters. Conversely, nine children used the multi-touch technology. Those children who only added individual counters in the first attempt were encouraged to add several counters simultaneously, as described in Section 3, in order to represent the respective amount in as few steps as possible. For this second attempt, the data show that all children used the multi-touch technology after explicit proposal in the process of representation. In light of the empirical findings, some children have already used multi-touch in their first intuitive approaches to represent quantities. According to explicit proposal, all children used the multi-touch study described at least once. This finding underlines the special importance of suitable tasks and impulses. 1 The investigation results shown in this Section are taken from Walter (2017/ in preparation). 454 9 counters 15 counters Number of students, First Second First Second who ... attempt attempt attempt attempt ... only added single 9 0 6 0 counters . ... added counters by 6 9 9 6 means of multi-touch. Table 1: Usage of multi-touch technology for representing numbers Detail analysis of the representation processes In order to learn more about the processes and not only about the question of whether multi-touch was used, three specific particularities are described below which have often been observed in children's methods of use. 1. Significance of the fingers as a primary representation medium The quantity representation on the 'Math Tablet' is done as described by creating counters with the input of fingers. Accordingly, the children work with two different media: their fingers and the 'Math Tablet'. A first particularity in the representation of quantities was that the simultaneous addition of multiple counters on the 'Math Tablet', which corresponds to a cardinal representation of quantities, is no guarantee that the representation of quantities with fingers was previously cardinal, too. As the following part of an interview shows, children can first display a quantity with their fingers sequentially and then try to touch the screen with all their fingers at the same time. The scene starts when the child has shown a set of nine elements in the left field only sequentially and then asked to represent the quantity in as few steps as possible in the right field. 1 I Try again! 2 M (stretches nine fingers one by one and counts quietly) One, two, three, four, five, six, seven, eight, nine. 3 M (touches her chin with the nine outstretched fingers one by one and counts quietly again) One, two, three, four, five, six, seven, eight, nine. 4 M (moves the nine fingers simultaneously to the screen to place nine counters. Only six fingers touch the screen.) None of the children investigated in this study represented a quantity on the fingers sequentially and then touched the screen with all fingers. Accordingly, it is essential to emphasize the importance of the fingers as a primary medium of representation, as the visual representation processes on this medium tend to 455 reflect children's thinking rather than the use of the 'Math Tablet'. This thus represents the secondary medium of representation. 2. (Subjectively) more reliable quantity representation by using single-touch As explained, some children did not use multi-touch consistently and represented counters only individually with single-touch. A possible explanation for this may be that at least some children felt subjectively safer in the addition of individual counters. The following transcript describes a student's approach during the presentation of a quantity that consists of nine elements. 1 I Can you (.) put nine counters in the right field? (points to the right part of the ‚Math Tablet’) 2 Z (adds one single counter) Hmm (deletes the counter again) 3 Z (stretches out all five fingers of the right hand and four fingers of the left hand simultaneously. Then she folds the four fingers of the left hand one after the other and then stretches three fingers one after the other. Thus, a total of eight fingers are stretched.) 4 Z (moves all eight outstretched fingers towards the right field. Shortly before her fingers touch the screen, she stops and only extends the index finger of the right hand.) 5 Z (places nine counters one by one with the right index finger in the right field) After the task was given Zoe begins the representation process. Because of a count error, she does not stretch out nine, but only eight fingers. She moves them to the tablet to touch the screen with all her fingers simultaneously. But just before the fingers touch the tablet, she deviates from the intended procedure and represents all counters sequentially with the index finger. Zoe rejected her Figure 2: Zoe’s finger positioning planned approach with multi-touch in (see turn 4) favor of sequential representation with single-touch. One possible reason for this is that she did not have sufficient space in a single field. Possibly, she herself stated that she could not provide the desired result with the position of her fingers (see Figure 2), whereupon she switched to the sequential addition of individual counters, which in her opinion was a more accessible and safer variant of the representation of the required quantity. Similar 456 scenes were also observed during interviews with other children. Some of them explicitly pointed to the limited place of the app. 3. Difficulties in representing of quantities with multi-touch In the transcribed scenes so far, it became clear that some children had difficulties in representing multiple counters by means of multi-touch. It often occured that children represented more or less counters than originally intended. More counters were created by the children touching the screen with their stretched fingers as well as with other fingers or the heel of hand inadvertently. Less counters than wanted were often represented because the children held two fingers too close together. In this case, the software generated only one counter for two touch inputs. Furthermore, several children did not manage to position their hands in such a way that only the fingertips touch the screen after repeated attempts to add several counters. As a result, students often produced two incompatible representations. Some children recognized their mistakes and fixed them immediately. However, other children saw the incompatible representations as belonging to one another. Even though they were initially sceptical. They often tended to rely on the representations produced by the "Math Tablet" rather than on their own mathematical concepts. Closing remarks The multi-touch technology opens up new mathematical-didactical possibilities, which lead to promising opportunities for the promotion of basic mathematical competences. The empirical findings described show, on the one hand, that all children, after a brief introduction, integrated this potential in their methods of use. On the other hand, it became clear that some children needed specific stimulus to use the multi-touch potential. To display nine counters nine children used single-touch, while six children did so for fifteen counters. However, in the case of the use of multi-touch, it must also be considered that the potential is not automatically exhausted. It could often be observed that quantities were displayed by stretching fingers sequentially and then added via multi-touch. Likewise, difficulties with multi-touch were identified that led children to use less appropriate approaches. A proper handling of the fingers as a mathematical hands- on material as well as the awareness of possible technically evoked difficulties is necessary. Under these conditions, there seem to be opportunities to enrich mathematics teaching and learning with this potential. References Baccaglini-Frank, A., Maracci, M. (2015). Multi-Touch Technology and Preeschooler's Development of Number-Sense. Digital Experiences in Mathematics Education, 1, 7–27. Barendregt, W., Lindström, B., Rietz-Leppänen, E., Holgersson, I., Ottosson, T. (2012). Development and Evaluation of Fingu: A Mathematics iPad Game Using Multi-touch Interaction. IDC 2012, Germany, 1–4. 457 Glaser, B. G., Strauss, A. L. (2005). Grounded theory. Strategien qualitativer Forschung (Vol. 2). Bern: Verlag Hans Huber. Krauthausen, G. (2012). Digitale Medien im Mathematikunterricht der Grundschule [Digital media in primary school mathematics]. Heidelberg: Springer Spektrum. Ladel, S. (2016). Lehren und Lernen von Mathematik mit digitalen Medien – ein Blick in die (nahe) Zukunft. In M. Peschel, T. Irion (Eds.), Neue Medien in der Grundschule 2.0: Grundlagen – Konzeption – Perspektiven (pp. 247–258). Frankfurt am Main: Grundschulverband. Ladel, S., Kortenkamp, U. (2011). Finger-symbol-sets and multi-touch for a better understanding of numbers and operations. In M. Pytlak, T. Rowland, E. Swoboda (Eds.), Proceedings of CERME 7 (pp. 1792–1801). Rzeszów: University of Rzeszów. Ladel, S., Kortenkamp, U. (2014). Number concepts – processes of internalization and externalization by the use of multi-touch technology. In C. Benz, B. Brandt, U. Kortenkamp, G. Krummheuer, S. Ladel, R. Vogel (Eds.), Early Mathematics Learning. Selected Papers of the POEM 2012 Conference (pp. 237–256). New York: Springer. Mayring, P. (2015). Qualitative Inhaltsanalyse [Qualitative content analysis] (Vol. 12). Weinheim und Basel: Beltz. Moyer-Packenham, P. S., Shumway, J. F., Bullock, E., Tucker, S. I., Anderson-Pence, K. L., Westenskow, A., Jordan, K. (2015). Young Children’s Learning Performance and Efficiency when Using Virtual Manipulative Mathematics iPad Apps. Journal of Computers in Mathematics and Science Teaching, 34(1), 41–69. Paek, S., Hoffman, D. L., Black, J. B. (2013). Using Touch-Based Input to Promote Student Math Learning in a Multimedia Learning Environment. Paper presented at the EdMedia 2013. Resnick, L. B. (1983). A Developmental Theory of Number Understanding. In H. P. Ginsburg (Ed.), The Development of Mathematical Thinking (pp. 109–151). New York: Academic Press. Sarama, J., Clements, D. H. (2016). Physical and Virtual Manipulatives: What is "Concrete"? In P. S. Moyer-Packenham (Ed.), International Perspectives on Teaching and Learning with Virtual Manipulatives (pp. 71–94). Switzerland: Springer. Segal, A. (2011). Do gestural interfaces promote thinking? Embodied interaction: Congruent gestures and direct touch promote performance in math: ProQuest Dissertations Publishing. Sinclair, N., Baccaglini-Frank, A. (2016). Digital Technologies in the Early Primary School Classroom. In L. D. English, D. Kirshner (Eds.), Handbook of International Research in Mathematics Education (pp. 662–686). New York: Taylor & Francis. Sinclair, N., Heyd-Metzuyanim, E. (2014). Developing number sense with TouchCounts. In S. Ladel, C. Schreiber (Eds.), Von Audiopodcast bis Zahlensinn (pp. 125–150). Münster: WTM-Verlag. Urff, C. (n. y.). Digitale Lernmedien für die Grundschulstufe [Virtual working equipment in primary stage]. Retrieved from www.lernsoftware-mathematik.de. Walter, D. (2017/ in preparation). Nutzungsweisen bei der Verwendung von Tablet-Apps [Methods of use when using apps]. Dissertation: TU Dortmund. 458 PRACTICAL COMPONENT OF MATHEMATICS EDUCATION OF PRE-SERVICE PRIMARY SCHOOL TEACHERS: STUDENTS’ PERSPECTIVE Renáta Zemanová,Darina Jirotková and Jana Slezáková Abstract The paper presents findings from the first part of the research related to the student teacher professional development and focused at this stage on primary school student teachers and their teaching practice in mathematics. The goal of this first part of our research is to gain data that would allow to bring about an effective change in the conception of primary school teacher education at two universities – University in Ostrava and Charles University in Prague. However, the findings could be of interest to the broader community of teacher educators as it cast light on student teachers’ need in their professional training. Keywords: Teacher training, teaching practice, linking theoretical and practical components of teacher training, mentor Introduction Educators from the departments of mathematics and didactics of mathematics at two faculties of education (Faculty of Education, University of Ostrava, PdF OU, and Faculty of Education, Charles University, PedF UK) have been collaborating in the area of primary teacher training for a long time. Apart from collaborations on the content of courses of mathematics for future primary teachers, they conduct research in several areas, e.g. research in student teacher’s empathy for pupils’ thinking processes when solving problems (Jirotková and Zemanová, 2013). Presently, the joint empirical research focuses on the conception of practical training of future teachers. The goal of this research is to look for ways of improvement of mathematics education of future primary school teachers within the frame of their continuous and one-block teaching practice. We are convinced that the results of two parallel surveys and their comparison will allow us to get background information for conceptions of effective changes in professional undergraduate primary teacher training. It is now the right time for curricular and content changes at both faculties, as the study programmes for future primary teachers are now in the process of re-accreditation. The here presented research is a part of the developmental project Innovation of courses of didactics of mathematics with teaching practice – developing close links between theoretical components of teacher training and teaching practice, which is currently being solved at PedF UK. Within this project we cooperate closely with the teachers who work as supervisors of our student teachers on their University of Ostrava, Czech Republic; e-mail: [email protected] Charles University, Czech Republic; e-mail: [email protected], [email protected] 459 teaching practice or just as mentor teachers. Supervisors from the faculty are responsible for individuals or pairs of student teachers during their teaching practice in all subjects. Mentor teachers cooperate with the supervisor from the faculty by allowing student teachers to carry out teaching experiments in their lessons, by assigning topics to be taught by the students, by giving student teachers brief information about their pupils, by revising lesson plans with the student teachers, by giving the student teachers a brief feedback. However, the responsibility for the Teaching practice course is in the hands of supervisors. Mentor teachers and supervisors often have considerable influence on students, their educational style or their beliefs about the objectives of mathematics education. This is what we plan to focus on in the broader perspective of our research. This paper has a narrower focus on students who go on their teaching practice, their points of view, attitudes and needs. 2. Theoretical background Researchers have paid a lot of attention to the area of student teacher professional development, teaching practice and its various aspects. Teaching practice is the topic of a number of conferences (e.g. III. National conference Teaching practice, 2003, or CERME conference whose one working group focuses on Mathematics Teacher Education and Professional Development). We could cite here a great number of studies focusing on teacher education and its practical component. Educators pay much attention to the quality of experience of future teachers with teaching and are aware of the fact that this has major impact on shaping future teachers (Lane et al., 2003). The author respected in the Czech Republic, Korthagen (2011) provides a detailed and wide account of the relationship between practice and theory in teacher education. From Czech authors we can mention e.g. Zvírotský (2011), who discusses the role and competences of supervisor educators, i.e. university lecturers in charge of students’ teaching practice. The two year project Clinical school (Spilková et al., 2015) focused on the issue of quality of the school where students’ teaching practice takes place and on the cooperation between supervisor educator, mentor teacher and student. Mazáčová (2004) focuses on connections between theoretical preparation and teaching practice, Filová (2002) discusses the selection of teachers to be in charge of teaching practice, Marková and Urbánek (2008) focus on mentor teachers’ reflection and their opinions on students. As far as foreign research is concerned we mention here the research from Sayeski and Puisen (2012), which analyses data from almost 400 teachers’ evaluations of cooperating teachers. From this data set those practices are identified that student teachers acknowledged as having a positive influence on their professional development. In Mouilding et al. (2014) teacher efficacy has been shown to play a role in teacher effectiveness. This study shows that a significant correlation exists between efficacy scores and perceptions of support by mentors during 460 student teaching. Izadinia (2016) examines similarities and differences between mentor teachers’ and student teachers’ perceptions of the components of a positive mentoring relationship and its impact on the identity formation of student teachers. Definition of the role of a mentor is also discussed in Butler and Cuenca (2012). However, our work must come out of the current possibilities which change every year – the choice of available schools and classes, possibilities of mentor teachers and supervisors from the faculty. The issue of evaluating supervisors and mentor teachers is the subject of our further research which will be conducted after this year’s teaching practice is over. In this study we focus on students’ requirements and expectations. This will allow us to take into consideration students’ expectations and requirements when selecting schools where teaching practices take place and when selecting mentor teachers who will help us to link the theoretical and practical part of teacher education in elementary mathematics efficiently. The conception of the practical part of teacher training at the two faculties is different. The teaching practice at PdF OU is now newly guaranteed by different departments and there is not one unifying concept. The Department of Mathematics with Didactics (KMD) is currently working on a significant change in the conception of continuous teaching practice, both as far as its content and extent are concerned. The goal of these changes is to deepen the link between theoretical and practical training in elementary mathematics with an emphasis on the practical component. Students’ subject continuous teaching practice at PedF UK is guaranteed by individual departments. There is no platform on which the participating departments would cooperate. In order to bring about effective changes in primary teacher training it is essential to describe the current situation. This means we have to know more about the diversity of students, their opinions, their pedagogical beliefs, their different needs, expectations and areas in which progress is needed. Data collection and their processing KMD at PdF OU looked for inspiration in the project Innovation currently solved at KMDM at PedF UK. Within the project the following materials are studied: 1) reflections of teachers in whose classes the teaching practice is conducted. Semi-structured interviews were conducted with these mentor teachers. They were asked about their experience with students, the reasons for being willing to become mentor teachers and spend time with students, about their cooperation with the supervisor-educator, about their expectations before a new teaching practice, about their strengths they want to influence the students with. 461 2) reflections of students before and after continuous teaching practice. Students answered in writing. 3) students’ portfolios after the extension module in didactics of mathematics (see Slezáková et al., 2017). The goal of analyses in materials 2) is the identification of incentives, e.g. of stimuli, impulses, motives contributing to shifts in both the teacher’s and student’s approaches to teaching mathematics and supervision of teaching practices. The tool for gaining these data are questionnaires. Questionnaires of both departments are in many items identical, which allows a comparison of the results. This will allow us to separate phenomena dependent on a particular university environment from phenomena that are universal. On PdF OU the first part of the questionnaire survey before the teaching practice was conducted from December 2016 until January 2017. 49 students were involved (24 students from the 4th year, 25 students from the 3rd year). 4th year students had spent one semester observing lessons, two semesters teaching episodes and had finished all mathematical courses with the exception of didactics of arithmetic. 3rd year students had not been to any lesson observations, had not conducted teaching episodes and had not attended any courses of didactics. The questionnaire on PdF OU consisted of the following questions: 1. What is your experience from teaching practice of mathematics? 2. In the 5th (or the 4th) year you will have one-block (or continuous) teaching practice. What do you expect from this teaching practice? What would you like to gain from this teaching practice? When do you expect to need help? What are you afraid of / looking for before this teaching practice? 3. How do you imagine your cooperation with the mentor-class teacher / your supervisor educator from the faculty? What do you expect from them? 4. Assess your strengths and weaknesses in teaching primary mathematics. At PedF UK the first part of the questionnaire survey was conducted in the beginning of October 2016 with 4th year students who had not had any teaching practice supervised by a mathematics educator. The students had finished one semester of the course Didactics of Mathematics I with a very small number of lectures and seminars (45 min lecture and 45 min seminar a week). The questionnaire was completed in writing by 53 students. The students signed an informed consent with participation in a research project in which all data will be processed and published anonymously. The questionnaire at PedF UK consisted of the same questions as the questionnaire administered at PdF OU with the only difference that it did not ask 462 about the one-block teaching practice that is included in the 5th year (question No. 2) but about the running practice which is in the 4th year of the study programme. The first question was related to any kind of teaching practice they had undergone during their undergraduate studies at the faculty. Results The following are some examples of students’ answers, which illustrate the most frequent answers to the posed questions. ad 1. “I think it was difficult for students to teach mathematics using constructivistic methods because the teaching practice is before the students have a course of the necessary methodology at the 4th year. However, we did appreciate having the chance to go to schools that actually use this method and see that children learning mathematics in this way find it much more enjoyable and less stressing than when taught traditionally” ad 2. Expectation: “I hope to gain more experience for my future work. I want to get a good feeling that I have shown the children something, that mathematics can be fun. I think I will need help and support especially in the area of how to not disclose something they should discover on their own.” Fears: “Probably that the teacher in the class will not like the new method if she is used to the traditional one.” Looking forward to: “To meet children, we do not get enough teaching practice in the course of our studies.” ad 3. “Critical feedback to know what I still need to improve.” “Tips for good websites that can help me, ideas on how I can create my own teaching aids if they are not in the school.” ad 4. Weakness: “… self-consciousness because I have never excelled in mathematics.” “…. that I did not enjoy mathematics at school. And this can be probably seen in my way of approaching it now.” “Impatience.” Strength: “… posing problems and creating aids even with children in the different environments.” “…the effort to provide to the pupils as many ways and models that will allow them to grasp the problem.” Discussion In the answers from the questionnaires we could identify phenomena that we classified into groups. They will become the basis for further analyses. These phenomena were: 1. Experience. This group was further divided with respect to what the experience was related to: stories with children, being taught a lesson, encouraging experience, source of doubts or fears. 2. Phenomena of personality-social development (PSD). These were further divided to doubts, fears, weaknesses, strengths. 463 3. Phenomena connected with the teaching practice itself. These were further classified according to a) what the phenomenon is related to: general, supervisor, class teacher, peer, other person, and b) their relation to the competences the phenomenon is connected with: does not manage, offers, manages. This is where the first stage of our research ends. In the next stage with more questionnaires from a new cohort of students we will analyze the data electronically in a process that will allow us to match all relevant answers for each identified phenomenon. For each phenomenon we will identify its frequency which will indicate its significance. A data set like this will allow us to compare different groups of respondents quite easily, e.g. at different stages of their professional practice, in different study groups and at different faculties. We can thus identify phenomena that are shared and universal and phenomena that are unique and characteristic for some groups of respondents. This will allow us to formulate incentives in their dependence on external conditions and to compare them. Conclusion The goal of the ongoing research is, as stated above, to look for ways to improvement of mathematics education of pre-service primary school teachers. The partial results of the survey at the two faculties were compared by a skilled insight at this first stage. The answers shared several strong common phenomena: 1. Students’ anxiety to teach mathematics using constructivist methods, since they have a lack of experience with this method from their school years. What we find important with respect to the planned innovation of the content and extent of the teaching practice curriculum are reactions of students from PedF who selected the extension module of didactics of mathematics. If this group of students (there are 20 students from this group in the research sample) turn out to be significantly more positive in our experiment, we will try hard to include more possibilities for gaining more experience with teaching of this type in the programme. 2. The fear of teaching mathematics using constructivist methods, which is much stronger in Ostrava than in Prague. The students are worried they will not be perfect in the role of a teacher – they will not be patient enough, they will not be able to suppress the typical teacher’s need to explain everything to their pupils, to guide them through the “best” way of solving a problem. We are able to identify both phenomena that are common at both faculties as well as specific for each of the faculty. We can study the influence of the environment, e.g. working with portfolios in different mathematical courses, teaching styles of university lecturers, methods used at schools where students do their teaching practice or of supervisors at this practice. Obviously if the situation is to change, 464 the selection of the mentor teacher is of major importance with respect to students on their teaching practice. In Prague we cooperate with a number of teachers who have used constructivist approaches to teaching for years. This gives students enough opportunities to observe constructivist teaching in different classrooms and different subjects and to compare this approach to transmissive teaching. When selecting suitable teachers we will build on experience from important studies (cf. Sayeski, Puisen, 2012; Izadinia, 2016). The gained results will be implemented into the planned organization changes of undergraduate primary teacher training at both faculties. Acknowledgement The research was supported by the research project PROGRES Q17 Teacher preparation and teaching profession in the context of science and research. References Butler, B., Cuenca, A. (2012). Conceptualizing the roles of mentor teachers during student teaching. Action in Teacher Education, 34, 296–308. Filová, H. (2000). Intervence cvičných učitelů v procesu vytváření pedagogických dovedností studentů učitelství primární školy. In V. Švec (Ed.), Monitorování a rozvoj pedagogických dovedností (pp. 213–219). Brno: Paido. Izadinia, M. (2016). Student teachers’ and mentor teachers’ perceptions and expectations of a mentoring relationship: do they match or clash? Professional Development in Education, 42(3), 387–402. Jirotková, D., Zemanová, R. (2013). Student-teachers’ empathy for pupils’ thinking process when solving problems. In J. Novotná, H. Moraová (Eds.), SEMT´13 (pp. 155–162). Prague: Charles University, Faculty of Education. Lane, S., Lacefield-Parachini, N., Isken, J. (2003). Developing novice teachers as change agent: Student teacher placement “against the grain.” Teacher Education Quarterly, 30(2), 55–68. Mazáčová, N. (2014). Pedagogická praxe. Prague: Charles University, Faculty of Education. Marková, K., Urbánek, P. (2008). Praktická příprava učitelů všeobecně vzdělávacích ředmětů: realita, problémy a perspektivy. In J. Vašutová et al. (Eds.), Vzděláváme budoucí učitele. Nové trendy v pedagogicko-psychologické přípravě studentů učitelství (pp. 79–109). Prague: Portál. Mouilding, L.R., Stewart, P.W., Dunmeyer, M.L. (2014). Pre-service teachers’ sense of efficacy: Relationship to academic ability, student teaching placement characteristics, and mentor support. Teaching and Teacher Education, 41, 60–66. Korthagen, F. et al. (2011). Jak spojit praxi s teorií: didaktika realistického vzdělávání učitelů. Brno: Paido. Sayeski, K.L., Puisen, K.J. (2012). Student Teacher Evaluations of Cooperating Teachers as Indices of Effective Mentoring, Teacher Education Quarterly, Spring, 117–130. 465 Slezáková, J., Jirotková, D., Kloboučková, J. (2017). Student portfolio as a tool for development of pre-service primary teachers’ competences in teaching mathematics. In J. Novotná, H. Moraová (Eds.), SEMT 2017 Proceedings: Equity and Diversity (pp. 383–393). Prague: Charles University. Faculty of Education. Spilková, V. et al. (2015). Klinická škola a její role ve vzdělávání učitelů. Prague: Charles University, Faculty of Education. Zvírotský, M. (2011). Náplň práce fakultních učitelů. In S. Bendl et al. (Eds.), Klinická škola: místo pro výzkum a praktickou přípravu budoucích učitelů (pp. 489–498). Prague: Charles University, Faculty of Education. SIMILARITIES AND DIFFERENCES IN IDENTIFICATION OF RECTANGLES OF FOURTH-GRADE PUPILS Katarína Žilková,Janka Kopáčováand Ľubomír Rybanský Abstract Within the project of Scientific Grant Agency of the Ministry of Education, Science, Research and Sport of the Slovak Republic and Slovak Academy of Sciences, hereinafter VEGA 1/0440/15, we examine geometric conceptions and misconceptions of pre-school and school age children. The paper deals with a part of the results regarding the identification of rectangles by fourth-grade pupils of elementary schools. In analysing the results, we will focus on the overall results of selected populations of children, mostly from northern Slovakia, while we bear in mind the aspect of the impact of socially disadvantaged backgrounds of pupils on their educational results. Keywords: Conceptions, fourth-grade children, children with low socio-economic status, misconceptions, models and non-models of shapes, rectangles INTRODUCTION AND THEORETICAL BASIS The problem of the impact of the pupils´ backgrounds on their mathematics education is serious and highly topical, it is therefore necessary not only to examine the causes of (no) success of pupils in the context of the background they come from, but also to search for the potential ways of reducing the impact of the background on pupils´ performance. Clements, Sarama, and Kitchen (2013) state: „some children come to school far less prepared to learn than others. For most, these differences do not disappear, and, in fact, they increase”. Analogous experiences in Slovakia show, that all children do not generally have the same starting position for their further education. The National Institute of Certified Measurement (NUCEM) in the published analysis notes: "international studies show a strong link between student performance and family background. The Comenius University in Bratislava, Slovakia; e-mail: [email protected] Catholic University in Ružomberok, Slovakia; e-mail: [email protected] 466 TIMSS study describes this relationship as an index Sources home environment (SEI - socioeconomic index), which describes the social, economic and cultural capital of the family "(NUCEM, 2015a). According to the results presented in the study it was proved that the better mathematical skills the first-grader has, the higher performance he/she achieves in mathematics. Similarly, the positive attitude was shown with pre-school children. In the context of the situation in Slovakia NÚCEM (2015b) notes, that "the impact of socioeconomic status on the pupil´s performance in Slovakia is significantly stronger than the OECD average". This means that "the Slovak Republic still ranks among the countries with a relatively high degree of influence of socio-economic background on pupils´ performance" (NUCEM, 2015b). Clements and Sarama (2015) in their study focused on "equity in mathematics achievement and education, including students who live in poverty and who are members of ethnic and linguistic minority groups." They stress that pupils with lower socio-economic status (hereinafter SES children) groups, respectively communities have significantly lower levels of educational attainment, and this is particularly true for math education. At the end of the study they formulate a request: "Children who live in poverty and who are members of linguistic and ethnic minority groups need more and better math programs. They need programs that Emphasize the higher-order concepts and skills at each level, as well as knowledge and skills base" (Clements and Sarama, 2015). The situation in Slovakia has required the implementation of several intervention programs to mitigate the impact of socially disadvantaged backgrounds for the educational results of children, including mathematics. One of these projects was the PRINED project (Project of inclusive education). From the final evaluation report (2015) concerning the evaluation of the impact of the intervention program including the area of mathematics shows that: The rate of improvement in the results of SES children has been, after the implementation of the intervention program, significantly higher than the rate of improvement in the results of intact children. Individual progress of children who attended the stimulus program was significantly stronger than the individual progress of children who did not attend the program (Liba et al., 2015). The positive impact of the implementation of target created intervention program is obvious. Through the program we will take into account the specificities of the target groups and to eliminate discrimination resulting from the environment in which the children grow up. For this, it is necessary to identify the current state of the knowledge of the children of that target group, their needs and interests, and based on these results, to modify educational approaches in mathematics. 467 RESEARCH A. Objectives of the research Clements and Sarama (2015) describe the results of mathematical competencies of pre-school children, while in geometry they state: “there were no significant differences on the simple tasks involving shape and comparison of shapes. There were significant differences on representing shapes, composing shapes, and patterning.” These results suggest that SES children can have problems in dealing with tasks that require higher cognitive processes. In this context and with regard to the main theme of our project of geometrical concepts and misconceptions of pupils of elementary education, we focused on finding the differences between the groups of intact children and SES children. We examined whether the significant differences in the perception of plain geometrical shapes at the primary level of van Hiele level of geometrical thinking, the level of visualization. The main objective of the research (in general) was to identify the most common misconceptions of fourth-grade pupils on geometrical shapes and their properties. The aim of this paper is to describe those parts of results which relate to the identification of rectangles in pictures, while we want to compare the results of pupils of intact population to pupils from socially disadvantaged backgrounds (SES children). On that basis mentioned above, we formulate the following research questions: 1. What problems do fourth-grade pupils have in simple tasks aimed at identifying models and non-model of the rectangles? 2. What are the differences between models and non-models of rectangles in terms of difficulty of their identification for fourth-grade pupils of elementary school? 3. What is the difference between the level of geometric thinking of intact children and SES children in identifying rectangles by pictures? The answers to the above questions will enable us not only to diagnose the level of geometric thinking of individual children, but in particular, provide information about which misconceptions of rectangles are the most common. Subsequently, it will be possible to implement such approaches into mathematical learning and also such models and non-models shapes to eliminate the possibility of misconceptions, respectively to create conditions for the formation of correct ideas. B. Research design Selection of research method and research body To detect the images of fourth-grade pupils of elementary schools for plain shapes (triangle, square, rectangle and circle), we used a knowledge test. The tasks in the test were designed to correspond with the first two Van Hiele levels - visualization and analysis. 468 The first part of the test focused on the detection of the capability of fourth-graders to name planar shapes correctly (triangle, square, rectangle and circle) by using graphic models. The tasks in this part of the test matched only the visualization level. In the second part of the test, the tasks were aimed at determining the characteristics of the shapes, and thus consistent with the second level of van Hiele level (analysis). Defined the van Hiele levels are sequential and hierarchical. This means that the transition from one (the lower) to the next (higher) level is determined by mastering of each of the preceding levels. Therefore, it is necessary at first, to diagnose the current level of fourth-graders in the ability of identifying rectangles and find out if they are already able to perceive rectangles not only holistically, but can distinguish important features of rectangles and their elemental properties. Diagnostic results should not be just a statement of capabilities of fourth-graders, but mainly to draft the sequence of tasks that enable a shift to more abstract van Hiele levels. The research sample consisted of 345 fourth-grade pupils of elementary schools, mainly in the areas situated in the north and the north-east of Slovakia (Fig. 1). Research subjects were selected based on their availability and a single set consisted of both the non-disabled (intact) population (205 students) and the population of children with lower socio-economic status (140 pupils). Figure 1: Localization of the research sample Methods of data collection, research tools A research tool was a knowledge test. The test was carried out between April and June 2015, in a traditional pencil-paper form. Pupils circled or wrote in short answers. According to the results that we have, in the paper we will focus only on the three selected tasks of the first part of the test, which focused on rectangles1. In the first task we expected pupils to write the correct name of the shape based on the picture given. In the other tasks we investigated whether a fourth-grader can distinguish between models and non-models of plane shapes (also based on the pictures). 1 In the primary school in Slovakia we classify quadrilaterals to 3 disjoint sets: squares, oblongs (rectangles, which are not squares), other quadrilaterals. The concept rectangle is not used in primary education. 469 Task 1. Write the correct name for each shape (Fig. 2). Figure 2: Task 1 Task 2. Is the shape in the picture a square? Circle the answer (Fig. 3). Task 3. Circle, which of the following shapes are not oblongs (Fig. 4). Figure 3: Task 2 Figure 4: Task 3 We understand that the formulation of the task 3 in the form of negation is more demanding for pupils than the formulation of the task 2. Both tasks (Task 2 and Task 3) were aimed essentially to the dichotomous classification lines, which are models of rectangles, and rectangles that are non-models. We analyzed the data obtained by using standard statistical methods. We were interested in basic statistical characteristics of the individual tasks, as well as the test as a whole. The reason is that with good characteristics of the test, or after its revision, it could serve a diagnostic function and become a useful tool for identifying individual pupils' skills, moreover, after diagnostic it would be possible to set for each pupil's individual plan of development to enable it to shift within each van Hiele levels. C. Results and discussion Analysis of the results was carried out in several stages, respectively from multiple perspectives. At first, we describe the overall results of the whole research group at the three tasks and then comparison of the groups of children from different backgrounds. The name of shapes Reliability of the subtest "Task 1", estimated by Cronbach's alpha, is 0.60, which is less than the minimum acceptable value of 0.7 (Kiln, 2001). However, this is caused by a low value of item Sq1A (Fig. 5). If it did not appear in the subtest, the value of Cronbach's alpha would be of 0.75, which indicates a good internal consistency of items. Apart from this item, no other component reduces the overall reliability of the subtest. 470 Based on the values of the parameter of difficulty and cluster analysis, we found out that the shapes were divided into three clusters, while the first cluster consisted of the most difficult objects and the objects in the third cluster were the least challenging (Fig. 5). Figure 5: Classification of items of the subtest “Task 1” into three clusters It is obvious that the biggest problem with the display name of bodies have fourth grade with the square in position with horizontal-vertical diagonals. If the body could not be identified as a square, usually considered him a rhombus or called him a "diamond". Models of rectangles When analyzing the other two tasks, we decided to evaluate the results separately for models rectangles and separately for the units in the picture that are not models of rectangles. This approach enables us to identify the demands of different shapes for fourth-graders. Figure 6: Classification of items of "models of squares” into two clusters The results we have obtained for rectangle models confirm the Van Hiele theory of an important factor in positioning the shape in the process of identifying shapes. In the original image they were located only three items, which represent squares models, and the results are shown in Fig. 6, with a cluster of two time forms a more severe level. 471 Analogously, we investigated the differences in the perception of the three oblongs models, and the results are shown in Fig. 7. Figure 7: Classification of items of "models of oblongs” into two clusters Non-models of rectangles In the task 3, 13 non-models of squares were shown. The reliability of the subtest, estimated by Cronbach's alpha reached the level of 0.89, which represents a very good internal consistency of items. The methodology of statistical analysis was the same as in the previous tasks, therefore we present only the final figure of the classification of non-models of squares on disjoint classes according to the results of cluster analysis (Fig. 8). Figure 8: Classification of items of “non-models of squares” into five clusters It is obvious, that the most difficult shapes to identify were those of square shape, but in fact are not squares, in accordance with van Hiele theory. Analogously, we focused on the ability of fourth-graders to identify the non- models of oblongs. The number in the picture was 12. In this case, the cluster analysis also showed 5 levels of the difficulty of the shapes, while the shapes in the fifth cluster were the easiest for the fourth-graders to determine the non- models of a oblong and a parallelograms showed in the first cluster caused pupils the most problems. This geometric shape pupils often considered an oblong. The 472 reason may also be that, according to the national curriculum, fourth-grade pupils of elementary schools in Slovakia have no information about the right angle and according to the additional information obtained from interviews in identifying the pupils compared only the length of the opposing sides. Figure 9: Classification of items of the „Oblongs subtest“ into five clusters Overall, we can say that the results of the whole research group showed a holistic perception of rectangles without any deeper analysis of elementary important elements. Comparison of success of solutions based on diversity of backgrounds Low socio‐economic background Yes No n average SD median min max n average SD median min max ‐ ‐ U1 140 ‐0,07 0,77 0,40 2,54 0,40 205 0,05 0,60 0,22 2,60 0,40 ‐ ‐ SqM 140 0,00 0,69 0,33 1,93 0,33 205 0,00 0,63 0,33 1,83 0,33 ‐ ‐ ReM 140 ‐0,09 0,72 0,25 2,08 0,25 205 0,06 0,53 0,25 2,08 0,25 ‐ ‐ SqN1 140 0,40 1,19 ‐0,08 0,72 2,77 205 0,10 0,91 ‐0,08 0,72 2,77 ‐ ‐ SqN2 140 ‐0,36 1,21 0,19 2,50 0,84 205 ‐0,06 0,90 0,19 2,50 0,84 ‐ ‐ ReN1 140 0,59 1,13 0,00 0,33 3,41 205 0,14 0,80 ‐0,30 0,33 3,33 ‐ ‐ ReN2 140 ‐0,42 1,10 ‐0,20 2,51 0,78 205 ‐0,10 1,00 0,09 2,57 0,78 Table 2: Descriptive statistics of latent variables 473 We want to find out whether SES children achieved comparable results with intact children. For this reason we have created a SES variable, which takes two values: yes, if a pupil comes from a low socio-economic background, and no, if a pupil does not come from a low socio-economic background. General statistics of both groups are listed in the Tab. 2. In terms of examining the statistical significance of the differences between the intact population and SES children (Tab. 3), we found that in all tasks SES children achieved worse results. Statistically significant difference appeared in case of REM, SqN1, SqN2, ReN1, ReN2, which represent all non-models of rectangles and but even models of oblongs. latent variable F(1,343) p úspešnejší (SES) U1 2,43 0,119 "no" (non‐significantly) SqM 0,001 0,973 "no" (non‐significantly) ReM 5,48 0,02 "no" (significantly) SqN1 7,05 0,008 "no" (significantly) SqN2 6,61 0,011 "no" (significantly) ReN1 19,12 <0,001 "no" (significantly) ReN2 7,96 0,005 "no" (significantly) Table 3: Results of ANOVA Taking into account the fact that the reported tasks are only task to identify the body by the original image, corresponding to the lowest van Hiele levels and finding that SES children achieved significantly worse results than children of intact population is alarming finding and does not make sense to verify tasks requiring higher thought processes. We therefore consider it necessary not only to present the findings of the expert community in Slovakia, but especially to develop a set of tasks with appropriate methodology to develop geometric thinking of that target group. CONCLUSIONS In the context of the comparison of our results with Clements and Sarama (2015) we can conclude that the differences of pupils´ backgrounds, in our sample, has a significant impact on the educational achievements of the fourth-grade pupils in Slovakia, namely the identification and classification of geometrical shapes. We have also confirmed the findings of national measurements, which had shown a strong influence of socio-economic background on pupils´ performance. It is therefore necessary to prepare programs, which will reflect the above findings and help develop the concepts of geometric shapes. We are pleased to note that during the project we have managed to transform some of the research results into teaching recommendations and these have been reflected in the preparation of new mathematics textbooks for primary education. In accordance with the statement "there are examples of schools that serve poor communities in the US that have 474 demonstrated that high achievement in mathematics does not occur only in high- resourced communities and schools" (Clements, Sarama and Kitchen, 2013), we believe that socio-economic background does not automatically mean poor results in mathematics education. There are many other factors to consider even not always related to the public policy of the state. Acknowledgement This study was supported by VEGA 1/0440/15 "Geometric conceptions and misconceptions of pre-school and school age children" and APVV-15-0378 "Optimalization of teaching materials for mathematics based on analysis of current needs and abilities of primary school pupils". References Clements, D. H, Sarama, J. Kitchen, R. (2013). Education and Equity. [online] Available from http://www.du.edu/kennedyinstitute/media/documents/equity-paper1.pdf. Clements, D. H, Sarama, J. (2015). Equity and Mathematics Education. [online] Available from http://www.du.edu/kennedyinstitute/media/documents/equity_2_kennedy_institute_ pages.pdf. NUCEM (2015a). TIMSS 2015. Prvé výsledky medzinárodného výskumu vedomostí a zručností žiakov 4. ročníka ZŠ v matematike a prírodných vedách. [online] Available from http://www.nucem.sk/documents/27/medzinarodne_merania/timss/publikacie/Prve_ vysledky_Slovenska_v_studii_IEA_TIMSS_2015.pdf NUCEM (2015b). PISA 2015. Prvé výsledky výskumu 15-ročných žiakov z pohľadu Slovenska. [online] Available from http://www.nucem.sk/documents/27/medzinarodne_merania/pisa/publikacie_a_dise minacia/4_ine/Prve_vysledky_Slovenska_v_studii__OECD_PISA_2015.pdf Liba, J., Kosová, B., Žilková, K., Matulayová, T., Janoško, P. (2015). Evalvačná správa z projektu PRINED – PRojekt INkluzívnej EDukácie. Bratislava: Metodicko- pedagogické centrum, 2015. ISBN 978-80-565-1415-3. van Hiele, P. M. (1999). Developing Geometric Thinking through Activities that Begin with Play. Teaching Children Mathematics, 5(6), 310–16. 475 476 W ORKSHOPS SPEAKING IN AND ABOUT THE MATHEMATICS CLASSROOM David Clarke, Carmel Mesiti ,Jarmila Novotná, Hana Moraová and Man Ching Esther Chan Abstract Learning can be conceptualised in terms of progressively enhanced participation in forms of institutionalised social practice, where discourses form key components of that practice. Students are initiated into the discourse of the mathematics classroom: a discourse with its own technical vocabulary and discursive and social conventions. Mathematics teachers similarly participate in a discourse community in which the mathematics classroom and its objects, agents and events provide the subjects of professional discourse and for which language mediates the experience of the classroom and the professional learning that experience engenders. This workshop addresses theoretical, methodological and practical issues associated with language use and the mathematics classroom. It draws on the video-based research program led by the International Centre for Classroom Research (ICCR) at the University of Melbourne, Australia. University of Malbourne, Australia; e-mail: [email protected], [email protected], [email protected] Charles University, Czech Republic; e-mail: [email protected], [email protected] 477 The workshop is presented in three parts, each highlighting a distinct discourse community: students; teachers; workshop participants. Session 1. Students speaking inside the classroom Session 2. Teachers speaking about the classroom Session 3. Workshop participants speaking about video from mathematics classrooms Keywords: lexicon, discourse, professional vocabulary, video research 1. Students speaking inside the classroom (Clarke and Chan) The first session of the workshop concerns data types and draws on an investigation of the social interaction of Year 7 students during collaborative problem solving in mathematics. This project collected multiple forms of data including student written products and video and audio recordings of students and their teacher in the classroom. Sample data collected from the project will be used to stimulate discussion regarding the analytical options afforded by different data types. 2. Teachers speaking about the classroom (Mesiti) Our interactions with classroom settings, whether as teachers or researchers are mediated through the activities that we see and experience, and significantly by those for which we also have a name. The lexicon of the teaching community, that is, terms that are familiar and in widespread use, are valuable resources of that community. In particular, we can ask: What practices (terms) are valued by the act of naming and how might these named practices be improved? In this session, participants will reflect on their own community’s naming system and learn about sophisticated classroom practices named by teachers in nine different communities worldwide. 3. How do we perceive lessons of mathematics? (Novotná and Moraová) When observing lessons, it is common that each of the observers focuses on and stresses out different phenomena. In consequence, joint reflection of lessons brings new points of views of other professionals and is very inspiring (Novotná, Moraová, Hošpesová, Žlábková, Bureš, 2016). Within the workshop, the participants will be reflecting on segments of video recordings of several lessons of mathematics in the 8th grade in different countries and will try to describe the interactions in the lessons. Since the participants of the workshop will come from different countries, a rich discussion on what terms to use to describe specific interactions can be expected. The participants will share their experiences and will also be focusing on the differences between their descriptions, the difficulties they come across when describing a lesson, and on the need to use expert, technical rather than evaluative terms for the descriptions. 478 References Clarke, D. J., Dı́ez-Palomar, J., Hannula, M., Chan, M. C. E., Mesiti, C., Novotná, J., Žlábková, I., Cao, Y., Yu, G., Hollingsworth, H., Roan, K., Jazby, D., Tuohilampi, L., Dobie, T. (2016). Language mediating learning: The function of language in mediating and shaping the classroom experiences of students, teachers and researchers. In C. Csíkos, A. Rausch, J. Szitányi (Eds.), Proceedings of the 40th Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 349–374). Szeged, Hungary: PME. Novotná, J., Moraová, H., Hošpesová, A., Žlábková, I., Bureš, J. (2016). How do we understand each other when we describe classroom activities – Lexicon. In M. Flégl, M. Houška, I. Krejčí (Eds.), Proceedings of the 13th International Conference Efficiency and Responsibility in Education 2016 (pp. 440–447). Prague: Czech University of Life Sciences Prague. ELICITING STUDENT EXPLANATIONS IN INTERACTIVE WHOLE- CLASS DISCUSSIONS Brian Doigand Susie Groves Abstract In this workshop, participants will experience both the student and teacher roles in a Japanese Structured Problem- Solving Mathematics Lesson. The aim is to participate in an extended professional development experience focused on the whole-class discussion (neriage) phase of Japanese structured problem-solving mathematics lessons, and use these experiences to explore anticipated teacher-student interactions and ways to elicit rich student explanations. Keywords: Classroom discourse, Japanese Lesson Study, problem solving This workshop focusses on the whole-class discussion (neriage) phase of Japanese structured problem-solving mathematics lessons where students compare, polish and refine solutions in the lead up to the final matome (or climax) of the lesson, where the mathematical goal of the lesson becomes clear. Leading such whole- class discussions is an area where adopters of Lesson Study outside of Japan have often experienced difficulties in moving to interactive discussion models beyond “show and tell” (Takahashi, 2008). The first session will focus on the key features of structured problem-solving mathematics lessons (Doig, Groves and Fujii, (2011) and will include participants taking part in a “mock” lesson, followed by an analysis, focused on the discussion phase of the lesson, during which solution strategies are shared and discussed. The emphasis in this session will be on avoiding the "show and tell" typical of many classrooms, and in contrast, promoting a community of inquiry, where learners drive their own learning. Deakin University, Australia; e-mail: [email protected] 479 In the second session attention will turn to the types of interaction that promote mathematical reasoning and the development of mathematical ideas (e.g. Groves and Fujii, 2008; Okazaki, Kimura and Watanabe, 2016). Participants will view short video extracts, from Australian elementary school classrooms, where structured problem-solving lessons have taken place, and analyse the ways in which the teachers elicited students’ explanations during whole class discussion phase of the lesson. In the final session, a group of participant volunteers will act as “teachers” and plan, and conduct, a structured problem-solving lesson, based on a similar problem to the one used in the first session. Other participants will play the rôle of elementary school students for this lesson. The workshop will end with a plenary discussion, where possibilities for future research collaboration between participants will be canvassed. References Doig, B., Groves, S., Fujii, T. (2011). The critical role of task development in Lesson Study. In Hart, L., Alston, A., Murata, A. (Eds.), Lesson study research and practice in mathematics education (pp. 181–199). Dordrecht, The Netherlands: Springer. Groves, S., Fujii, T. (2008). Progressive discourse, Neriage, and some underlying assumptions. In Proceedings of 41st Annual Conference of JSME (pp. 525–530). Japan Society of Mathematical Education (JSME), Japan. Okazaki, M., Kimura, K., Watanabe, K. (2016). Types of interaction that promote or hinder the narrative coherence of a mathematics lesson. In C. Csíkos, A. Rausch, J. Szitányi (Eds.), Proceedings of the 40th Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 395–402). Szeged, Hungary: PME. Takahashi, A. (2008). Beyond show and tell: Neriage for teaching through problem solving – Ideas from Japanese problem-solving approaches for teaching mathematics. Paper presented at Topic Group 19: Research and development in problem solving in mathematics education, ICME–11, Monterey, Mexico. A PATHWAY TO INQUIRY Libuše Samková Abstract The presented workshop will serve as a practical guide to inquiry based mathematics education at elementary school level. After an introduction on historical and theoretical background, it will inform about various types of tasks that may lead to inquiry activities of elementary school pupils, and involve the participants of the workshop in solving and posing these tasks. University of South Bohemia, Czech Republic; e-mail: [email protected] 480 Keywords: Inquiry based mathematics education, inquiry task, open approach to mathematics, problem solving, problem posing This workshop aims to supplement the presentation Planning and conducting inquiry based mathematics course for future primary school teachers (Samková, 2017) with first-hand practical experience of inquiry activities. In particular, the workshop will focus on inquiry based mathematics education (IBME) at elementary school level. After a necessary introduction devoted to historical and theoretical perspectives of IBME, we will cast around for possible sources of mathematical inquiry in elementary school, and debate various characteristics of inquiry tasks, i.e. of the tasks that might lead to inquiry activities of pupils. As a guide on our way to inquiry we will employ issues related to open approach to mathematics, i.e. to the approach based on open problems whose starting situation, process, and/or goal situation are not exactly given. Within this approach, open problems have multiple levels of grasping, multiple correct ways of solving, multiple correct answers, and/or multiple ways to transform the problem into a new one (Nohda, 1995, 2000). To get a closer idea of inquiry activities, the participants of the workshop will solve several inquiry tasks on their own, each of the solved tasks will have a slightly diverse characteristic. Some of the tasks will be arithmetic, some geometric. Afterwards the participants will discuss an educational potential of such tasks. One of the inquiry tasks that will be assigned to the participants is inspired by an open problem presented by Pehkonen at one of the previous SEMT conferences: With twelve matchsticks one can make a square the area of which is 9 au (au = area units). Can you use twelve matches to make a polygon with an area 5 au? How many different polygons of 5 au can you make with twelve matches? Can there be more than ten different solutions? Is it possible to use twelve matches to make a 1 au (or 2, 3, 4 au) polygon? (Pehkonen, 1999, pp. 57-58) This task allows a wide range of inquiry activities, particular parts of the task correspond to different levels of mathematical knowledge, and to different mathematical contents. Focusing just on squares and rectangles, and with a square grid as a tool, the task can be assigned to primary school pupils. Focusing on other polygons included triangles, parallelograms, regular polygons, and with Pythagorean theorem in background, the task offers wide inquiry opportunities for lower-secondary pupils. As a preparatory activity, the pupils may look for all triangles that can be made from the 12 matches. As a supplementary activity, the pupils may look for answers to questions such as whether is it possible to make a 481 trapezoid using the 12 matches, or may propose and assign similar questions to their classmates. After the experience with solving inquiry tasks, the participants will also attempt to pose their own inquiry tasks that could be employed in school, e.g. by modifying some tasks that do not allow inquiry, or by creating brand new tasks on a chosen/given topic. The participants will reflect the activity and share their work. During the reflection we shall also discuss other related issues, e.g. the results of the research on how primary school teachers posed inquiry tasks that was referred on previous SEMT conference (Hošpesová et al., 2015). Acknowledgement Supported by the Czech Science Foundation, project No. 14-01417S. References Hošpesová, A., Samková, L., Tichá, M., Roubíček, F. (2015). How primary school teachers posed problems for inquiry based mathematics education. In J. Novotná, H. Moraová (Eds.), Proceedings of SEMT ʻ15 (pp. 156–165). Prague: Charles University, Faculty of Education. Nohda, N. (1995). Teaching and evaluating using "open-ended problems" in the classroom. ZDM Mathematics Education, 27(2), 57–61. Nohda, N. (2000). Teaching by open-approach method in Japanese mathematics classroom. In T. Nakahara, M. Koyama (Eds.), Proceedings of PME 24 (Vol. 1, pp. 39–53). Hiroshima: Hiroshima University. Pehkonen, E. (1999). Open-ended problems: A method for an educational change. In M. Hejný, J. Novotná (Eds.), Proceedings of SEMT ʻ99 (pp. 56–62). Prague: Charles University, Faculty of Education. Samková, L. (2017). Planning and conducting inquiry based mathematics course for future primary school teachers. In J. Novotná, H. Moraová (Eds.), Proceedings of SEMT ´17 (pp. 354–364). Prague: Charles University, Faculty of Education. SHIFTING PARADIGM IN PROBLEM SOLVING: ALGEBRAIC VERSUS ARITHMETIC THINKING Annie Savardand Elena Polotskaia Abstract Students’ difficulties in solving word problems continue to attract the attention of researchers all over the world (e.g. Lesh and Zawojewski, 2007; Westwood, 2011). In arithmetic, successfully solving word problems means applying the appropriate arithmetic operations. Researchers also highlight the importance of relational reasoning McGill University, Canada; e-mail: [email protected] Université du Québec en Outaouais, Canada; e-mail: [email protected] 482 and modelling (e.g. Bednarz and Janvier, 1993; Novotná, 2003; Riley, Greeno and Heller, 1984; Thevenot, 2010). The Relational Paradigm attributes the leading role to the knowledge of mathematical relationships, giving way to a new perspective on mathematical reasoning development and problem solving (Davydov, 1982, 2008). Keywords: Relational paradigm, word problem solving, additive structures Working within the Relational Paradigm for three years, we implemented a new approach to teaching additive word problem solving in elementary school. We tried simultaneously developing two ways to understand a problem: as sequence of events and as system of quantitative relationships. Our data shows: a) on average, the experimental group performed significantly better in problem solving than the control group; b) in the control group, there was a considerable lack of success in solving problems requiring relational reasoning – there was no such effect in the experimental group. In this workshop, we want to support our participants in shifting paradigm in problem solving by creating an awareness of the limits of the operational paradigm; presenting the relational paradigm in relation to algebraic and arithmetic thinking; and by theorizing the two paradigms in problem solving; algebraic versus arithmetic thinking. Contents: Operational Paradigm & Relational Paradigm Session Aims Activities 1 Creation of an awareness of the Captain game (representing some limits of the operational problems in team of four); paradigm Discussion on the role of representation in problem-solving, reasoning and learning. 2 Introduction of the relational Presentation of short videos; paradigm in relation to Discussion on algebraic and algebraic and arithmetic arithmetic thinking in solving thinking simple arithmetic problems. Developmental perspective. 3 Theorization of the two Group work on classification of paradigms in problem solving; excerpts of professional and algebraic versus arithmetic academic texts. thinking Discussion on theory-to-practice issues. References Bednarz, N., Janvier, B. (1993). The arithmetic-algebra transition in problem solving: Continuities and discontinuities. In Proceedings of the 15th Annual Meeting of the International Group for the Psychology of Mathematics Education (North American chapter PME-NA (Vol. 2, pp. 19–25). Asilomar, California. 483 Lesh, R., Zawojewski, J. (2007). Problem solving and modeling. In F. K. J. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (Vol. 1, pp. 763–787). IAP. Davydov, V. V. (1982). Psychological Characteristics of the formation of elementary mathematical operations in children. In T. P. Carpenter, J. M. Moser, T. A. Romberg (Eds.), Addition and subtraction: cognitive perspective (pp. 224–238). Lawrence Erlbaum Associates. Davydov, V. V. (2008). Problems of developmental instruction: a theoretical and experimental psychological study. Hauppauge, NY: Nova Science Publishers. Novotná, J. (2003). Étude de la résolution des « problèmes verbaux » dans l’enseignement des mathématiques. Strategies. Prague : Charles University. Riley, M. S., Greeno, J. G., Heller, J. L. (1984). Development of children’s problem- solving ability in arithmetic. In H. P. Ginsburg (Ed.), The development of mathematical thinking (pp. 153–196). Orlando, FL: Academic Press, Inc. Thevenot, C. (2010). Arithmetic word problem solving: Evidence for the construction of a mental model. Acta Psychologica, 133(1), 90–5. Westwood, P. (2011). The problem with problems: Potential difficulties in implementing problem-based learning as the core method in primary school mathematics. Australian Journal of Learning Difficulties, 16(1), 5–18. PRIMARY STUDENTS EXPLORE MATHEMATICS AT THE UNIVERSITY – ACTIVITIES FOR ALL Petra Scherer, Kristina Hähn, Christian Rütten and Stephanie Weskamp Abstract Out-of-school learning can serve as an interesting learning offer in addition to regular instruction. The project ‘Good Noses Mathematics’ (‘Mathe-Spürnasen’; www.uni- due.de/didmath/mathe-spuernasen) offers university visits for primary classes (grade 4; 9- to 10-year-old students) for exploring and doing mathematics. It is also connected to teacher education. The concept of the project covering design research as well as action research will be presented and will be illustrated with regard to different focal points. Keywords: Substantial learning environments, natural differentiation, out-of-school learning, heterogeneity The project ‘Good Noses Mathematics’ offers university visits for primary classes (grade 4) for experimenting and doing mathematics (cf. Rütten and Scherer 2015). Following a design-based research approach for this project substantial learning environments have been developed that represent central mathematical topics (cf. Wittmann 2001). The activities should not substitute regular instruction but University of Duisburg-Essen, Germany; e-mail: [email protected], [email protected], [email protected], [email protected] 484 should be an additional learning offer. The tasks and problems within a learning environment should offer multiple strategies, multiple levels and solutions; so it should be suitable for all students (cf. Krauthausen and Scherer 2010). The workshop will present the underlying concept and research and will discuss some of the project’s learning environments. The selected learning environments cover arithmetical as well as geometrical topics and context-related as well as context-free problems. The participants will have the opportunity to explore and reflect selected tasks and problems. Moreover, documents of fourth graders will be discussed to illustrate the potential for heterogeneous learning groups and to show different research perspectives. Session 1: Introduction of the Project and Learning Environment ‘The Square’ • workshop introduction and overview • theoretical background of the project, illustration of concept and objectives • general design and structure of the learning environments used in the project • presentation of the learning environment ‘The Square’ (introduction, tiding up art quadratically, tetrominos, squares in squares) • exploration of ‘Tiding up Art Quadratically’ (cf. Hähn & Scherer 2017) by the workshop participants • presentation students’ work from empirical studies; discussion of the relevance for inclusive mathematics education Session 2: Learning Environment ‘Fibonacci Sequence’ presentation of the learning environment ‘Fibonacci Sequence’ (introduction, number chains, golden spiral, building towers) exploration of ‘Building Towers’ by the workshop participants, discussion of different strategies discussion of ‘Building Towers’ as a learning environment for primary students as well as for teacher students reconstruction and comparison of students’ and teacher students’ strategies and perspectives of structuring (cf. English 1991; Rütten and Weskamp 2015) Session 3: Learning Environment ‘Pascal’s Triangle’ and Perspectives presentation of the learning environment ‘Pascal’s Triangle’ (introduction, arithmetical patterns, galton board, ways in Mannheim, cf. Weskamp 2015) exploration of ‘Ways in Mannheim’ by the workshop participants, discussion of strategies and categories analysing documents from empirical studies with regard to strategies of structuring and different types of argumentation, discussion of students’ work with respect to different levels of demands project perspectives and final discussion 485 References English, L. D. (1991). Young children’s combinatoric strategies. Educational Studies in Mathematics, 22(5), 451–474. Hähn, K., Scherer, P. (2017, in press). Kunst quadratisch aufräumen. Eine geometrische Lernumgebung im inklusiven Mathematikunterricht. To appear in U. Häsel-Weide, M. Nührenbörger (Eds.), Gemeinsam Mathematik lernen – mit allen Kindern rechnen. Frankfurt/M.: Arbeitskreis Grundschule. Krauthausen, G., Scherer, P. (2010). Ideas for Natural Differentiation in Primary Mathematics. Arithmetical environments. Rzeszów: University of Rzeszów. Rütten, C., Scherer, P. (2015). ‘Throwing dice’ versus ‘Passing the Pigs’ – Fourth- graders’ reasoning about probability. In J. Novotná, H. Moraová (Eds.), SEMT 2015. International Symposium Elementary Maths Teaching. Proceedings: Developing mathematical language and reasoning in elementary mathematics (pp. 284–292). Prague: Charles University, Faculty of Education. Rütten, C., Weskamp, S. (2015). Türme bauen – Eine kombinatorische Lernumgebung für Grundschulkinder und Lehramtsstudierende. In F. Caluori, H. Linneweber- Lammerskitten, C. Streit (Eds.), Beiträge zum Mathematikunterricht 2015 (Vol. 2, pp. 772–775). Münster: WTM. Weskamp, S. (2015). Einsatz von substanziellen Lernumgebungen in heterogenen Lerngruppen im Mathematikunterricht der Grundschule. In F. Caluori, H. Linneweber-Lammerskitten, C. Streit (Eds.), Beiträge zum Mathematikunterricht 2015 (Vol. 2, pp. 996–999). Münster: WTM. Wittmann, E. C. (2001). Developing mathematics education in a systemic process. Educational Studies in Mathematics, 48(1), 1–20. MATHEMATICAL TASKS AND MATHEMATICAL CREATIVITY: THEORY AND PRACTICE Dina Tirosh,Pessia Tsamir, Esther Levenson and Ruthi Barkai Abstract Today, it is common to see children playing on tablets and iPads. This workshop invites participants to explore applications that aim to promote young students’ knowledge of number concepts and repeating patterns and investigate the affordances and constraints of different applications. We will also consider the place of tablets and their applications in children’s mathematics education. Keywords: Tablet applications; number concepts; repeating patterns Tel Aviv University, Israel; e-mail: [email protected] Kibbutzim College of Education and Tel Aviv University, Israel; e-mail: [email protected], [email protected] 486 Introduction From an early age we see young children playing with their parents’ (or other adults’, or even their own!) tablets and iPads. Can we harness this relatively new technological tool, and the often free games that may be downloaded to these devices, to promote young children’s mathematics knowledge? This workshop will explore free downloadable mathematics applications (i.e., apps), appropriate for children ages 4-8. The overall aim of the workshop is to discuss with participants the affordances and constraints of using apps to develop young children’s mathematical knowledge. Participants are invited to bring their own tablets, iPads, or smartphones to each session of the workshop. The workshop The general goal of this workshop is to develop criteria for choosing apps and to discuss ways in which mathematics apps can be used in promoting early number and patterning knowledge. The apps we will explore make use of touch-screen technology, offering a multi-modal perspective on learning mathematics (Sinclair and de Freitas, 2014). Together, we will examine how this perspective can offer new insights and opportunities for learning number concepts and patterning skills. The first session will focus on number apps. During the early years, children develop number skills such as counting, enumerating, comparing quantities, ordering numbers and quantities, and addition. Participants will be asked to download apps on to their tablets, iPads, or smartphones, “play” with those apps, and analyse them in terms of their potential to promote the learning of number concepts. Specifically, we will ask: what do we look for in an app that makes it suitable for young students to learn number concepts? Participants will be asked to consider if all apps have the potential to promote learning number concepts. The second session will focus on repeating pattern tasks. Repeating patterns are patterns with a cyclical repetition of an identifiable 'unit of repeat' (Zazkis and Liljedahl, 2002). For example, the pattern ABBABBABB… has a minimal unit of repeat of length three. We will begin by exploring repeating pattern tasks discussed in previous studies (e.g., Papic, Mulligan and Mitchelmore, 2011) without the use of apps, and what elements of pattern knowledge these tasks promote. Participants will then be asked to download repeating pattern apps on to their tablets, iPads, or smartphones, “play” with those apps, and analyse them in terms of their potential to promote patterning skills. We will discuss the affordances and constraints of engaging students with patterning tasks without and with apps. Participants will be asked to consider how the touch-screen technology can affect children’s learning of number and pattern concepts. The third session will revolve around the question of using apps with and without adult intervention. Should we encourage teachers to incorporate this tool into their teaching? Should we encourage parents to download apps for their children with the aim of enhancing, or supporting, what their children are learning in more 487 formal environments? Together we will examine ways in which a significant adult (parent, teacher, older sister, uncle, etc.) may purposely and interactively engage a child with a mathematical app. Acknowledgement This workshop addresses research supported by The Israel Science Foundation (grant No. 1270/14). References Papic, M., Mulligan, J., Mitchelmore, M. (2011). Assessing the development of preschoolers' mathematical patterning. Journal for Research in Mathematics Education, 42(3), 237–269. Sinclair, N., de Freitas, E. (2014). The haptic nature of gesture: Rethinking gesture with new multitouch digital technologies. Gesture, 14(3), 351–374. Zazkis, R., Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49(3), 379–402. 488 P OSTERS EVOKING AND EXAMINING GEOMETRIC HABITS OF MIND WHILE CONSTRUCTING TETRAHEDRON BY MODULAR ORIGAMI Asuman Duatepe-Paksuand Burçak Boz-Yaman Abstract This study aimed to use modular origami to evoke and examine preservice teachers’ geometric habits of mind. Folding steps of one of the platonic object, tetrahedron, were analyzed by considering categories of geometric habits of mind. A sample of preservice teachers were guided to construct a tetrahedron by paper folding. While they were following the folding instructions, they were asked to respond to questions aimed to evoke geometric habits of mind. In the poster, we explain how we relate geometric habits of mind with the folding instruction of tetrahedron in detail. Keywords: modular origami, tetrahedron, geometric habits of mind, preservice teacher. In academic life, every course or experience should be used as an opportunity to help students develop general habits of mind (Cuoco, Goldenberg and Mark 1996). In their study, Cuoco et. al (1996) identified characteristics of habits of mind specific to mathematics. Among these characteristics, some are related to geometric approaches, such as the use of proportional reasoning and the use of several languages to describe things. On the other hand, by considering these characteristics of mathematical habits of mind, Driscoll, DiMatteo, Nikula and Pamukkale University, Turkey; email: [email protected] 489 Egan (2007) constructed a framework highlighting productive mental habits about geometric thinking called Geometric Habits of Mind (GHOM) based on four-year project titled “Fostering Geometric Thinking in the Middle Grades”. They defined four categories of habits of mind; (1) Reasoning with Relationship, (2) Generalizing Geometric Ideas, (3) Investigating Invariants, and (4)Balancing Exploration and Reflection. The first one is about finding relationships like congruency, similarity or parallelism within geometric figures, and thinking about how these relationships help understanding of concepts. Generalizing Geometric Ideas is about understanding and describing the geometric phenomena regarding whether it occurred every time or always. The third habit is about understanding and identifying some properties that stay the same while some others change. Finally, using multiple ways to understand and solve the geometric problem and think about the learned concept periodically is considered as the last habit. Using manipulative materials in teaching geometry is helpful for effective learning of abstract concepts and relationships. Driscoll et al. (2007) present the power of paper-folding exercises on geometric thinking in the “Fostering Geometric Thinking Toolkit”. So, origami, which is the art of paper folding, can be used to identify and trigger geometric thinking and geometric habits of mind. Moreover, according to research studies, origami helps students to comprehend geometric basics and principles (e.g. Boakes, 2009; Pearl, 2008; Sze, 2005). Therefore, in the current study, we aimed to use a specific modular origami to evoke and examine students’ geometric habits of mind. This study is part of the bigger project which aimed to examine preservice teachers’ geometric habits of minds while they were constructing platonic objects by modular origami. For the examination of student teachers’ geometric habits of mind, tetrahedron was the preferred object to construct by using origami because it is easy to fold and also by questioning each folding step all GHOMs could be triggered. Tetrahedron is one of the five platonic solids with all faces being congruent equilateral triangles and having other geometrical concepts that could be observed while constructing it by using origami instructions. The tetrahedron constructed here has two symmetrical modules and folding instructions involves ten steps. The folding instructions of the tetrahedron were implemented in a 2-hour classroom session with seven (5 female, 2 male) preservice teachers. An origami paper and worksheet were provided to each preservice teacher at the beginning of the implementation. The worksheet involves questions related to each folding steps. The questions were generally focused on the shape the participants got in each folding step and mainly on the concepts of triangle, quadrilateral, length, area, volume and transformations. The aim of these questions were to evoke geometric habits of mind. We had questions for every geometric habit of mind. While the preservice teachers were following the instructions, they were asked to answer each question in the worksheet. Each folding step will be described in detail in the poster. We will explain how we 490 relate geometric habits of mind with the folding instructions of the tetrahedron. Moreover, we will give some observations from the implementation with reflections regarding future studies. Acknowledgement This study was supported by PAU-ADEP. References Boakes, N. (2009). Origami-Mathematics Lessons: Researching its Impact and Influence on Mathematical Knowledge and Spatial Ability of Students. Retrieved from: http://math.unipa.it/~grim/21_project/Boakes69-73.pdf on 12.5.2017. Cuoco, A. A., Goldenberg, E. P., Mark. J. (1996). Habits of mind: An organizing principle for mathematics curriculum. Journal of Mathematical Behavior, 15(4), 375–402. Driscoll, M., DiMatteo, R. W., Nikula, J. E., Egan, M. (2007). Fostering geometric thinking: A guide for teachers grades 5-10. Portsmouth, NH: Heinemann. Pearl, B. (1994). Math in Motion: Origami in the Classroom (K-8). Langhorne, PA: Math in Motion, Incorporated. Sze, S. (2005). Math and mind mapping: Origami construction. Dunleavy: Niagara University. (ERIC Document Reproduction Service No. ED490352.) PROBLEM POSING IN PROSPECTIVE PRIMARY SCHOOL TEACHERS’ EDUCATION: CASE OF CONCEPT CARTOONS Alena Hošpesováand Marie Tichá Abstract The poster will present creation of Concept Cartoons as a possible way to support the development and refinement of prospective primary teachers’ pedagogical content knowledge. Keywords: Problem posing, Concept Cartoons, PCK of prospective teachers Background This poster is a follow up to of our work connected with problem posing and its influence on the creation of pedagogical content knowledge (PCK) of prospective teachers (e.g., Samková, Tichá and Hošpesová 2015). Mathematics is taught primarily through problem solving. In prospective teachers´ education, problem posing seems to be a beneficial learning/teaching method to lead to (a) a deeper understanding of mathematical content, (b) the creation of a basis of PCK and, in particular, (c) consciousness of necessity to University of South Bohemia, Czech Republic; e-mail: [email protected] 491 hold precise ideas. Based on several investigations carried out with prospective teachers, we have found that problem posing has a significant educational, motivational, and diagnostic role in teacher education (Hošpesová and Tichá, 2015). It highlights misconceptions and motivates prospective teachers to eliminate shortcomings. As one prospective teacher explained: “... [when] posing a problem we have to think in a completely different way, which leads to a greater understanding, and consequently results in easier problem solving.” In this study, we asked prospective primary teachers to create a “Concept Cartoon” (CC). CCs are a special type of tasks in which several correct as well as incorrect solutions of a problem or statements related to the problem are given. (more details on the types of CCs can be found in Samková and Hošpesová, 2015). In primary school teachers’ education, we use this tool in the framing of both solving and posing problems to stress pedagogical view on the problems and necessity to used valid arguments in its analysis. In both cases a good knowledge of mathematical content is necessary because the solver has to explain which solutions/statements are correct/wrong and why. Posing of CCs requires awareness of possible misunderstandings and misconceptions of pupils and highlights the importance of experience. Methodology of the study In this study, 47 prospective teachers created problems in the form of CCs and made their didactical analysis (i.e., what is the objective of solution of the task and estimation of the correct and incorrect (but plausible) solution of the problem). The study has a qualitative design as we aim to identify unknown phenomena in the problems. The created CCs were coded in respect to (a) type of problem, (b) mathematical content, and (c) proper formulation of statements in bubbles. We focused on answering the question: What types of CCs prospective teachers posed? Selected findings The prospective teachers who posed an open problem that allowed for more correct and incorrect solving procedures/solutions succeeded in the creation of a CC (for example, the following problem for 2nd graders: How many children can share 12 bonbons so that no bonbon is left?) In some cases, the prospective teachers described the real situation that was the source of the problem (for example, shopping and ski resort). They did not formulate the question, but there were several statements for which correctness in the given environment could be decided. Several of the prospective teachers met the requirement of formulating several erroneous solutions by posing an arithmetical task based on the order of operations (for example: calculate 42 – (7 + 8 – 6) 3 + 9), or a word problem 492 with a complicated structure. In the wrong solutions, a number or a step of the solution was skipped or calculated incorrectly. Formulation of the learning objective of concrete CC showed that the prospective teachers were mostly able to write them in the “language of the intended pupil’s performance”; but they sometimes formulated the learning objectives in a general manner (e.g., support of logical thinking, recognizing arithmetical operations, and finding a plan of solution), or failed to realize a thought-provoking idea hidden in the problem. Acknowledgement The contribution was supported by project GAJU 121/2016/S and by RVO 67985840. References Samková, L., Tichá, M., Hošpesová, A. (2015). Error patterns in computation in Concept Cartoons. In J. Novotná, H. Moraová (Eds.), International Symposium on Elementary Mathematics Teaching (SEMT) (pp. 390–391). Prague: Charles University, Faculty of Education. Samková, L., Hošpesová, A. (2015). Using Concept Cartoons to investigate future teachers' knowledge. In K. Krainer, N. Vondrová (Eds.), CERME 9 – Ninth Congress of the European Society for Research in Mathematics Education. Retrieved from https://hal.archives-ouvertes.fr/hal-01289873/document. Hošpesová, A., Tichá, M. (2015). Problem Posing in Primary School Teacher Training. In F. M. Singer, N. Ellerton, J. Cai (Eds.), Mathematical Problem Posing: From Research to Effective Practice (pp. 433–447). New York: Springer. PRIMARY SCHOOL MATHEMATICAL KANGAROO: MISTAKES IN SELECTED WORD PROBLEMS Eva Nováková Abstract In our poster we analyze mistakes in two Mathematical kangaroo (closed) word problems. In the Czech environment, closed word problems are rather scarce; primary school pupils often do not have enough experience with them. We believe that the quantitative analysis of the choice of incorrect distractors enables us to estimate the reasons of pupils’ mistakes. Keywords: Mathematical Kangaroo, primary school mathematics, word problems, mistakes in pupils’ solutions International Math Kangaroo competition is designed for pupils aged 8-18 in six categories. The competition does not have multiple rounds – on the same day all Masaryk University in Brno, Czech Republic; e-mail: [email protected] 493 participants solve the same 24 tasks in appropriate age categories. In the Czech Republic, the competition has taken place since 1995. Tasks of Écolièr 2015 and their solutions by Czech pupils aged 9-11 years were analysed by quantitative methods in a broadly designed research (Nováková, 2016). Each year the Kangaroo tasks are selected by a group of experts from the countries which take part in the contest. They are sorted out into three levels. All tasks are designed as multiple-choice one, each with five possible answers. Our research was performed on the sample of 430 respondents, pupils of 4th and 5th grade of primary school. We chose two “real-life” word problems (Palm, 2008). 1) There are 5 ladybirds (see g.). Two ladybirds are friends with each other if the numbers of spots that they have differ exactly by 1. On Kangaroo Day each of the ladybirds was sent to each of her friends one SMS greeting. How many SMS greetings were sent? A) 2 B) 4 C) 6 D) 8 E) 9 Task with unreal, yet for pupils "realistic" semantic background (non-real word problem) depicted by means of a picture, was assigned medium level of difficulty (4 points). Successful solution requires linguistic understanding, reader skills, as well as thorough "reading" of a picture: there are two ladybirds with three spots (1 + 2, 2 + 1). 25.1% of pupils marked the correct answer - answer "C", 16.2% of pupils did not provide any answer for the task at all. Most frequent distractor was "B", chosen by 27.9% pupils who obviously did not realize that whereas there were two ladybirds with three spots, the SMS greetings were sent in both directions. The distractor "A" was chosen by 16.8% pupils who did not realise that whereas there were two ladybirds with three spots, there was only one SMS greeting in each direction. 2) Anna, Berta, Charlie, David and Elisa were baking cookies during the weekend. Over the whole weekend Anna made 24 cookies, Berta 25, Charlie 26, David 27 and Elisa 28. After the whole weekend one of them had twice as many cookies as after Saturday, one 3 times, one 4 times, one 5 times and one 6 times as many. Who baked the most cookies on Saturday? (A) Anna (B) Berta (C) Charlie (D) David (E) Elisa A realistic task, a real-life word problem, called “application” (Tom, 1999). Successful solution requires linguistic understanding and ability to apply real-life experience. In the test, it was the last task from the most difficult ones. 494 Only 10.9% of pupils marked the correct answer "C", representing the lowest success rate of all tasks in the test. 26.0% of pupils did not provide any answer, representing the highest share of all tasks in the test. Concerning distractors, "E" significantly prevails, with 56.9% of pupils choosing it – since Eliška baked the most cookies over the weekend, they deduced (incorrectly) that she baked the most also on Saturday, without taking into account other conditions. Our poster presents histograms of choices of distractors in both tasks and samples of pupils' solution records. Research data seem to confirm the expectation that the analysis of chosen distractors is useful for primary school practise (Hejný, 2013), because it makes the estimate of reason of pupils' failure to a large extent possible. References Hejný, M. et al. (2013). Čtenářské, matematické a přírodovědné úlohy pro první stupeň základního vzdělávání: náměty pro rozvoj kompetencí žáků na základě zjištění šetření TIMSS a PIRLS 2011. Praha: ČŠI. Nováková, E. (2016). Analýza úloh ze soutěže Matematický klokan a jejich řešení žáky základní školy. Brno: Masarykova univerzita. Palm, T. (2008). Impact of authenticity on sense making in word problem solving. Educational Studies in Mathematics, 67(1), 37–58. Toom, A. (1999). Word problems: Applications or mental manipulatives. For Learning of Mathematics, 19(1), 36–38. http://www.mathkang.org/ http://www.aksf.org/ LEARNING ENVIRONMENTS FOR DIVERSE LEARNERS – SUBSTANTIAL MATHEMATICS FOR ALL Petra Scherer, Kristina Hähn, Christian Rütten and Stephanie Weskamp Abstract Out-of-school learning can serve as an interesting learning offer in addition to regular instruction. The project ‘Good Noses Mathematics’ (‘Mathe-Spürnasen’; www.uni- due.de/didmath/mathe-spuernasen) offers university visits for primary classes (grade 4; 9- to 10-year-old students) for exploring and doing mathematics and is also connected to teacher education. The con-cept of the project covering design research as well as action research will be presented and will be illustrated with two mathematical topics “The Cube” and “Fibonacci Sequence”. Keywords: substantial learning environments, natural differentiation, out-of-school learning, heterogeneity University of Duisburg-Essen, Germany; e-mail: [email protected], [email protected], [email protected], [email protected] 495 The Project ‘Good Noses Mathematics’ – the Concept The project aims at offering learning situations, tasks and problems that should be suitable for all students in the sense of a natural differentiation enabling students to use multiple strategies, multiple levels and finding individual solutions. Moreover, out-of-school learning could offer specific learning experiences and might increase the interest in mathematics. Following a design-based research approach substantial learning environments have been developed that represent central mathematical topics in the sense of fundamental ideas. The focal point of this theory-oriented approach is the linking of mathematical topics and the connection of content related and process related competences. Connections should be made explicit between different inner- mathematical topics as well as to context-related topics (everyday experiences, historical developments etc.). The topics developed so far are: Platonic Solids, Fibonacci Sequence, Pascal’s Triangle, The Square, The Cube, and The Circle. The learning environments have been tried out, analyzed, and revised (cf. Baltes et al. 2014). According to the action research approach the concrete learners’ activities and learning processes were analyzed within different studies: reconstructing concrete strategies (cf. Rütten and Weskamp 2015), levels of argumentations (cf. Rütten and Scherer 2015), interactions in inclusive settings (cf. Hähn 2017), and analyzing different topics in heterogeneous groups (cf. Weskamp 2015), but also with respect to low achievers. Another central objective is the implementation in teacher education by different formats (cf. Rütten and Weskamp 2015). Since 2012 (project start) more than 50 classes (about 1250 students) visited the university to explore one of the topics in small groups under different perspectives. The learning environments were also tried out in schools as well as in teacher education. Two projects will be presented in the following more detailed. “The Cube” – reconstructing stochastic reasoning The learning environment should enable students to understand stochastic phenomena when throwing the dice by the geometrical characteristics of the cube. Hence, two random generators for explorations in elementary stochastic are compared and students’ explanations of occurring phenomena and their arguments were analyzed. The learning environment could initiate the construction of a conceptual layer that links geometrical and stochastic topics. “Fibonacci Sequence” – reconstructing combinatorial reasoning After an introductory unit dealing with the concrete numbers and construction of the Fibonacci Sequence according to the rabbits problem, a follow-up unit “Building Towers”, which was designed for primary students as well as for teacher students, focuses on the relations within the sequence. When 496 reconstructing the strategies, horizontal and vertical perspectives of structuring became obvious. It could be shown that there exist similarities but also differences between students and teacher students when solving one and the same problem. References Baltes, U., Rütten, C., Scherer, P., Weskamp, S. (2014). Mathe-Spürnasen – Grundschulklassen experimentieren an der Universität. In J. Roth, J. Ames (Eds.), Beiträge zum Mathematikunterricht 2014 (pp. 121–124). Münster: WTM-Verlag. Hähn, K. (2017). Analyses of learning situations in inclusive settings: a coexisting learning situation in a geometrical learning environment. In J. Novotná, H. Moraová (Eds.), SEMT 2017 Proceedings: Equity and Diversity (pp. 187–196). Prague: Charles University. Faculty of Education. Rütten, C., Scherer, P. (2015). ‘Throwing dice’ versus ‘Passing the Pigs’ – Fourth- graders’ reasoning about probability. In J. Novotná, H. Moraová (Eds.), SEMT 2015 Proceedings: Developing mathematical language and reasoning in elementary mathematics (pp. 284–292). Prague: Charles University, Faculty of Education. Rütten, C., Weskamp, S. (2015). Türme bauen – Eine kombinatorische Lernumgebung für Grundschulkinder und Lehramtsstudierende. In F. Caluori, H. Linneweber- Lammerskitten, C. Streit (Eds.), Beiträge zum Mathematikunterricht 2015 (Vol. 2, pp. 772–775). Münster: WTM. Weskamp, S. (2015). Einsatz von substanziellen Lernumgebungen in heterogenen Lerngruppen im Mathematikunterricht der Grundschule. In F. Caluori, H. Linneweber-Lammerskitten, C. Streit (Eds.), Beiträge zum Mathematikunterricht 2015 (Vol. 2, pp. 996–999). Münster: WTM. A TRANSDISCIPLINARY APPROACH TO ELEMENTARY MATH LITERACY LEARNING THROUGH VISUAL ART Barbara Ann Temple, Kathryn Bentley, Natalie Blundelland David K. Pugalee Abstract Design research is a perfect model to develop curriculum due to the iterative nature of the process. This poster reports on the first cycle in this research process: design. Specifically, participants learn about design research in mathematics education, are active participants in providing feedback on a curriculum idea by considering initial development of math activities (the first phase in the research model), and connect these processes to the work of developing curriculum for diverse audiences. Keywords: Math literacy, transdisciplinary learning, visual art Arts and Science Council Institution, USA; e-mail: [email protected], [email protected], [email protected] University of North Carolina at Charlotte, USA, e-mail: [email protected] 497 Introduction Developing relevant math curriculum that focuses on infusing scientific inquiry and visual art provides alternate ways of promoting math learning. Design research “goes beyond simply observing and actually involves systematically engineering contexts in ways that allow [researchers] to improve and generate evidence-based claims about learning” (Barab and Squire, 2004, p. 2). The efforts of the researcher to “systematically change” the curriculum is a positive choice, providing better results with every iteration of the experiment (Barab and Squire, 2004). Collins, Beranek and Newman (1990) describe the importance of “flexible design revision” as one of eight major components of design research. This pilot design research study proposes an instructional intervention in second grade math lessons using scientific inquiry, documentary film clips, and visual thinking strategies. Nature Matters (Blunk and Temple, 2016) is a documentary film that illustrates how using the natural world engages and motivates as well as personalizes learning for diverse learners. Based on the successful transdisciplinary learning evidenced in this film, specific scientific-based film clips are used in teaching math concepts. Aim of the research What happens when second grade students learn math concepts through scientific inquiry and visual art? Sample Learning Module for Measurement Examining the scientific context of waterfalls through visual art, students explore the following math standards: Students think like a mathematician, represent and solve problems involving addition and subtraction, represent and interpret data, understand place value, & measure, and estimate lengths in standard units. To achieve these goals, students engage in the following: make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, model with mathematics, use appropriate tools strategically, attend to precision, & look for and make use of structure. Sample Pre-Test: Students view two waterfalls in the film from different locations in the US. Students are asked to study these pictures and answer the following questions. 1) Which waterfall do you think is taller? 2) Estimate how tall you think that waterfall is. 3) Describe the different waterfalls. What makes you think that? What else do the film clips tell you about the waterfalls? What makes you think that? What other mathematics concepts are you thinking about the waterfalls that may or may not be visible in the film clips? The level of student competence of the particular contexts and mathematical concepts are assessed after the learning segment (arts-based) intervention to determine two data points: 1) How has student knowledge and understanding of 498 particular contexts and math concepts changed since the intervention strategies were implemented? 2) What particular impact did the visual art-based texts have on student knowledge and understanding of particular contexts and math concepts? Sample Post-Test: Students analyze pictures of four different waterfalls. They are asked to study the four pictures and perform the following tasks: 1) Estimate the heights of each waterfall. 2) Create a graph showing the waterfalls from shortest to tallest. 3) How wide are the waterfalls? 3) How are these waterfalls like the first two waterfalls in the film clips? How are they different? What makes you think that? References Barab, S., Squire, K. (2004). Design-based research: Putting a stake in the ground. The Journal of the Learning Sciences, 13(1), 1–14. Retrieved from http://www.gerrystahl.net/teaching/winter12/reading3a.pdf. Blunk, C., Temple, B.A. (2016). Nature Matters. Charlotte, NC.: Through a Glass Productions. Retrieved from http://naturemattersfilm.com. Collins, A., Beranek, B., Newman, D. (1990). Toward a design science in education. In E. Scalon, T. O´Shea (Eds.), New Directions in Educational Technology (pp. 15-22). Berlin, Heidelberg: Springer. 499 SEMT ‘17 was organised in cooperation with Klub přátel didaktiky matematiky na Univerzitě Karlově (Club of Friends of Didactics of Mathematics at Charles University) Univerzita Karlova, Pedagogická fakulta Katedra matematiky a didaktiky matematiky International Symposium Elementary Maths Teaching SEMT ‘17 Proceedings Editors: Jarmila Novotná and Hana Moraová Grafická úprava: Jarmila Novotná a Hana Moraová Tisk: Retida, spol. s r.o., Dělnická 54, 170 00 Praha 7 1. vydání ISBN 978-80-7290-955-1