Conference Proceedings

Contents

Foreword v Keynote papers Professor David Clarke 3 Speaking in and about mathematics classrooms internationally: The technical vocabulary of students and teachers. Mr Phil Daro 8 Standards, what’s the difference?: A view from inside the development of the Common Core State Standards in the occasionally United States. Professor Kaye Stacey 17 Mathematics teaching and learning to reach beyond the basics. Professor Paul Ernest 21 The social outcomes of school mathematics: Standard, unintended or visionary?

Concurrent papers Professor Robyn Jorgenson 27 Issues of social equity in access and success in mathematics learning for Indigenous students. Professor Tom Lowrie 31 Primary students’ decoding mathematics tasks: The role of spatial reasoning. Professor John Pegg 35 Promoting the acquisition of higher order skills and understandings in primary and secondary mathematics. Associate Professor Rosemary Callingham 39 Mathematics assessment in primary classrooms: Making it count. Dr David Leigh-Lancaster 43 The case of technology in senior secondary mathematics: Curriculum and assessment congruence? Associate Professor Joanne Mulligan 47 Reconceptualising early mathematics learning. Professor Peter Sullivan 53 Learning about selecting classroom tasks and structuring mathematics lessons from students. Mr Ross Turner 56 Identifying cognitive processes important to mathematics learning but often overlooked. Associate Professor Robert Reeve 62 Using mental representations of space when words are unavailable: Studies of enumeration and arithmetic in Indigenous Australia. Professor Merrilyn Goos 67 Using technology to support effective mathematics teaching and learning: What counts? Dr Shelley Dole 71 Making connections to the big ideas in mathematics: Promoting proportional reasoning. Dr Sue Thomson 75 Mathematics learning: What TIMSS and PISA can tell us about what counts for all Australian students.

Poster presentations 81

Conference program 85

Crown Conference Centre map and floorplan 89

Conference delegates 93 Research Conference 2010 Planning Committee Professor Geoff Masters CEO, Conference Convenor, ACER Dr John Ainley Deputy CEO and Research Director National and International Surveys, ACER Professor Kaye Stacey Professor Mathematics Education, University of Melbourne Dr David Leigh-Lancaster Mathematics Manager, Victorian Curriculum and Assessment Authority Mr Ross Turner Principal Research Fellow, ACER Ms Kerry-Anne Hoad Director ACER Institute, ACER Ms Lynda Rosman Manager Programs and Projects, ACER Institute, ACER

Copyright © 2010 Australian Council for Educational Research 19 Prospect Hill Road Camberwell VIC 3124 AUSTRALIA www.acer.edu.au ISBN 978-0-86431-958-6 Design and layout by Stacey Zass of Page 12 and ACER Project Publishing Editing by Carolyn Glascodine and Kerry-Anne Hoad Printed by Print Impressions

Research Conference 2010 iv Foreword

Geoff Masters Australian Council for Educational Research Research Conference 2010 is the fifteenth national Research Conference. Through our research conferences, ACER provides significant opportunities at the national Geoff Masters is Chief Executive Officer and a level for reviewing current research-based knowledge in key areas of educational member of the Board of the Australian Council policy and practice. A primary goal of these conferences is to inform educational for Educational Research (ACER) – roles he has policy and practice. held since 1998. He has a PhD in educational measurement from Research Conference 2010 brings together key researchers, policy makers and the University of Chicago and has published teachers from a broad range of educational contexts from around Australia and widely in the fields of educational assessment and overseas. The conference will explore the important theme of teaching and learning research. mathematics. The conference will draw together research-based knowledge about Professor Masters has served on a range of effective teaching and learning of mathematics and explore approaches to teaching bodies, including terms as founding President of that develop the mathematical proficiency of students and catch their interest in the Asia-Pacific Educational Research Association; President of the Australian College of Educators; mathematics from the early years through to post-compulsory education. Chair of the Technical Advisory Committee for We are sure that the papers and discussions from this research conference will the International Association for the Evaluation of Educational Achievement (IEA); Chair of make a major contribution to the national and international literature and debate on the Technical Advisory Group for the OECD’s key issues related to the effective teaching and learning of mathematics. Programme for International Student Assessment (PISA); member of the Business Council of We welcome you to Research Conference 2010, and encourage you to engage Australia’s Education, Skills and Innovation in conversation with other participants, and to reflect on the research and its Taskforce; member of the Australian National connections to policy and practice. Commission for UNESCO (and Chair of the Commission’s Education Network); and member of the International Baccalaureate Research Committee. He has undertaken a number of reviews for governments, including a review of examination procedures in the New South Wales Higher School Certificate (2002); an investigation of options for the introduction of an Australian Certificate of Education (2005); a national review Professor Geoff N Masters of options for reporting and comparing school Chief Executive Officer, ACER performances (2008); and a review of strategies for improving literacy, numeracy and science learning in Queensland primary schools (2009). Professor Masters was the recipient of the Australian College of Educators’ 2009 College Medal in recognition of his contributions to education.

vii Research Conference 2010 viii Keynote papers

Speaking in and about mathematics classrooms internationally: The technical vocabulary of students and teachers

Abstract Presentation summary This presentation takes patterns of Classroom discourse (and professional language use as the entry point for discourse about classrooms) is a form the consideration of discourses in and of social performance undertaken about the mathematics classroom. within affordances and constraints These patterns of language take the that can be both cultural and linguistic. form of discourses performed within The nature of these discourses, as mathematics classrooms around the performed in mathematics classrooms, world and among the international provides a key indicator of pedagogical David Clarke mathematics education community principles underlying classroom practice University of Melbourne about the mathematics classroom. and the theories of learning on which Cross-cultural comparisons reveal these principles are implicitly founded. David Clarke is a Professor of Education and how discourses in and about the The discourses about mathematics the Director of the International Centre for Classroom Research (ICCR) at the University mathematics classroom have developed classrooms give expression to these of Melbourne. Over the last 15 years, Professor in different cultures. Research is used pedagogical principles sometimes Clarke’s research activity has centred on capturing to explore the role of spoken language explicitly and sometimes through the complexity of classroom practice through a in mathematics classrooms situated embedding privileged forms of program of international video-based classroom research. The ICCR is unique in the facilities in Asian and Western countries. In practice in the naming conventions by it offers for the manipulation and analysis of conceptualising effective learning, which the mathematics classroom is classroom data and provides the focus for researchers, teachers and curriculum described. From research undertaken in collaborative activities among researchers from developers need to locate proficiency classrooms situated in different cultures, China, the Czech Republic, Germany, Hong Kong, Israel, Japan, Korea, New Zealand, Norway, the with mathematical language within it appears that both mathematical Philippines, Portugal, Singapore, South Africa, their framework of valued learning discourse and professional discourse Sweden, the United Kingdom and the United outcomes. Further, different cultures, take different forms and are differently States of America. Under Professor Clarke’s employing different languages, have valued in different communities. This direction the ICCR has developed a system for web-mediated, secure, high-speed data chosen to name and therefore privilege presentation draws on and connects entry, retrieval and analysis on an international different classroom activities. Research research into these two discourses. scale (videoPortal). Other significant research is reported into how language is has addressed teacher professional learning, and might be used to describe the The spoken metacognition, problem-based learning, and events of mathematics classrooms assessment (particularly the use of open- mathematics study ended tasks for assessment and instruction in in different cultures. Research and mathematics). Current research activities involve theorising undertaken in and about Research was conducted into the multi-theoretic research designs, cross-cultural those mathematics classrooms must situated use of mathematical language analyses and the challenge of research synthesis be sensitive to the participants’ in selected mathematics classrooms in education. Professor Clarke has over 120 internationally. The major concern research publications, including 8 books, 35 book conceptions of classroom practice, chapters, 41 refereed journal articles, and 39 as performed in classroom discourse of this study was to document the refereed papers in conference proceedings. and as expressed in the professional opportunity provided to students in discourse of mathematics educators in each classroom for the oral articulation those communities. of the relatively sophisticated mathematical terms that formed the conceptual content of the lesson and to distinguish one classroom from another according to how such student mathematical orality was afforded or constrained in both public and private classroom contexts. This research was undertaken as a sub- project within the Learner’s Perspective Study, in which data generation used

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 3 three video cameras, supplemented class. All three types of oral interactions whole class or teacher–student by the reconstructive accounts of were transcribed, although type (iii) interactions, both of which were classroom participants obtained in interactions could only be documented considered to be public from the point post-lesson video-stimulated interviews. for the selected focus students in each of view of the student. The complete research design has lesson. Where necessary, all transcripts The average number of public been detailed elsewhere (Clarke, were then translated into English. utterances per lesson provides an 2006). For the analysis reported here, The analysis determined the number indication of the public oral interactivity the essential details relate to the of utterances occurring in whole class of a particular classroom. Figure 1 standardisation of transcription and and teacher–student interactions in a distinguishes utterances by the teacher translation procedures. Since three sequence of five lessons from each of (light grey), individual students (black) video records were generated for the classrooms studied (a total of 105 and choral responses by the class each lesson (teacher camera, student lessons from 21 classrooms in Berlin, (e.g. in Seoul) or a group of students camera and whole class camera), it was Hong Kong, Melbourne, San Diego, (e.g. in San Diego) (dark grey). Any possible to transcribe three different Seoul, Shanghai, Singapore and Tokyo), teacher-elicited, public utterance types of oral interactions: (i) whole together with the frequency of public spoken simultaneously by a group class interactions, involving utterances statement of mathematical terms and, of students (most commonly by a for which the audience was all or most in a separate analysis, the number of majority of the class) was designated of the class, including the teacher; utterances and spoken mathematical a ‘choral response’. Lesson length (ii) teacher–student interactions, terms in the context of student–student varied between 40 and 45 minutes and involving utterances exchanged (rather than public) interactions. An the number of utterances has been between the teacher and any student utterance was taken to be a single, standardised to 45 minutes. Each bar or student group, not intended to be continuous oral communication of in Figure 1 represents the average over audible to the whole class; and (iii) any length by an individual or group five lessons for that classroom. Figure 2 student–student interactions, involving (choral). Private student–student shows the number of publicly spoken utterances between students, not interactions were distinguished from mathematical terms (as defined earlier) intended to be audible to the whole

700 Teacher 600 Choral Student 500

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Seoul 1Seoul 2Seoul 3 Tokyo 1Tokyo 2Tokyo 3Berlin 1Berlin 2 ShanghaiShanghai 1 Shanghai 2 3 SingaporeSingapore 2 3 Hong KongHong 1 KongHong 2 Kong 3 San DiegoSan 1DiegoMelbourne 2 Melbourne 1 Melbourne 2 3

Figure 1: Average number of public utterances per lesson in whole class and teacher–student interactions (public oral interactivity)

Research Conference 2010 4 350 Teacher 300 Choral Student 250

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Average number of key mathematical terms per lesson of key number Average 0

Seoul 1Seoul 2Seoul 3 Tokyo 1Tokyo 2Tokyo 3Berlin 1Berlin 2 ShanghaiShanghai 1 Shanghai 2 3 SingaporeSingapore 2 3 Hong KongHong 1 KongHong 2 Kong 3 San DiegoSan 1DiegoMelbourne 2 Melbourne 1 Melbourne 2 3

Figure 2: Average number of key mathematical terms per lesson in public utterances (whole class and teacher–student interactions) (mathematical orality) per lesson, averaged over five lessons for both public and private Oral spoken mathematics (e.g. Seoul 1, for each classroom. Interactivity and Mathematical Orality 2 and 3). On the other hand, if the and expressed as per focus student teacher subscribes to the view that The classrooms studied can be also per lesson, effectively averaged over student understanding resides in the distinguished by the use made of the the spoken contributions of at least capacity to both justify and explain choral recitation of mathematical terms 10 students per classroom. Detailed the use of mathematical procedures, or phrases by the class. This recitation findings are reported elsewhere (e.g. in addition to technical proficiency included both choral response to Clarke & Xu, 2008). in carrying out those procedures in a teacher question and the reading solving mathematics problems, then the aloud of text presented on the board It is clear that some mathematics nurturing of student proficiency in the or in the textbook. The most striking teachers valued spoken mathematics spoken language of mathematics will be difference between first and second and some did not. Some teachers prioritised, both for its own sake as a stage analyses (Figures 1 and 2) was orchestrated the public rehearsal of valued skill and also because of the key the reversal of the order of classrooms spoken mathematics, but discouraged role that language plays in the process according to whether one considers private (student-student) talk (e.g. whereby knowledge is constructed. public oral interactivity (Stage One) or Shanghai 1, 2 and 3), while other Despite the frequently assumed mathematical orality (Stage Two). teachers utilised student–student similarities of practice in classrooms mathematical conversations as a key In considering student-student characterised as Asian, differences instructional tool (e.g. San Diego utterances, only focus students’ ‘private’ in the nature of students’ publicly 2 and Melbourne 1). If the goal of utterances could be recorded. The spoken mathematics in classrooms in classroom mathematical activity classrooms in Shanghai and Seoul were Seoul, Hong Kong, Shanghai, Singapore was fluency and accuracy in the use characterised by the almost complete and Tokyo were non-trivial and of written mathematics, then the absence of this form of interaction. suggest different instructional theories teacher may accord little priority to Frequency counts were constructed underlying classroom practice. students developing any fluency in

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 5 The international of different classroom activities in order unnamed activity will be absent from classroom Lexicon to stimulate participants’ recall of the any catalogue of desirable teacher largest possible number of pedagogical actions and consequently denied Project terms. specific promotion in any program of mathematics teacher education. The Lexicon Project is based on It might be expected that the Actions considered as essential the premise that the international internationalisation of the mathematics components of the mathematics dominance of English runs the risk education community would afford an teacher’s repertoire in one country – of denying researchers, theoreticians expansive re-conception of the practice for example, mise en commun (France), and practitioners access to many of mathematics teaching reflective pudian (China), ucitelskᡠozvenaˇ (Czech sophisticated, technical classroom- of the wide diversity of classroom Republic) or matome (Japan) – may related terms in languages other practices found in mathematics be entirely absent from any catalogue than English, which might otherwise classrooms around the world. Ironically, of accomplished teaching practices contribute significantly to our internationalisation has strengthened the in English. Yet each of these same understanding of classroom instruction establishment of English as the lingua pedagogical activities may well reward and learning. The intended product of franca of the international mathematics independent research, offering novel this research is a ‘Classroom Lexicon’ of education community and thereby instructional and learning opportunities such terms, with English definitions and restricted international use of some of (see, for example, Shimizu, 2008). descriptive detail, supported by video the subtle and sophisticated constructs exemplars. Such a video-illustrated by which mathematics teachers and Mise en commun – a whole-class lexicon has the potential to be a major teacher educators in non-English activity in which the teacher elicits resource in teacher pre-service and speaking countries would describe and student solutions for the purpose in-service programs and to offer new evaluate the practices occurring in their of drawing on the contrasting insights to classroom researchers. The mathematics classrooms. approaches to synthesise and lexicon is produced by face-to-face highlight targeted key concepts. negotiation with researchers from If an activity is named, it can be more than 10 countries, through the recognised and it becomes possible Pudian – an introductory activity in collaborative coding of a selection of to ask ‘how well is it done?’ and ‘how which the teacher elicits student video material of mathematics lessons might it be done better?’ Not only is prior knowledge and experience drawn from classrooms in Cesky an unnamed activity less accessible for the purpose of constructing Budejovice, Hong Kong, Melbourne, San for research analysis, but practising connections to the content to be Diego, Shanghai, Tokyo and Uppsala. teachers are denied recognition of covered in the lesson. an activity that at least one culture The particular lessons were chosen in Ucitelská ozvenaˇ – the ‘teacher’s feels is sufficiently important to have consultation with local researchers in echo’ when the teacher each country to provide a wide variety been given a specific name. An

Figure 3: Video stimulus layout (key elements are: three synchronized camera views – teacher camera, whole class camera, student camera; classroom dialogue in English subtitles; timecode)

Research Conference 2010 6 reformulates a student’s answer to languages are different classrooms. increase its clarity or mathematical In the same way that the differential correctness; ideally, without promotion of fluency in spoken appropriating the student’s mathematics in different classrooms intellectual ownership of the around the world enacts a different response. classroom mathematics, teachers, other educators, and researchers in different Matome – a teacher-orchestrated countries have at their disposal very discussion, drawing together the different linguistic tools by which to major conceptual threads of a conceptualise, theorise about, and lesson or extended activity – most research the mathematics classroom. commonly a summative activity at Our capacity to study, understand the end of the lesson. and enact classroom practice must be We, as researchers, select our enhanced rather than constrained by theoretical tools because the actions our growing internationalisation. and outcomes they privilege resonate with educational values that we already References hold. These educational values find their embodiment in the forms of Clarke, D. J. (2006). The LPS research classroom activity that our culture has design. In D. J. Clarke, C. Keitel & Y. chosen to name. This reproductive Shimizu (Eds.), Mathematics Classrooms process can only amplify our pre- in Twelve Countries: The Insider’s existing assumptions regarding what Perspective, pp. 15–37. Rotterdam: is to be valued and what is to be Sense Publishers. discarded. Research-based advocacy Clarke, D. J., & Xu, L. H. (2008). of instructional practice runs the risk Distinguishing between mathematics of only entrenching the vision of the classrooms in Australia, China, Japan, classroom enshrined in the researcher’s Korea and the USA through the lens language and culture. Language does of the distribution of responsibility not just mediate the researcher’s for knowledge generation: Public categorisation of what occurs in the oral interactivity versus mathematical classroom. Language was there before orality. ZDM – The International Journal us, determining which classroom in Mathematics Education, 40(6), activities are conceptualised and 963–981. enacted by the participants. Further, the theories we construct are constrained Shimizu, Y. (2006). How do you to those constructs and relationships conclude today’s lesson? The form and we are capable of naming. And our functions of ‘Matome’ in mathematics ‘evidence-based’ instructional advocacy lessons. In Clarke, D., Emanuelsson, J., reproduces this chain of compounded Jablonka, E., & Ah Chee Mok, I. (Eds.), constraints, leading us to ignore other, (2006). Making Connections: Comparing potentially effective, instructional mathematics classrooms around the alternatives. world. Rotterdam: Sense Publishers. Summative remarks The professional discourse of the international mathematics education community is constrained by the dominance of English. The classrooms experienced and described by teachers and researchers speaking non-English

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 7 Standards, what’s the difference?: A view from inside the development of the Common Core State Standards in the occasionally United States

Abstract may be, neither is in plain sight. What is in plain sight are standards, tests, Standards sequence as well as express textbooks and students. priority. On what basis? Learning trajectories sequence through empirical LTs are too complex and too investigation and theory. The sequence, conditional to serve directly as as far as it goes, has empirical validity, standards. Still, LTs point the way to but only some sequences have been optimal learning sequences and warn developed. Standards, in contrast, must against hazards that could lead to choose what students need to learn as a sequence errors. Teachers and students Philip Daro matter of policy. This article will discuss need time within the lesson and across Chair, Common Core Standards issues of sequence, focus and coherence the unit to pull students from PLoTs Mathematics Workgroup USA in mathematics standards from the along LTs to the SSTs. This requires perspective of the Common Core State standards to be within reach. Currently, Phil Daro Chairs the State-led Standards (CCSS) for Mathematics in Common Core Standards Mathematics The types of errors in the way Workgroup in the USA which is drafting common the United States of America. standards might be sequenced are College and Career Readiness Standards on Decisions about sequence in standards reviewed. behalf of 48 US States and was a member of the lead writing team for the Common Core State must balance the pull of three Standards. important dimensions of progression: Introduction Phil is a Senior Fellow for Mathematics for cognitive development, mathematical One sees the difficulty with this America’s Choice where he focuses on programs coherence, and the pragmatics of standards business. If they are for students who are behind and algebra for all; instructional systems. Standards are taken too literally, they don’t go he also directs the partnership of University of written as though students in the California, Stanford and others with San Francisco far enough, unless you make them Unified School District for the Strategic Education class have learned approximately 100 incredibly detailed. You might give a Research Partnership (SERP), with a focus on per cent of preceding standards. This discussion of a couple of examples, mathematics and science learning among students is wild fiction in any real classroom. to suggest how the standards should learning English or developing academic English, This difference between the genre develops research agenda and projects which be interpreted in spirit rather than address priorities identified in the school district. convention of ‘immaculate progression’ by the letter. But of course, this is a in standards and the wide distribution slippery slope. He has directed, advised and consulted to a of student readiness in real classrooms range of mathematics education projects in Roger Howe, Yale,  the USA. The most extensive and intensive is a dangerous difference to ignore. engagements include NAEP Validity studies, Each student arrives at the day’s lesson March 15, 2010  ACHIEVE, FAM (Foundations of Mathematics) with his or her own mathematical input to common core standards program development for America’s Choice, biography, whatever the student Balanced Assessment Project (co-Director), … the “sequence of topics and Mathematics Assessment Resources (MARS), learned on their personal trajectory performances” that is outlined in the El Paso Collaborative (consultant), school through mathematics. A spectacular a body of mathematics standards districts and states, the New Standards Project. diversity of such personal learning must also respect what is known From the mid 1980s until the 1990s, Phil was trajectories (PLoTs) faces the teacher the state Director of the California Mathematics about how students learn. As Project for the University of California. He has at the beginning of each lesson. There Confrey (2007) points out, also worked with reading and literacy experts and are two related manifolds in play developing “sequenced obstacles panels on problems related to academic language during each lesson: the manifold of and challenges for students… development, especially in mathematics classroom PLoTs (personal learning trajectories) discourse. absent the insights about meaning in the classroom and the manifold of that derive from careful study of learning trajectories (LTs) that enable learning, would be unfortunate and the learning of the mathematics being unwise.” In recognition of this, the taught. As real as these trajectories development of these Standards

Research Conference 2010 8 began with research-based learning This article will look at the general coherence, and the pragmatics of progressions detailing what is issues of sequence, focus and instructional systems. The situation known today about how students’ coherence in mathematics standards differs for elementary, middle and high mathematical knowledge, skill, and from the perspective of the Common school grades. In brief: elementary understanding develop over time. Core State Standards (CCSS) for standards can be more determined Mathematics. I was a member of the by research in cognitive development Common Core  small writing team for the CCSS. and high school more by the logical State Standards,  As such, I was part of the design, development of mathematics. Middle 2010 deliberation and decision processes, grades must bridge the two, by no including especially reviewing and means a trivial span. Sequence, Coherence and making sense of diverse input solicited For example, the Common Core Focus in Standards and and unsolicited. Among the solicited State Standards (CCSS) incorporate a Learning Trajectories input were synthesised ‘progressions’ progression for learning the arithmetic from learning progressions researchers. Learning trajectories sequence levels of of the base ten number system. A cognitive actions and objects through logical development mathematically empirical investigation and theory. Grade level vs. development would begin with sums of terms which As result the sequence has empirical Standards sequence for grade levels; are products of a single digit number validity. However, the question of that is, the granularity of the sequence and a power of ten, including rational what is being sequenced is a matter is year-sized. Standards do not explicitly exponents for decimal fractions. Yet no of researcher choice, often driven by sequence within grade level, although one thinks this is the way to proceed. theoretical considerations related to a they are presented in some order that Instead, the CCSS for grade 1 ask trajectory of interest to the researcher. makes more or less sense. Sometimes students to, Some researchers (Clements and this order within grade is compelling, 2. Understand that the two digits Sarama, 2010 {this report}) suggest thus luring users to over interpret the of a two-digit number represent these choices include consultation with within grade presentation as teaching amounts of tens and ones. mathematicians and educators to obtain sequence. valid focus. Still, the choice of what Understand the following as special mathematics gets research attention is From the start, we encounter a cases: problematic convention: standards are not, in itself, a valid basis for deciding a. 10 can be thought of as a written as though students have learned what to teach. Standards, in contrast, bundle of ten ones—called a everything (100% ) in the standards begin with choices about what students “ten.” need to learn as a matter of policy. for the preceding grade levels. No one thinks most students have learned b. The numbers from 11 to 19 Standards, perforce, sequence as 100%, but this genre convention for are composed of a ten and one, well as express priority. On what standards seems a sensible approach two, three, four, five, six, seven, basis? By design, at least, one hopes. to avoiding redundancy and excessive eight, or nine ones. … To what extent can and has the linguistic nuance. But how does this design of mathematics standards The relative weight to give cognitive mere genre convention drive the development vs. mathematical been informed by research and management of instruction? Test empirically well founded theories of coherence gets more tangled with construction? Instructional materials and multiplication, the number line and learning trajectories? This article will their adoption? Teaching? Expectations contemplate that question for the especially fractions. In third grade, the and social justice? Ah…the letter or the CCSS introduces two concepts of recently developed Common Core spirit and the slippery slope. State Standards in mathematics, the fractions: closest this nation has ever come to Cognitive development, 1. Understand a fraction 1/b as the national standards. It is an interesting mathematical coherence and quantity formed by 1 part when a tale that leads to fundamental, pedagogic pragmatics whole is partitioned into b equal perhaps very productive, questions parts; understand a fraction a/b as about standards and trajectories, and Decisions about sequence in standards the quantity formed by a parts of their consequences for instruction, must balance the pull of three size 1/b. curriculum, assessment and the important dimensions of progression: management of instruction. cognitive development, mathematical

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 9 2. Understand a fraction as a number as number lines are as mathematical Measure and estimate lengths in on the number line; represent objects of study, number lines confused standard units. fractions on a number line diagram. students when used to teach other • Measure the length of an ideas like operations and fractions. In a. Represent a fraction 1/b on object by selecting and using other words, include the number line a number line diagram by appropriate tools such as rulers, as something to learn, but don’t rely on defining the interval from 0 to yardsticks, meter sticks, and it to help students understand that a 1 as the whole and partitioning measuring tapes. fraction is a number. it into b equal parts. Recognize • Measure the length of an object that each part has size 1/b and The difference in advice on fractions twice, using length units of that the endpoint of the part on the number line was not easy to different lengths for the two based at 0 locates the number sort through. In the end, we placed measurements; describe how 1/b on the number line. the cognitively sensible understanding the two measurements relate to first and the mathematical coherence b. Represent a fraction a/b on the size of the unit chosen. with the number line second. We a number line diagram by included both and used both to • Estimate lengths using units of marking off a lengths 1/b from build understanding and proficiency inches, feet, centimeters, and 0. Recognize that the resulting with comparing and operations with meters. interval has size a/b and that its fractions. endpoint locates the number a/b • Measure to determine how on the number line. Does the number line appear out of much longer one object is the blue in third grade? No. We looked than another, expressing the The first concept relies on student to the research in learning trajectories length difference in terms of a understanding of equal partitioning. for measurement and length to see standard length unit. Jere Confrey (2008) and others have how to build a foundation for number detailed the learning trajectory of Relate addition and subtraction to lines as metric objects (Clements, children that establishes the attainability length. 1999c; Nührenbörger, M., 2001; Nunes, of this concept of fraction. Yet by itself, T., Light, P., and Mason, J.H. 1993). The • Use addition and subtraction this concept is isolated from broader Standards from Asian countries like within 100 to solve word ideas of number that, for the sake of Singapore and Japan were also helpful problems involving lengths that mathematical coherence, are needed in encouraging a deeper and richer are given in the same units, early in the study of fractions. These development of measurement as a e.g., by using drawings (such ideas are established through the foundation for number and quantity. as drawings of rulers) and second standard that defines a fraction equations with a symbol for the as a number on the number line. This Clements and Sarama (2009) unknown number to represent definition has a lot of mathematical emphasize the significance of the problem. power and connects fractions in a measurement in connecting geometry simple way to whole numbers and, and number, and in combining skills • Represent whole numbers as later, rational numbers including with foundational concepts such lengths from 0 on a number negatives (Wu, H., 2007). Simple as conservation, transitivity, equal line diagram with equally looking forward, but mysterious coming partitioning, unit, iteration of standard spaced points corresponding from prior knowledge. units, accumulation of distance, and to the numbers 0, 1, 2, …, and origin. By around age 8, children can represent whole-number sums The Writing Team of CCSS received use a ruler proficiently, create their own and differences within 100 on a wide and persistent input from units, and estimate irregular lengths number line diagram. teachers and mathematics educators by mentally segmenting objects and that number lines were hard for This work in measurement in 2nd counting the segments. young students to understand and, grade is, in turn, supported by 1st grade as an abstract metric, even harder The CCSS foundation for the use of standards: to use in support of learning other the number line with fractions in 3rd • Express the length of an object concepts. Third grade, they said, is grade can be found in the 2nd grade as a whole number of length early for relying on the number line Measurement standards: units, by laying multiple copies to help students understand fractions. of a shorter object (the length We were warned that as important

Research Conference 2010 10 unit) end to end; understand the negation of ‘can’ negates ‘should’. real as these trajectories may be, none that the length measurement Standards serve a different purpose. are in plain sight. of an object is the number of They map stations through which …teaching is like riding a unicycle same-size length units that span students are lead from wherever they juggling balls you cannot see or count. it with no gaps or overlaps. start. Limit to contexts where the What is in plain sight are standards, Immaculate progression literalism has object being measured is tests, textbooks and students. A contributed to confusion about what spanned by a whole number of teacher cannot actually know the “proficient” means as a test result. Most length units with no gaps or students’ PLoTs. Nor has research state tests have “proficient” cut scores overlaps. mapped the territory of the standards at 60% or less (with guessing allowed with LTs.. And the MTs are themselves This sequence in the CCSS was guided on multiple choice, [usually 4 choices], a matter of considerable choice in by the learning trajectory research. This items that make up close to all of the starting point, and often beyond the research informed the CCSS regarding test). Thus even the distribution of mathematical education of the teacher. essential constituent concepts and skills, ‘proficient’ students lacks large chunks What is real is hard to see, while appropriate age and sequence. Yet the of learning of the standards, at least as standards flash brightly from every test, goal of having number line available assessed by the standards based test. text and exhortation that comes the for fractions came from the need for teacher’s way. mathematical coherence going forward The rough terrain of prior from 3rd grade, rather than from learning where lessons live Learning trajectory research develops learning trajectory research. evidence and evidence based The standards based curriculum is a trajectories (LTs). Evidence establishes Instructional Systems and sequence through the calendar: year that LTs are real for some students, to year, month to month, day to day. Standards a possibility for any student and Think of this as a horizontal path possibly modal trajectories for the Perhaps the most important of concepts and skills. Such a path distribution of students. LTs are too consequence of standards is their can match textbooks and tests, but complex and too conditional to serve impact on instruction and instructional never the distribution of students in a directly as standards. Still, LTs point systems. This impact is often mediated classroom. Beneath the surface of the the way to optimal learning sequences by high stakes assessments which standards sequence trajectory (SST) and warn against hazards that could will be dealt with later. Two crucial is the underwater terrain of prior lead to sequence errors (see below). instruction issues will be discussed that knowledge. Each student arrives at The CCSS made substantial use of are too often buried in comforting the day’s lesson with his or her own LTs, but standards cannot simply be cushions of unexamined assumptions. mathematical biography, whatever LTs; standards have to include the The first issue is, how do the structure, the student learned on their personal essential mathematics, MTs, whether properties and behavior of mathematics trajectory through mathematics. A we know anything about its location knowledge interact with instruction? spectacular diversity of such personal in an LT or not, and standards have to learning trajectories (PLoTs) faces the The second issue arises from the accommodate the variation in students, teacher at the beginning of each lesson way standards are written, as though if not teachers, at each grade level. (Murata, A., & Fuson, K. C., 2006). students in the middle of grade 5 have How do and could these four learned approximately 100% of what is The teacher, on the other hand, trajectories (LTs, MTs PLoTs, and SSTs) in the standards for grade k-4 and half brings to this diversity an ambition interact? A system could just leave it of 5. This is never close to true in any for some mathematics to be learned. to individual teachers to reckon the real classroom. This difference between The mathematics has a location in yet optimization among them. It could the genre convention of “immaculate another trajectory: the logical sequence impose strong SSTs as pressure in an progression” in standards and the of ideas which reflects the deductive accountability system, without providing wide distribution of student readiness structure of mathematics (MTs). Thus, for PLoTs or taking advantage of LTs. in real classrooms has important there are three related manifolds in It could name the territory between consequences. It means, for one thing, play: the PLoTs (personal learning what students bring (PLoTs) and the that standards are not a literal portrayal trajectories) in the classroom, the MTs what standards demand (SST) the of where students are or can be at and the learning trajectories (LTs). As “achievement gap”, a dark void that a given point in time. And, for me, only explains steps not taken rather

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 11 than which way to go. It could tell the LTs and MTs. That is, LTs and MTs 2. Recognize and represent teachers to keep turning the pages can provide the map from PLoTs to proportional relationships between of the textbook based on standards SSTs . The map, alas, is of a territory covarying quantities. according to the planned pace, and rely that is only partially explored. There a. Decide whether two quantities on the shear force of expectation to are still unknown seas and fears of sea are in a proportional pull students along. At least this would monsters and dreams of gold to frighten relationship, e.g., by testing for create the opportunity to learn, however and distract us from the voyage. Still, we equivalent ratios in a table or fleeting and poorly prepared students know enough in elementary grades to graphing on a coordinate plane might be to take advantage of it. While do what is needed to make LTs a part and observing whether the this is better than denial of opportunity, of teacher knowledge and a feature in graph is a straight line through it is a hollow, if not cynical, response to tools for teachers. the origin. the promise standards make to students. Teachers need knowledge of how Shouldn’t we do better? b. Identify the constant of LTs work and the specifics of LTs that proportionality (unit rate) What would be better? Some nations, will help them understand the most in tables, graphs, equations, including high performing nations, common PLoTs they will find among diagrams, and verbal assume in the structure of their their students (Murata, A., & Fuson, K. descriptions of proportional instructional systems that students C., 2006). They need knowledge of relationships. differ at the beginning of each lesson. the relevant MTs. And they need tools Asian classrooms, K-5, and mostly that illuminate rather than obscure c. Represent proportional 6-9, follow a daily trajectory of initially the PLoTs. They need instructional relationships by equations. projecting the divergence of students’ programs and lesson protocols that For example, total cost, t, is development (refracted through the pose SSTs as the finish line, but proportional to the number, n, day’s mathematics problem/s) into accommodate PLoT variation. They purchased at a constant price, the classroom discourse and pulling need time within the lesson and across p; this relationship can be the divergence toward a convergent the unit to pull students from PLoTs expressed as t = pn. learning target. The premise is: each along LTs to the SSTs. This requires d. Explain what a point (x, y) on lesson begins with divergence and standards to be within reach. the graph of a proportional ends with convergence. Such a system The crucial issue in this situation is relationship means in terms requires enough time to achieve how well the standards driven texts of the situation, with special convergence each day, enough time and tests improve the performance attention to the points (0, 0) and on a small number of problems. A of the instructional system in moving (1, r) where r is the unit rate. hurried instructional system cannot the PLoTs along the LTs. It is quite ‘wait’ for students each day. Standards This standard is the culmination of a possible for standards to be out of must require less to learn rather than manifold of progressions and, itself, whack with LTs and PLoTs so that they more each year to make time for daily the beginning of more advanced diminish performance. Standards are convergence. A system which optimises progressions. Pat Thompson has only a good idea when they usefully daily convergence will be more robust remarked (2010, advice to standards) map underlying LTs and MTs so they and accumulate less debt in the form of that proportionality cannot be a single can help teachers see and respond to students unprepared for the next lesson. progression because it is a whole city PLoTs. If the sequence in the standards Such debt compounds. Unlike the of progressions. This standard, which conflicts seriously with LTs or are too national debt, it does not compound stands along side other standards far removed from PLoTs, they can quietly, but makes all the noises of on ratios and rates, explicitly draws steer the instructional systems away childhood and adolescence scorned. on prior knowledge of fractions, from teaching and learning, toward equivalence, quantitative relationships, Start by understanding the task and then statuesque poses facing out and the coordinate graph, unit rate, tables, the people in place who can do their same waste of chances inside. ratios, rates and equations. Implicitly, parts to accomplish the task. The task is For example, the CCSS at grade 7 this prior knowledge grows from to take the domain of PLoTs, the given have a standard for proportional even broader prior knowledge. The rough terrain of what the distribution of relationships. sequence supporting this Standard students bring, and transform the PLoTs in the SST barely captures the peaks to SSTs, give or take. The function that of a simplification of the knowledge can take PLoTs to SSTs is mapped by

Research Conference 2010 12 structure. The complexity of the express this kind of complexity; they If the field had a well understood manifold of LTs guarantees that the refer to some observable surface of corpus of cognitive actions, situations, distribution of PLoTs in a classroom will learning. But this linguistic convenience knowledge etc. then these names have splendid variety. can lead to logical fallacies when we could refer to parts of this corpus. attribute unwarranted ‘thinginess’ But the field, school mathematics, What could help the teacher properties to what we actually want has no such widely understood confronted with the variety of students to learn. corpus (indeed, it is an important readiness? Certainly not pressure to hope that common standards will “cover” the standards in sequence The important point is that learned lead to common understandings like (SST), keep moving along at a good things are not things or topics (names) this). What the names refer to, in pace to make sure all students and not just standards. A sequence of effect, are the familiar conventions have an ‘opportunity’ to see every topics or standards skims the surface of what goes on in the classrooms. standard flying by. Perhaps some and misses the substance and even The reference degenerates to the old knowledge of the LTs would help the form of a subject. Compare, for habits of teaching: assignments, grading, teachers understand the variety of example, the Standard, assessment, explanation, discussion. PLoTs and what direction to lead the • Add and subtract fractions with The standards say, ‘Do the usual students from wherever they begin unlike denominators (including assortment of classroom activities for the lesson. Even hypothetical LTs can mixed numbers) by replacing some content that can be sorted into do more good than harm because given fractions with equivalent the names in the standards. We will they conceptualize the student as a fractions in such a way as to call this “covering the standards” with competent knower and learner in produce an equivalent sum or instructional activity. the process of learning and knowing difference of fractions with like more (Clements, 2004a). Perhaps a “Covering” has a very tenuous denominators. For example, 2/3 system of problems and assignments relationship with learning. First, there + 5/4 = 8/12 + 15/12 = 23/12. (In with the diagnostic value of revealing are many choices within a topic about general, a/b + c/d = (ad + bc)/bd.) how different students see the focus, coherence within and between mathematics…how they think about to what the student must actually topics, what students should learn to do it…where they are along the LT. A know and do to “meet” the standard with knowledge, how skillful they need teacher needs the thinking itself, not a (for example, Steffe, 2004,2009; to be at what, and so on endlessly. score that evaluates the thinking. Confrey et al, 2008, 2009; Wu, 2007; Teachers make these choices in many Saxe et al, 2005). The standard gives different ways. Too often, the choices How do standards express the a goal, but does not characterize the are made in support of a classroom form and substance of what knowledge and competencies needed behavior management scheme relied students learn? to achieve the goal. While this point on by the teacher. Second, different may seem obvious, it gets lost in the students will get very different learning What is the nature of the ‘things’ compression chambers where systems from the same offered activity. Third, students learn? Sometimes what is are organized to manage instruction for the quality of the discussion, the wanted is a performance, as in learn school districts. Devices are installed to assigned and produced work, the to ride a bike. Standards, instruction manage “pacing” and monitor progress feedback given to students will vary and assessment can happily focus on with “benchmark assessments”. widely by teacher working under the the visible performance in such cases. These devices treat the grade level blessing of the same standard. But often, in mathematics anyway, is standards as the form and substance of Covering is at best weak. When a mental action on a mental object, instruction. That is, students are taught combined with standards that are too far reasoning maneuvers and rules, grade level “standards” instead of from the prior knowledge of students, representational systems and languages mathematics. This nonsense is actually and too many; the chemistry gets nasty for mathematical objects and relations, widespread, especially where pressures in a hurry. Teachers move on without cognitive schema and strategies, webs to “meet standards” are greatest. of structured knowledge, and social the students; students accumulate representations, and so on. Many of Standards use conventional names and debts of knowledge (knowledge these learned things are systems that phrases for topics in a subject. To what owed to them) and opportunities for interact with other systems in thinking, do these refer? understanding the next chapter, the next knowing and doing. Standards cannot course are undermined.

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 13 The foregoing discussion of instructional basis of the size of the other b. Coherence requires progression systems illustrates the importance (and factor, without performing the ABC, but standards only have potential for mayhem) in sequencing indicated multiplication. AC standards. What constituents are b. Explaining why multiplying c. Term is used that has insufficient necessary and sufficient as prior a given number by a fraction definition for that use. knowledge for a given concept or greater than 1 results in a action, and how can the constituents 3. Cognitive prematurity: product greater than the be arranged to lead up to the target given number (recognizing a. B depends on cognitive actions concept? This question has many multiplication by whole and structures that have not local answers that have to be fitted numbers greater than 1 as a developed yet. together into regions that make some familiar case); explaining why sense, if not harmony. Standards are b. B is a type of schema or multiplying a given number by further constrained by how much can reasoning system, learner has a fraction less than 1 results be learned at any one grade level, and not developed that type of in a product smaller than the by the coherence within a grade level. schema or system. given number; and relating the These questions are not only design principle of fraction equivalence c. Student develops immature choices, but potential sources of error a/b = (n×a)/(n×b) to the effect of version of B and carries it with consequences for the viability of multiplying a/b by 1. forward (see 4) instruction. The next sections examine the types of errors that could menace a In grade 6 and 7 rate, proportional 4. Contradiction: standards based system. relationships and linearity build upon a. Cognitive development entails this scalar extension of multiplication. ABC, mathematical logic entails Types of Sequence Errors Students who engage these concepts CBA. with the unextended version of There are several types of errors with multiplication (a groups of b things) 5. Missing connection: B is about or serious consequences for students and will have PLoTs that do not support depends on connection between teachers in the way standards might be the required MTs. This burdens the X-Y , but X-Y connection not sequenced. For example, a common teacher and student with recovering established. type of sequence error occurs when a through LTs. This will be taxing enough concept, B depends on A2 version of 6. Interference: without ill sequenced standards concept A, more evolved than the A1 causing instructional systems to neglect a. B depends on A2 version of A, version; Standards have only developed extending multiplication. more evolved than A1 version; A1. Student tries to learn B using Standards have only developed A1 instead of A2. Rate, proportional Major types of sequence errors follow: A1. Student tries to learn B using relationships and linearity (B) depend 1. Unrealistic: A1 instead of A2. on understanding multiplication as a scaling comparison (version A2), but a. Too much too fast so gaps in b. B belongs nestled between A students may have only developed learning create sequence issues and C, but D is already nestled version A1 concept of multiplication, for students, system cannot there. When learning B is the total of things in a groups of b each. deliver students who are in attempted, D interferes. sequence. In the CCSS, multiplication is defined in 7. Cameo: grade 3 as a x b = c means a groups of b. Distribution of prior a. B is learned but not used for b things each is c things. In grade 4, the mathematics knowledge and a long time. There is no C concept of multiplication is extended to proficiency in the student and such that C depends on B for comparison where c = a x b means c teacher population is too far a long time. B makes a cameo is a times larger than b. In grade 5, the from the standards; no practical appearance and then gets lost in CCSS has: way to get students in a good the land of free fragments. enough sequence. 5. Interpret multiplication as scaling 8. Hard Way: (resizing), by: 2. Missing ingredient: a. C needs some ideas from B, a. Comparing the size of a product a. A is an essential ingredient of B, but not all the difficult ideas and to the size of one factor on the Standards sequence B before A. technical details that make B

Research Conference 2010 14 take more time than it is worth that should be explicit and 8. Waste: and make it hard for students to defended.) a. Invest time and cognition on B find the needed ideas from B, so b. Too fine: complex ideas are and B is not important. C fails. chopped up so the main idea 9. Resolution of hierarchy: b. There are multiple possible is lost; the coherence may be routes to get from A to E, evoked, but not illuminated. a. The hierarchal relationship standards take an unnecessarily Alignment transactions in between standards is not difficult route test construction, materials explicated. Details are confused development miss the main with main ideas. 9. Aimless: point but ‘cover’ the incidentals. b. The hierarchy of standards does a. Standards presented as lists that Students can perform the not explain relationships among lack comprehensible progression. vertical line test but do not ideas, it just collects standards know what a function is or how into categories. Types of Focus and Coherence functions model phenomena. 10. Excessively literal reading: Errors c. Too broad: includes whatever The issues of focus and coherence in and focuses on nothing in a. This error is in the reading as standards deserves more attention particular. much as the writing; it leads to fragmented interpretation of the than we will give it here. Nonetheless, 3. Wrong focus learning trajectories interact with subject, losing the coherence coherence and focus in standards. The a. Focus on answer getting between the standards. methods, often mnemonic following are critical types of error of b. Reading individual standards as devices, rather than focus and coherence: individual ingredients of a test. mathematics. 1. Sprawl: when the explicit goal is to 4. Narrow focus have the ingredients cook into a. Mile wide, inch deep. Collection a cake, tasting the uncooked a. Just skills, or just concepts or of standards dilutes the ingredients is a poor measure of just process; or just two out of importance of each one. how the cake tastes (although it three. b. Standards demand more than is related). The goal, as stated in is possible in the available time 5. Priorities do not cohere: the grade level introductions and the practices standards is for the for many students and teachers, a. Fragments that have large gaps students to cook. so teachers and students forced between them; to edit on the fly. This is the opposite of focus. b. grain size too fine What are Standards? c. Standards are just lists without 6. Congestion: Standards are promises. Standards enough organisational cues in a. Some grade levels are congested promise the student, “Study and learn relation to hierarchy of concepts with too much to be learned; what is here, do your assignments and and skills density precludes focus we promise you will do well on the test.” We need tests and examinations 2. Wrong grain size b. B, C, D are all being learned designed to keep that promise. We a. The granularity is too specific at once, but cognitive actions need school systems designed to keep or too general. The important needed for learning can only the promises. understanding is at a certain handle one or two at a time. level of specificity where the Only BC and CD are learned, Bibliography structure and the cognitive but the essential point is learning Baroody, A. J., Cibulskis, M., Lai, M.-l., handles are, more specific or BCD and the system BC-BD-CD. & Li, X. (2004). Comments on more general; grain size will not 7. Inelegance: the use of learning trajectories in match up to prior knowledge, curriculum development and research. mental objects and actions on a. AXBYCZ is equivalent to ABC Mathematical Thinking and Learning. them (see Aristotle Ethics: the and wasted time and cognition 6(2), 227–260. choice of specificity is a claim on –X-Y-Z.

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 15 Black, P., & Wiliam, D. (1998). Fuson, K.C. (2004). Pre-K to grade mathematical development. Journal Assessment and classroom learning. 2 goals and standards: Achieving of Educational Computing Research, Assessment in Education, 5(1), 21st-century mastery for all. In D.H. 27(1&2), 93-110. Clements, J. Sarama, and A. DiBiase Black, P., & Wiliam, D. (1998). “Inside Sarama, J., & Clements, D. H. (2009a). (Eds.), Engaging Young Children in the Black Box: Raising Standards Early childhood mathematics education Mathematics Mahwah, NJ: Erlbaum. Through Classroom Assessment,” Phi research: Learning trajectories for young Delta Kappan 80. Kilpatrick, J., Swafford, J., & Findell, B. children. New York: Routledge. (2001). Adding it up: Helping children Case, R., & Okamoto, Y. (1996). The Saxe, G., Taylor, E., McIntosh, C., & learn mathematics. Washington, DC: role of central conceptual structures in Gerhart, M. (2005). Representing National Academy Press. the development of children’s thought. Fractions with standard notions: A Monographs of the Society for Research Murata, A., & Fuson, K. C. (2006). developmental analysis. Journal for in Child Development. Serial No. 246, Teaching as assisting individual research in Mathematics Education. 36 61, (1-2), constructive paths within an (2), 137-157. interdependent class learning zone: Clements, D.H. (1999c). Teaching length Sherin, B., & Fuson, K. (2005). Japanese first graders learning to measurement: Research challenges. Multiplication Strategies and the add using 10. Journal for Research in School Science and Mathematics, 99(1), Appropriation of Computational Mathematics Education, 37, 421-456. 5-11. Resources. Journal for Research in NCTM. (2006). Curriculum focal points Mathematics Education., 36 (4), 347- Clements, D. H., & Sarama, J. (2004a). for prekindergarten through grade 8 395. Hypothetical learning trajectories. mathematics: A quest for coherence. Mathematical Thinking and Learning, Simon, M. A. (1995). Reconstructing Reston, VA: National Council of 6(2). mathematics pedagogy from a Teachers of Mathematics. constructivist perspective. Journal for Clements, D. H., & Sarama, J. (2004b). Nührenbörger, M. (2001). Insights into Research in Mathematics Education, Learning trajectories in mathematics children’s ruler concepts—Grade-2 26(2), 114-145. education. Mathematical Thinking and students’ conceptions and knowledge Learning. 6(2), 81-89. Steffe, L. P. (2004).”On the construction of length measurement and paths of learning trajectories of children: Clements, D. H., & Sarama, J. (2009). of development. In M.V.D. Heuvel- The case of commensurate fractions”. Learning and teaching early math: The Panhuizen (Ed.), Proceedings of the Mathematical Thinking and Learning. 6 learning trajectories approach. New 25th Conference of the International (2), 129-162 York: Routledge. Group for the Psychology in Mathematics Steffe, L. P. & Olive, J (2009). Children’s Common Core State Standards for Education, 3, 447-454. Utrecht, The fractional knowledge. Boston, Springer. Mathematics, 2010; Corestandards.org Netherlands: Freudenthal Institute. Wilson, P. H. (2009). Teachers’ Uses of Confrey, J., Maloney, A., Nguyen, Nunes, T., Light, P., and Mason, J.H. a Learning Trajectory for Equipartitioning. K., Mojica, G., & Myers, M. (2009). (1993). Tools for thought: The North Carolina State University, Equipartitioning/Splitting as a Foundation measurement of length and area. Raleigh, NC. of Rational Number Reasoning Learning and Instruction, 3, 39-54. Using Learning Trajecories. Paper Park, J., & Nunes, T. (2001). The Wu, H. (2007), Fractions, decimals and presented at the 33rd Conference development of the concept of rational numbers”, http://math.berkeley. of the International Group for the multiplication. Cognitive Development, edu/-wu/ Psychology of Mathematics Education, 16, 763-773. Thessaloniki, Greece. Peterson, P. L., Carpenter, T. P., & Confrey, J. (2008). A synthesis of the Fennema, E. H. (1989). Teachers’ research on rational number reasoning: knowledge of students’ knowledge A learning progressions approach to in mathematics problem solving: synthesis. Paper presented at the 11th Correlational and case analyses. Journal International Congress of Mathematics of Educational Psychology, 81, 558-569. Instruction. Sarama, J., & Clements, D. H. (2002). Building Blocks for young children’s

Research Conference 2010 16 Mathematics teaching and learning to reach beyond the basics

Abstract Because of their abstractness, learning about the objects with which The purpose of this presentation is mathematics is concerned is difficult. to paint a broadbrush picture of the Because mathematics is a doing challenge of providing mathematics subject, transforming and combining teaching that encourages learning these objects is central, so developing that goes beyond ‘the basics’. The the relevant skills to a high degree presentation focuses on mathematical of fluency is central. The difficulty reasoning and suggests ways in which of the learning is heightened by the Kaye Stacey it can be given a more secure place hierarchical nature of mathematics, in Australian mathematics classrooms. where skill is built on skill and concept University of Melbourne Two studies are reported, both of is built on concept. No wonder that which arose from concern about the learning ‘the basics’ (the concepts, the Kaye Stacey is Foundation Professor of ‘shallow teaching syndrome’ evident Mathematics Education at the University of skills and how to use them in standard Melbourne and the leader of the Science and in many Australian classrooms where ways to solve problems that relate Mathematics Education cluster. She works as there is very little mathematical directly to real-world situations) can a researcher, primary and secondary teacher reasoning in evidence. One study easily fill all the time in school devoted educator, supervisor of graduate research and examined Year 8 textbooks, finding as an adviser to governments. She has written to mathematics. Listing the concepts, many practically oriented books and articles for that very few presented ‘rules without the skills and their direct applications mathematics teachers, as well as producing a reasons’ and taken overall generally could also easily fill a whole national large set of research articles. Professor Stacey’s presented a good array of explanations curriculum. research interests centre on mathematical involving reasoning of several distinct problem solving and the mathematics types to help students understand Important as the content above is, curriculum, particularly the challenges which and despite the tendency for it to are faced in adapting to the new technological why results were true. It was evident, environment. She is currently a member of the however, that these explanations appear to define what mathematics is, Australian Research Council College of Experts. were generally only used to justify mathematics is only partially described Her research work is renowned for its high the rule, and were not called upon by such concepts, skills and standard engagement with schools. Her doctoral thesis applications. The less visible aspect from the University of Oxford, UK, is in number in any way once it was established. A theory. She has been the mathematics expert second study interviewed about 20 of mathematics is its process side on the Australian Advisory Committee for the leaders in mathematics education to (how mathematics is done) which OECD PISA project since its inception and is explore their opinions on the shallow for the past nearly 20 years has been now Chair of its international Mathematics Expert labelled ‘Working Mathematically’ in Group. Kaye Stacey was awarded a Centenary teaching syndrome (most – but not Medal from the Australian government for all – felt it was a real effect of disturbing Australia. In the presentation, I will give outstanding services to mathematical education. prevalence), and the teaching of a brief overview of the various ways mathematical reasoning and problem in which this strand has been treated solving. The presentation includes some in Australian mathematics in the past, suggestions for strengthening the place leading up to the current first cycle of of mathematical reasoning in Australian the Australian curriculum. Here the classrooms and the new Australian elements of Working Mathematically curriculum. most clearly appear as two of the four proficiency strands: problem solving Introduction and reasoning. Neither of these strands seems to be yet operationalised as The purpose of this paper is to paint clearly as will be required if teachers a broadbrush picture of the challenge are to be encouraged to pay serious of providing mathematics teaching that attention to them. This presentation encourages learning that goes beyond will present ideas on the development ‘the basics’. The paper focuses on of the reasoning strand. mathematical reasoning and suggests ways in which it can have a more Reasoning in mathematics is a cognitive secure place in Australian mathematics process of looking for reasons and classrooms. looking for conclusions. To learn mathematics, students need to learn

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 17 about the reasons which others have us to see the nature of the reasoning resources was the only reason given found to support conclusions (for that they display and promote. The more than once for suggesting that example, why the angle sum of any study’s focus was on explanations of there might have been positive change. triangle is 180 degrees) and they also why important mathematical results are The first observation from the textbook need to engage in their own reasoning true, not explanations of what or how study is that mathematical results are both when working on what Polya calls (e.g. What does NNW mean?, How do established using a variety of different ‘problems to prove’ and ‘problems to you make a stem-and-leaf plot?). These modes of reasoning. Most of the find’. These two sides are connected. why explanations involve mathematical textbooks made some attempt to Learning about the reasoning of experts reasoning at its best. explain every rule rather than simply should assist in fostering your own In the second study, also carried out presenting ‘rules without reason’. reasoning abilities; it should establish with Dr Jill Vincent, we interviewed Textbooks, and good lessons, build a feeling that mathematics makes about 20 mathematics education an understanding of mathematical sense and is not just a set of arbitrary leaders around Australia to explore results by offering a range of ‘didactic rules; and more generally, it should their responses to the notion of explanations’, including but not demonstrate the uniquely deductive the shallow teaching syndrome and restricted to age-appropriate versions character of mathematics. the place of elements of working of ‘proper’ mathematical proofs. The I will report on two related studies mathematics (including reasoning) phrase didactic explanation does not that are relevant to the question of in classroom teaching. They were imply a verbal demonstration provided how students in Year 8 learn about education department officers, by the teacher or textbook in a reasoning. The starting point for both mathematics association leaders and colloquially ‘didactic’ manner, but is these studies is an international study, textbook writers. Although the sample intended to recognise that there are the TIMSS 1999 video study, which was too small to draw firm conclusions, many useful explanations for students analysed a random sample of Year there were few obvious differences in addition to formal proofs. A didactic 8 Australian lessons and compared in responses by employment type, explanation may be evident through them with lessons from six other although the education department guided discovery, use of a manipulative countries. The video study (http:// officers were more aware of system model, a data gathering activity, or a www.acer.edu.au/research; http://www. level initiatives and the daunting scale teacher presentation. lessonlab.com/timss1999) revealed of the task of reaching all schools with Many textbooks provide more than many positive features of Australian in-depth assistance. one explanation for a result. While classrooms. However, the Australian For the textbook study, we selected multiple mathematical proofs of a result mathematics lessons displayed a cluster nine popular textbooks from four are in a sense redundant (one good of features which I call the ‘shallow Australian states, and within that chose proof suffices to prove), in teaching teaching syndrome’ (Stacey, 2003): seven topics where there was a result it is beneficial to offer multiple ways a predominance of low complexity of mathematical importance that of establishing the same result. Seven problems, which are undertaken with needed some justification or proof. different modes of explanations were excessive repetition, and an absence Examples include the angle sum of identified. In a few cases, results are of mathematical reasoning and triangles, multiplication of two negatives, proved by deduction using a general connections in classroom discourse. To the area of a circle and the rule for case, in a way that closely approximates give just one example, only 2 per cent division of fractions. For each topic and standard mathematical proofs, although of the problem solutions presented by each textbook, we examined all the at a low level of formality. Deductive teachers or students in the Australian explanations of the result presented reasoning is also evident in other ways. lessons demonstrated ‘making explicitly in the explanatory text or Since students at Year 8 do not speak connections’, i.e. showed some linking the associated electronic material algebra fluently, deduction is often not between mathematical concepts, facts devoted to that topic. The explanatory from a general case, but from a special or procedures. text typically occupied half a page, but case that is intended to be general. The first study (Stacey & Vincent, 2009) sometimes only one or two lines. We So, for example, students learned that examined the way in which textbooks asked the 20 mathematics education multiplying two negatives results in a present explanations of mathematical leaders whether they thought the positive by cleverly extending the 5 results. It is often reported that amount of classroom reasoning had times table to negative integers. Such secondary teaching is dominated by changed since the 1999 study. The expectation that students will see textbooks, and so it was of interest to introduction of better electronic the general in the particular is very

Research Conference 2010 18 common in all mathematics teaching reasoning in explanations, and it is does teachers can present these elaborations (e.g. demonstrating how to carry out not seem that prevalence of ‘textbook’ from the material provided, so this an algorithm), but the textbooks did teaching is an adequate explanation finding further highlights the often not draw any attention to the need to for the lack of reasoning evident in cited need for teachers to possess think of the specific case in a general Australian classrooms in the video study sufficiently strong mathematical way. This is one simple way in which (although related factors such as a knowledge and deep mathematical students’ appreciation of the unique prevalence of low complexity problems pedagogical content knowledge. This features of mathematical reasoning in the textbooks certainly contribute). highlights another strong theme of the could be improved, even before they However, apart from offering examples interview study, where many of the have the formal mathematical language of reasoning, there were few instances respondents expressed strong concern to deal with it well. of instruction in mathematical reasoning. that teachers teaching out-of-field Amongst the 69 instances examined, needed considerably more support Didactic explanations using inductive one exception was that two textbooks to do a good job on the working reasoning that is more appropriate to explicitly rejected measuring for finding mathematically themes. science than mathematics, are common. the angle sum of a triangle in favour Sometimes a rule is confirmed by For establishing a firmer place for of a deductive proof. In the other showing that in specific instances the mathematical reasoning in Australian exception, a textbook mentioned rule would give the same result as classrooms than it has at present, I that an explanation presented for a could be predicted from a model (for suggest the following. specific case could also be applied in all example, the result of sharing a quarter other cases, explicitly pointing to the 1 Although all aspects of working of a pizza between three people could generality that was required. Attention mathematically are taught during be shown to be the same as the to instruction in reasoning, and to engagement with the content of answer obtained by following the to-be- pointing out key elements of reasoning, mathematics, this does not mean learned rule). At other times, students would enrich the didactic explanations that they should not ever receive measure or count to empirically given. explicit attention. This applies at the discover a rule from data, such as the level of classroom tasks, classroom angle sum of a triangle is 180 degrees. We found that the nature of the discourse, unit planning and In a few instances, the textbooks made reasoning depends on the result being curriculum description. In classroom it clear that testing a few cases was explained. All textbooks had at least teaching, as in the textbooks, there not an adequate mathematical proof, one deductive explanation of the are many opportunities where but this could certainly be done more formula for the area of a trapezium, instruction in reasoning is simple to often to improve student awareness but only half contained deductive add. of reasoning. Many of the empirical explanations for the angle sum of a activities seem to us to have substantial triangle. The nature of the reasoning 2 A description is needed of a pedagogical value (as noted above, also varies from textbook to textbook developmental path in mathematical having multiple methods adds to since different books are written with reasoning across the grades, that learning), but textbooks could comment different student audiences in mind. In would give teachers, textbook that their role is in mathematical the interview study, one of the most authors and curriculum writers a discovery rather than in proof. common explanations for all features of sense of what type of reasoning the shallow teaching syndrome was the they can expect and encourage at In some cases, the ‘explanations’ difficulty of providing suitable material each level and in what directions made no contribution to developing of this nature to a mixed ability class. students’ reasoning should be mathematical thinking at all. Sometimes, Overcoming this difficulty is not as developed. This could not be as there was simply a statement or appeal simple as some people claim. specific as in the content strands, to authority (e.g. Euclid or a computer), but it could still be helpful in and others discussed loose qualitative In the textbooks, explanations were developing a shared vocabulary, analogies which may have had some generally very curtailed and usually clear goals and expectations. mnemonic value but were not omitted basic reasoning (for example, modelling the mathematical essence. stating that a finding about a specific 3 Guidance for teachers be provided case also applies in general). Hence the on the usefulness of didactic Looking over the results, it was clear explanations are unlikely to stand alone, explanations, the distinction (in that these textbooks generally paid and students must rely on teachers some cases) with age-appropriate reasonable attention to mathematical to elaborate. It is unlikely that all proof, and ways of evaluating them.

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 19 4 The major purpose of explanations in the textbooks seemed to be to derive a rule in preparation for using it in the exercises, rather than to give explanations that might be used as a thinking tool in subsequent problems. Changing this practice could give reasoning more prominence.

Acknowledgement I thank the survey participants for generously giving their time and sharing their expertise and acknowledge the financial support of the Australian Research Council Discovery Grant DP0772787 ‘The Shallow Teaching Syndrome in School Mathematics’ for part of this work. References Polya, G. (1945) How to solve it. Princeton, NJ: Princeton University Press Stacey, K. (2003). The need to increase attention to mathematical reasoning. In H. Hollingsworth, J. Lokan & B. McCrae Teaching Mathematics in Australia: Results from the TIMSS 1999 Video Study. (pp 119–122). Melbourne: ACER. Stacey, K., & Vincent, J. (2009). Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks. Educational Studies in Mathematics, 3, 271–288.

Research Conference 2010 20 The social outcomes of learning mathematics: Standard, unintended or visionary?

Abstract industry and work-centred problems. This is not necessary for all, for the Why teach mathematics? Why should depth and type of problems vary students in school learn mathematics? across employment types, and most What are our intended aims and the occupations requiring specialist outcomes of teaching and learning mathematics also provide specialist mathematics in school? To offer my training. However, a strong case can answers to these questions I find it be made for school providing the basic useful to distinguish three groups of understanding and capabilities upon Paul Ernest aims/outcomes: which further specialist knowledge and skills can be built. University of Exeter, UK 1 Standard aims of school mathematics – what are generally 3. Advanced specialist knowledge Paul Ernest is emeritus professor of philosophy agreed to be the basic or standard of mathematics education at Exeter University, reasons for teaching the subject? This knowledge, learned in high school United Kingdom, visiting professor in Oslo and or university, is not a necessary goal Trondheim, Norway, and adjunct professor at 2 Unintended outcomes of Hope University, Liverpool, United Kingdom. school mathematics – are there for all adults, but such advanced study His main research interests concern fundamental unexpected and unintended leads to a highly numerate professional questions about the nature of mathematics and outcomes of the process for some class, as exists in France, Hungary, etc., how it relates to teaching, learning and society. where all students study mathematics He has lectured and published widely on these or all students? subjects and his most cited books are The to around 18 years of age minimum. Philosophy of Mathematics Education, Routledge, 3 Visionary aims for school Advanced specialist knowledge is 1991, and Social Constructivism as a Philosophy mathematics – what do we as needed by a minority of students as a of Mathematics, SUNY Press, 1998. In 2009 he mathematics educators wish to foundation for a broad range of further was keynote speaker at the world class PME 33 see as both aims and outcomes research conference in Greece. Professor Ernest studies at university, including STEM founded and edits the Philosophy of Mathematics of school maths teaching/learning? subjects, as well as medical and social Education Journal, accessed via http://www.people. What new emphases would science studies. Clearly this option must ex.ac.uk/PErnest/. Recent special issues have enhance our students and indeed be available in an advanced technological focused on mathematics and social justice, and society beyond what we do now? mathematics and art. society, and indeed more students should be encouraged to pursue it, but The standard aims of school it should not dominate or distort the mathematics school mathematics curriculum for all. These are basic and functional goals These three categories constitute that aim to develop the following useful or necessary mathematics capabilities: for all or some, primarily for the benefit of employment and society 1. Functional numeracy from an economic perspective, as This involves being able to deploy well as sustaining mathematics and mathematical and numeracy skills mathematical interests themselves. They adequate for successful general also benefit the recipient students in employment and functioning in society. terms of functioning in society, work This is a basic and minimal requirement and further study. for all at the end of schooling, excluding only those few with some preventative Unintended outcomes of disability. school mathematics What could the unintended outcomes 2. Practical, work-related knowledge of school mathematics be? What I have This is the capability to solve practical in mind are the values, attitudes and problems with mathematics, especially beliefs that students develop during

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 21 their years of schooling that are not learnt from sport, attitudes are vital content and its use. Instead I want planned or intended, outcomes of what to success, and for students a lack to move away from content and is known as the ‘hidden’ curriculum of of confidence in their mathematical propose aims for mathematics that schooling. These concern beliefs about abilities becomes a self-fulfilling are empowering and broadening for the nature of mathematics, about what prophecy – a failure cycle (Figure 1). students. Students should develop: is valuable in mathematics, and about 4 Mathematical confidence who can be successful in mathematics. Poor confidence and maths These beliefs include: self-concept; possible 5 Mathematical creativity through problem posing and solving • Mathematics is intrinsically difficult maths anxiety and inaccessible to all but a few. 6 Social empowerment through maths ➚ ➘ (critical citizenship) • Success in mathematics is due to fixed inherited talent rather than to Reduced 7 Broader appreciation of effort. Failure at persistence mathematics. mathematical & learning • Mathematics is a male domain, and These four aims are less directly tasks opportunities is incompatible with femininity. utilitarian since they are more to maths avoidance do with personal, cultural and social • Mathematics is an abstract relevance, although ultimately I believe theoretical subject disconnected Figure 1: The failure cycle they have powerful incidental benefits from society and day-to-day life. for society, as well as for individual Take another example. Despite • Mathematics is abstract and students. progress, mathematics is still widely timeless, completely objective and seen as a male domain, and although absolutely certain. 4. Mathematical confidence girls now equal boys in mathematical • Mathematics is universal, value-free achievement at 16 years of age or Elevating this to an aim should come and culture-free. so, too many women still doubt their as no surprise given the importance own abilities and choose not to pursue I attach to attitudes as part of the Every one of these beliefs is wrong, and mathematics related studies or careers incidental outcomes of school many of my writings over the past 30 after this age, mathematics. Mathematical confidence years have been devoted to showing includes being confident in one’s this (Ernest 1991). The good news is In my view, values, images, beliefs and personal knowledge of mathematics, that a growing number of researchers attitudes about maths underlie many feeling able to use and apply it, and and teachers have come to reject these of the differences in learning outcomes being confident in the acquisition of beliefs. Furthermore, their acceptance observed across different groups new knowledge and skills when needed. has always varied greatly by country of students defined in terms of sex, This is the most directly personal and culture, so for example Asian socio-economic status and ethnicity. outcome of learning mathematics, it countries typically subscribe to the For example, in Australia, mathematics uniquely involves the development belief that mathematical success is due performance of Indigenous Australians of the whole person in a rounded to effort rather than intrinsic ability. can lag over two years behind that of way, encompassing both intellect non-Indigenous students (Queensland The bad news is that such beliefs are and feelings. Effective knowledge and Studies Authority, 2004). But a full still held by many students and parents. capabilities rest on freedom from account of such inequalities requires Such beliefs are still communicated negative attitudes to mathematics, more complex explanations involving through popular images of mathematics and the feelings of enablement and such notions as Bourdieu’s cultural widespread in society and the media, empowerment, as well as enjoyment in capital and structural inequalities and in the image of mathematics learning and using mathematics. These present in society, as well as the maths presented in some classrooms. latter lead to persistence in solving related misconceptions discussed here. difficult mathematical problems, as well One widespread outcome, although far as willingness to accept difficult and from universal, is that many students Visionary goals for school challenging tasks. Matching but inverting develop negative attitudes about mathematics the failure cycle I discussed above (see mathematics and about their own Figure 1) is the virtuous, upwardly mathematical capabilities. As we have The traditional mathematics curriculum is defined in terms of mathematical spiralling success cycle (see Figure 2).

Research Conference 2010 22 Pleasure, confidence, sense of questions and problems to be movement has spring up to deal with of self-efficacy, motivation solved, has been more neglected in theory and practice in this area. There in maths maths. But it enables the seeing of are many relevant publications such mathematical connections between as Skovsmose (1994), Ernest (2001) ➚ ➘ superficially diverse questions and and the special issue of The Philosophy topics, and the framing of questions by of Mathematics Education Journal Success at Effort, analogy. It involves seeking models for forthcoming summer 2010. maths tasks persistence, different aspects of life or mathematical and maths choice of more patterns as discovered or chosen by 7. Appreciation of mathematics students themselves. This is where overall demanding tasks The last of my proposed seven aims full creativity flowers through student or capabilities is the development of choices at every stage: problem or Figure 2: The success cycle mathematical appreciation. There is model formulation, the choice of an analogy between capability versus This cycle is one of the intrinsic methods to apply, and the construction appreciation in mathematics, on the mechanisms which draws us to of solutions. one hand, and the study of language the pleasures of success and self- versus that of literature, on the other. enhancement like a light draws a moth. 6. Social empowerment through Mathematical capability is like being Indeed we can potentially turn a failure mathematics able to use language effectively for oral cycle into a success cycle by subtracting Contrary to popular belief, mathematics and written communication, whereas risk and making success achievable. is a political subject. Mathematics should mathematical appreciation parallels In school this means reducing the be taught in order to socially and the study of literature, concerned with importance of examinations and paying politically empower students as citizens the significance of mathematics as an more attention to the quality of student in society. It should enable learners to element of culture and history, with learning experiences. function as numerate critical citizens, its own stories and cultural pinnacles, In my view this domain of attitudes, able to use their knowledge in social so that the objects of mathematics are beliefs and values is one of the most and political realms of activity, for the understood in this way, just as great important psychological dimensions betterment of both themselves and books are in literature. of learning mathematics and we for democratic society as a whole. The appreciation of mathematics itself, need to pay much more attention This involves critically understanding and its role in history, culture and to it in school. Seemingly insignificant the uses of mathematics in society: to society in general, involves a number incidents can switch a learner on or off identify, interpret, evaluate and critique of dimensions and roles, including the mathematics, and we need to be more the mathematics embedded in social, following. sensitive to this in our teaching. commercial and political systems and claims, from advertisements, such as • Having a sense of mathematics as 5. Mathematical problem posing and in the financial sector, to government a central element of culture, art solving and interest-group pronouncements. and life, present and past, which Economics is applied mathematics and permeates and underpins science, Mathematics is too often seen as a this is the main language of politics, technology and all aspects of non-creative and mechanical subject, power and personal functioning human culture. This extends from but deploying mathematical knowledge in society. Every citizen needs to symmetry in appreciating elements and powers in both posing and solving understand the limits of validity of such of art and religious symbolism, problems is the area of greatest uses of mathematics, what decisions to understanding how modern potential for creativity in school maths. it may conceal, and where necessary physics and cosmology depend Students choose which models and reject spurious or misleading claims. on algebraic equations such as approaches to use in their solutions. Ultimately, such a capability is a vital Einstein’s E = mc2. It must include Problem solving is widely endorsed, but bulwark in protecting democracy and understanding how mathematics too often focused on routine problems. the values of a humanistic and civilised is increasingly central to all aspects True problem solving, the creative use society. of daily life and experience, of mathematics, requires non-routine through its import in commerce, problems, in which new methods and Critical citizenship through mathematics economics (e.g., the stock market), approaches must be created. Problem is a major topic on its own and telecommunications, ICT, and posing, the articulation and formulation the Critical Mathematics Education

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 23 the role it plays in representing, • Understanding the ways that References coding and displaying information. mathematical knowledge is However, it must be recognised established and validated through Bourdieu, P. and Passeron, J. C. (1977) that mathematics is becoming proof is also important, as well Reproduction in Education, Society and invisible as it is built into the social the limitations of proof. I believe Culture, London: Sage. systems that both control and this should include introduction Ernest, P. (1991). The Philosophy of empower us in our increasingly to the philosophy of mathematics: Mathematics Education. London: complex societies and lives. understanding that there are big Falmer Press. questions and controversies about • Being aware of the historical whether mathematics is discovered Ernest, P. (2001). ‘Critical Mathematics development of mathematics, or invented, about the certainty of Education’. In Gates, P. (Ed.), Issues the social contexts of the origins mathematical knowledge and about in mathematics teaching, pp. 277-293. of mathematical concepts, its what type of things mathematical London: Routledge/Falmer.. symbolism, theories and problems. objects are. Being aware of such The evolution of mathematics Queensland Studies Authority. controversies supports a more is inseparable from the most (2004). Overview of statewide student critical attitude to the social uses of important developments in performance in aspects of literacy and mathematics, as well as withstanding history, from ancient societies numeracy: Report to the Minister for attributions of certainty to anything in Mesopotamia, Egypt, India Education and Minister for the Arts. mathematical. and Greece (number and tax Brisbane, QLD: Queensland Studies and accounting, geometry and • Learners should gain a qualitative Authority. surveying) via medieval Europe and intuitive understanding some of Skovsmose, O. (1994). Towards a and the Middle East (algorithms the big ideas of mathematics such as philosophy of critical mathematics and commerce, trigonometry pattern, symmetry, structure, proof, education. Dordrecht: Kluwer. and navigation, mechanics paradox, recursion, randomness, and ballistics) to the modern chaos, infinity. Mathematics contains Philosophy of Mathematics Education era (statistics and agriculture- many of the deepest, most powerful Journal (2010). Special Issue on Critical biology-medicine-insurance, logic and exciting ideas created by Mathematics Education, no. 25, and digital computing-media- humankind. These extend our Summer 2010. Accessed from http:// telecommunications). This includes thinking and imagination, as well as people.exeter.ac.uk/PErnest/ May 2010 being aware of ethnomathematics, providing the scientific equivalent of which studies informal culturally poetry, offering noble, aesthetic, and embedded mathematical concepts even spiritual experiences. and skills from cultures around the Are these aims concerning appreciation globe, both rural and urban, past feasible for school? Even big ideas and present. like infinity can be appreciated by • Having a sense of mathematics schoolchildren. Many an interested as a unique discipline, with its 8-year-old will happily discuss the central branches and concepts infinite size of space, or the never- as well as their interconnections, ending nature of the natural numbers. interdependencies, and the overall In mathematics we are privileged to unity of mathematics. This includes have around 2000 hours of compulsory its central roles in many other school time over the years – surely disciplines as applied mathematics. we can afford to spend some time After many years spent studying on these visionary aims – they have mathematics learners should have the potential to help build more some conception of mathematics as confident and knowledgeable students a discipline, including understanding and citizens, and dare I say it, a better that there is much more to society? mathematics than number and what is taught in school.

Research Conference 2010 24 Concurrent papers

Issues of social equity in access and success in mathematics learning for Indigenous students

Abstract to address the systemic marginalisation of Indigenous Australians. If the field On Western measures of education continues to research and theorise performance, such as NAPLAN, about mathematics education divorced students living in remote areas of from the reality of the teaching context, Australia are over-represented in the the field will remain impoverished and tail of performance. The gap between unable to address the systemic failure Indigenous and non-Indigenous learners of generations of Indigenous learners. in numeracy widens as students progress through school (ACARA, Robyn Jorgensen Planning 2009). This presentation explores for Attendance Griffith University the context within which this gap is Learning created and offers some suggestions Robyn Jorgensen is Professor of Education at to teachers, educational researchers Language/ Mathematics Griffith University. Professor Jorgensen has culture worked in the area of equity in mathematics and policy makers on reasons for this education for more than two decades. Her gap, but also on how the gap may be work explores how the social, political and addressed. cultural contexts contribute to the exclusion Figure 1: Planning for learning of some students as they come to learn mathematics school mathematics. The particular foci of her Introduction work have been in the areas of social class, Provision of quality learning for To develop a more holistic sense of the geographical location (rural and remote) and issues of teaching mathematics in some Indigenous contexts and learners. She recently Indigenous learners, particularly for took leave from the university sector to work students whose home culture is still of the most disadvantaged contexts in with Anangu communities in Central Australia. very strong and not contiguous with the Australian educational landscape, The immersion in the lived worlds of remote Western culture, remains an elusive I propose a model that incorporates, Aboriginal education has provided key insights but is not limited to, a number of key into the delivery of Western education in remote challenge. Developing quality learning Australia. environments for Indigenous students issues impacting on the development of requires a holistic approach to practice quality learning for Indigenous students. and policy. Keeping mathematics In this paper I contend that without education isolated from the complex regular attendance and subsequent milieu in which learning occurs fails to engagement in mathematics learning, incorporate and address the competing the issues of culture and language must demands faced by teachers and also be considered as part of the nexus education providers. In this session of mathematics education. Failure to do I consider three key elements that so, will result in the continued practices impact on mathematics teaching and that have for generations dealt failure learning: attendance, language/culture to too many students. and mathematics. All of these variables impact on how teachers and education Attendance systems plan for quality learning. Attendance is the most challenging In the model proposed in this aspect of education delivery in remote presentation, I wish to extend the communities. The need to attend thinking of mathematics educators (and engage) is perhaps the biggest to encourage a greater awareness, challenge for teachers – of mathematics recognition and embodiment of the and other subjects – in creating quality wider issues that shape, constrain learning. The pressure on schools to and enable mathematics learning. have good attendance figures means Without consideration of these other that there is a range of techniques used variables, the field of mathematics to record student attendance. Typically education is impoverished and unable students may appear to be marked as

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 27 Table 1: Secondary school attendance by Indigenous status and age, 2006 understandings are not evident, so holding high expectations may be a Age in Years Indigenous % Non-Indigenous % worthy ideal, the practical ramifications 15 73 89 for secondary-aged students requires 16 55 81 a primary level of work. This renders the ‘high expectations’ as misplaced in 17 36 66 terms of benchmarking activities. (Source: ABS, 2010) Language and culture Table 2: Secondary school attendance by remoteness area by age, Indigenous persons, 2006 In many remote areas, home culture is still a strong part of the life worlds Age in Major Inner Outer Very of Indigenous students. These cultural Remote years cities regional regional remote activities impact on learning in many 15 % 77 77 76 67 53 ways. First, cultural events can demand time out of school. In Central Australia, 16 % 60 58 60 49 34 Men’s Business may require many 17 % 44 38 37 29 16 young fellas to be out of school for a (Source: ABS, 2010) month or more, as well as the impact on the community members through attending, but the reality is that they outcomes so that for any cohort of which Men’s Business is undertaken. may have appeared for only a short students, the variance in performance Other cultural events, such as Sorry time in the day. As such, attendance levels is considerable. This makes Business, similarly impact on attendance. figures are often significantly inflated in planning for learning complex and In Northern Arnhem land there have terms of the real number of students unpredictable. The frustration caused been moves to shift school terms to attending. This rolling attendance to teachers by non- or irregular allow for the extended cultural activities presents unique problems for the attendance has a devastating effect over the wet season which may go teaching of mathematics. Not only is for many teachers on their sense of for several months. Collectively, these attendance irregular over a period of identity. As one teacher commented, events take priority over schooling, time, but also over the day. As such, ‘I did not spend four years training to thus resulting in substantive periods of both short-term and long-term planning have a class with no students turning missed school. are compromised. up.’ At a more local level, culture impacts As can be seen in Table 2, for With overall poor attendance, teachers on the interactions in classrooms. This secondary Indigenous students, in remote areas are faced with may be in the way that the students attendance rates at school decreases substantive issues in how to address the interact with the teacher and/or with the level of remoteness. Similar significant gaps in learning. While there community. The styles of interaction trends occur for primary school is a considerable push from Indigenous and questioning are often different from students. For example, for 17-years- educators such as Chris Sarra (1995) those of mainstream education. For olds living in major cities, 44 per cent to have high expectations of learners, students coming into school, there is of Indigenous students attend school. In this goal can be somewhat misplaced. a need to constitute their Indigenous contrast, only 16 per cent of 17-year- The issues around attendance means habitus to enable them to access the old Indigenous students living in remote that while the teachers may hold dialogic patterns in order to ‘crack areas attend school. high expectations of learning in the code’ of classroom practice. For mathematics, the levels of achievement example, posing questions in classrooms Teacher morale is seriously and understandings are quite limited – such as ‘What is the sum of 15 and compromised by poor attendance. for students. This makes the high 23?’ – is met with a barrage of answers. Never sure if there will be 1 or 2 expectations mantra difficult due to the Students play a different game to the students or 20 students, teachers are very limited achievement and need for teacher. While the teacher’s game is required to be professional and prepare backfilling of mathematical ideas. The one in which he/she is seeking the as if there will be a full contingent gaps for many Indigenous learners are students to add two numbers and of students attending. However, the profound. Many basic concepts and come to a total of 38, the students’ poor attendance is reflected in learning game is one of responding with any

Research Conference 2010 28 answer. These two dialogic patterns are differences in meaning are significant. differences make for very different quite different in goal so that there is As has been identified in other learners assumptions that underpin learning considerable scope for misrecognition of mathematics (Zevenbergen, Hyde, & activities. of the outcome. Power, 2001), the skills learnt in reading In many remote communities, the texts mean that skimming is a well Language and culture are intrinsically absence of number in their world views developed strategy, yet in mathematics intertwined so that the culture is is obvious. The need for number is the highly contracted language means represented through language. As the relative to the region. As Wittgenstein that such a strategy is very misplaced. language game above indicates, the (1953) argued strongly, our knowledge goals of the teachers may be different systems derive from and are shaped from those of the students but these Temporality by the language games that are played goals are intrinsically interwoven with Many Indigenous cultures live in out in a particular system. The need the cultures. In Pitjantjatjara, language the here and now so that long- for number in remote areas is limited. use is very frugal so that there is often term planning is a foreign/elusive For coastal mobs, where trading was little said and what is said is very concept. Yet planning underpins more likely a keener sense of number is contracted. The language structure much of Western thought. There are more relevant, but this is not the case is one with brevity in speech. This is considerable examples of how the non- in remote areas. Many students do not evident in the language developed planning of Indigenous practices and know their age or birthday; few have within the context of desert people. events are at loggerheads with Western phones in the home; streets are not ways of thinking. The need to plan a named or numbered; there is no need Prepostions long trip in the desert is undertaken for large numbers. Their life worlds with a strong sense of gravity as it can shape the need for number (or other In Pitjantjatjara, there are less than 10 mean life and death. Yet, for many mathematical ideas/concepts). prepositions, whereas English has more Indigenous people, the trip is one of than 60. If the language of mathematics While number may not be a strong opportunity as the sense of life and is considered in concert with the aspect of many Indigenous cultures, death is not as paramount due to their pedagogic relay where concepts are the sense of space is acute. In a intimate knowledge of the desert and taught/learned through language, the comprehensive study of Yolngu life survival. These two very different world use of prepositions in coming to learn worlds, Watson and Chambers (1989) views impact on the primary goal of mathematics is profound. As has been documented the complex ways in much of what is taught in schools and argued elsewhere (Zevenbergen, 2000, which land was signed. For Yolngu, the home cultures. 2001), coming to learn mathematics the land was marked by cultural and is heavily associated with the use of historical events. These landmarks prepositions. How one learns number Mathematics were ‘sung’ to younger generations sense is through comparisons and place. In drawing together absenteeism and who internalised these stories and so Consider the following statements culture, the impact on mathematics developed a sense of their land. These – Which number is bigger than 4?; becomes obvious. In remote stories are markedly different from Which number is 2 more than 6?; communities, there is a lack of number those of Western conventions, yet Which number comes before 3?; Which and text so that immersion in number serve to make strong connections to number comes after 11? These little is difficult in remote communities. the land. words are significant in how students Some of the fundamental assumptions learn the value and order of numbers. made in Western world views are Planning for quality very different from those of the Imagine the difficulties of Indigenous learning bush. Travelling along a dirt road learners, who often have hearing may be measured in kilometres, In order to create environments that problems, differentiating between off with particular markers at particular support access and success in school and of. In Pitjantjatjara for example, distances. However, travel in outback mathematics for Indigenous learners, there is no ‘f’ sound, so terms such as roads is marked by other significant the three key factors that have been ‘football’ is pronounced as ‘pootball’. In bearings – such a landmarks or man- identified in this paper must be trying to hear the difference between made markers rather than a particular considered in concert with an emphasis off and of when there is no sound distance. Similarly, the quality of roads on planning for learning. The learning in the home language would be very at a point in time is more profound is for both teachers and students. The difficult. Yet, in mathematics, these than the distance to be travelled. These reality for teaching in remote areas is

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 29 that the teaching force is predominantly develop innovative models of planning and learning (pp 201–223). Westport, early career teachers who have had for diversity in learning needs and CT: Ablex. little or no exposure to remote demands of remote education. Working Zevenbergen, R. (2002). Mathematics, education, to working with Indigenous within the existing dominant paradigms social class and linguistic capital: An students and communities and to will not yield the outcomes required analysis of a mathematics classroom. teaching as a profession. Collectively for successful Indigenous education In B. Atweh & H. Forgasz (Eds.), these experiences contribute to the participation and/or outcomes. Social-cultural aspects of mathematics identified difficulties with retaining education: An international perspective teachers in remote areas. The high References (pp. 201–215). Mahwah, NJ: Erlbaum. turnover rates can be seen to be indicative of the challenges of remote Australian Bureau of Statistics. (2010). Zevenbergen, R., Hyde, M., & Power, education. This claim is not new and Indigenous statistics for schools. http:// D. (2001). Language, arithmetic word the issues have been recognised for www.abs.gov.au/websitedbs/cashome. problems and deaf students: Linguistic some time as can be seen in the nsf/4a256353001af3ed4b2562 strategies used by deaf students to Human Rights and Equal Opportunities bb00121564/be2634628102566bc solve tasks. Mathematics Education Commission report: a25758b00116c3d!OpenDocument Research Journal, 13(3), 204–218. Accessed May 15, 2010. … schools may suffer from high teacher turnover, a lack of Australian Curriculum, Assessment and specialist services, a restricted Reporting Authority. (2009). http:// range of curriculum options www.naplan.edu.au/reports/national_ and a high proportion of young report.html. Accessed May 15, 2010. inexperienced teachers. Sarra, C. (2005). Strong and smart: (Commonwealth Schools Commission, Reinforcing aboriginal perceptions 1975: 75–79) of being aboriginal at Cherbourg Coming into remote contexts to teach state school. Unpublished PhD: Indigenous students whose attendance Murdoch University http://wwwlib. is often low, who have gaps in their murdoch.edu.au/adt/browse/view/adt- mathematical understandings, whose MU20100208.145610 culture and languages are significantly Stokes, H. Stafford, J. & Holdsworth, different from mainstream schools, R. (unknown). Rural and Remote creates a set of challenges that need school education: A survey for the to be addressed. Teachers need to Human Rights and Equal Opportunity develop skills that will enable them Commission. Melbourne: Youth to learn to plan and adapt to these Research Centre, University of circumstances. Appropriate access Melbourne. http://www.hreoc.gov. to such skill development is critical if au/pdf/human_rights/rural_remote/ successful change is to be implemented. scoping_survey.pdf. Accessed May 12, However, this must also be considered 2010. within the constraints imposed by economics, geography and available Watson, H. & Chambers, W. (1989) resources for such skill development. Singing the land, Signing the land. Further compounding the issue of Geelong: Deakin University Press. professional development is the risk of Wittgenstein, L. (1953). Philosophical investment in staff where there is a high investigations. Oxford: Blackwell turnover. Zevenbergen, R. (2000). ‘Cracking the Planning for quality learning must take code’ of mathematics classrooms: into consideration these multiple factors School success as a function in order to enable access and success of linguistic, social and cultural for Indigenous learners. Neophyte and background. In J. Boaler (Ed.), Multiple established teachers need to be able to perspectives on mathematics teaching

Research Conference 2010 30 Primary students decoding mathematics tasks: The role of spatial reasoning

Abstract society and the increasing challenge of representing burgeoning amounts Representation is an important aspect of information in visual and graphic of mathematics. In recent years forms. The amount of information graphics representations have become at an individual’s disposal and the increasingly widespread as society extent to which this information can comes to terms with the information be manipulated and directed toward age. Although the mathematics curricula specific purposes has also increased have not varied to any recognisable (e.g., the detailed information available Tom Lowrie degree in the past decade or so, the for weather forecasts). From a young assessment procedures associated age, children are exposed to visual Charles Sturt University with mathematics education certainly forms of communication with more have. This presentation highlights the intensity and engagement, whether Tom Lowrie is Director of the Research Institute changing nature of students’ spatial for Professional Practice, Learning and Education playing computer games, navigating web (RIPPLE) at Charles Sturt University. Professor reasoning as they engage with different pages, or interpreting the rich design Lowrie’s previous positions included working as types of mathematics representations. features of more traditional pictorial a primary school classroom teacher, teaching A case is presented which describes representations, and as a consequence mathematics education and research method the shift from students’ use of courses to undergraduate and postgraduate different forms of sense making are students at CSU and working with classroom encoding techniques to represent required. teachers on curriculum frameworks. Previous mathematical ideas to an increasing administrative positions include being the Head, reliance on students decoding graphical Within education contexts increased School of Education and acting Dean of the representations constructed by others. attention has been given to the role of Faculty of Education at CSU. The presentation analyses a number representation in school mathematics A substantial body of Professor Lowrie’s research of student work samples as they were (e.g., National Council of Teachers is associated with spatial sense, particularly videotaped completing assessment of Mathematics [NCTM] Yearbook, students’ use of spatial skills and visual imagery to 2001). Mathematical representations solve mathematics problems. He has co-authored items from the National Assessment Mathematics for children: Challenging children to Plan for Literacy and Numeracy have always been viewed as an integral think mathematically (now in its third edition) (NAPLAN). Implications from the study component of the ideas and concepts and has been the Editor of the Australian Primary include the recognition that students used to understand and engage with Mathematics Classroom Journal. Professor Lowrie’s mathematics (NCTM, 2000); however, current research projects include Australian need to acquire different spatial- Research Council grants which examine young reasoning skills which allow them to the structure of these representations students’ ability to decode information graphics consider (and navigate) all the elements continue to evolve. In this presentation in mathematics and Mathematics in the digital of a mathematics task, including I argue that the nature and degree of age: Reframing learning opportunities for influence mathematical representations disadvantaged Indigenous and rural students. specific features of a graphic and the surrounding text. have on teaching and learning contexts have changed and these changes have Introduction emerged almost unnoticed. Although mathematics curricula has Representations tend to fall under two changed little in the past ten years systems, namely internal and external the way in which mathematical ideas representations. Internal representations are represented and communicated are commonly classified as pictures ‘in has shifted dramatically. Until recently, the mind’s eye’ (Kosslyn, 1983) and most mathematics tasks that primary- include various forms of concrete and aged students were required to solve dynamic imagery (Presmeg, 1986) were heavily word based, whereas the associated with personalised, and current practice, from both curriculum often idiosyncratic, ideas, constructs and assessment perspectives, is to and images. External representations have more graphics embedded include conventional symbolic systems into task representation (Lowrie & of mathematics (such as algebraic Diezmann, 2009). This is unsurprising notation or number lines) or graphical given the increased use of graphics in representations (such as graphs and maps).

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 31 Although these two systems do not Mandatory assessment practices, such have different purpose, structure and exist as separate identities (Goldin as the National Assessment Plan for orientation. & Shteingold, 2001), there is some Literacy and Numeracy (NAPLAN) One of our current investigations scope (and benefit) for thinking of (MCEETYA, 2009), foster this change (Lowrie & Logan) has set out to these two forms of representation in in student information processing. The consider the influence encoding and different ways. Internal representations structure and nature of NAPLAN-like decoding processes have on primary- often involve the process of encoding tasks promote decoding, especially in aged students’ mathematical thinking as information. Encoding generally occurs situations where students are required they complete tasks in the NAPLAN. when students construct their own to generate a multiple-choice solution. Grade 3 and 5 students (N = 45) representations in order to solve a Our studies (e.g., Lowrie & Diezmann, who sat the 2010 NAPLAN were task. Encoding techniques include 2009) have shown that students are interviewed on the 2009 NAPLAN drawing diagrams, visualising and reluctant to actually draw on their before attempting this year’s paper. spatial reasoning. These techniques test booklets when they complete Students were videotaped as they provide students with the opportunity questions in the NAPLAN. Other solved the tasks and explained their to understand all the elements of forms of encoding, including internal solutions to ten items from the any given problem in a way that is representations, are seldom evoked respective grade NAPLAN tests. The meaningful to them, for example, since the answer to the questions interview protocol encouraged the drawing a circle and dividing it into generally appear on the page and this students to verbalise their thinking segments in order to better understand thus reduces the likelihood of students and to represent their thinking in ways a fraction problem. By contrast, utilising other forms of imagery. they felt appropriate (i.e., writing down decoding techniques are used to Moreover, the types of questions numbers or drawing a picture). The make sense of information within a posed typically require students to semi-structured interview allowed given task, when the information has decode information from the graphics students the opportunity to reflect been represented visually for others embedded in the task. By providing upon an experience that is otherwise to solve, for example, interpreting a graphical representation to scaffold only a quantitative measure of a map to determine the coordinate thinking, a whole new set of skills and performance. position of a specific street crossing. practices is brought to the fore. The Ten years ago, a high proportion capacity to interpret various forms Representation and sense of mathematics tasks were word- of information is now required for problem based and teachers explicitly students to solve tasks and these skill making with graphic-based taught heuristics which included ‘draw sets are quite different to those needed tasks a diagram’, or ‘imagine the problem when encoding information. Of the 75 items across the Grade 3 scene’. These approaches required and Grade 5 tests, few items would encoding of information. Currently, a Encoding and decoding be classified as traditional word-based high proportion of tasks have a diagram information in mathematics problems. In fact, only 13 of the 35 embedded in the representation. As a Grade 3 items (37%) and 15 of the 40 With colleagues I have been consequence, it is hard for students to Grade 5 items (38%) did not contain investigating students’ encoding (Lowrie think beyond the diagram to construct a graphic within the task. Moreover, & Logan) and decoding (Diezmann representational meaning and thus only 15 items (20%) across the two & Lowrie, 2008; Lowrie & Diezmann, approaches to problem solving now are tests would be considered traditional 2007; Logan & Greenlees, 2008) more likely to require decoding skills. word problems. The students seldom skills as they solve mathematics tasks utilised encoding skills to solve the This presentation considers the commonly used as assessment items. tasks, especially internal representations changing nature of mathematics The work on encoding has focused like drawing a diagram and constructing representation in classroom on the extent to which students utilise personal images or representations. practices, and an evolution in student pictures or diagrams to make sense When students did construct such engagement – where students are of tasks and the extent to which they representations, they were almost increasingly required to decode evoke imagery to contextualise the entirely on tasks for which a graphic information but at the same time are problem. The studies that investigate was not embedded within the task (see less likely to experience situations in students’ decoding skills have Figure 1). Thus, when a task contained which they are challenged to encode considered the extent to which children an external graphic representation, mathematics ideas and representations. make sense of information graphics that

Research Conference 2010 32 students were unlikely to create a Given the high proportion of the tasks With regard to Figure 2, the student personalised internal representation as in each test containing graphics, it was located the position of the library as part of their sense making. not surprising that students frequently the starting point. In order to complete utilised decoding techniques to solve the task, the student rotated the With regard to Figure 1, the student the tasks. In these situations, the map to the right (see Figure 3) as a drew circles to represent the cakes students did not have any markings way of ensuring she could follow the and enclosed each group of five circles and thus did not draw diagrams or subsequent directions. This meant she with a square to represent a box. He pictures to scaffold their understandings. was facing the library as opposed to then proceeded to keep a tally (in In relation to the students decoding standing in front of the library. She then his head) of the number of ‘cakes’ he (see Figure 2), the graphics generally turns right along High Street, which is had represented until he reached 34. had an important part to play in the in fact left of the library. Consequently, He then argued that 7 boxes were task solution. In some situations, the she answered this task incorrectly. She required. This type of procedure graphic merely provided a context for had her hands on the page following represents a common encoding the task; however, in most situations, the route with her fingers as she technique utilised by students to solve the information contained within the proceeded to work out the task. This word problems. graphic was indeed influential. example highlights the necessity of correctly decoding the graphic (in this instance a map task) in order to generate an appropriate solution. The presentation will provide a number of examples which highlight the ways children encode and in particular, decode graphical representations in mathematics tasks. Implications Several practical implications emerge from the study.

Figure 1: Example of a student using an encoding technique

Figure 2: An example of a task that requires decoding using spatial Figure 3: The same task represented in the reasoning and mental imagery orientation the student used to solve the item

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 33 • The movement away from a graphic and the surrounding text, Ministerial Council on Education, traditional word-based problem when solving mathematics tasks. Employment, Training and Youth solving limits students’ opportunities Affairs [MCEETYA] (2009). National to utilise encoding techniques References assessment program: Literacy and to make sense of mathematics numeracy. Grade 3 and 5 Numeracy. Diezmann, C. M., & Lowrie, T. (2008). ideas. If these encoding skills are Retrieved 6 February 6, 2010 from: Assessing primary students’ knowledge not encouraged and promoted http://www.naplan.edu.au/tests/ of maps. In O. Figueras, J. L. Cortina, elsewhere, students’ general naplan_2009_tests_page.html reasoning skills will be restricted S. Alatorre, T. Rojano, & A. Sepúlveda, National Council of Teachers of since such techniques are necessary (Eds.), Proceedings of the Joint Meeting Mathematics. (2000). Principles and when students encounter novel or of the International Group for the standards for school mathematics. complex problems. Psychology of Mathematics Education 32, and the North American chapter Reston, VA: Author. • Conversely, the introduction of XXX (Vol. 2, pp. 415–421). Morealia, Presmeg, N. C. (1986). Visualisation mathematics tasks rich in graphics Michoacán, México: PME. in high school mathematics. For the requires a different skill base. Learning of Mathematics, 6(3), 42–46. Explicit attention needs to be given Goldin, G., & Shteingold, N. (2001). to specific types of graphics since Systems of representations and they have different structure and the development of mathematical conventions. Teaching map-based concepts. In A. A. Cuoco (Ed.), graphics, for example, requires The roles of representation in school different approaches and techniques mathematics (pp. 1–23). Reston, than graph-based graphics. Indeed VA: National Council of Teachers of bar graphs and line graphs require Mathematics. specific and independent attention. Kosslyn, S. M. (1983). Ghosts in the • Given the increasing reliance mind’s machine. New York: Norton. of graphics in society, it is Logan, T., & Greenlees, J. (2008). not surprising that graphic Standardised assessment in representations hold a prominent mathematics: The tale of two items. place in current forms of In M. Goos, R. Brown & K. Makar assessment. And since assessment (Eds.), Navigating currents and charting tends to influence and even directions. Proceedings of the 31st drive practice, the way in which annual conference of the Mathematics mathematics ideas and conventions Education Research Group of are represented impact greatly Australasia, Vol. 2, pp. 655–658. on teaching practices and student Brisbane, QLD: MERGA. learning. Lowrie, T., & Logan, T. (2007). Using • Students are required to decode spatial skills to interpret maps: external representation with more Problem solving in realistic contexts. regularity than the process of Australian Primary Mathematics evoking internal representations Classroom, 12(4), 14-19. through encoding. Although Lowrie, T., & Diezmann, C. M. (2007). both require high levels of spatial Solving graphics problems: Student reasoning, most representations are performance in the junior grades. The now ‘teacher’ generated rather than Journal of Educational Research, 100(6), student constructed. 369–377. • Students need to acquire different Lowrie, T., & Diezmann, C. M. (2009). spatial-reasoning skills which allow National numeracy tests: A graphic them to consider all the elements of tells a thousand words. Australian a task, including specific features of Journal of Education, 53(2), 141–158.

Research Conference 2010 34 Promoting the acquisition of higher-order skills and understandings in primary and secondary mathematics

Abstract ‘constructive alignment’ (Biggs, 1996). It is the position of the author that What do we mean by higher-order the SOLO (Structure of the Observed skills? How do students develop higher- Learned Outcome) model (Biggs & order skills, and utilise abstract ideas Collis, 1982; 1991; Pegg, 2003) meets or concepts? How can we promote these requirements and provides a the acquisition of higher-order theoretical underpinning for assessment understandings in a classroom situation? and instruction decisions taken by This session considers these questions teachers. and the reasons for the difficulties and John Pegg challenges teachers face in addressing The ideas reported here draw on data University of New England the need to promote higher-order from three large-scale longitudinal understandings in their students. The studies, involving the SOLO framework, John Pegg began his career as a secondary research reported draws on data from with primary and secondary teachers mathematics teacher. Currently he is Professor three large-scale longitudinal studies in NSW. This paper draws from and Director of the National Centre of Science these studies ideas associated with ICT and Mathematics Education for Rural and carried out with primary and secondary Regional (SiMERR) Australia at the University teachers. The approaches are consistent the development of higher-order skills of New England, Armidale. SiMERR programs with recent research findings on and understandings. The use of SOLO identify and address important educational issues cognition and brain functioning, and emphasises the integral role assessment of (i) specific concern to education in rural and practices play as part of normal regional Australia, and (ii) national concern to provide insight into how such skills educators across Australia but ensuring rural and are developed in students. Participants classroom activity with the information regional voices are strongly represented. will consider practical ways to create obtained being used to inform, monitor His work is far ranging, and is particularly conditions that increase the likelihood and promote student learning (Black & known internationally and nationally for its of higher-order skills and understandings Wiliam, 1998). contribution to theory-based cognition research in their students. in mathematics education and assessment. The findings of these studies Recently he has been involved in many large- illustrated dramatically the value such scale nationally significant projects linked to: Introduction a framework plays when groups of underachieving students in literacy and basic teachers interpreted student responses Mathematics, statewide diagnostic testing There is little evidence of systematic programs in science, developmental-based use of cognitive-based research to assessment tasks and plan how assessment and instruction, the validation of to influence wide-scale curriculum responsive instruction might proceed. the NSW professional teaching standards, and developments, or their associated Without a framework such as SOLO, the ÆSOP study investigating faculties achieving teachers could offer little guidance on outstanding student learning outcomes. assessment and instruction practices (Pegg & Panizzon, 2001). Significantly, how they might decide consistently and central to this paper, if assessment and across a range of activities whether and teaching practices are to improve, assessment items were appropriate, then such practices must rest on whether student responses to theoretical bases for learning which assessment items were adequate, provide useable information to what skills and understandings students teachers to guide their thinking and possessed, and where instruction might subsequent teaching actions (Pellegrino, be directed most profitably in the Chudowsky, & Glaser, 2001). future. Further, any theoretical position In this paper we consider: What is adopted must be empirically based meant by higher-order skills? How will and not simply rely on ‘logic’ for students acquire higher-order skills its rationale. The theory must offer and utilise abstract ideas or concepts? teachers the opportunity to achieve In what ways can we promote the the synchronisation of the three arms acquisition of higher-order skills and of curriculum – assessment, pedagogy, understandings in a classroom? and syllabus content – thus achieving

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 35 Higher-order skills and many hundreds of published articles. the stimulus that are typical of the next understandings SOLO is a model for categorising mode of understanding. the responses of students in terms of The strength of the SOLO model is What do we mean by higher-order structural characteristics. skills and understandings? Probably the linking of the hierarchical nature the best-known description is offered The focus of the SOLO categorisation of cognitive development through by Bloom’s Taxonomy, named is on cognitive processes rather the modes and the cyclical nature after the leader of the group of than the end products alone. The of learning through the levels. Each academics in 1956 that released the task of the teacher is to analyse level provides building blocks for the Taxonomy of Educational Objectives. the pattern of ideas presented by next higher level. SOLO also provides There are six categories to Bloom’s the student. SOLO facilitates the teachers with a common and shared Taxonomy. These are: knowledge, successful completion of this task by language that enables them to describe comprehension, application, synthesis, providing a balance between structural in a meaningful way their observations analysis and evaluation. Knowledge and complexity and content/context. In of student performance. This is comprehension are seen as important SOLO, development is dependent particularly important when teachers lower-level skills and are concerned upon the nature or abstractness of try to articulate differences between with remembering information and the task (referred to as the mode) lower-order and higher-order skills and basic understanding. Higher-order skills and a person’s ability to handle, with understandings. involve application (using knowledge), increased sophistication, relevant cues analysis, synthesis and evaluation. (referred to as the level of response). SOLO and higher-order functioning While Bloom’s Taxonomy has come SOLO comprises five modes of under increasing criticism leading to functioning referred to as sensori-motor, The most common modes for review (Anderson et al., 2001), the iconic, concrete symbolic, formal and instruction for primary and secondary basic ideas still offer help to teachers, post formal. Learning can occur in one mathematics are the concrete symbolic in advance of testing, to identify of these modes or be multi-modal. mode (becoming available on average assessment items that target different Within each mode are series of three about 5–6 years of age) and the formal categories of quality. The issue here levels of response. A unistructural mode (becoming available around is that the category of a particular response is one that includes only one 15–16 years of age). In SOLO the question does not usually provide relevant piece of information from levels are ordered within a mode, insight into the level of a student’s the stimulus; a multistructural response with students entering the field picking response. is one that includes several relevant up single aspects, then multiple but independent pieces of information from independent aspects, and finally SOLO adopts a different position, the stimulus; and a relational response integrating these separate aspects into a namely, that ‘there are “natural” stages is one that integrates all relevant pieces cohesive whole. in the growth of learning any complex of information from the stimulus. These material or skill’ (Biggs & Collis, 1982, three levels comprise a U-M-R cycle of It is the answers coded at the p. 15). The model seeks to describe development. unistructural and multistructural levels this growth sequence through a series that are seen as lower-order responses. of modes of understanding and levels Having achieved a relational level Here the students recall single or of performance within these modes. response in one cycle, students move multiple ideas, know basic facts, and SOLO levels provide teachers with a to the next level that represents a are able to undertake routine tasks by convenient way to label portions of the new unistructural level in a new cycle. applying standard algorithms. continuum for practical purposes. This enhanced unistructural response represents (i) a consolidation of the Higher-order skills commence at the relational level. This arises through the SOLO model previous relational response into a single more succinct form within the ability to integrate information and The relevance of SOLO to higher-order same mode, or (ii) a new unistructural make personal connections resulting functioning is that it is an empirically response that not only includes all in using this knowledge in related verifiable assessment framework relevant pieces of information, but but new areas. Here students are designed for use in classrooms. Over also extends the response to integrate able to: demonstrate some flexibility the past 30 years, SOLO has built a relevant pieces of information not in in their work; undertake problems substantial empirical base involving without relying on step-by-step learnt numerous research studies resulting in algorithms; see novel connections not

Research Conference 2010 36 previously taught; have an overview of memory. The current consensus is • Freeing up of resources at lower the concept under consideration and that working memory and short- levels allows students to focus on how different aspects of the concept term memory are distinct. Short-term inherently attention-demanding are linked; show insight – able to memory is associated with information higher-order cognitive activities. undertake ‘new’ questions; and provide that is held for short periods of time reasonable evidence of understanding. and reproduced in an unaltered Implications for learning III fashion. Long-term memory is where The relational level response is a • At the unistructural and permanent knowledge is stored for long precursor to more abstract thinking that multistructural levels relevant periods of time. Individuals access and occurs in the subsequent mode (the information can be ‘taught’ in the work on this stored knowledge through formal mode) where students are able traditional sense. their working memory. to work with relationships between • At the relational level, ‘teaching’ in concepts as their thought processes Implications for learning I a traditional sense is problematic as become more abstract and they move students need to develop their own away from the need for concrete • Human intelligence comes from connections – their own way. referents. They are able to formulate stored knowledge in long-term their own hypotheses, develop their memory, not long chains of • Language development is own models, work in terms of general reasoning in working memory. important in developing students’ principles, and construct their own understanding and reducing • Skilled performance consists of mathematical arguments. working memory demands at the building chains of increasingly multistructural level – establishing a complex schemas in long-term strong basis for relational responses. Ideas about cognitive memory by combining elements architecture consisting of low-level schemas into • Students can respond by rote What determines the SOLO levels high-level schemas. at relational levels without understanding and hence give for particular students? The answer • A schema can hold a huge amount the impression of having attained seems to encompass six main ideas. of information as a simple unit in higher-order skills. These are: general cognitive abilities of working memory. the student; familiarity of the content; presentation of the task; degree of • Higher-order processing occurs Implications for teaching when there is ‘sufficient space’ interest or motivation of the student; Once students can respond in working memory so that amount of relevant information that consistently at the multistructural appropriate schemas can be can be retained simultaneously for this level, with appropriate language skills, accessed from long-term memory task; and the amount of information teachers should focus on creating and worked upon. processing required for a solution. an environment to promote SOLO relational responses. Such an approach These last two points are particularly Implication for learning II important to this discussion as they encourages students to integrate their lead to the notion of working memory. • Improved automaticity in understanding of individual ideas and Working memory is a theoretical fundamental/basic skills, such as see connections and elaborations construct and is usually defined as the calculating, at lower levels frees not previously met. Attempting non- ability to hold information in the mind up working memory resources for routine problems is one important while transforming or manipulating it. processing higher-order skills and way in achieving high-order skills and Working memory is used to organise, understandings. understandings as, in general, these contrast, compare, or work on • Deliberate practice at the questions require at least relational information. Working memory is limited unistructural level reduces the responses. Generally, with non-routine in capacity and duration. As we become demands of working memory on questions, there are no prescribed more expert in a task, our working these concepts. algorithmic approaches. memory capacity does not increase but Examples of how to generate such it does become more efficient. • If at the unistructural level, working memory demands are reduced, the environments include providing There is some conjecture about the growth of multistructural responses students with: relationship between working memory is facilitated. and both short-term and long-term

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 37 • the answer to a problem and strategies as students move through Bloom, B.S., (Ed.) (1956). Taxonomy of having them generate questions, i.e., levels acquiring new knowledge. educational objectives: The classification reversibility of educational goals: Handbook I, An implication of the SOLO hierarchy cognitive domain. New York: Longman. • more information than the question/ is that higher-order skills and problem requires understandings in the mathematics Pegg, J. (2003). Assessment in classroom are built upon the acquisition Mathematics: a developmental • less information than the question/ of lower-order skills and understandings. approach. In J.M. Royer (Ed.), Cognition problem requires. They have a symbiotic association in and mathematics teaching and which: (i) the relational level represents learning. New York: Information Age Conclusion the start of higher-order functioning; Publishing. Higher-order skills and understandings and (ii) the unistructural level Pegg, J., & Panizzon, D. (2001). are more difficult to learn and represents higher-order functioning Determining levels of development in to teach, as they require more for an earlier growth cycle and at the outcomes-based education: Nice idea, cognitive processing and different same time the beginning of lower-order but where is the research-base for the forms of instruction. Such skills and functioning in the current cycle. decisions taken? Paper presented at understandings are prized as they Finally, working from a developmental the American Educational Research allow knowledge to be owned by the cognitive perspective, such as the Association Conference in Seattle, on individual and, hence, applied in novel SOLO model, exposes as fanciful and 10–14th April, 1–5. ways to different situations. Teachers counter productive ‘commonsense’ should orchestrate, at the appropriate Pellegrino, J. W., Chudowsky, N., & expectations of teachers: ‘that almost times, environments for higher-order Glaser R. (Eds.) (2001). Knowing what all the time their students should be mathematical thinking activities to take students know: The science and design engaged in higher-order thinking’. place on the syllabus content being of educational assessment. Washington: covered in class. References National Academies Press. For the successful development of higher-order skills and understandings, Anderson, L. W., Krathwohl, D. R., activities of instruction and assessment Airasian P. W., Cruikshank, K. A., need to be closely intertwined. In Mayer, R .E., Pintrich, P. R., Raths, particular, formal testing and informal J., & Wittrock, M. C. (Eds.) (2001). formative assessments need to inform A taxonomy for learning, teaching, teaching. Considering assessments and assessing: A revision of Bloom’s this way will help teachers understand taxonomy of educational objectives. where students are in their learning Addison Wesley Longman. journey, and better facilitate the focus Biggs, J. (1996). Enhancing teaching of instruction to meet the actual needs through constructive alignment. Higher of students. Education, 32, 347–364. Important in this movement from Biggs, J., & Collis, K. (1982). Evaluating lower-order to higher-order skills the quality of learning: The SOLO and understandings is the use of an taxonomy. NY: Academic Press. evidence-based cognitive framework. Biggs, J., & Collis, K. (1991). Multimodal This paper advocates the SOLO learning and the quality of intelligent model as one suitable framework. behaviour. In H. Rowe (Ed.), With such a model, teachers have Intelligence: Reconceptualization and at their disposal signposts along a measurement (pp. 56–76). Melbourne: continuum of cognitive development. ACER. One obvious consequence is that such a framework helps explain when it is Black, P., & Wiliam, D. (1998). most appropriate to address higher- Assessment and classroom learning. order skills and understandings, and Assessment in Education, 5(1), 7–74. when to consider different instructional

Research Conference 2010 38 Mathematics assessment in primary classrooms: Making it count

Abstract Alternatively, it may be formative and used to change teaching and learning Much has been written about approaches. assessment of learning, assessment for learning and assessment as learning. Consider this scenario observed in a These three conceptions of assessment Tasmanian primary school: are examined in relation to primary The teachers are meeting in grade mathematics. Drawing on research teams. They are sharing the ‘big from Australia and overseas, effective books’ about mathematics that Rosemary Callingham practices in mathematics assessment the children in their class have in the primary classroom are identified produced. The discussion centres University of Tasmania and the implications for teaching and on what the books demonstrate learning considered. about the children’s understanding, Rosemary Callingham is Associate Professor in Mathematics Education at the University of and what the teachers need to Tasmania. She has an extensive background in Introduction do to move that forward. In the mathematics education in Australia, at school, Assessment practice has been an discussion, teachers compare system and tertiary levels. Her experience the work samples and make includes classroom teaching, mathematics ongoing focus of educational research curriculum development and implementation, for over a quarter of a century. In that judgements about their own and large-scale testing and pre-service teacher time new tools have been developed other teachers’ students. They education within two universities. She has and the curriculum focus has shifted to refer frequently to the state worked on projects in Hong Kong and North curriculum documents, NAPLAN Korea, as well as studies in many parts of the outcomes of the learning process Australia. Her research interests include statistical (Black & Wiliam, 2003). The promise results, the school policies and literacy, mental computation and assessment of raising students’ learning outcomes ‘throughlines’ that have been of mathematics and numeracy, and teachers’ through targeted assessment stimulated developed collaboratively to pedagogical content knowledge. Australian and other education systems ensure a common language and to introduce large-scale and costly focus across the school. These assessment programs such as NAPLAN, throughlines, along with specific as part of a ‘pressure and support’ strategies for computation, are approach to educational reform (Fullan, prominent in every classroom. 2000). Despite this activity, the promise By the end of the meeting, all of improved outcomes from changed teachers have a commitment to assessment practices has not been some action for their class, and achieved on a large scale (Stiggins, to increase the school focus on 2007). specific aspects of mathematics at which the students appeared to In this paper, aspects of quality do less well on the NAPLAN. This assessment practice in primary school is in a middle-lower socio- mathematics are explored, based economic range and is one of the on local and international research. most successful in the state on Assessment is regarded as more than NAPLAN numeracy, particularly the task or method used to collect when value-added measures are data about students. It includes the considered. process of drawing inferences from the data collected and acting upon The picture painted above is of a those judgements in effective ways. real school in which mathematics Such actions may occur at many assessment is used productively. The levels, but the key focus considered teachers were using a complex mix here is the school and, particularly, of assessment information to develop the classroom. The assessment focus teaching plans. NAPLAN data was may be summative in nature providing discussed to identify where, as a school, a snapshot in time of mathematical there were identified strengths and competence or achievement. weaknesses. This use of NAPLAN

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 39 assessment data provided a formative difficulty is choosing what to use. The Assessment as learning function at a school level. The work second, predicting likely responses, is If teachers find it difficult to articulate that students had produced in their also one that teachers can do relatively meaningful activities that would move classrooms was being used both well, and is now supported by a their students forward, what does formatively and summatively. Teachers plethora of work samples and examples this suggest about assessment as referred to the curriculum standards to from publishers, education systems and learning, that is assessment completely make judgements about their students’ professional bodies. Identifying what to indistinguishable from the learning progression and understanding, do next, however, is difficult (Wiliam, activity? Such assessment is informal, moderating their decisions against 2000a). undertaken as part of the teacher’s work samples from other teachers’ Recent work on identifying and ‘normal’ activity. It often involves classrooms through deep professional measuring teachers’ mathematical a teacher recognising a ‘teachable discussion. These conversations pedagogical content knowledge, moment’ and acting on this. For supported teachers in making choices however, indicates that although example, in a Korean kindergarten class for their own classrooms. primary teachers can recognise children were using blocks to explore The classroom is the powerhouse of and predict students’ responses to the number nine by putting them into learning. Teachers make a difference questions, both correct and incorrect groups of five and four. One girl had (Hattie, 2009) and efforts to improve ones, they have considerable difficulty taken ten blocks and had organised students’ learning outcomes must focus in identifying the next steps to take these into two groups of five. The on teacher practice. It is impossible, to develop students’ understanding teacher noticed this and set up the however, to talk about assessment (Watson, Callingham, & Donne, 2008a, next task to rearrange the blocks into divorced from pedagogy. The approach 2008b). groups of six and three. This next step that the teacher uses underpins the For example, one primary teacher provided the child with the chance quality and nature of learning in the participating in a study relating to self-correct, and she put the extra classroom (Wiliam & Thompson, 2007). to developing students’ statistical block back into the container. Clearly Such approaches include the use of understanding in response to a question the teacher made an assessment of the assessment for learning – identifying a showing information about market child and gave an immediate response student’s ‘readiness to learn’ (Griffin, share among large supermarkets using a that provided feedback to her in a way 2000) so that planned learning pie graph that added up to more than that changed her actions. It seems that experiences are maximally effective. 100 per cent, suggested that students this kind of teaching activity meets the The notion of assessment for learning might respond in the following ways: requirements indicated by Black and implies that teachers will not only be Wiliam (1998) for effective feedback. able to identify what students can do, *What percentage of the retail Classroom assessment, both assessment but also what activities and learning market Coles has. *Some might for and as learning, relies on dialogue experiences need to be planned to notice (a) that it doesn’t add up to between the child and the teacher develop students’ thinking. 100%, *(b) 61% should be more than half the graph, *(c) the whole (Callingham, 2008). Primary teachers know this and when asked about what Assessment for learning graph is inaccurate (not measured using a protractor etc.) they would do with their students often What does this look like in practice? reply in terms of the questions they First a task is needed that addresses In her response to the follow-up would pose or the discussions they the desired mathematical concept question, ‘How would/could you use would have. Teachers in the statistics and also provides for a wide range this item in the classroom? For example, study were asked, for example, how of different levels of understanding. how would you intervene to address they would respond to a child who Teachers then predict likely responses, the inappropriate responses?’, the same had read a pictograph about how and maybe group these into categories teacher answered ‘As a critical literacy/ children came to school and had given of similar understanding. The final maths activity’. Although this teacher the incorrect response ‘Bike, because action, and this is the key, is to develop demonstrated a depth of understanding the majority of boys ride to school’. A strategies for extension for each level of the mathematics involved, and about typical response was this one from a of understanding. The first of these what her Year 6 students might do, she South Australian primary teacher: actions, providing a task, is relatively was unable or unwilling to suggest any That’s interesting isn’t it? I would easy. There is an abundance of quality real follow-up activity. be asking what his reasoning material available to teachers – the

Research Conference 2010 40 behind that would be and obviously he would say, well they’re all boys and Tom’s a boy, therefore he will come to school because that’s where most of the boys come along. And I would discuss with that child, and talk about his reasoning why he discounted the bus, car, walking and train. What was the reasoning behind you discounting the fact that he couldn’t come by bus, car, walk or train? And that would be how I would move him forward. Figure 1: Kindergarten children’s attempts at copying a pattern Teachers perceive this kind of activity as the process of teaching, rather attention to the order of the symbols. applying models such as the NSW than feedback from assessment, and The bottom example, however, orders Quality Teaching model, more this perception has implications for the symbols but appears to be reading productive professional learning might professional learning (Callingham, Pegg the pattern from right to left, making be focused on addressing students’ & Wright, 2009). a mistake as the pattern runs onto specific, identified learning needs, using a second line. If these samples were the many work samples now available Assessment of learning collected at the end of a teaching and asking the question ‘where to So far there has been little in sequence, they perform a summative now’? function, providing a record at one this discussion about the place of Mathematics learning is idiosyncratic point in time of what a child can do. summative assessment: assessment – no two children learn mathematics In contrast, collected during a teaching of learning. In recent years it seems in the same way. It is also non-linear sequence, the same task could provide that teachers have rejected the – proceeding in jumps as a group of formative information helping to inform notion of summative assessment. ideas coalesce into a new cognitive the teacher’s planning. Biggs (1998), however, argued that it framework. Assessment needs to has an important place in classroom accommodate these variations so assessment, and should be seen as Assessment in the primary that feedback to students can directly part of a comprehensive assessment mathematics classroom: change what they do, such as the subtle plan. He advocated, for example, using Making it count feedback given by the Korean teacher graded portfolios as an ‘information- Assessment is arguably the most described earlier. Educating teachers rich’ form of summative assessment and powerful element in teaching and about effective feedback, however, suggested that whether an assessment learning. Quality assessment can may be more efficacious within a was summative or formative was provide information to students, pedagogical perspective than one that is largely a matter of timing. Assessment teachers, parents and systems in directed at assessment. of learning does not have to be effective and useful ways. To be helpful, test-based, and work samples that Perhaps the time has come to stop however, it must be broad ranging, demonstrate a student’s mathematical worrying about the nature of the collecting a variety of information using understanding are affirming and assessment activity, its summative or a range of tasks before, during and after powerful demonstrations to the child, formative purpose and the political a teaching sequence. and others, of what he or she has ends for which the information may, or learned. The two work samples shown To make assessment count, the focus may not, be used. Instead, all educators in Figure 1, for example, demonstrate of professional learning for primary need to get ‘back to basics’ and two kindergarten students’ attempts mathematics teachers might need to remember that it is quality teachers, to copy a pattern. The child who shift. Rather than developing teachers’ making rapid professional judgements produced the top example appears mathematical content knowledge, on the run in busy classrooms that to understand that the design has to changing pedagogical approaches create the ‘meanings and consequences’ run across the page, but doesn’t pay through rich mathematical tasks, or (Wiliam, 2000b) that affect children’s

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 41 interest and involvement in matters Stiggins, R. (2007). Assessment through mathematical. the student’s eyes. Educational Leadership, 64(8), 22–26. References Watson, J. M., Callingham, R., & Donne, Biggs, J. (1998). Assessment and J. (2008a). Establishing pedagogical classroom learning: A role for content knowledge for teaching summative assessment? Assessment in statistics. In C. Batanero, G. Burrill, C. Education: Principles, Policy & Practice, Reading & A. Rossman (2008). Joint 5(1), 103–110. ICMI/IASE Study: Teaching Statistics in School Mathematics. Challenges Black, P., & Wiliam, D. (1998). for Teaching and Teacher Education. Assessment and classroom learning. Proceedings of the ICMI Study 18 and Assessment in Education, 5(1), 7–74. 2008 IASE Round Table Conference. Black, P., & Wiliam, D. (2003). ‘In praise Monterrey: ICMI and IASE. Online: of educational research’: Formative www.stat.auckland.ac.nz/~iase/ assessment. British Educational publicatons Research Journal, 29(5), 623–637. Watson, J. M., Callingham, R., & Donne, Callingham, R. (2008). Dialogue and J. (2008b). Proportional reasoning: feedback: Assessment in the primary Student knowledge and teachers’ mathematics classroom. Australian pedagogical content knowledge. Primary Mathematics Classroom, 13(3), In M. Goos, R. Brown, & K. Makar 18–21. (Eds.), Navigating currents and charting directions. Proceedings of the 31st Callingham, R., Pegg, J., & Wright, T. annual conference of the Mathematics (2009). Changing teachers’ classroom Education Research Group of Australasia practice through developmental (Vol. 2, pp. 555–562). Adelaide: assessment: Constraints, concerns and MERGA. unintended impacts. In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing Wiliam, D. & Thompson, M. (2007). divides (Proceedings of the 32nd Integrating assessment with instruction: annual conference of the Mathematics What will it take to make it work? Education Research Group of In C. A. Dwyer (Ed.) The future of Australasia), 8pp. [CDROM]. assessment: Shaping teaching and Palmerston North, NZ: MERGA. learning. Mahwah, NJ: Lawrence Erlbaum Associates. Fullan, M. (2000). The return of large- scale reform. Journal of Educational Wiliam, D. (2000a). Integrating formative Change, 1, 5–28. and summative functions of assessment. Paper presented to the WGA 10 Griffin, P. (2000). Students! Take for the International Congress on your marks, get set, learn. Identifying Mathematics Education 9, Makuhari, ‘Readiness to Learn’ as a benefit Tokyo. Available from http://www. of outcomes based education. dylanwiliam.net/ Keynote address delivered at the Education Queensland Mount Wiliam, D. (2000b). The meanings Gravatt Symposium, Assessment and consequences of educational and Reporting in an Outcomes assessments. Critical Quarterly, 42(1), Framework, July 17, 2000. 105–127. Hattie, J. A. C. (2009). Visible learning: A synthesis of meta-analyses relating to achievement. Abingdon: Routledge.

Research Conference 2010 42 The case of technology in senior secondary mathematics: Curriculum and assessment congruence?

Abstract 2000) and strategies and processes that lead to certain directions and This paper outlines how curriculum and approaches being taken within and assessment congruence considerations across jurisdictions. The re-energising have been addressed in the context of discussions on the role of digital of the incorporation of computer technologies in the school mathematics algebra system (CAS) technology into curriculum arising from the emerging Victorian senior secondary mathematics Australian national curriculum initiative curriculum and assessment, in particular is a good example of a contemporary David Leigh-Lancaster examinations, over the period context for these considerations 2000–2010. The role of some related (ACARA, 2009). Victorian Curriculum and Assessment research is discussed. Authority (VCAA) It has been common to associate Introduction mathematical functionality with certain David Leigh-Lancaster is the Mathematics devices; for example, numerical Manager at the Victorian Curriculum and The relationship between curriculum with scientific calculators; statistical Assessment Authority (VCAA), former Head and assessment is central to discourse with spreadsheet based applications; of Mathematics P– 12 at Kingswood College, in mathematics education. It is a Victoria, and has taught secondary mathematics geometry with dynamic geometry for about 20 years. During this time Dr Leigh- focus of close attention in the senior software; graphing with graphics Lancaster has been extensively involved in secondary years where there is a strong calculators; and symbolic manipulation curriculum development, teacher professional connection to matters of certification with computer algebra systems (CAS). learning, resource development, examination and pathways into post-secondary setting and marking and the development These associations have been used as and verification of school-based assessment education, training and work. A key the basis of jurisdiction specifications in mathematics. He has longstanding interests aspect of mathematics is the role of for proscribed, permitted or prescribed in mathematical logic, computability theory, technology in working mathematically. technology access in formal assessment, foundations of mathematics, history and How this is reflected in senior philosophy of mathematics and mathematics especially examinations. Over the past education, the nature of mathematical inquiry, secondary mathematics curriculum and half-decade they have become less curriculum design and teaching, learning and assessment is one of the big issues of distinctive with multiple functionalities assessment in mathematics. Dr Leigh-Lancaster’s our time, especially as various software available on a single platform, for research interests focus on meta-mathematics and hand-held devices that support education, the interface between mathematics example CASIO Classpad or Texas and school mathematics, and the notion of and integrate powerful numerical, Instruments Nspire hand-held devices congruence between curriculum, assessment and statistical, graphical, geometric and and general purpose CAS software pedagogy – in particular with respect to the role symbolic functionality have become such as Maple and Mathematica. of enabling technology. readily available for widespread use These technologies can also be in school mathematics. The notion of used for developing documents that congruence is used here as a metaphor integrate text with ‘live’ mathematical for effective alignment between the use computations (calculations, tables, of technology as an enabling tool in graphs, diagrams, symbolic expressions) the curriculum and its use in related and as presentation tools. assessment. The term technology will be understood to indicate a synergy In their complementary relationship, between an artefact and the knowledge curriculum and assessment are key and understanding of how it can be indicators of educational beliefs, values used as a tool for a purpose. Relevant and preferences; for example, what is, research includes philosophical studies or is not to be done, and how it may or meta-analyses of beliefs and values be done, by and for whom, and in what (see, for example, Bishop, 2007; Ernest, contexts. If curriculum is to say what 1991), rationales, policies, trials and students should, as a consequence of pilot studies (see, for example, Stacey, their learning, know and be able to do McCrae, Chick, Asp & Leigh-Lancaster, (concepts, skills, processes and the like) and assessment is the means by which

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 43 judgments are made about progress Table 1: Assumed technology for end of year 12 final examinations in Victoria from and achievement, then a curriculum 1970 that sets expectations for the active use of technology as an enabling Stage Assumed technology for end of Year 12 examinations in Victoria tool for working mathematically Pre-1978 Four-figure logarithm tables and/or an approved slide rule. requires congruent expectations and practices for assessment. This is 1978– Scientific calculator. Until 1990 there was a single 3-hour typically informed by inter- jurisdiction 1996 examination. From 1991 there were two 1½-hour examinations. benchmarking research of curriculum and/or assessment routinely carried 1997 Scientific calculator – approved graphics calculator permitted but out by education authorities as part of not assumed. the development – evaluation – review 1998– Approved graphics calculator assumed for Mathematical Methods cycle (see, for example, Coupland, 1999 and Specialist Mathematics (both examinations). Scientific calculator 2007). with bivariate statistical functionality or approved graphics calculator assumed for Further Mathematics (both examinations). A brief historical background 2000– Approved graphics calculator for Further Mathematics, Mathematical Over the past few decades, various 2005 Methods and Specialist Mathematics (both examinations). technologies have been used in senior secondary mathematics curricula and Approved CAS (calculator or software) for Mathematical Methods related Year 12 final examinations in CAS pilot study, 2002–2005 (both examinations). Victoria. While different models have 2006– Approved graphics calculator or CAS for Further Mathematics been used to design and develop 2009 (both examinations). these curricula, there have been essentially three main types of final year Mathematical Methods and Mathematical Methods (CAS) were mathematics courses: alternative but like studies with a common technology free Examination 1 (worth 40 marks) and a separate technology • a practically oriented statistics and assumed Examination 2 (worth 80 marks), with around 70% – discrete mathematics course (e.g. 80% common material, approved graphics calculator assumed for networks), often with a business/ Mathematical Methods Examination 2, approved CAS assumed for financial mathematics component/ Mathematical Methods (CAS) Examination 2. option Specialist Mathematics – technology free Examination 1. Approved • a mainstream function, algebra, graphics calculator or CAS assumed for Examination 2 (technology calculus and probability course active but graphics calculator/CAS neutral). • an advanced mathematics functions 2010– Approved CAS or graphics calculator assumed for Further and relations, algebra, calculus, 2013 Mathematics (both examinations). vectors, complex numbers, differential equations and mechanics Mathematical Methods (CAS) and Specialist Mathematics each have course (this course assumes a 1-hour technology free examination. concurrent or previous study of the Mathematical Methods (CAS) and Specialist Mathematics each mainstream calculus based course). have a 2-hour technology active examination. An approved CAS In Victoria, from 1993 these have (calculator or software) is the assumed technology. been called Further Mathematics, 2014 and (Draft) Australian curriculum has four senior secondary Mathematical Methods/Mathematical beyond mathematics studies: Essential mathematics (Course A); General Methods CAS and Specialist mathematics (Course B); Mathematical methods (Course C) and Mathematics respectively, and their Specialist mathematics (Course D), currently under consultation. If corresponding assumed technologies things proceed well, 2014 could be the first year of implementation for examinations are shown in Table 1. in Victoria. Assessment remains the province of states and territory jurisdictions for the interim.

Research Conference 2010 44 The extent to which a technology such Mathematical Methods – gender, and performance of the two as CAS is actively used in curriculum, Mathematical Methods (CAS) cohorts with respect to assessment pedagogy and assessment has much 2006–2009 in concurrent advanced mathematics variation across jurisdictions (see, study – Specialist Mathematics. The for example, Leigh-Lancaster, 2000). The Victorian model for trialling, performance of the two cohorts A curriculum may specify expected development and implementation of on common assessment items in student use of CAS in working Mathematical Methods (CAS), has been examinations has been monitored mathematically, while precluding, substantially informed by experience closely by the VCAA and reported in permitting or assuming its use in and expertise from other jurisdictions Assessment Reports (see, for example, components of school-based or – the College Board, Denmark, VCAA, 2010a, 2010b) and papers examination assessment. Decisions France, Austria and Switzerland. It is, (see, for example, Evans, Jones, Leigh- about possible or required use (or however, quite unique. Victoria is the Lancaster, Les, Norton & Wu, 2008). not) may rest with the class teacher, only jurisdiction to have moved from Facility with traditional ‘by-hand’ or be partly or wholly prescribed by an established study, Mathematical skills is an area of some interest – the relevant authority. With respect Methods (1992–2009) to concurrent mean score data on the technology to the use of CAS in examination piloting of a related equivalent and free Examination 1 for 2006–2009 assessment, it may be the case that alternative study, Mathematical Methods consistently indicate that, in general, the use of technology is precluded CAS (2001–2005); then concurrent the Mathematical Methods (CAS) for some components (College Board implementation of both fully accredited cohort perform at least as well as AP Calculus, Denmark, Sweden, and studies as equivalent but alternative the Mathematical Methods cohort on Victoria, Western Australia, New (2006–2009) with a transition to the related questions. In particular for 2009 Zealand) and permitted (College Board CAS version replacing the ‘parent’ (where the size of the cohorts was AP Calculus, Sweden) or assumed version of the study from 2009 (Units around 7000–8000), the distribution of (Denmark, Victoria, Western Australia, 1 and 2 – Year 11 level) and 2010 student scores for each cohort across New Zealand) for other components. (Units 3 and 4 – Year 12 level). During the mark range from 0 to 40 shows Other jurisdictions permit but do the concurrent implementation phase, that at the top end, the performance not require CAS for all examination both studies had a common technology of the two cohorts is essentially the assessment (France, Tasmania). Some free examination; and each had its own same; at the very bottom end, the jurisdictions do not have externally technology assumed examination with performance of the Mathematical set examinations, with only school- 70 % – 80 % questions common to Methods (CAS) cohort tends to be based assessment (Ontario Canada, the two papers. The first phase of the better, while from the low to high mark Queensland), but have a curriculum VCAA Mathematical Methods (CAS) range the Mathematical Methods (CAS) that explicitly incorporates the use pilot study was founded in the work cohort consistently achieves a slightly of CAS while teachers decide locally of the Computer Algebra System – higher score than the Mathematical what technology is to be used in Curriculum Assessment and Teaching Methods cohort. This pattern persists assessment (typically with at least (CAS-CAT) project (2000 – 2002) when the data is controlled for graphics calculator functionality an Australian Research Council grant general mathematical ability using the assumed). A summary of jurisdictions funded research project partnership Mathematics, Science and Technology which permit or require student access between the VCAA, the University of component of the General Ability to CAS for some components of Melbourne, and calculator companies. Test (which has moderate correlation their senior secondary curriculum and The expanded pilot (2001–2005) also with respect to study specific ability) assessment can be found at Computer incorporated the use of CAS software. conducted in the middle of the same Algebra in Mathematics Education Questions of interest include year. When Examination 1 results (see CAME, 2010). Thus there will be consideration of matters such as are used to control for ability on multiple assessment models, and their potential and actual curriculum gains, the common Examination 2 extended efficacy with respect to the aims of the perceived and actual impact of regular response questions (that is, technology corresponding curriculum is a rich area student access to CAS on student facility independent or graphics calculator/ for research. with traditional ‘by-hand’ skills, changes CAS functionality neutral) comprising in teacher pedagogy and student 21 items for a score of 35 marks out of approaches to working mathematically, a total of 80 marks, a similar pattern is use of technology with respect to observed, as shown in Figure 1.

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 45 Scores on Maths Methods exam 1 and exam 2 by CAS and non-CAS groups 2009 assessment and qualifications. London: CAME. Retrieved May 25, 2010 from http:// 35 www.lkl.ac.uk/research/came/curriculum.html Coupland, M. (2007). A critical analysis 30 non-CAS CAS of selected Australian and international mathematics syllabuses for the post- 25 compulsory years of secondary schooling. Sydney: Board of Studies. 20 Ernest, P. (1991). Philosophy of Mathematics 15 Education. London: Falmer.

10 Evans, M., Jones, P., Leigh-Lancaster, D., Les, M., Norton, P., & Wu, M. (2008). The 2007 items (extended answers) 5 Common Technology Free Examination for Victorian Certificate of Education (VCE)

Average Raw Score on exam 2 common Score Raw Average 0 Mathematical Methods and Mathematical 0 5 10 15 20 25 30 35 40 45 Methods Computer Algebra System Raw Score on Maths Methods exam 1 (short answers) (CAS and non-CAS groups) (CAS). In M. Goos, R. Brown & K. Makar (Eds.), Navigating currents and charting Figure 1: Average score with respect to Examination 1 (technology free) score directions. Proceedings of the 31st annual conference of the Mathematics Education Research Group of Australasia, University This is perhaps not surprising – there function, and the table of values for the of Queensland, Brisbane (pp. 331–336). is an a priori argument that use of derivative, can be generated together. Brisbane: MERGA. CAS as an enabling technology which Students could then employ this to Leigh-Lancaster, D. (2000). Curriculum and provides numerical, graphical and compare their perception of the gradient assessment congruence – Computer Algebra algebraic representation of functions of the function across its domain (and Systems (CAS) in Victoria. Ohio: Ohio State and relations (and can move smoothly subsets of the domain) with what they University. Retrieved May 25, 2010 from between these representations) are seeing as the point at which the http://www.math.ohio-state.edu/~waitsb/ affords additional support for learning derivative is being evaluated is moved papers/t3_posticme2000/leigh-lancaster.pdf compared to technology that provides along the curve that forms the graph of Stacey, K., McCrae, B., Chick, H., Asp, G., for only numerical and graphical the function. Naturally, the general result & Leigh-Lancaster, D. (2000). Research- representation such as a graphics is established by a proof of suitable level led policy change for technologically-active calculator. If one wishes to develop of formality for the student cohort. senior mathematics assessment. In J. Bana & student facility with the product rule A. Chapman (Eds.), Mathematics Education for differentiation (fg)′ = fg′ + gf ′ then References Beyond 2000. Proceedings of the 23rd this is assisted by being able to readily annual conference of the Mathematics Australian Curriculum Assessment and generate and analyse correct patterns, Education Research Group of Australasia Reporting Authority. (2009). The Shape (pp. 572–579). Freemantle: MERGA. for example, moving from the general of the Australian Curriculum: Mathematics. form of the product rule to a form Melbourne: Author. Retrieved May 25, Victorian Curriculum and Assessment where f is left undetermined, and a 2010 from http://www.acara.edu.au/ Authority. (2010a). Mathematical Methods variety of specific function rules for g verve/_resources/Australian_Curriculum_-_ Examination 1 Assessment Report 2009. used, to the form where the rule of f is Maths.pdf Melbourne: VCCA. Retrieved May 25, specified, for example ex and the same 2010 from http://www.vcaa.vic.edu.au/ Bishop, A. J. (2007). Values in mathematics variety of specific function rules used. vcaa/vce/studies/mathematics/methods/ and science education. In U. Gellert assessreports/2009/mm1_assessrep_09.pdf In this context, evaluation of the & E.Jablonka (Eds.) Mathematisation derivative can be related directly to the demathematisation: Social, philosophical and Victorian Curriculum and Assessment gradient of the tangent to the graph of educational ramifications (pp. 123–139). Authority. (2010b). Mathematical Methods the product function at a particular point Rotterdam: Sense Publishers. Examination 2 Assessment Report 2009. Melbourne: VCCA. Retrieved 25 May and represented graphically. Where Computer Algebra in Mathematics 2010 from http://www.vcaa.vic.edu.au/vcaa/ dynamic functionality is also utilised, the Education. (2010). Some senior secondary vce/studies/mathematics/cas/assessreports/ graph of the corresponding derivative mathematics CAS active/permitted curriculum, mmcas2_assessrep_09.pdf

Research Conference 2010 46 Reconceptualising early mathematics learning

Abstract complex mathematical knowledge and abstract reasoning much earlier Over the past decade a suite of than previously considered. A range studies focused on the early bases of studies prior to school and in early of mathematical abstraction and school settings indicate that young generalisation has indicated that an children do possess cognitive capacities awareness of mathematical pattern which, with appropriately designed and and structure is both critical and salient implemented learning experiences, can to mathematical development among enable forms of reasoning not typically Joanne Mulligan young children. Mulligan and colleagues seen in the early grades (e.g., Clarke, have proposed a new construct, Clarke, & Cheeseman, 2006; Papic, Macquarie University Awareness of Mathematical Pattern and Mulligan, & Mitchelmore, 2009; Perry & Structure (AMPS), which generalises Dockett, 2008). Joanne Mulligan is an Associate Professor of across mathematical concepts, can be Education and Associate Director of the Centre reliably measured, and is correlated On the other hand, finding more for Research in Mathematics and Science effective ways of establishing the Education (CRiMSE) at Macquarie University, with structural development of Sydney. Her background in educational mathematics. root causes of learning difficulties in psychology, primary teacher education and mathematics is a key concern. The mathematics education psychology is combined A current large evaluation study was gap between achievers and non- with early teaching and administrative experience designed and implemented to measure achievers in mathematics begins in in NSW primary schools. Over the past 25 and describe young children’s structural years her research has focused primarily on early childhood and becomes wider as the development and assessment of number development of mathematics in the first students grow older, and there is still concepts and processes, word problems, year of schooling, Reconceptualising Early insufficient research evidence and little multiplicative reasoning, and pattern and structure Mathematics Learning: The Fundamental consensus about the underlying causes with 4- to 9-year-olds. She has made a significant Role of Pattern and Structure. An contribution to large-scale Australian government of underachievement. Despite initiatives and state-funded numeracy projects since the intervention was implemented to and reforms in mathematics education 1990s (e.g., Count Me In Too; Counting On; the evaluate the effectiveness of the Pattern many children do not seem to access Numeracy Research in NSW Primary Schools’ and Structure Mathematical Awareness the deep ideas and key processes that Project; the Early Years Numeracy Research Program (PASMAP) on kindergarten Project (Victoria) and the Mathematical Thinking lead to success beyond school. of Preschoolers in Rural and Regional Australia students’ mathematical development. (DEST). She has also contributed to the Four large schools (two from Sydney The Pattern and Structure Project, development and analysis of numeracy items in and two from Brisbane), 16 teachers initiated in 2001, aims to meet this the NSW Basic Skills Testing Program and quality and their 316 students participated challenge through a different approach assessment tasks for the NSW Quality Teacher to mathematics learning, beginning Program. in the first phase of a two-year longitudinal study. This paper provides with very young children, that reaches As chief investigator of a current ARC Discovery an overview of the background studies beyond basic numeracy to one that project, her research aims to reconceptualise cultivates mathematical patterns and traditional views and practices of early that informed the development of mathematical development and learning. PASMAP, describes aspects of the relationships. Over the past decade, a Associate Professor Mulligan has developed a assessment and intervention, and suite of studies focused on the early range of interview-based assessment instruments provides some preliminary analysis bases of mathematical abstraction based on frameworks of learning that enable and generalisation, has found that an in-depth analysis of mathematical growth. of the impact of PASMAP on Her techniques have potentially significant students’ representations of structural awareness of mathematical pattern implications for addressing students’ learning development. and structure is both critical and salient difficulties. Current research encompasses a to mathematical development among range of projects focused on early mathematical young children. Mulligan and colleagues development and professional learning such Introduction as the role of technological tools, the use of have proposed a new construct, children’s literature, preschoolers’ mathematical One of the most fundamental Awareness of Mathematical Pattern and patterning and mathematics education in challenges for mathematics education Structure (AMPS), which generalises Indigenous early childhood contexts. She is also today is to inspire young children across mathematical concepts, can be currently leading a NSW DET project, Enhancing to develop ‘mathematical minds’ Success in Mathematics (ESiM), focused on reliably measured, and is correlated middle schooling. and pursue mathematics learning with increasingly developed structural in earnest. Current research shows features of mathematics (Mulligan & that young children are developing Mitchelmore, 2009). Finding reliable

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 47 and consistent methods for describing early mathematics are now regarded as research with young children focused the growth of children’s mathematical critical to future success in the subject on data modeling and statistical structures and relationships, and utilising (Thomson, Rowe, Underwood, & Peck, reasoning also provide an integrated children’s ideas to develop quantitative 2005). approach to studying structural reasoning at an optimum age, when development (e.g., English, 2010; they are eager to learn, is central to this Research on pattern and Lehrer, 2007). project. structure The Pattern and Structure Research on early mathematics learning What is pattern and Project has often been restricted to an analysis structure? of children’s developmental levels Early studies on the structure of A mathematical pattern may be of single concepts such as counting, multiplication and division (Mulligan described as any predictable regularity, but has not provided insight into & Mitchelmore, 1997), the number usually involving numerical, spatial or common underlying processes that system (Thomas, Mulligan, & Goldin, logical relationships. In early childhood, develop mathematical generalization 2002), and area measurement the patterns children experience include (Mulligan & Vergnaud, 2006). However, (Outhred & Mitchelmore, 2000) repeating patterns (e.g., ABABAB recent initiatives in early childhood focused on analysing and describing …), spatial structural patterns (e.g., mathematics education, for example, structural development in studies of geometrical shapes), growing patterns the Building Blocks Project (Clements & 5- to 12-year-olds. Further research (e.g., 2, 4, 6, 8, …), units of measure Sarama, 2009), the Big Maths for Little on children’s representations of or transformations. Structure refers to Kids Project (Ginsburg, Lee & Boyd, mathematics found that a lack the way in which the various elements 2008) and the Mathematics Education of structural awareness impedes are organised and related including and Neurosciences (MENS) Project mathematical development and relates spatial structuring (see Mulligan et al., provide frameworks to promote ‘big to poor representational capacity. Low 2003). Structural development can ideas’ in early mathematics and science achievers consistently produced poorly emerge from, or underlie mathematical education (van Nes & de Lange, 2007). organised representations lacking in concepts, procedures and relationships structure, whereas high achievers used This trend is reflected in the increasing and is based on the integration of abstract notations with well-developed body of research into young children’s complex elements of pattern and structures. Essentially, low-achieving structural development of mathematics structure that lead to the formation students did not focus on structural and early algebraic reasoning. Algebraic of simple generalisations. For example, features when learning mathematics thinking is thought to develop from the recognising structural features of (see Mulligan, 2010). ability to see and represent patterns equivalence, 4 + 3 = 3 + 4 may reflect and relationships such as equivalence A suite of studies that followed, the the child’s perceived symmetrical and functional thinking from the early Pattern and Structure Project, indicated structure (see Mulligan & Mitchelmore, childhood years (Papic, Mulligan, & that young children who understand 2009). Mitchelmore, 2009; Warren & Cooper, the underlying structure of one 2008). Research in number (Hunting, mathematical concept are also likely to Background 2003; Mulligan & Vergnaud, 2006; perceive the structure underlying other There is increasing evidence that Thomas, Mulligan & Goldin, 2002; quantitative concepts, and can learn structural development is crucial to van Nes & de Lange, 2007; Young- to abstract and generalise concepts mathematical reasoning and problem- Loveridge, 2002), patterning and at an early age. The assessment of solving among young children. Failure reasoning (Clements & Sarama, 2009; first graders found their responses to to perceive pattern and structure English, 2004), spatial measurement a range of mathematical tasks could may also provide an explanation for (Outhred & Mitchelmore, 2000; Slovin be categorised into four stages of poor mathematical achievement. Early & Dougherty, 2004), and early algebra structural development – pre-structural, assessment of, and intervention in (Blanton & Kaput, 2005; Carraher, emergent, partial and structural, with a mathematics learning, is considered Schliemann, Brizuela, & Earnest, 2006; fifth stage, advanced structural, added preventative of later learning difficulties Warren & Cooper, 2008), have all with the progression of high-achieving (Clements & Sarama, 2009; Wright, shown how progress in students’ students (Mulligan & Mitchelmore, 2003). The quality, scope and depth of mathematical understanding depends 2009). The student’s stage of structural both the teaching and assessment of on a grasp of underlying structure. development was highly consistent Significant concentrations of new

Research Conference 2010 48 overall and reflected their level of 316 students from a diverse range learning experiences are scaffolded, mathematical understanding. of socio-economic and cultural where children are encouraged to contexts, participated in the evaluation seek out and represent pattern and The Pattern and Structure Mathematics throughout the 2009 school year. Two structure across different concepts Awareness Program (PASMAP) was different mathematics programs were and transfer this awareness to other then developed to raise students’ implemented: in each school, two concepts. It focuses on fundamental awareness of pattern and structure kindergarten teachers implemented processes such as simple and complex through a variety of well-connected the PASMAP and two implemented repetitions, growing patterns and pattern-eliciting experiences. Studies their standard program. The PASMAP functions, unitising and multiplicative have included an extensive, whole- framework was embedded into the structure also common to units of school project across Kindergarten to standard kindergarten mathematics measure; spatial structuring, the spatial Year 6; two year-long, design studies curriculum. A researcher/teacher visited properties of congruence and similarity, in Years 1 and 2; and an intensive, a each teacher on a weekly basis and and transformation (see Mulligan, 15-week empirical evaluation of an equivalent professional development Mitchelmore, English, & Robertson, individualised program with a small for both pairs of teachers was provided. 2010). Emphasis is also laid on counting group of kindergarten children (see Incremental features of the program through patterns and measures, the Mulligan, 2010). were introduced by the research team structure of operations, equivalence and In related studies, Papic found that gradually, at approximately the same commutativity. preschoolers who are provided with pace and with equivalent mentoring for opportunities to engage in mathematical each teacher, over three school terms. Discussion experiences that promote emergent All students were pre- and post- Preliminary analysis indicates that both generalisation (an intervention tested with I Can Do Maths (ICDM) groups of students made significant program) are capable of abstracting (Doig & de Lemos, 2000); from pre- progress in mathematics learning complex patterns before they start test data two ‘focus’ groups of five outcomes as described by the state formal schooling (Papic, Mulligan, & children in each class were selected syllabus and measured by the ICDM Mitchelmore, 2009). from the upper and lower quartiles, test. It was not expected that significant These studies indicate that young respectively. These 160 students were differences would be found between children can learn complex pre- and post- interviewed using a PASMAP and regular students on mathematical concepts very quickly new version of a 20-item Pattern pre- and post-tests scores on this and effectively by focusing on crucial and Structure Assessment (PASA). standardised measure. However, initial features of mathematical pattern Intervention-based data included analysis of qualitative data, tracking of and structure; visual memory, observation notes, digital recordings of the ‘focus’ students, indicated marked constructing and representing structures their learning experiences and a range differences between groups in students’ independently of models, and the of work samples. Student profiles of level of structural development (AMPS). articulation of ‘sameness and difference’ learning aim to (i) describe the ‘tracked’ Students participating in the PASMAP was central to this process. However, developmental pathway(s) of their program showed higher levels of these findings also supported those mathematical concepts and processes, AMPS than the regular group, made of earlier studies in that low achievers (ii) analyse the quality of the underlying connections between mathematical failed to perceive structure even in structural characteristics, (iii) describe ideas and processes, and formed simple mathematical forms such as the salient features or relationships built by emergent generalisations. Some of the properties of a square. the student between components or more able students used one aspect concepts, and (iv) provide evidence of of pattern and structure to build new Reconceptualising Early emergent generalisations and reasoning and more complex concepts. Gradually Mathematics Learning to support these. these connections became more like systems of learning that had common This new study was designed to The Pattern and Structure structural features. Goldin in his work evaluate the effectiveness of PASMAP with Thomas and colleagues refers to on students’ mathematical development Mathematics Awareness these as autonomous powerful systems in the first year of formal schooling. Program Intervention that become independent over time A purposive sample of four large The program is innovative in its (Thomas, Mulligan, & Goldin, 2002). primary schools, two in Sydney conceptual framework and the way and two in Brisbane, representing

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 49 Some exemplars of students’ use of simple grid patterns introduced indicate that consistent methods for developing structural features are now early in the program and the counting analysing students’ AMPS are indeed described. Students used ten frame patterns of multiples. Figures 7, 8 and possible and this process provides a cards to promote the structure of ten, 9 show attempts to draw the pattern rich basis for assessing and scaffolding spatial and counting patterns, grouping from memory, but the structure students’ mathematical development. and addition combinations. As an of increasingly larger squares is not Our goal is a reliable, coherent model assessment task, they were required to generalised and the number of units is for categorising and describing structural draw the frame from memory, describe counted or added on individually. Figure development with aligned pedagogical how they did this and why the frame 9 shows units aligned but extended frameworks. was used. Figures 1 to 6 show typical uni-dimensionally; this is adding a In the forthcoming Australian National examples of ten frames that have column rather than recognising the Curriculum (ACARA, 2010), Number been drawn by six individuals at the multiplicative structure. Figure 10 shows and Algebra strands are aligned same point in the learning sequence. the student’s structural development of with Problem Solving and Reasoning Each figure reflects developmental the pattern of increasingly larger arrays Proficiencies. ‘An algebraic perspective features of students’ awareness and as squares using the alignment of the can enrich the teaching of number … use of the structure of the ten-frame: ‘growing squares’. He also explains the and the integration of number and the use of 2-wise or 5-wise patterns numerical sequence as multiplicative. algebra, especially representations of (quinary-based structure), the use of relationships can give more meaning to co-linearity (row and column structure) Implications the study of algebra in the secondary and the construction of addition pairs. One outcome of the project is to years. This combination incorporates Figures 1 to 3 show no recognition validate alternative developmental pattern and/or structure and includes of the structure of the ten-frame and paths for young children’s mathematics functions, sets and logic’. Further, its facility, although these students learning. Ultimately this research the integration of measurement and were using ten frames regularly; these may provide better pathways for geometry, and statistics and probability students had poor AMPS across a range those children who may be prone to brings new opportunities to develop of tasks. Figure 4 shows awareness of difficulties in learning mathematics; a structural approach. The proposed the pattern of fives and Figures 5 and 6 that is, those who lack AMPS. PASMAP will enable professionals to strong structural features. Tracking, describing and classifying develop and evaluate a new approach In another task the children had children’s models, representations and with flexibility – one that integrates to recall their use of pattern cards explanations of their mathematical patterns and structural relationships in depicting the pattern of squares i.e., 1, ideas, and analysing the structural mathematics across concepts so that a 2 × 2, 3 × 3, 4 × 4, 5 × 5 square grid features of this development are more holistic outcome is achieved. cards. This pattern was linked to prior fundamentally important. Our studies

Figure 1: Pre- Figure 2: Figure 3: Figure 4: Partial Figure 5: Partial Figure 6: structural image Emergent Emergent structure shown structure: aligned Structural of ‘tall buildings structural images structural images by 2 x 5 unequal single units ten features showing with bridges’. of single units. of ‘single and units. frame structure. 5-wise pattern. double’ frames.

Research Conference 2010 50 Figure 7: Emergent Figure 8: Partial structure: Figure 9: Partial structure: Figure 10: Structural structure: pattern of squares pattern of squares using pattern of squares limited response showing pattern using single units equal-sized units; lack of to 5x5 and array structure structure of ‘square’

Mathematics learning for the future Clements, D., & Sarama, J. (2009). Mathematics: Essential for learning, will require young children to reason Learning and teaching early maths: essential for life (Proceedings of mathematically in creative and flexible The learning trajectories approach. NY: the 30th annual conference of the ways in order to solve multi-disciplinary Routledge. Mathematics Education Research problems. Focusing on pattern and Group of Australasia, Hobart, Vol. 1, Doig, B., & de Lemos, M. (2000). I can structure may not only lead to pp. 22–41). Adelaide: AAMT. do maths. Melbourne: ACER improved generalised thinking, but can Mulligan, J. T., Prescott, A., & also create opportunities for developing Ellemor-Collins, D. & Wright, R., (2009). Mitchelmore, M. C. (2003). Taking cognitive capacities commensurate with Structuring numbers 1–20:Developing a closer look at young students’ the abilities of young learners and the facile addition and subtraction, visual imagery. Australian Primary demands of mathematics learning for Mathematics Education Research Mathematics Classroom, 8(4), 23-27. the future. Journal, 21(2), 50–75. Mulligan, J. T., Mitchelmore, M.C., English, L. D. (2010). Modeling with English, L., & Robertson, G. (in press). References complex data in the primary school. Implementing a Pattern and Structure In R. Lesh, P. Galbraith, C. R. Haines, Australian Curriculum, Assessment and Mathematics Awareness Program in & A. Hurford (Eds.), Modeling students’ Reporting Authority. (2010). Shape of kindergarten. Shaping the future of mathematical modeling competencies: the Australian curriculum: Mathematics. mathematics education, (Proceedings ICTMA 13. Springer. http://www.acara.edu.au/verve/_ of the 33rd annual conference of resources/Australian_Curriculum_-_ English, L. D. (2004). Promoting the the Mathematics Education Research Maths.pdf development of young children’s Group of Australasia), Fremantle, WA: Blanton, M., & Kaput, J. (2005). mathematical and analogical reasoning. MERGA. In L. D. English (Ed.), Mathematical and Characterizing a classroom practice Mulligan, J. T., & Mitchelmore, M. C. analogical reasoning of young learners. that promotes algebraic reasoning. (2009). Awareness of pattern and Mahwah, NJ: Lawrence Erlbaum. Journal for Research in Mathematics structure in early mathematical Education, 36, 412–446. Ginsburg, H. P., Lee, J. S., & Boyd, J. S. development. Mathematics Education Carraher, D. W., Schliemann, A. D., (2008). Mathematics education for Research Journal, 21(2), 33–49. young children: What it is and how to Brizuela, B. M., & Earnest, D. (2006). Mulligan, J. T., & Vergnaud, G.(2006). promote it. Social Policy Report, 22(1), Arithmetic and algebra in early Research on children’s early 3–11 and 14–23. Available online mathematics education. Journal for mathematical development: Towards from: http://www.srcd.org/spr.html Research in Mathematics Education, 37, integrated perspectives. In A. 87–115. Hunting, R. (2003). Part–whole number Gutiérrez & P. Boero (Eds.), Handbook Clarke, B., Clarke, D., & Cheeseman, J. knowledge in preschool children. of research on the psychology of (2006). The mathematical knowledge Journal of Mathematical Behavior, 22(3), mathematics education: Past, present and understanding young children 217–235. and future (pp. 261–276). London: bring to school. Mathematics Education Lehrer, R. (2007). Introducing students Sense Publishers. Research Journal, 18(1), 78–103. to data representation and statistics. In J. Watson & K. Beswick (Eds.),

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 51 Mulligan, J. T. (2010). The role of relationships. Australian Journal of Early representations in young children’s Childhood, 27(4), 36–42. structural development of van Nes, F., & de Lange, J. (2007). mathematics. Mediterranean Journal of Mathematics education and Mathematics Education, 9(1), 163–188. neurosciences: Relating spatial Outhred, L. N., & Mitchelmore, M. structures to the development of C. (2000). Young children’s intuitive spatial sense and number sense. The understanding of rectangular area Montana Mathematics Enthusiast, 2(4), measurement. Journal for Research in 210–229. Mathematics Education, 31, 144–167. Warren, E., & Cooper, T. J. (2008). Papic, M., Mulligan, J. T., & Mitchelmore, Generalising the pattern rule for visual M. C. (2009). The growth of growth patterns: Actions that support mathematical patterning strategies in 8 year olds thinking. Education Studies preschool children. In M. Tzekaki, M. in Mathematics. 67(2), 171–185. Kaldrimidou, & H. Sakonidis (Eds.), Wright, R. J. (2003). Mathematics Proceedings of the 33rd conference Recovery: A program of intervention of the International Group for the in early number learning. Australian Psychology of Mathematics Education Journal of Learning Disabilities, 8(4), (Vol. 4, pp. 329–336). Thessaloniki, 6–11. Greece: PME. Perry, B., & Dockett, S. (2008). Acknowledgements Young children’s access to powerful mathematical ideas. In L. D. English The research reported in this paper (Ed.), Handbook of international was supported by Australian Research research in mathematics education (2nd Council Discovery Projects grant No. ed). NY: Routledge. DP0880394, Reconceptualising early mathematics learning: The fundamental Slovin, H., & Dougherty, B. (2004). role of pattern and structure. The authors Children’s conceptual understanding express their thanks to Dr Coral of counting. In M. J. Høines & A. Kemp; research assistants – Nathan B. Fuglestad (Eds.), Proceedings Crevensten, Susan Daley, Deborah of the 28th conference of the Adams and Sara Welsby; participating International Group for the Psychology teachers, teachers aides, students and of Mathematics Education (Vol. 4, pp. school communities for their generous 209–216). Bergen, Norway: PME. support of this project. Thomas, N., Mulligan, J. T., & Goldin, G. A. (2002). Children’s representations and cognitive structural development of the counting sequence 1–100. Journal of Mathematical Behavior, 21, 117–133. Thomson, S., Rowe, K., Underwood, C., & Peck, R. (2005). Numeracy in the early years. Melbourne: Australian Council for Educational Research. Young-Loveridge, J. (2002). Early childhood numeracy: Building an understanding of part–whole

Research Conference 2010 52 Learning about selecting classroom tasks and structuring mathematics lessons from students

Abstract with large samples, with the goal of understanding behaviour. As part of a larger project1, students’ views on their preferences for particular This research perspective also adopted types of mathematical tasks were a similar perspective to that of Daniels, sought, as well as how they describe Kalkman and McCombs (2001), who their ideal mathematics lesson, and argued that even though students are their responses to specifically prepared able to articulate coherent views on tasks from sequences of lessons. The issues of pedagogy they are seldom students had particular views about asked to do so, and that students Peter Sullivan both tasks and lessons and were able are particularly able to comment on Monash University to articulate their views. Teachers classroom and school environments. would do well to seek to find out the Allen (2003) similarly argued that there Peter Sullivan is Professor of Science, Mathematics types of tasks and lessons that particular has been too little attention to students’ and Technology Education at Monash University. students prefer, and to be more explicit perspectives of aspects of teaching He is the author of the shape paper for the new and class organisation. It is recognised national mathematics curriculum, editor of the about what they are intending to do in Journal of Mathematics Teacher Education, for four every one of their lessons. that teaching involves much more than years was a member of the Australian Research finding ways to present the content, Council College of Experts, and is president Introduction and is connected to relationships, of the Australian Association of Mathematics student self-regulation (Dweck, 2000) Teachers. There are many sets of and motivation (Middleton (1995), recommendations about characteristics so it is relevant to seek students’ of effective teaching, which are perspectives on these issues. generally compiled theoretically, or from surveys, or from descriptions In terms of seeking students’ views of exemplary teachers (see Clarke & about tasks the project chose to focus Clarke, 2004; Hattie & Timperley, 2007; data collection on the extent to which Education Queensland, 2010). The they felt they learned, and whether research summarised here attempted they liked particular types of tasks since to examine the views of students on these seemed to be main determinants the types of tasks they value, and the of their decisions on engagement. In the structure of lessons that they prefer. piloting of our instruments we found that the students were able to respond While there have been many studies to both types of prompts without seeking students’ attitudes, values, requiring further clarification. Our beliefs and motivation, the approach approach was to seek some responses reported here aligns with Zan and to predetermined scales as well as di Martino (2010) who argued that some free format narratives by the emphasis should move from measuring students to allow their real concerns attitudes to describing them. They to emerge. We collected three argued for more narrative approaches complementary sets of data, giving a to describing student attitudes, including breadth of types of data and therefore greater insights into the views of students. The three separate data sets are not presented here due to space 1 TTML is an Australian Research Council funded limitations but will be presented in the research partnership between the Victorian workshop. A summary of the findings Department of Education and Early Childhood are described in the following sections. Development, the Catholic Education Office (Melbourne), Monash University and Australian Catholic University. Barbara Clarke and Doug Clarke were also researchers on the project.

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 53 Responses of students to emphasise the role of the tasks and the Write a story about your ideal predetermined prompts about nature of the challenge they offer. maths class. Write about the sorts tasks and pedagogies of questions or problems you Narrative descriptions of like to answer, what you like to A survey was designed to gather students’ perceptions of be doing and what you like the responses on aspects of lessons and characteristics of desired teacher to be doing in your ideal tasks from a cross-section of students. maths class. As well as seeking information on mathematics lessons various aspects of lessons, we also Using a different approach, we The intention was to gain insight into included specific items asking students also sought insights into students’ what the students recalled about their to compare different types of tasks and perceptions of the desired mathematics classes, and it can be to indicate their preferences. characteristics of mathematics lessons assumed that these responses can be through their narrative responses. It taken as indicative of the lesson features The items on general aspects of that the students like. The following is pedagogy were adapted from Clarke was hoped in this way to gain insights into the ways students described their an example of a typical student’s essay, et al. (2002) and Sullivan et al. (2009), presented as it was written: and the items on tasks were written desired characteristics, rather than by specifically. There were 930 students rating lesson characteristics prepared My favorite maths would start in 96 classes across 17 schools who by us. We did this through open-ended with a 10 min introduction were completed the survey. responses to particular prompts on the the teacher explains the game overall survey. to all of us and still allowing time To summarise the results from the In summary, the main impression for questions. The games would survey, it seems that at each of these be 2+ people for a competition middle years’ levels there is a range from their responses is their diversity, and there are clearly many ways in and people will split into groups of student satisfaction and confidence, and will organize who plays who and teachers should be aware of the which students respond to lessons. There were two trends in their lesson 5 min every one will be playing views of each of their students. It also at all times unless there is an odd seems that teachers make a difference descriptions of, on one hand, students recalling effective teaching of a content amount of people we will play for to students’ responses and teachers 25 min. at the end of the Lesson need support not only to find out topic, whereas there were others who remembered interesting aspects the groups will figure out who students’ levels of satisfaction and was the winner and people can confidence, but also on strategies to of the pedagogy. In explaining their choice of lesson, the main category of share what they Learnt Liked and address negative responses. Each of the strategies they used. Sharing is for task types presented were liked most responses related to fun, but learning something new was also frequently 10 min.for my second option I by some students, and likewise each of would do real life problems Like the types was rated as the one from cited. We note that the descriptions of hated lessons also referred to particular 250 grams of sugar for $10.50 or which they can most learn; this suggests 750 grams for $33.15. I like real that teachers need to use all types of topics. So while recognising that some students dislike some topics, teachers life problems because they could task in their teaching. A related issue is help me one day and its set out that students may need support to gain are advised to focus on the students’ learning of content, and to choose differently than math. for this the benefits from tasks that they do not like explanation is for 5 min this is or do not feel that they can learn from. interesting and fun ways to engage students in that learning. because you don’t need to explain It seems important that teachers make the rules. students aware of the purpose of tasks and what it is the teachers are hoping Students’ essays on their ‘ideal In this response there were two key the students will learn from them. The maths class’ elements: the use of a game, and the use of real-life problems, but the real students seem to like tasks that are We also sought students’ views implication is that this is indicative easy yet feel they learn best from tasks on lessons and teaching through a of the detail that students used to that are challenging. Of course, we particular prompt seeking narrative describe the ideal class. would hope that students can also learn responses. We asked the students in from tasks they find easy, and like tasks two of the schools that completed a In summary, it seems that the responses that are challenging. Again, it may be lesson sequence to write an essay, the to this prompt about an ideal lesson important for teachers to illustrate or particular prompt of which was: seemed dependent on the teacher. In

Research Conference 2010 54 synthesising the responses, students project: Final report. Australian Catholic Journal of Mathematics Teacher liked lessons that used materials University and Monash University. Education, 13(1), 27–48. (although these were not structured Clarke, D. M., & Clarke, B. A. (2004). materials), were connected to their Mathematics teaching in Grades lives, involved games, were practical K–2: Painting a picture of challenging, with some emphasis on measurement, supportive, and effective classrooms. in which they worked outside, In R. N. Rubenstein & G. W. Bright with the method of grouping being (Eds.), Perspectives on the teaching important, and over half of the students of mathematics (66th Yearbook of claim to like to be challenged. An the National Council of Teachers of interesting result was that, contrary to Mathematics, pp. 67–81). Reston, VA: expectations, many students claimed to NCTM. like help from the teacher only after a period of effort. Daniels, D. H., Kalkman, D. L., & McCombs, B. (2001). Young children’s Conclusion perspectives on learning and teacher practices in different classroom It is clear that there is much that can contexts: Implications for motivation. be learned from the responses of Early Education and Development, students. The students who responded 12(2), 253–272. to these instruments are clearly aware of aspects of teaching, including those Dweck, C. S. (2000). Self-theories: Their aspects that are subtle. While most role in motivation, personality, and of their comments are not surprising, development. Philadelphia: Psychology they do endorse strongly many of Press. the pedagogies that some teachers Education Queensland. (2010). seem reluctant to adopt. One clear Productive pedagogies. Downloaded in implication is the need for teachers January 2010 from http://education. to use a variety of tasks and lesson qld.gov.au/corporate/newbasics/html/ structures, a recommendation that one pedagogies/pedagog.html suspects has particular significance for secondary teachers. Another implication Hattie, J., & Timperley, H. (2007). is that, since not all tasks or lessons can The power of feedback. Review of be those preferred by students, teachers Educational Research, 77(1), 81–112. need to make efforts to explain the Middleton, J. A. (1995). A study of choice of task and its purpose, and to intrinsic motivation in the mathematics explain the goal of particular pedagogies classroom: A personal construct that they might use. approach, Journal for Research in Mathematics Education, 26(3), 254– References 279. Allen, B. (2003). Pupils’ perspectives on Sullivan, P. Prain, V., Campbell, C., learning mathematics. In B. Allen & S. Deed, C., Drane, S., Faulkner, M., Johnston-Wilder (Eds.), Mathematics McDonough, A., Mornane, A., & education: Exploring the culture of Smith, C. (2009). Trying in the middle learning (pp. 233–241). London: years: Students’ perceptions of their Routledge Falmer. aspirations and influences on their Clarke, D., Cheeseman, J., Gervasoni, A., efforts. Australian Journal of Education, Gronn, D., Horne, M., McDonough, 5(2), 176–191. A., Montgomery, P., Roche, A., Zan, R. & di Martino, P. (2010). ‘Me and Sullivan, P., Clarke, B., & Rowley, maths’: Toward a definition of attitude G. (2002). Early numeracy research grounded on students’ narrative.

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 55 Identifying cognitive processes important to mathematics learning but often overlooked

Abstract Introduction This presentation introduces a set The OECD’s Programme for of mathematical competencies that International Student Assessment deserve to be given more attention (PISA) aims to measure how effectively in our mathematics classrooms, on 15-year-olds can use their accumulated the grounds that the possession of mathematical knowledge to handle these competencies relates strongly ‘real-world challenges’. The measures to increased levels of mathematical we derive from this process are literacy. The presenter argues that referred to as measures of mathematical Ross Turner widespread under-representation of literacy. The literacy idea seems to have Australian Council for Educational these competencies among the general really taken hold among those countries Research populace contributes to unacceptably that participate in PISA. It is generally large measures on the mathematics regarded as very important that people Ross Turner manages ACER’s International terror index. can make productive use of their PISA project, coordinating the ACER team mathematical knowledge in applied and and international consortium partners to meet The argument in support of these practical situations. the requirements of ACER’s contract with the competencies comes out of the OECD. He has filled this role as a Principal In this presentation I will demonstrate Research Fellow since 2007, and before that OECD’s Programme for International provided general leadership and management to Student Assessment (PISA). It is based some illustrative PISA items as a way the PISA project and to other ACER projects as on the results of research conducted of introducing a set of mathematical a Senior Research Fellow since 2000. Ross also by members of the PISA mathematics competencies that are fundamental to provides leadership in the mathematics area, the possession and development of having led PISA mathematics framework and expert group. That research will be test development and being responsible for PISA described, the competencies under mathematical literacy, and will propose mathematics implementation throughout his time discussion will be defined, and the that these deserve a stronger place in at ACER. case for greater emphasis on these our mathematics classes. For 13 years prior to that Ross was employed in competencies will be made. various roles at the Victorian Board of Studies. He was seconded in 1987 to contribute to redevelopment of the mathematics curriculum and assessment arrangements in the Victorian Certificate of Education. He was appointed to the position of Manager, Mathematics in 1989 and led the implementation of the VCE mathematics study. He was appointed as Manager, Research and Evaluation in 1993. In that role he monitored annual VCE outcomes, and oversaw development and implementation of statistical procedures employed in the processing and reporting of VCE data.

Research Conference 2010 56 Illustrative PISA items Table 1: Per cent correct for three illustrative PISA mathematics questions

Two items from the unit titled Exports Per cent correct  Per cent correct  Question involve interpreting data presented (all students) (Aus students) in a bar graph and a pie chart. The Exports Q1 67.2 85.8 first question calls for the direct interpretation of a familiar graph form: Exports Q2 45.6 46.3 identifying that the bar graph contains Carpenter 19.4 23.3 the required information, locating the bar for 1998 and reading the required number printed above the bar. 100.0 The second question is more involved, 90.0 since it requires linking information from 80.0 67.2 the two graphs presented: applying 70.0 the same kind of reasoning required in 60.0 the first question to each of the two 50.0 45.6 graphs to locate the required data, then 40.0 30.0 performing a calculation using the two 19.4 figures found from the graphs (find 9% 20.0 of 42.6 million). 10.0 0.0 A further question Carpenter is presented, which requires some Figure 1: International per cent correct of all PISA 2003 mathematics questions geometrical knowledge or reasoning. Familiarity with the properties of basic geometric shapes should be sufficient to establish that while the ‘horizontal’ components of the four shapes are equivalent, the oblique sides of Design B are longer than the sum of the ‘vertical’ components of each of the other shapes. What do we find when problems such as these are given to random samples of 15-year-olds across over 60 countries around the world? Table 1 presents the per cent correct data for all students internationally and all Australian students who were given the listed questions in the PISA 2003 survey. The chart in Figure 1 shows where these publically released questions fit in the context of the whole PISA 2003 survey instrument. The international per cent correct for the illustrative items are labelled, amidst the 84 items used in the survey (with a bar for each item, ordered by their international percent correct value). Exports Q1 was one of the easier items in the test, while Exports Q2 was a moderately difficult

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 57 item. Carpenter was one of the most or ‘factorise these expressions’. The factors that explain so much of what difficult items. objectives were clearly mathematical. makes mathematics items difficult is an important finding. In the real world, that’s not normally Is there a problem? how mathematics comes to us. We Those competencies can be thought We could speculate about differences have to make the judgments and of as a set of individual characteristics in performance levels between decisions about what mathematical or qualities possessed to a greater or Australian and international students, knowledge might be relevant, and how lesser extent by individuals. However, but for my immediate purpose, I might to apply that knowledge. That assumes we can also think about these simply suggest that as a mathematics we are motivated enough in the first competencies from the ‘perspective’ teacher, I would have hoped that place to even notice that mathematics of a mathematics problem, or a survey most 15-year-olds could answer might be relevant. question: to what extent does the questions like these correctly. This question call for the activation of This brings us back to one of the most also has implications for what happens each of these competencies? In the important and influential ideas that to those 15-year-olds when they following section the six competencies underpins the PISA project: its emphasis leave school, since the mathematical are defined, and the task–level demand on what is called literacy. PISA measures capabilities students demonstrate for activation of each competency at and reports the degree to which the by the time they are nearing school different levels is described. 15-year-olds in participating countries leaving age foreshadows the approach have developed their literacy skills in those individuals will take to using mathematics and the other survey Communication mathematics later in life. domains so that they can apply their Mathematical literacy in practice Is the problem that many students knowledge to solve contextualised involves communication. Reading, don’t know the required mathematical problems – problems that are more decoding and interpreting statements, concepts; that they have not learned like the challenges and opportunities questions, tasks or objects enables the required mathematical skills? Or we meet in our work, leisure, and in the individual to form a mental model could it be that too many 15-year- our life as citizens. But what are the of the situation, an important step in olds are simply unable to activate the capabilities that equip adults to meet understanding, clarifying and formulating required knowledge when it could such challenges? a problem. During the solution process, be useful; that there is a disconnect which involves analysing the problem between the way in which many of us Mathematical competencies – using mathematics, information may have been taught, and the opportunities the research need to be further interpreted, and to use mathematics in life outside intermediate results summarised and The frameworks that governed the school? presented. Later on, once a solution mathematics part of the PISA surveys has been found, the problem solver Usually the opportunities to use conducted in 2000, 2003, 2006 may need to present the solution, and mathematics that we come across are and 2009 describe a set of eight perhaps an explanation or justification, not packaged in quite the way they mathematical competencies. For the to others. were in school. There, you knew when purposes of a research activity we you were going to a mathematics class. have carried out, these have been Various factors determine the level When you went to that class, you did configured as a set of six competencies and extent of the communication so expecting that you would do things that are fundamental to the concept demand of a task. For the receptive related to mathematics. You had a of mathematical literacy that PISA aspects of communication, these factors mathematics teacher who taught and espouses, namely the capacity to include the length and complexity of demonstrated mathematical ideas and use one’s mathematical knowledge the text or other object to be read skills, gave you some examples, and to handle challenges that could be and interpreted, the familiarity of the then pointed you to a set of exercises amenable to mathematical treatment. ideas or information referred to in the more or less like those used to Our research has shown that these text or object, the extent to which demonstrate the idea or skill you were competencies can be used to explain a the information required needs to be learning. You were given instructions very large proportion of the variability disentangled from other information, like ‘count these objects’, or ‘add in the difficulty of PISA mathematics the ordering of information and these numbers’, or ‘draw this graph’, test items, possibly as much as 70 whether this matches the ordering per cent of that variability. To identify of the thought processes required to

Research Conference 2010 58 interpret and use the information, and defined, and to check that the model required involves simply following the extent to which different elements satisfies the requirements of the task; or direct instructions. At a slightly higher (such as text, graphic elements, graphs, to evaluate or compare models. level of demand, items require some tables, charts) need to be interpreted reflection to connect different pieces in relation to each other. For the Representation of information in order to make expressive aspects of communication, inferences (for example, to link This competency can entail selecting, the lowest level of complexity is separate components present in the devising, interpreting, translating observed in tasks that simply demand problem, or to use direct reasoning between, and using a variety of provision of a numeric answer. As within one aspect of the problem). At representations to capture a situation, the requirement for a more extensive a higher level, tasks call for the analysis interact with a problem, or to present expression of a solution is added, for of information in order to follow or one’s work. The representations example when a verbal or written create a multi-step argument or to referred to include equations, formulas, explanation or justification of the result connect several variables; or to reason graphs, tables, diagrams, pictures, textual is required, the communication demand from linked information sources. At descriptions and concrete materials. increases. an even higher level of demand, there This mathematical ability is called on is a need to synthesise and evaluate Mathematising at the lowest level with the need information, to use or create chains to directly handle a given familiar of reasoning to justify inferences, or Mathematical literacy in practice representation, for example translating to make generalisations drawing on can involve transforming a problem directly from text to numbers, or and combining multiple elements of defined in the real world to a strictly reading a value directly from a graph information in a sustained and directed mathematical form (which can include or table. More cognitively demanding way. structuring, conceptualising, making representation tasks call for the assumptions, formulating a model), or selection and interpretation of one Devising strategies interpreting a mathematical solution or standard or familiar representation a mathematical model in relation to the Mathematical literacy in practice in relation to a situation, and at a original problem. frequently requires devising strategies higher level of demand still when they for solving problems mathematically. The demand for mathematisation arises require translating between or using This involves a set of critical control in its least complex form when the two or more different representations processes that guide an individual problem solver needs to interpret and together in relation to a situation, to effectively recognise, formulate infer directly from a given model; or to including modifying a representation; and solve problems. This skill is translate directly from a situation into or when the demand is to devise a characterised as selecting or devising mathematics (for example, to structure representation of a situation. Higher a plan or strategy to use mathematics and conceptualise the situation in a level cognitive demand is marked by to solve problems arising from a relevant way, to identify and select the need to understand and use a non- task or context, as well as guiding its relevant variables, collect relevant standard representation that requires implementation. measurements and make diagrams). substantial decoding and interpretation; The mathematisation demand increases to devise a representation that captures In tasks with a relatively low demand with additional requirements to modify the key aspects of a complex situation; for this ability, it is often sufficient or use a given model to capture or to compare or evaluate different to take direct actions, where the changed conditions or interpret representations. strategy needed is stated or obvious. inferred relationships; to choose a At a slightly higher level of demand, familiar model within limited and clearly Reasoning and argument there may be a need to decide on articulated constraints; or to create a a suitable strategy that uses the This skill involves logically rooted model for which the required variables, relevant given information to reach a thought processes that explore and relationships and constraints are explicit conclusion. Cognitive demand is further link problem elements in order to and clear. At an even higher level, the heightened with the need to devise make inferences from them, check a mathematisation demand is associated and construct a strategy to transform justification that is given, or provide a with the need to create or interpret given information to reach a conclusion. justification of statements. a model in a situation in which many Even more demanding tasks call for the assumptions, variables, relationships In tasks of relatively low demand for construction of an elaborated strategy and constraints are to be identified or activation of this ability, the reasoning to find an exhaustive solution or a

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 59 generalised conclusion; or to evaluate symbolic concept. Increased cognitive demanded, with the others demanded or compare different possible strategies. demand is characterised by the need little or not at all. The communication for explicit use and manipulation of demand lies in the need to interpret Using symbolic, formal symbols (for example, by algebraically reasonably familiar nevertheless slightly and technical language and rearranging a formula), or by activation complex stimulus material, and the operations and use of mathematical rules, representation demand lies in the need definitions, conventions, procedures to handle two graphical representations This involves understanding, or formulas using a combination of of the data. For Q2, the representation manipulating, and making use of multiple relationships or symbolic demand is even higher because of symbolic expressions within a concepts. And a yet higher level of the need to process the two graphs mathematical context (including demand is characterised by the need in more detail. Each of the other arithmetic expressions and operations) for multi-step application of formal competencies is also called on to some governed by mathematical conventions mathematical procedures; working degree, with the need for reasoning, and rules. It also involves understanding flexibly with functional or involved some strategic thinking, and calling on and utilising formal constructs based on algebraic relationships; or using both some low-level procedural knowledge to definitions, rules and formal systems and mathematical technique and knowledge perform the required calculation. also using algorithms with these entities. to produce results. The symbols, rules and systems used For Carpenter, the reasoning required will vary according to what particular The research on these competencies comprises the most significant demand, mathematical content knowledge is saw a group of experts assign ratings but each of the other competencies is needed for a specific task to formulate, to PISA mathematics items according demanded to some degree. solve or interpret the mathematics. to the level of each competency demanded for successful completion of The message? The demand for activation of this each item. Sets of items were rated by Of course this research has further ability varies enormously across tasks. several experts, and the ratings were to go; nevertheless, the results of this In the simplest tasks, no mathematical analysed: the average ratings were used work are encouraging enough for me rules or symbolic expressions need as predictors in a regression on the to make some conjectures about the to be activated beyond fundamental empirical difficulty of the items. The importance of this set of competencies, arithmetic calculations, operating with level of demand for activation of these and about how this information might small or easily tractable numbers. More six competencies is an extremely good be used in mathematics classrooms: demanding tasks may involve direct predictor of the difficulty of the test use of a simple functional relationship, item. • Possession of these six either implicit or explicit (for example, competencies is crucial to the In Table 2 the competency ratings of familiar linear relationships); use of activation of one’s mathematical the illustrative items presented earlier, formal mathematical symbols (for knowledge. example, by direct substitution or assigned by three experts, are reported. • The more an individual possesses sustained arithmetic calculations For Exports Q1, a relatively easy item, these competencies, the more able involving fractions and decimals); or an the communication and representation he or she will be to make effective activation and direct use of a formal competencies are the most strongly mathematical definition, convention or use of his or her mathematical

Table 2: Competency ratings of three experts for the four illustrative PISA items

Rating  (from raters Competency 1/2/3) Commun- Repres- Reasoning and Devising Symbols and Item Mathematising ication entation argument strategies formalism Exports Q1 1/1/2 1/0/0 1/1/1 0/1/0 0/0/0 0/1/0 Exports Q2 1/1/2 1/0/1 2/2/2 1/1/1 2/0/1 0/1/1 Carpenter 2/2/1 1/0/1 1/1/1 2/3/2 2/1/1 1/1/1

Research Conference 2010 60 knowledge to solve contextualised problems. • These competencies should be directly targeted and advanced in our mathematics classes. In general, not enough time and effort is devoted in the mathematics classroom to fostering the development in our students of these fundamental mathematical competencies. Moreover, the curriculum structures under which mathematics teachers operate do not provide a sufficient impetus and incentive for them to focus on these competencies as crucial outcomes, alongside the development of the mathematical concepts and skills that typically take centre stage.

What actions can be taken to improve this situation? We must recognise the importance of the fundamental mathematical competencies that I have referred to. These competencies must be given a conscious focus in our mathematics classes, through teaching and learning activities, and through assessment. In my view, a key place to start is with the nature of discussion that is facilitated in mathematics classrooms. Students need to be given opportunities to articulate their thinking about mathematics tasks and about mathematical concepts. Obviously teachers play a central role in orchestrating that kind of discussion in class and this provides the basis for encouraging students to take the next key step, writing down their mathematical arguments. Giving emphasis to the communication of mathematical ideas and thinking, both in oral and written forms, is essential both to improving communication skills, but also to developing the mathematical ideas communicated and the capacities to use them.

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 61 Using mental representations of space when words are unavailable: Studies of enumeration and arithmetic in Indigenous Australia

Abstract special role in the emergence of exact arithmetic during child development’ Here we describe the nature and use (Pica et al., p. 503). This is a Whorfian of spatial strategies in a standard non- position: concepts of exact number are verbal addition task in two groups of impossible without counting words. children, comparing children who speak That is, one cannot possess the concept only languages in which counting words of exactly fiveness, without having a are not available with children who word corresponding to five. were raised speaking English. We tested speakers of Warlpiri and Anindilyakwa This view is not universal. Gelman Robert A. Reeve aged between 4 and 7 years old at and Gallistel (1978) argue that the University of Melbourne two remote sites in the Northern child’s development of verbal counting Territory of Australia. These children is a process of mapping a stably Robert Reeve completed his honours degree in used spatial strategies extensively, and ordered sequence of counting words psychology at the University of Sydney in 1976. were significantly more accurate when (CW) onto an ordered sequence of After completing a PhD at Macquarie University mental marks for numerosities they in 1984, under the supervision of Jacqueline they did so. English-speaking children Goodnow, he moved to the University of Illinois used spatial strategies very infrequently, call ‘numerons’. This system is shared to take up a postdoctoral position with Ann but relied an enumeration strategy with non-verbal species such as crows Brown. In 1986 he was awarded a National supported by counting words to do the and rats, and is implemented in an Academy of Education fellowship to study the ‘accumulator’ system that accumulates origins of children’s mathematical difficulties. He addition task. The main spatial strategy was also offered, and accepted, a faculty position exploited the known visual memory a fixed amount of neural energy or at the University of Illinois in the same year. He strengths of Indigenous Australians, and activity for each item enumerated. Each moved to the University of Melbourne in the involved matching the spatial pattern of numeron corresponds to a level of the early 1990s where he is currently an associate accumulator. professor in the Department of Psychological the augend set and the addend. These Sciences, in the Faculty of Medicine, Dentistry findings suggest that counting words, One can think of the mental number and Health Sciences. He runs the Developmental far from being necessary for exact Math Cognition group in Psychology Sciences, line (MNL) as being a scale that is members of which study the nature and origins arithmetic, offer one strategy among calibrated against the accumulator. of children’s mathematical learning difficulties. In others. They also suggest that spatial Similarly, one can think of the count collaboration with Brian Butterworth of University models for number do not need to be list as being lined up against points or College London, he has been awarded research one-dimensional vectors, as in a mental grants to investigate (1) indigenous mathematics, regions on the MNL. Spatial metaphors and (2) the nature of developmental dyscalculia. number line, but can be at least two- of abstract concepts and relations Since 2003, he has been working with Indigenous dimensional. are extremely widespread in human groups in the Northern Territory, studying cognition: emotions are described as ethnomathematics. He has also recently Introduction completed a six-year longitudinal study designed high or low, personal relationships to identify early markers of dyscalculia in young Indigenous Amazonians, whose can be close or distant, most people children. He serves on the editorial board of languages lack our kind of ‘count-list’, go forward into the future, backward several international child development journals. appear unable to accurately carry into the past, etc. It is not therefore out tasks that require ‘the capacity to surprising that cardinal numbers, which represent integers’ (Gordon, 2004; Pica, are abstract properties of sets, should Lemer, Izard, & Dehaene, 2004). The attract spatial models. The unconscious Amazonian researchers, therefore, claim spatial representation of numbers, that ‘Language would play an essential revealed in number bisection tasks, is role in linking up the various nonverbal usually thought of as one-dimensional representations to create a concept of vectors – a line with a single large exact number’ (Pica et al., p. 499) direction. However, where individuals and conclude ‘Our results thus support have automatic and conscious the hypothesis that language plays a representations of number – Galton’s

Research Conference 2010 62 ‘number forms’ (Galton, 1880) – these 2008). Here we focus on a non-verbal and 9 English-speaking children from are indeed lines, but more complex, in exact addition task. Addition is typically Melbourne. Approximately half the two or even three dimensions (Seron, acquired in stages using counting Northern Territory children were 4 to Pesenti, Noël, Deloche, & Cornet, procedures. Where two numbers or 5 years old and half were 6 to 7 years 1992; Tang, Ward, & Butterworth, two disjoint sets, say 3 and 5, are to old. 2008). be added together, in the earliest stage In Willowra and Angurugu, bilingual the learner counts all members of the Here we ask the question: what will Indigenous assistants were trained by an union of the two sets – that is, will individuals do when they do not have interviewer to administer the tasks, and count 1, 2, 3, and continue 4, 5, 6, 7, counting words in tasks that require all instructions were given by a native 8, keeping the number of the second exact calculation? The Whorfian speaker of Warlpiri or Anindilyakwa. set in mind. In a later stage, the learner position would entail that exact To acquaint helpers with research will ‘count-on’ from the number of the calculation is impossible. On the other practices and to familiarise children first set, starting with 3 and counting hand, the position espoused by Locke with test materials (e.g., counters), just 4, 5, 6, 7, 8. At a still later stage, (Locke, 1690/1961) and Whitehead familiarisation sessions were conducted. the child will count on from the larger (Whitehead, 1948), and subsequently Children played matching and sharing of the two numbers, now starting at 5, by Gelman and Butterworth (2005), games using test materials (counters and counting just 6, 7, 8. (Butterworth, is that ‘Distinct names conduce to our and mats). For the matching games, 2005). It is probably at this stage that well reckoning’ because, as Whitehead the interviewer put several counters addition facts are laid down in long- notes, ‘By relieving the brain of all on her mat, and children were asked term memory (Butterworth, Girelli, unnecessary work, a good notation to make their mat the same. Children Zorzi, & Jonckheere, 2001). If the sets it free to concentrate on more had little difficulty copying the number learner does not have access to these advanced problems, and in effect and location of counters on the strategies, because his or her language increases the mental power of the race’ interviewer’s mat. lacks the CW, what will they do? (Whitehead, 1948). (Note: Many learners during these In the basic memory task, identical Are CWs the only ‘good notation’? stages use their fingers – a handy 24-cm × 35-cm mats and bowls Here we examine the ability of set – to help them count, especially containing 25 counters were placed Indigenous Australian children of 4 to when the addition involves numbers in front of a child and the interviewer. 7 years to carry out simple non-verbal rather than sets of objects. That is, they The interviewer sat beside the child, as addition problems. These children will represent the 3 by raising three recommended in Kearins (1981), rather lived in remote sites in the Northern fingers, and then count on using the than opposite as is typical in testing Territory, and were monolingual in one five fingers of the other hand. Now, European children. The interviewer of two Australian languages, Warlpiri despite the fact that many cultures with took counters from her bowl and or Anindilyakwa. These languages have no specialised number words use body- placed them on her mat, one at a time, very limited number vocabularies. parts and body-part names to count, in pre-assigned locations. Four seconds Although these languages contain this is not what happens in Australia. after the last item was placed on the quantifiers such as few, many, a lot, Although gestural communications mat, all items were covered with a several, etc., these are not relevant are very widespread there (Kendon, cloth and children were asked by the number words, since they do refer to 1988), there is no record of body- Indigenous assistant to ‘make your exact numbers, and the theoretical part counting or of showing numbers mat like hers’. Following three practice claim is about exact numbers. Our using body-parts. This seems to be a trials in which the interviewer and an comparison group was a school in conventional form of communication Indigenous assistant modelled recall Melbourne. that is lacking in Australia. Indeed, none using one and two counters, children of our Northern Territory children used completed 14 memory trials comprising We have already shown that these their fingers to help them with these two, three, four, five, six, eight, or children perform accurately as English- tasks. nine randomly placed counters. In speaking children on tasks that required modelling recall, counters were placed remembering the number of objects on the mat without reference to their in an array and on matching the Method initial location. Number and locations number of sounds with a number of We tested 32 children aged 4 to 7 of children’s counter recall were objects (Butterworth & Reeve, 2008; years: 13 Warlpiri-speaking children, recorded. In earlier analyses we found Butterworth, Reeve, Reynolds, & Lloyd, 10 Anindilyakwa-speaking children, that Indigenous children tended to

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 63 use spatial strategies to reconstruct and whether these differences, if they pattern strategies more often, χ2 (1, the numerosities of random memory exist, affect problem-solving success. N = 56) = 18.08, p < .001. Similarly, arrays (Butterworth & Reeve, 2008). The strategy used to solve each when correct, older Northern Territory Of interest is whether they would problem was classified as either an children used an enumeration strategy use similar strategies in the non-verbal enumeration or a pattern strategy. more often than younger NT children, addition task. For a problem-solving attempt to be χ2 (1, N = 57) = 4.30, p < .05. For classified an enumeration strategy, the incorrectly solved problems, the results The same materials (mats and tokens used to convey answers were were reversed for Melbourne and counters) were used in the non-verbal placed by the child on his or her mat young Northern Territory children: addition task. The interviewer placed in a random or linear arrangement young Northern Territory children one counter on her mat and, after 4 (often with audible enumeration). For a tended to err when they used an seconds, covered her mat. Next, the problem-solving attempt to be classified enumeration strategy, χ2 (1, N = 62) = interviewer placed another counter a pattern strategy, a child appeared 14.91, p < .001. beside her mat and, while the child to concatenate the two patterns (the watched, slid the additional counter Figures 1 and 2 show strategy use original token pattern, and the pattern under the cover and onto her mat. for correct and incorrect answers as of added tokens). The pattern strategy Children were asked by the Indigenous a function of age and test location. reflects an attempted reproduction assistant to ‘make your mat like hers’. Figure 1 shows that Melbourne children of the spatial layout of the initial and Nine trials comprising 2 + 1, 3 + 1, are more likely to obtain the correct added arrays. In this case, no audible 4 + 1, 1 + 2, 1 + 3, 1 + 4, 3 + 3, 4 answer if they used an enumeration enumeration accompanied token + 2, and 5 + 3 were used. Children’s strategy (p < .01), and that this effect placement. These two strategies appear answers were recorded. We were is reversed for the younger Northern to reflect two meaningfully different particularly interested in the ways Territory children (p < .05). However, computation processes. in which computed answers to the older Northern Territory children’s non-verbal addition problems were When problems were solved correctly, correct non-verbal addition problem- approached, and in whether Indigenous Melbourne children used enumeration solving ability does not seem to depend children would use spatial strategies in strategies more often than their young on strategy use. However, Figure 2 computing answers. Northern Territory peers, who used shows that older Northern Territory

Results 100 The patterns of findings are reasonably clear. Compared to their Melbourne 90 Pattern peers, the younger Northern Territory 80 Enumeration children solved marginally more non- verbal addition problems correctly 70 (means = 2.3 and 3.2 problems correct 60 respectively, F (1, 20) = 3.27, p < .09). Further, the older Northern Territory 50 children solved more problems 40 correctly than the younger Northern Territory children (means = 3.2 and 30

4.5 problems respectively, F (1, 23) = of StrategyProportion Use 10.15, p < .01). 20 10 Strategies 0 Of interest are differences in the Melbourne1 Younger NT2 Older NT3 strategies used to solve the non-verbal addition problems by the different Children’s Location and Age groups of children (Melbourne vs 1 p < .01, 2 p < .05, 3 n.s. Northern Territory, and younger vs older Northern Territory children) Figure 1: Proportion of strategy use for correct nonverbal addition responses as a function of children’s location and age

Research Conference 2010 64 100 participants. It may well be that naming the number of objects in the array 90 Pattern to be remembered is the preferred strategy for the English-speaking 80 Enumeration children, but not for the Northern 70 Territory children. 60 Kearins (1986) considers two possible explanations for this. One is a genetic 50 hypothesis proposed by Lockard 40 (1971). According to this, there is selection of abilities according to 30 niche, especially where a population

Proportion of StrategyProportion Use is relatively isolated. Desert dwellers, 20 of the sort that Kearins tested, are 10 hunter-gatherers who are ‘possessor of unusual knowledge and skills in the 0 natural world. They can live off the land 1 1 2 Melbourne Younger NT Older NT where almost no Westerners can do Children’s Location and Age so, finding water and food in apparently arid country.’ People began to occupy 1 n.s, 2 p < .05 Australia at least 40 000 years ago Figure 2: Proportion of strategy use for incorrect nonverbal addition responses as a (Flood, 1997) and have been relatively function of children’s location and age isolated from other populations during that time. Thus, survival in this hostile environment may have favoured children are more likely to err if they responses (5 vs 13) used the pattern those who could acquire these special used an enumeration strategy (p < .05). strategy. skills. The ability to retain spatial and These results suggest that a pattern- topographical information could make Discussion matching strategy is an effective spatial the difference between life and death It is clear that English-speaking children heuristic when CWs to support in the desert. By contrast, the invention in Melbourne almost never use the enumeration are not available. Notice of agriculture 10 000 years ago put pattern strategy, but perform the task that the patterns used here are two- an emphasis on different kinds of using an enumeration strategy. By dimensional, suggesting that a one- skills, and also resistance to animal- contrast, Northern Territory children dimensional oriented number line originated diseases that are pandemic matched in age with the English- is not the only way for children to in Europe and Asia, such as smallpox, speakers, use pattern strategies nearly represent numbers. One might ask measles etc. (Diamond, 1997). It is twice often as enumeration. What is why pattern matching is the preferred striking therefore that in Kearins’s study, of particular interest is the fact that strategy for the Northern Territory both semi-traditional participants who the pattern strategy is more effective children. One possible reason is that lived in the desert and non-traditional for them, and that attempting to Indigenous Australians are very good participants who lived on the desert enumerate leads to a preponderance at remembering spatial patterns. In fringe performed equivalently, and of errors. Indeed, even for the English- a version of Kim’s game, where one better on all tasks than non-indigenous speakers, the only four documented has to recall the location of a variety participants from a forestry and farming uses of pattern were all correct. The objects on a tray, Kearins (1981) area. These results appear to support older Northern Territory children have showed that Indigenous adolescents the genetic hypothesis since it is not begun to use the pattern strategy more and children were superior to their where you live but your ancestry that often, now making up about half of all non-Indigenous counterparts. Moreover, is critical. strategies used. However, the majority Kearins found that the nameability However, Kearins (1986) raises of their correct responses (30 vs 24) of the objects in the array to be another possibility: differences in and the minority of their incorrect remembered, affected non-indigenous child-rearing practices. Indigenous participants but not Indigenous Australians, like other hunter-gatherers,

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 65 rarely transmit information or skills by from Indigenous Australian children. Pica, P., Lemer, C., Izard, V., & Dehaene, verbal instruction (‘All that nagging’). Proceedings of the National Academy S. (2004). Exact and approximate Rather children are encouraged of Sciences of the USA, 105, 13179– calculation in an Amazonian indigene to learn by observation. This may 13184. group with a reduced number lexicon. mean that children acquire skills of Science, 306, 499–503. Diamond, J. (1997). Guns, germs and remembering what they see earlier or steel: The fates of human societies. Seron, X., Pesenti, M., Noël, M.-P., better than non-indigenous children. London: Jonathan Cape. Deloche, G., & Cornet, J.-A. (1992). This is supported by several studies Images of numbers, or ‘When 98 is that Kearins cites. Thus, parents and Flood, J. (1997). Rock art of the upper left and 6 sky blue’. Cognition, the general learning environment Dreamtime: Images of ancient Australia. 44, 159–196. of Indigenous Australian children Sydney, Australia: Harper Collins encourage those skills particularly Publishers. Tang, J., Ward, J., & Butterworth, B. useful for the desert niche, of which (2008). Number forms in the brain. Galton, F. (1880). Visualised numerals. good spatial memory and routine Journal of Cognitive Neuroscience, 20(9), Nature, 21, 252–256. dependence on it are a part. Of 1547–1556. course, genetic factors and child-rearing Gelman, R., & Butterworth, B. (2005). Whitehead, A. N. (1948). An practices may not be unrelated. Number and language: How are they introduction to mathematics ((Originally related? Trends in Cognitive Sciences, We do not doubt that a good notation published in 1911) ed.). London: 9(1), 6–10. is helpful for carrying out mental Oxford University Press. work, in this case, carrying out simple Gelman, R., & Gallistel, C. R. (1978). addition. However, our results suggest The child’s understanding of number. that counting words are not the only (1986 ed.). Cambridge, MA: Harvard good notation, and that a strategy for University Press. mapping items to be enumerated onto Gordon, P. (2004). Numerical cognition a spatial representation could also be without words: Evidence from effective when counting words are not Amazonia. Science, 306, 496–499. available. The relationship between an accumulator mechanism and a two- or Kearins, J. (1981). Visual spatial memory three-dimensional mental spatial array is of Australian Aboriginal children of still to be elucidated. desert regions. Cognitive Psychology, 13, 434–460. References Kearins, J. (1986). Visual spatial memory Butterworth, B. (2005). The in Aboriginal and White Australian development of arithmetical abilities. Children. Australian Journal of Journal of Child Psychology & Psychiatry, Psychology, 38(3), 203–214. 46(1), 3–18. Kendon, A. (1988). Sign languages of Butterworth, B., Girelli, L., Zorzi, M., & Aboriginal Australia: Cultural, semiotic Jonckheere, A. R. (2001). Organisation and communicative perspectives. of addition facts in memory. Quarterly Cambridge: Cambridge University Journal of Experimental Psychology, 54A, Press. 1005–1029. Lockard, R. B. (1971). Reflections on Butterworth, B., & Reeve, R. (2008). the fall of comparative psychology – Is Verbal counting and spatial strategies there a message for us all? American in numerical tasks: Evidence from Psychologist, 26, 168–179. Indigenous Australia. Philosophical Locke, J. (1690/1961). An essay Psychology, 21, 443–457. concerning human understanding (Based Butterworth, B., Reeve, R., Reynolds, F., on Fifth Edition, J. W. Yolton (Ed.). & Lloyd, D. (2008). Numerical thought London: J. M. Dent. with and without words: Evidence

Research Conference 2010 66 Using technology to support effective mathematics teaching and learning: What counts?

Abstract policy and practice might usefully inform each other in supporting effective What counts when it comes to mathematics teaching and learning in using digital technologies in school Australian schools. mathematics? Is technology there to help students get ‘the answer’ more The first part of the presentation quickly and accurately, or to improve considers key messages from research the way they learn mathematics? The on learning and teaching mathematics way people answer this question is with digital technologies. The second illuminating and can reveal deeply held part offers some snapshots of Merrilyn Goos beliefs about the nature of mathematics practice to illustrate what effective The University of Queensland and how it is best taught and learned. classroom practice can look like when This presentation considers the extent technologies are used in creative Merrilyn Goos is Director of the Teaching to which technology-related research, ways to enrich students’ mathematics and Educational Development Institute at The policy and practice might usefully inform learning. The third part analyses the University of Queensland. From 1998–2007 technology messages contained in the Professor Goos co-ordinated pre-service and each other in supporting effective postgraduate courses in mathematics education at mathematics teaching and learning in draft Australian curriculum – Mathematics UQ. Her research in mathematics education has Australian schools. The first part of the and the challenges of aligning curriculum investigated secondary school students’ learning, policy with research and practice. teaching approaches that promote higher order presentation considers key messages thinking, mathematics teachers’ learning and from research on learning and teaching development, and the professional learning of mathematics with digital technologies. Key messages from research mathematics teacher educators. This work has The second part offers some snapshots on learning and teaching been supported by two ARC Large Grants and two ARC Discovery Grants. Professor Goos has of practice to illustrate what effective mathematics with digital also led large-scale, cross-institutional research classroom practice can look like when technologies projects commissioned by the Australian and technologies are used in creative Fears are sometimes expressed that the Queensland Governments in numeracy education ways to enrich students’ mathematics and school reform. In 2004 she won an Australian use of technology, especially hand-held learning. The third part analyses the Award for University Teaching, followed in 2006 calculators, will have a negative effect technology messages contained in the by an Associate Fellowship of the Carrick Institute on students’ mathematics achievement. for Learning and Teaching in Higher Education draft Australian curriculum – Mathematics However, meta-analyses of published (now the Australian Learning and Teaching and the challenges of aligning curriculum Council). Professor Goos is currently President of research studies have consistently found policy with research and practice. the Mathematics Education Research Group of that calculator use, compared with non- Australasia. Introduction calculator use, has either positive or neutral effects on students’ operational, Digital technologies have been available computational, conceptual and in school mathematics classrooms since problem-solving skills (Ellington, 2003; the introduction of simple four-function Hembree & Dessart, 1986; Penglase & calculators in the 1970s. Since then, Arnold, 1996). A difficulty with these computers equipped with increasingly meta-analyses, however, is that they sophisticated software, graphics select studies that compare treatment calculators that have morphed into ‘all- (calculator) and control (non-calculator) purpose’ hand-held devices integrating groups of students, with the assumption graphical, symbolic manipulation, that the two groups experience statistical and dynamic geometry otherwise identical learning conditions. packages, and web-based applications Experimental designs such as this do offering virtual learning environments not take into account the possibility have changed the mathematics teaching that technology fundamentally changes and learning terrain. Or have they? students’ mathematical practices and This presentation considers the extent even the nature of the mathematical to which technology-related research, knowledge they learn at school.

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 67 Technology and mathematical knowing and doing come together is servant if used by students or teachers knowledge evident in the mathematical practices only as a fast, reliable replacement for of the classroom. For example, school pen and paper calculations without In their contribution to the 17th ICMI mathematical practices that, in the past, changing the nature of classroom Study on Mathematics Education and were restricted to memorising and activities. Technology is a partner when Technology, Olive and Makar (2010) reproducing learned procedures can be it provides access to new kinds of tasks analysed the influence of technology on contrasted with mathematical practices or new ways of approaching existing the nature of mathematical knowledge endorsed by most modern curriculum tasks to develop understanding, explore as experienced by school students. documents, such as conjecturing, different perspectives, or mediate They argued as follows: justifying and generalising. Technology mathematical discussion. Technology If one considers mathematics to can change the nature of school becomes an extension of self when be a fixed body of knowledge mathematics by engaging students in seamlessly integrated into the practices to be learned, then the role of more active mathematical practices of the mathematics classroom. such as experimenting, investigating and technology in this process would Pierce and Stacey (2010) offer an problem solving that bring depth to be primarily that of an efficiency alternative representation of the ways their learning and encourage them to tool, i.e. helping the learner to do in which technology can transform ask questions rather than only looking the mathematics more efficiently. mathematical practices. Their for answers (Farrell, 1996; Makar & However, if we consider the pedagogical map classifies ten types of Confrey, 2006). technological tools as providing pedagogical opportunities afforded by access to new understandings of Olive and Makar (2010) argue a wide range of mathematical analysis relations, processes, and purposes, that mathematical knowledge and software. Opportunities arise at three then the role of technology relates mathematical practices are inextricably levels that represent the teacher’s to a conceptual construction kit. linked, and that this connection can be thinking about: (p. 138) strengthened by the use of technologies. • the tasks they will set their students They developed an adaptation of Their words encapsulate the contrasting (using technology to improve speed, Steinbring’s (2005) ‘didactic triangle’ purposes of technology that were accuracy, access to a variety of that in its original form represents the foreshadowed in the opening paragraph mathematical representations) of this paper. For learners, mathematical learning ecology as interactions between knowledge is not fixed but fluid, student, teacher and mathematical • classroom interactions (using constantly being created as the learners knowledge. Introducing technology technology to improve the display of interact with ideas, people and their into this system transforms the learning mathematical solution processes and environment. When technology is part ecology so that the triangle becomes a support students’ collaborative work) tetrahedron, with the four vertices of of this environment, it becomes more • the subject (using technology to student, teacher, task and technology than a substitute for mathematical work support new goals or teaching creating ‘a space within which new done with pencil and paper. Consider, methods for a mathematics course). for example, the way in which dynamic mathematical knowledge and practices geometry software allows students may emerge’ (p. 168). Snapshots of classroom to transform a geometric object by Within this space, students and teachers mathematical practice ‘dragging’ any of its constituent parts may imagine their relationship with Two snapshots are presented here to to investigate its invariant properties. technologies in different ways. Goos, illustrate how technology can be used Through this experimental approach, Galbraith, Renshaw and Geiger (2003) creatively to support new mathematical students make predictions and test developed four metaphors to describe practices. conjectures in the process of generating how technologies can transform mathematical knowledge that is new for teaching and learning roles. Technology Changing tasks and classroom them. can be a master if students’ and interactions teachers’ knowledge and competence Technology and Mathematical are limited to a narrow range of Geiger (2009) used the master-servant- Practices operations. Students may become partner-extension-of-self framework to Learning mathematics is as much about dependent on the technology if they analyse a classroom episode in which doing as it is about knowing. How are unable to evaluate the accuracy of he asked his Year 11 students to use the output it generates. Technology is a the dynamic geometry facility on their

Research Conference 2010 68 — CAS calculators to draw a line √45 Changing course goals and syntax was correct, but said they should units long. His aim was to encourage teaching methods think harder about their assumptions. students to think about the geometric Eventually, the teacher directed the representation of irrational numbers. Geiger, Faragher and Goos (in press) problem to the whole class and one The anticipated solution involved investigated how CAS technologies student spotted the problem: ‘You can’t using the Pythagorean relationship support students’ learning and social — have an exponential equal to zero’. This 62 + 32 = ( 45 )2 to construct a right- interactions when they are engaged in √ resulted in a whole class discussion of angled triangle with sides 6 and 3 units mathematical modelling tasks. In this — the assumption that extinction meant a long and hypotenuse 45 units long. snapshot, Year 12 students worked on √ population of zero, which they decided Figure 1 summarises the flow of the the following question: was inappropriate. The class then episode and how technology was used. When will a population of 50,000 agreed on the position that extinction In this episode, technology was initially bacteria become extinct if the was ‘any number less than one’. used as a servant to perform numerical decay rate is 4% per day? Students used CAS to solve this new calculations that did not lead to the One pair of students developed equation and obtain a solution. desired geometric solution. It became an initial exponential model for In this episode the teacher exploited a partner when students passed their the population y at any time x, the ‘confrontation’ created by the calculators around the group or x y = 50000 x (0.96) . They then CAS output to promote productive displayed their work to the whole class equated the model to zero in order interaction among the class (technology to offer ideas for comment and critique. to represent the point at which the as partner). Using this pedagogical As a partner it gave the student who bacteria would be extinct, with the opportunity allowed the teacher to found the solution the confidence he intention of using CAS to solve this refocus course goals and teaching needed to introduce his conjectured equation. When they entered the methods on promoting thinking about solution into a heated small group equation into their CAS calculator, the mathematical modelling process debate. In terms of Pierce and Stacey’s however, it unexpectedly responded rather than on practice of skills. (2010) pedagogical map, this episode with a false message. The students illustrates opportunities provided by a thought this response was a result Aligning curriculum with task that link numerical and geometric of a mistake with the syntax of their research and practice? representations to support classroom command. When they asked their interactions where students share and teacher for help, he confirmed their The brief research summary and discuss their thinking. classroom snapshots presented above show how digital technologies provide a ‘conceptual construction kit’ (Olive & — Makar, 2010, p. 138) that can transform Table 1: Draw a line √45 units long students’ mathematical knowledge and practices. To what extent does the Classroom interaction Role of technology Australian curriculum – Mathematics Students find the square roots of various numbers. Servant support this transformative view of Students pass calculators back and forth to share and Partner technology? critique each other’s thinking. The shape paper that provided the Teacher invites student to present calculator work to Master (prior group initial outline of the K–12 mathematics whole class. Audience identifies misconceptions about work) then partner curriculum (National Curriculum Board, how calculators display decimal versions of irrational (whole class display 2009) made it clear that technologies numbers. and discussion) should be embedded in the curriculum ‘so that they are not seen as optional Teacher hint: think about triangles. Students search Servant tools’ (p. 12). Digital technologies were for Pythagorean formulation without geometric seen as offering new ways to learn and representation. teach mathematics that helped deepen Teacher redirects students to consider geometry, not Partner students’ mathematical understanding. just numbers. Student interrupts group discussion to It was also acknowledged that students propose geometric solution; passes his calculator around should learn to choose intelligently group to share and defend his solution.

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 69 between technology, mental, and pencil (2010) taxonomy, in that technology on technology mediated learning and paper methods. may be used to make computation and in secondary school mathematics graphing quicker and more accurate classrooms. Journal of Mathematical The draft consultation version 1.0 and possibly to link representations. Behavior, 22, 73–89. of the K–10 mathematics curriculum expected ‘that mathematics classrooms Although the technology messages Hembree, R., & Dessart, D. (1986). will make use of all available ICT in contained in the Australian curriculum Effects of hand-held calculators in teaching and learning situations’. The – Mathematics do not do justice to pre-college mathematics education: A intention is that use of ICT is to be what research tells us about effective meta-analysis. Journal for Research in referred to in content descriptions teaching and learning of mathematics, Mathematics Education, 17, 83–99. and achievement standards. Yet this it is almost inevitable that there are Makar, K., & Confrey, J. (2006). is done superficially and inconsistently gaps between an intended curriculum Dynamic statistical software: How throughout the curriculum, with and the curriculum enacted by teachers are learners using it to conduct data- technology often being treated as and students in the classroom. Many based investigations? In C. Hoyles, J. an add-on that replicates by-hand teachers are already using technology Lagrange, L. H. Son, & N. Sinclair (Eds.), methods. This is seen, for example, in effectively to enhance students’ Proceedings of the 17th Study Conference the following content description from understanding and enjoyment of of the International Commission on the Year 8 Number and Algebra strand: mathematics. In their hands lies the Mathematical Instruction. Hanoi Institute ‘Plot graphs of linear functions and use task of enacting a truly futures-oriented of Technology and Didirem Université these to find solutions of equations curriculum that will prepare students Paris 7. including use of ICT’ (emphasis added). for intelligent, adaptive and critical citizenship in a technology-rich world. National Curriculum Board (2009). In the corresponding consultation Shape of the Australian curriculum: versions of the four senior secondary Mathematics. Retrieved May 29, 2010 mathematics courses, the aims for all References from http://www.acara.edu.au/verve/_ courses refer to students choosing Ellington, A. (2003). A meta-analysis of resources/Australian_Curriculum_-_ and using a range of technologies. the effects of calculators on students’ Maths.pdf Nevertheless, each course contains achievement and attitude levels in a common technology statement – precollege mathematics classes. Journal Olive, J., & Makar, K., with V. Hoyos, L. ‘Technology can aid in developing for Research in Mathematics Education, K. Kor, O. Kosheleva, & R. Straesser skills and allay the tedium of repeated 34, 433–463. (2010). Mathematical knowledge and calculations’ – that betrays a limited practices resulting from access to digital view of its role. Across the courses, Farrell, A. M. (1996). Roles and technologies. In C. Hoyles & J. Lagrange variable messages about the use of behaviors in technology-integrated (Eds.), Mathematics education and technology are conveyed in words like precalculus classrooms. Journal of technology – Rethinking the terrain. The ‘assumed’ and ‘vital’ in Essential and Mathematical Behavior, 15, 35–53. 17th ICMI Study (pp. 133–177). New General Mathematics to ‘should be Geiger, V. (2009). Learning mathematics York: Springer. widely used in this topic’, ‘can be used with technology from a social Penglase, M., & Arnold, S. (1996). The to illustrate practically every aspect perspective: A study of secondary graphics calculator in mathematics of this topic’, or no mention at all for students’ individual and collaborative education: A critical review of recent some topics in Mathematical Methods practices in a technologically rich research. Mathematics Education and Specialist Mathematics. mathematics classroom. Unpublished Research Journal, 8, 58–90. In both the K–10 and senior secondary doctoral dissertation, The University Pierce, R., & Stacey, K. (2010). Mapping mathematics curricula, uses of of Queensland, Brisbane, Australia. pedagogical opportunities provided technology, where made explicit, are Geiger, V., Faragher, R., & Goos, M. by mathematics analysis software. mostly consistent with the servant (in press). CAS-enabled technologies International Journal of Computers for metaphor of Goos et al. (2003), despite as ‘agents provocateurs’ in teaching Mathematical Learning, 15(1), 1–20. the more transformative intentions and learning mathematical modelling evident in the initial shaping paper. in secondary school classrooms. Steinbring, H. (2005). The construction Pedagogical opportunities afforded by Mathematics Education Research Journal. of new mathematical knowledge in the curriculum are restricted to the classroom interaction: An epistemological Goos, M., Galbraith, P., Renshaw, P., level of tasks in Pierce and Stacey’s perspective. New York: Springer. & Geiger, V. (2003) Perspectives

Research Conference 2010 70 Making connections to the big ideas in mathematics: Promoting proportional reasoning

Abstract sometimes with life-threatening or disastrous consequences, for example, The focus of this paper is on incorrect doses in medicine (Preston, proportional reasoning, emphasising 2004). Proportional reasoning therefore its pervasiveness throughout the is a major aspect of numeracy, yet it mathematics curriculum, but also is implicit in school curricula and often highlighting its elusiveness. Proportional limited to the study of rate and ratio in reasoning is required for students mathematics only. to operate successfully in many rational number topics (fractions, The development of proportional Shelley Dole decimals, percentages), but also other reasoning is a complex operation, and The University of Queensland topics (scale drawing, probability, ... [it] requires firm grasp of trigonometry). Proportional reasoning various rational number concepts Shelley Dole is a senior lecturer in mathematics is also required in many other school education at The University of Queensland. such as order and equivalence, Dr. Dole is Director of the Primary and Middle curriculum topics (for example, drawing the relationship between the Years Teacher Education Programs and teaches timelines in history; interpreting unit and its parts, the meaning in Bachelor and Master of Education courses. density, molarity, speed calculations and interpretation of ratio, and Dr. Dole is an experienced classroom teacher, in science). In this paper, an overview having taught in primary and secondary schools in issues dealing with division, Victoria, Northern Territory and Queensland. She of mathematics education research especially as this relates to has also been a tertiary educator in universities in on proportional reasoning will be dividing smaller numbers by larger Queensland, Tasmania and Victoria. Her research presented, highlighting the complex ones. A proportional reasoner interests include students’ mathematical learning nature of the development of difficulties, misconceptions and conceptual has the mental flexibility to change; assessment in mathematics; middle years proportional reasoning and implications approach problems from multiple mathematics curriculum; mental computation; for learning and instruction. Through perspectives and at the same time the development of proportional reasoning and presentation of results of a current has understandings that are stable multiplicative thinking within the study of rational research project on proportional number, and mental computation and numeracy. enough not to be radically affected Dr. Dole’s research interests focus particularly on reasoning in the middle years, teaching by large or ‘awkward’ numbers, promoting students’ conceptual understanding of approaches that have captured and or the context within which a mathematics to encourage success and enjoyment engaged students’ interest in exploring problem is posed. (Post, Behr & of mathematical investigations in school. proportion-related situations will be Lesh, 1988, p. 80) shared. Proportional reasoning is intertwined Background with many mathematical concepts. For example, English and Halford Proportional reasoning is a fundamental (1995) stated that: ‘Fractions are the cornerstone of mathematics knowledge building blocks of proportion’ (p. 254). (Lesh, Post, & Behr, 1988). Proportional Similarly, Behr et al. (1992) stated that reasoning is the ability to understand ‘the concept of fraction order and situations of comparison. Examples of equivalence and proportionality are everyday tasks that require proportional one component of this very significant reasoning include estimating the better and global mathematical concept’ (p. buy, interpreting scales and maps, 316). Also, Streefland (1985) suggested determining chances associated with that ‘Learning to view something ‘in gambling and risk-taking. Proportional proportion’, or ‘in proportion with ...’ reasoning has been described as precedes the acquisition of the proper one of the most commonly applied concept of ratio’ (p. 83). Developing mathematics concepts in the real students’ understanding of ratio and world (Lanius & Williams, 2003). proportion is difficult because the Underdeveloped proportional reasoning concepts of multiplication, division, potentially impacts real-world situations, fractions and decimals are the building

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 71 blocks of proportional reasoning, and students’ informal intuitive problem (Post et al., 1988). The literature students’ knowledge of such topics is solving procedures, guiding students provides various suggestions for generally poor (Lo & Watanabe, 1997). to ‘formulae and algorithmisation’ (p. activities and strategies for promoting 84). Such an approach was taken in a the proportion concept. The use of The development of proportional teaching experiment conducted by Lo ratio tables has been suggested as reasoning is a gradual process, and Watanabe (1997) where a Year one means for building students’ ratio underpinned by increasingly 5 child was exposed to proportional understanding (English & Halford, sophisticated multiplicative thinking and reasoning tasks to promote intuitive 1995; Middleton & Van den Heuvel- the ability to compare two quantities multiplicative reasoning skills and hence Panhuizen, 1994; Robinson, 1981; in relative (multiplicative), rather than develop proportional reasoning. Streefland, 1985). English and Halford absolute (additive) terms (Lamon, (1995) provided the following example 2005). The essence of proportional Research has indicated that students’ of a ratio table, which assists in the reasoning is on understanding the (and teachers’) understanding of comparison of the number of soup multiplicative structures inherent in proportion is generally poor (e.g., Behr cubes per person: proportion situations (Behr, Harel, et al., 1992; Fisher, 1988; Hart, 1981). Post & Lesh, 1992). Children’s intuitive Streefland (1985) stated that ‘Ratio is soup cubes 2 4 6 8 strategies for solving proportion introduced too late to be connected people 4 8 12 16 problems are typically additive (Hart, with mathematically related ideas 1981). The teacher’s role, therefore, is such as equivalence of fractions, scale, English and Halford stated, ‘A table of to build on students’ intuitive additive percentage’ (p. 78). English and Halford this nature provides an effective means strategies and guide them towards (1995) suggested that proportional of organising the problem data and building multiplicative structures. Strong reasoning is taught in isolation and thus enables children to detect more readily multiplicative structures develop as remains unrelated to other topics. Behr all the relations displayed, both within early as the second grade for some et al. (1992) stated, ‘We believe that and between the series ... it serves as a children, but are also seen to take time the elementary school curriculum is permanent record of proportion as an to develop to a level of conceptual deficient by failing to include the basic equivalence relation’ (p. 254). stability, often beyond fifth grade (Clark concepts and principles relating to & Kamii, 1996). Behr et al. (1992) multiplicative structures necessary for The MC SAM project suggested that exploring change will later learning in intermediate grades’(p. Promoting proportional reasoning has help students develop multiplicative 300). Behr et al. also added, ‘There is been the focus of a large research understanding. For example, students a great deal of agreement that learning project undertaken by The University can be encouraged to discuss the rational number concepts remains a of Queensland (2007–2010). Not only change to 4 which will result in 8. From serious obstacle in the mathematical did this project target proportional an additive view, 4 can change to 8 by development of children ... In contrast reasoning in mathematics but in science adding 4. From a multiplicative view, there is no clear argument about how as well, as proportional reasoning is 4 can change to 8 by multiplying by 2. to facilitate learning of rational number fundamental to many topics in both The difference between the additive concepts’ (p. 300). mathematics and science (Lamon, and multiplicative view can be seen by As the proportion concept is 2005). The MC SAM project, an looking at other numbers. The additive intertwined with many mathematical acronym for Making Connections: rule holds for 13 changing to 17, but concepts, this has implications for Science and Mathematics, brought not the multiplicative rule. According instruction. The development of a together middle years’ mathematics to Behr et al. (1992), ‘the ability to rich concept of rational number, and and science teachers around this represent change (or difference) in thus proportional relationships, takes important topic, providing an both additive and multiplicative terms a long time (Streefland, 1985). The opportunity for teachers to explore and to understand their behaviour proportional nature of various rational the proportional reasoning linkages under transformation is fundamental number topics must be the focus of between topics in both mathematics to understanding fraction and ratio instruction as these topics are revisited and science, and to create, implement equivalence’ (p. 316). Moving students continually throughout the curriculum, and evaluate innovative and engaging towards formal ratio and proportion in order to build and link students’ learning experiences to assist students principles and procedures is termed proportional understanding (Behr et al., to promote and connect essential by Streefland (1985) as ‘anticipating 1992). Building proportional reasoning mathematics and science knowledge. ratio’, where the teacher capitalises on must be through multiple perspectives The project had two major aims. First,

Research Conference 2010 72 it aimed to develop an instrument Developing the instrument was ninth graders averaged just 21 per cent to assess middle years students’ guided by literature and especially and the fourth graders averaged 5 per proportional reasoning knowledge. the American Association for the cent for the same item. The results Second, it aimed to use this data to Advancement of Science (AAAS) were a wake-up call to all teachers in develop and trial specific learning (2001) Atlas of Science Literacy. The the project: the fourth and fifth grade experiences in both mathematics and Atlas identifies two key components teachers realised that there were some science that may support students’ of proportional reasoning: Ratios very good proportional reasoners in access to particular topics in those and Proportion (parts and wholes, their grades, and the eighth and ninth subjects and promote proportional descriptions and comparisons and grade teachers realised that they were reasoning skills. computation) and Describing Change taking for granted the proportional (related changes, kinds of change, and reasoning skills of their students. Item There is a large corpus of existing invariance). The AAAS provided the analysis and students’ results provided research that has provided analysis framework for the development of direction for targeted teaching. of strategies applied by students the proportional reasoning assessment Collectively, results of the whole test to various proportional reasoning instrument. The test included items suggested that a much greater focus on tasks (e.g., Misailidou & Williams, on direct proportion (whole number proportional reasoning must occur in all 2003; Hart, 1981), Such research has and fractional ratios), rate and inverse classes at every opportunity. highlighted issues associated with the proportion items, as well as fractions, impact of ‘awkward’ numbers (that Throughout the project, a series of probability, speed and density items. is, common fractions and decimals integrated mathematics and science Guided by the words of Lamon (2005), as opposed to whole numbers), the tasks has been developed, shared and who suggested that students must be common application of an incorrect adapted by the teachers. One of the provided with many different contexts, additive strategy, and the blind simplest, and one that has been taken ‘to analyse quantitative relationships application of rules and formulae to up most widely by all fourth grade to in context, and to represent those proportion problems. Prior research ninth grade teachers, is an exploration relationships in symbols, tables, and has also emphasised the complexity into why penguins huddle, incorporating graphs’ (p. 3), the items included of the development of proportional the surface area to volume ratio. contexts of shopping, cooking, mixing reasoning and the need for further and By using three 2-cm cubic blocks, cordial, painting fences, graphing stories, continued work in the field to support penguins can be created. Focusing on saving money, school excursions students’ development of proportional one penguin, the surface area of the anddual measurement scales. For reasoning. In fact, it is estimated that penguin can be found by counting each item on the test, students were approximately only 50 per cent adults the faces of the cubes (14) and the required to provide the answer and can reason proportionately (Lamon, volume can be counted by counting explain the thinking they applied to 2005). In our study, we wanted to the number of cubes (3). A huddle solve the problem. take a snapshot of a large group of is formed by putting 9 penguins into students’ proportional reasoning on Approximately 700 students in the a cubic arrangement. A data table is tasks that relate to mathematics and middle years of schooling (Years 4–9) constructed and students can analyse science curriculum in the middle participated in this assessment. Initially, the results to consider how the surface years of schooling. This component of project teachers had mixed feelings area to volume ratio changes as the the project was concerned with the about the test’s capacity to assess their huddle gets bigger. development of an instrument that students’ proportional reasoning. The One of the capstone elements of the would provide a ‘broad brush’ measure ninth grade teachers stated that they project has been the development of of students’ proportional reasoning and thought the test would be too easy for a unit of work on density. Although their thinking strategies, and that would their students; the fourth grade teachers density is typically regarded as a have some degree of diagnostic power. stated that the test was too hard. The topic within the middle years science This challenge was undertaken with full highest average score however, for curriculum, conceptual understanding awareness of both the pervasiveness the ninth-graders on one item was just of density requires understanding of and the elusiveness of proportional 75 per cent, with the fourth-graders mathematics topics including mass reasoning throughout the curriculum averaging 15 per cent for that item. and volume, as well as number sense and that its development is dependent On several other items, the eighth and mental computation. It also upon many other knowledge and ninth graders scored less than 50 requires data gathering, data analysis, foundations in mathematics and science. per cent. On one particular item, the interpretation of data, graphing,

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 73 measuring, using measuring instruments, force equation) may be based on M. Behr (Eds.), Number concepts and problem solving, problem posing, assumptions of students’ proportional operations in the middle grades (pp. conducting experiments and controlling reasoning that are not stable. The 93–118). Hillsdale, NJ: Erlbaum. variables, which are components of significance of this project has been Lo, J-J., & Watanabe, T. (1997). both mathematics and science curricula. that it brought together mathematics Developing ratio and proportion The integrated unit on density was and science teachers to explore the schemes: A story of a fifth grader. developed and trialled in a number synergies between mathematics and Journal for Research in Mathematics of middle years mathematics and/or science curriculum through proportional Education, 28(2), 216–236. science classrooms. It was implemented reasoning. to varying degrees in most classes by Middleton, J., & Van den Heuvel- project teachers, but was specifically References Panhuizen, M. (1995). The ratio table. implemented by the project team in a Mathematics Teaching in the Middle fifth and seventh grade classroom. At American Association for the School, 1(4), 282–288. the beginning of the unit, the students’ Advancement of Science (AAAS). Misailidou, G., & Williams, J. (2003). had limited knowledge of density, with (2001). Atlas of Science Literacy:Project Diagnostic assessment of children’s developing understanding of mass and 2061. AAAS. proportional reasoning. Journal of volume. At the end of the unit, students Behr, M., Harel, G., Post, T., & Lesh, Mathematical Behaviour, 22, 335–368. could describe how an object might R. (1992). Rational number, ratio sink or float in water by simultaneously and proportion. In D. Grouws (Ed.), Post, T., Behr, M., & Lesh, R. (1988). considering both its volume and Handbook on research of teaching and Proportionality and the development mass. All students could verbalise learning (pp. 296–333). New York: of prealgebra understandings. In A. the concept of density and showed McMillan. F. Coxford & A. P. Shulte (Eds.), The greater conceptualisation of units of Ideas of Algebra, K–12 (pp. 78–90). Clark, F. & Kamii, C. (1996). measure for volume. Results of this Reston, VA: NCTM. Identification of multiplicative thinking study provide evidence of the capacity in children in Grades 1–5. Journal for Preston, R. (2004). Drug errors & of targeted, integrated mathematics Research in Mathematics Education, patient safety: The need for a change and science units for the development 27(1), 41–51. in practice. British Journal of Nursing, of connected mathematics and 13(2), 72–78. science knowledge and promotion of English, L., & Halford, G. (1995) proportional reasoning skills. Mathematics education: Models and Robinson, F. (1981). Rate and ratio: processes. Mahwah, NJ: Erlbaum. Classroom tested curriculum materials Concluding comments for teachers at elementary level. The Fisher, L. (1988). Strategies used by Ontario Institute for Studies in The development of proportional secondary mathematics teachers to Education, Ontario: OISE Press. reasoning is a slow process exacerbated solve proportion problems. Journal for by its nebulous nature and lack of Research in Mathematics Education, Streefland, L. (1985). Searching for the specific prominence in school syllabus 19(2), 157–168. roots of ratio: Some thoughts on the documents. Our project teachers have long term learning process (towards Hart. K. (1981). (Ed.). Children’s revisited their traditional work program ... a theory). Educational Studies in understanding of mathematics 11–16. and its two-week mathematics unit Mathematics, 16, 75–94. London: John Murray. on ratio and proportion. They have put greater emphasis on proportional Lamon, S. (2005). Teaching fractions reasoning and multiplicative thinking and ratios for understanding (2nd ed.). in the study of scale drawing, linear Mahwah: Erlbaum. equations, trigonometry, percentages, Lanius, C. S., & Williams, S. E. (2003). number study, mapping, ratio and Proportionality: A unifying theme rate situations. Science teachers in for the middle grades. Mathematics the project a greater awareness of Teaching in the Middle School, 8(8), the mathematical foundations of 392–396. proportional reasoning and how science topics and presentations of Lesh, R., Post, T., & Behr, M. (1988). equations (e.g., density equation and Proportional reasoning. In J. Hiebert &

Research Conference 2010 74 Mathematics learning: What TIMSS and PISA can tell us about what counts for all Australian students

Abstract Goals. The Measurement Framework for National Key Performance Measures Teachers and school leaders will be (MCEETYA, 2008) sets out the National familiar with NAPLAN – as a census Assessment Program as a basis for of students in Years 3, 5, 7 and 9 reporting ongoing progress towards the it involves all educators. However, goals by drawing on agreed definitions as part of the National Assessment of Key Performance Measures. The Program, Australia also participates in Framework is designed to be a living two international assessments, PISA document, in that it will be updated Sue Thomson and TIMSS, which are, by design, light to report on the most recent goals as sample assessments and involve only defined in the Melbourne Declaration on Australian Council for Educational a small proportion of schools. The Educational Goals for Young Australians, Research students we are educating today will allowing it to respond to new goals and compete in a global market, and we challenges. Sue Thomson is a Principal Research Fellow at have to be sure that the education the Australian Council for Educational Research we are providing them with is one The National Assessment Program in the National and International Surveys research encompasses all tests endorsed by program. that will provide them with a strong base, both in knowledge and skills MCEETYA, such as the national literacy Dr Thomson is the National Research and numeracy tests (NAPLAN), three- Coordinator for Australia in the Trends in and in the ability to apply those skills International Mathematics and Science Study to real-world problems. In addition yearly sample assessments in science (TIMSS), which measures achievement in to the assessments, PISA and TIMSS literacy, civics and citizenship, and ICT mathematics and science for students in grades collect a rich array of contextual literacy, and Australia’s participation in 4 and 8, the Progress in International Reading the international assessments PISA and Literacy Study (PIRLS), which measures reading information from students, teachers literacy of grade 4 students, and the National and schools – including background TIMSS. Project Manager for Australia for the OECD factors, and attitudes and beliefs about Teachers and school leaders are Programme for International Student Assessment learning mathematics. What should be (PISA), which examines reading, mathematical and familiar with NAPLAN – as a census scientific literacy of 15-year-old students. particularly interesting for educators is of students in Years 3, 5, 7 and 9 it not just how well students perform on Dr Thomson’s research at ACER has involved involves all educators. However, many extensive analysis of large-scale national and the international assessments, but how may not be aware of PISA and TIMSS, international data sets – the Longitudinal Surveys much the other information we gather as they are light sample assessments of Australian Youth (LSAY), as well as TIMSS and can tell them about what Australian which, by design, involve only a PISA. students can and can’t do. proportion of schools. In addition Dr Thomson was engaged as an expert writer to the assessments, PISA and TIMSS on the National Numeracy Review, and has Introduction consulted with DEEWR, FaHCSIA and the collect a rich array of contextual Victorian and ACT Departments of Education In 1999, the Ministers responsible information from students, teachers on a variety of data analysis projects related to for school education, the Ministerial and schools – including background TIMSS and PISA. Council on Education, Employment, factors, and attitudes and beliefs about Before joining ACER, Dr Thomson lectured at a Training and Youth Affairs, agreed to a learning mathematics. What should be number of universities in Statistics and Research new set of National Goals for Schooling particularly interesting for educators is Methodology while she completed her PhD not just how well students perform on focusing on students’ attributions and engagement in the Twenty-first Century (MCEETYA, in mathematics over the transition from primary 1999). The aim of these goals was the international assessments, but how to secondary school. to provide Australian students with much the other information we gather Dr Thomson has published a variety of articles high-quality schooling to provide can tell them about what Australian and research reports based on her work at them with the necessary knowledge, students can and can’t do. ACER (see also under Sue Fullarton), and has understanding, skills and values for presented findings at conferences internationally The presentation will be structured and nationally. a productive and rewarding life. around the questions teachers often MCEETYA also set in train a process to ask: enable nationally comparable reporting of progress against these National • What are PISA and TIMSS? Who participates?

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 75 • Why do we need these assessments The Programme for International a student’s individual progress against as well as NAPLAN? Student Assessment (PISA) is the other national standards. In contrast, a light major international assessment included sample (about 5% of all Australian • What can these studies tell me in the National Assessment Program, and students at each year or age level) of about what our students learn Australia been a participant since the students is tested in the international compared to other countries? study began in 2000. PISA is managed assessments. This sample is a nationally • What can they tell me about our by the Organisation for Economic representative random sample, stratified students’ motivation, engagement Co-operation and Development to ensure accurate data for each state, and self-efficacy – and how this (OECD); it tests competencies in each school sector (government, compares to other countries? reading, mathematics and scientific Catholic and independent) and each literacy, and occurs every three geographic location band (metropolitan, • What can these studies tell us about years. The underlying PISA model regional, rural). These data enable us to equity – both within Australia and aims to measure how well 15-year- examine our educational system against internationally? Are some students olds, approaching the end of their international standards. disadvantaged in Australia, and is compulsory schooling, are prepared for this common internationally? In terms of what is assessed, the meeting the challenges they will face in NAPLAN tests are informed by the their lives beyond school. With its goal TIMSS and PISA – some National Statements of Learning in of measuring competencies, the PISA English and Mathematics that underpin details assessment focuses on young people’s the current state and territory learning The Trends in International ability to apply the knowledge and skills frameworks; in contrast the TIMSS Mathematics and Science Study (TIMSS) they have learned throughout their and PISA assessments are developed is a long-running study of achievement school lives to real-life problems and against frameworks developed at in mathematics and science, managed situations. an international level. The TIMSS by the International Association for the In 2010/2011 more than 60 educational framework is developed after extensive Evaluation of Educational Achievement systems, from countries as diverse consultation between representatives (IEA). The assessments occur every as Ghana, Saudi Arabia, England, of all countries involved and an expert four years at Years 4 and 8, and Honduras, United States of America panel of mathematics educators, and Australia’s participation in TIMSS 2011 and Germany will participate in TIMSS. represents those goals of mathematics will be our fifth since the combined In the following year, 67 countries will education that are regarded as mathematics and science assessment participate in PISA, including all OECD important in a significant number of evolved from separate international countries plus a growing number of countries. Mathematics in the TIMSS assessments in 1985. Underpinning non-OECD or partner countries, again assessment is readily recognisable as TIMSS is a research model in which from locations as diverse as Shanghai, the mathematics in most curricula – the the curriculum, broadly defined, is Qatar and Azerbaijan. The growing content domains of number, algebra, used as the major organisational number of countries participating in measurement, geometry and data concept in considering how educational one or both studies reflects the value (data display, geometric shapes and opportunities are provided to students, that governments place on obtaining measures and number at Year 4), and and the factors that influence how international comparative data. the cognitive domains knowing, using students use these opportunities. The concepts, applying and reasoning are TIMSS curriculum model has three NAPLAN, PISA and TIMSS familiar territory to teachers. aspects: the intended curriculum (what society expects students to learn and So why do we need NAPLAN The PISA mathematical literacy how the system should be organised and PISA and TIMSS? The answers framework revolves around wider to facilitate this), the implemented lie in who are assessed, how the uses and applications of mathematics curriculum (what is actually taught in assessments are constructed, and the in people’s lives, and has three classrooms, who teaches it and how it additional information gained from the main dimensions: mathematical is taught) and the achieved curriculum international assessments. content, mathematical processes and (which is what the students have the situations or contexts in which In NAPLAN all students are tested, learned, and what they think about mathematics is used. Mathematical and the data provide results at the these subjects). content is defined in terms of Steen’s student level. NAPLAN is intended to (1990) deep mathematical ideas, provide diagnostic information about adapted as overarching ideas. These

Research Conference 2010 76 overarching ideas are quantity, space in New Zealand (or Kazakhstan1). mathematical literacy scale. While this is and shape, change and relationships, and There is, of course, a lot more that is clearly better than the OECD average uncertainty. The PISA framework also published in our national reports, and of 42 per cent of students, we can identifies a number of competencies this paper will present some of these aim to do better. In Hong Kong, for – labelled as the reproduction cluster results. Largely, this paper will report example, one of the highest performing (relatively familiar items that require result in terms of proficiency levels for countries, only 25 per cent of students essentially the reproduction of PISA and benchmarks for TIMSS. In did not achieve proficiency level 3. knowledge already acquired), the PISA, six proficiency levels have been At Year 8, in TIMSS 2007, Australian connections cluster (problems that described, representing a continuum of students performed at around the extend or develop from familiar mathematics achievement. MCEETYA international average in mathematics settings to a minor degree) and the have set proficiency level 3 as the overall. In the content domain of reflection cluster (builds further on the minimum standard for Australian data and chance, Australian students connections cluster – items require students. In TIMSS, there are four performed at a level significantly higher some insight or creativity in identifying benchmarks ranging from low to high, than the international average; however. solutions). also representing a continuum of in the content areas of algebra and mathematics achievement. While no So all three studies are embedded geometry, Year 8 students in Australia base levels have been set by MCEETYA in different models – NAPLAN and performed at a level significantly for TIMSS, students performing at the TIMSS in curriculum models, but one lower than the international average. low benchmark or not achieving the national and the other international, Thirty-nine per cent of Australian Year low benchmark must be thought of to and PISA as a yield study, looking at 8 students were either at the low be at risk, particularly at Year 8. whether students have in fact learned benchmark or did not achieve the low what we expect them to have learned benchmark in mathematics overall. over the cumulative years of education. Content Australian Year 4 students achieved It’s important that any assessment of The international assessments also at a level significantly higher than the mathematics should reflect the maths provide us with a wealth of contextual international average in TIMSS 2007, that it is most important for students information – because the focus is not with performance in data and chance to learn. What do PISA and TIMSS tell just on what a particular student is able significantly higher than the international us that our students know well, and to do, and because for such studies average, and performance in number in what areas are they lagging behind the context of learning is considered at a level significantly lower than the internationally? as important as the learning itself. Both international average. Around 30 per TIMSS and PISA collect background PISA results from 2003, which was the cent of Australian students achieved data on students – the educational last full assessment of mathematical at or below the low benchmark in resources to which they have access, literacy (enabling us to report on mathematics overall. the educational experience of their subscales), show that Australian Summing up, Australian students parents, and their attitudes towards and 15-year-old students have a generally perform better than the international beliefs about schooling and themselves high level of overall mathematical average at all levels in topics related as learners, in particular in relation to literacy, significantly higher than the to data and chance, while achievement mathematics. TIMSS collects data from OECD average. Australian students in the areas of number and algebra mathematics teachers as well, as TIMSS overall also scored at a level significantly are potentially weaker than in other is sampled on intact classes, whereas higher than the OECD average on each countries. However, these data indicate PISA samples 15-year-old students of the subscales – not quite as well in that there is a substantial proportion randomly across classes within a school. quantity but better in uncertainty. But of students exhibiting poor levels in terms of proficiency levels, one-third of mathematical understanding in of Australian students did not achieve What can we learn from PISA Australian schools at all year levels. and TIMSS? proficiency level 3 on the overall If you have heard of PISA and TIMSS Equity in Australia, it is most likely that you 1 Many of the headline reports (even in Mathematics is no longer just a will have heard where we rank, or broadsheets such as The Australian) for the prerequisite subject for science and which countries score higher than us, last release of the TIMSS 2007 results were engineering students, but a fundamental or how our scores compare to those along the lines of “Borat’s kids beat Aussie kids in maths and science” literacy requirement for the 21st

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 77 century. Equity implies that every 20 of the 37 participating countries. In PISA 2003, 15-year-old Australian student has an opportunity to learn the In 12 of those countries the gender girls reported significantly lower levels mathematics that is assessed. Can PISA differences were in favour of boys of instrumental motivation, self-concept and TIMSS help identify subgroups of and the remaining 8, in favour of girls. in maths, self-efficacy and interest in students who are not achieving as well Australia was one of the 18 countries maths, and significantly higher levels of as we would hope? What else can we in which there were no significant maths anxiety. This finding holds even find out about these groups of students gender differences in the composite when students achieving at the same that may provide some clues as to why mathematics score. Within the proficiency level are compared. It also achievement is lower than could be subscales, however, boys significantly held internationally – in all countries expected? outperformed girls in number, while girls (even Iceland) boys had higher levels of significantly outperformed boys in data self-concept and self-efficacy, and in the While the Australian PISA and TIMSS display. vast majority of countries (there were data are generally reported by gender, approximately two exceptions) interest Indigenous background, immigrant In 25 of the 49 countries participating in mathematics and lower levels of status, socio-economic background in TIMSS 2007 at Year 8 there were mathematics anxiety. and geographic location of school in no gender differences. In 16 of the the national and international reports, countries there were significant gender Similarly in TIMSS 2007 at Year 4 this paper will focus on two important differences in favour of girls, and in in Australia, there was a significantly factors. only 8 countries, of which Australia higher proportion of boys reporting was one (Algeria, Lebanon, Syria, El high levels of self-confidence in Gender Salvador, Tunisia, Ghana and Columbia mathematics (with no associated were the others), were there significant difference in score between male In PISA 2003, mathematical literacy differences in favour of boys. The and female students). At Year 8 just was in many countries a male- national TIMSS 2007 report (Thomson, 39 per cent of girls compared to 51 oriented subject, with boys in 28 Wernert, Underwood & Nicholas, per cent of boys reported high levels out of the 41 countries significantly 2008) noted that this was not because of self-confidence – and almost one- outperforming girls. Only in Iceland of an increase in the scores of boys, but quarter of girls (24%) reported low did girls outperform boys. In Australia a decline in the average score for girls. levels. This was broadly the case in no significant gender differences were Contrary to the findings internationally, most participating countries2. In further found on the overall mathematical in which girls performed significantly analysis (see Thomson, Wernert, literacy scale. Unpacking this a little better than boys in all domains other Underwood & Nicholas, 2008), the further, however, it was also found than number, Australian boys outscored effect of gender on achievement was that while there were no differences girls in data and chance, and number, found to be substantially explained by overall, or in the subscales for quantity while there was no significant difference the differences in self-confidence in or change and relationships, Australian in the other domains. More boys learning mathematics. In other words, boys performed significantly better than than girls were achieving at the higher it is not being a girl in and of itself that girls on the subscales space and shape benchmarks in both year levels (Year 4 makes the difference, but that being and uncertainty. There were no gender and Year 8) in TIMSS 2007. a girl means a student is less likely to differences in the lower proficiency have high levels of self-confidence that levels, with 33 per cent of both male To summarise, Australian boys can lead to higher levels of achievement and female students not achieving outperformed girls in PISA 2003 in in mathematics. proficiency level 3. At the higher levels the areas of space and shape and of achievement slightly more boys uncertainty, in TIMSS 2007 at Year 4 in (7%) than girls (4%) achieved the very number, and in Year 8 in number and highest proficiency level, but the same data and chance. Girls outperformed 2 However, at Year 8 in a number of Middle- proportion of male and female students boys in TIMSS 2007 at Year 4 in data Eastern countries (Oman, Qatar, Palestine, achieved at the next two highest display. There were no significant Bahrain, Saudi Arabia and Kuwait), girls significantly outperformed boys and in general achievement levels. gender differences on any other had higher levels of self-confidence than boys subscale. Given these few differences, Mathematics in TIMSS 2007 – significantly so in Qatar, Bahrain and Saudi it is interesting to look at students’ Arabia. There were only four countries in was generally not as gendered attitudes and beliefs about mathematics. which a significantly higher proportion of girls internationally. At Year 4 level, there reported high levels of self-confidence than were significant gender differences in boys, in contrast to the 26 countries in which the opposite was reported.

Research Conference 2010 78 These are important findings for non-Indigenous students (De Bortoli & students, there were some interesting teachers and researchers. Why is it Thomson, 2009). This represents more findings, recently described in DeBortoli that there are still gender differences in than one full proficiency level difference. & Thomson (2010). Amongst Australian favour of males in so many countries The score gap between Indigenous and 15-year-old students in PISA 2003, in all areas of mathematical literacy, non-Indigenous was similar across all as previously described, there were as shown in PISA, while a more subscales. significant gender differences in curriculum-based assessment such instrumental motivation, self-concept in In an international perspective, this as TIMSS finds gender differences in maths, self-efficacy and interest in maths, places our Indigenous students at a favour of boys in some countries and and maths anxiety. Amongst Indigenous level significantly lower than students girls in others? Why are boys more students, however, there were no in 30 other countries, the same self-confident and have higher levels significant gender differences in interest, as students in Greece and Serbia, of self-concept and lower levels of instrumental motivation or anxiety, and higher than students in Turkey, anxiety in mathematics, even when girls although Indigenous girls had very high Uruguay, Thailand, Mexico, Indonesia, outperform them? Conversely, why do scores on this latter construct, reflecting Tunisia and Brazil. girls still doubt their abilities even when levels of anxiety in mathematics they are clearly achieving at a high In terms of achievement at proficiency much higher than the OECD or the level? If girls do not see mathematics levels, 70 per cent of Indigenous Australian average. In self-concept in as an area of strength, despite their students, compared to 32 per cent maths, significant differences were achievement levels, and suffer from of non-Indigenous students were not found for Indigenous students, but they higher levels of anxiety, then it is achieving at the MCEETYA standard were smaller in magnitude than those unlikely that they will continue their of level 3 or above. Forty-three per for non-Indigenous students. studies through to university level. cent of Indigenous students were not In TIMSS 2007, there were significantly achieving at the basic OECD acceptable Indigenous students greater proportions of Australian boys standard of level 2 or above, that they than girls in the high levels of both A special focus of both PISA and TIMSS argue is a baseline level of proficiency self-confidence and valuing mathematics. in Australia has been to ensure that at which students begin to demonstrate However, amongst the Indigenous there is a sufficiently large sample of the type of skills that they need to population, this was not the case, with Indigenous students, so that valid and be able to fully participate in society similar proportions of boys and girls reliable comparisons can be made. In beyond school. About 5 per cent of reporting high levels of both. both studies, the random selection of Indigenous students were, however, students in PISA and classes in TIMSS achieving at the highest two proficiency Further investigation is needed to ensures that some Indigenous students levels. examine these findings – to find out are part of the main sample. In addition whether they reflect actual differences At both Year 4 and Year 8 in TIMSS to this, however, all eligible Indigenous in beliefs amongst Indigenous boys and 2007, non-Indigenous students scored students (i.e. 15-year-olds in PISA, and girls or whether it is simply an artefact at a substantially higher level than Year 4 or Year 8 students in TIMSS) of the sample size, since standard errors Indigenous students – 91 score points are sampled and asked to participate. are larger for the Indigenous sample. at Year 4 and 70 score points at Year The National Centre and the Education PISA 2012 will, we hope, provide 8. At Year 4, Indigenous students’ Ministers communicate with school some of these answers – the focus is scores were, on average, almost one principals to explain the purpose of again on mathematics, and Australia standard deviation lower than those of this extra sample and to convey to is implementing a different sampling non-Indigenous students in number, and them the importance of encouraging methodology which we hope will result around three-quarters of a standard Indigenous students to attend the in a much bigger sample of Indigenous deviation lower in data display and assessment session. students than ever before. geometric shapes and measures. At Year It has been widely reported that the 8 also, Indigenous students scored at In terms of factors influencing the achievement levels of Indigenous a significantly lower level (between achievement of Indigenous students, the students continue to lag well behind 54 and 67 score points) than non- effect of socio-economic background those of non-Indigenous students. In Indigenous students in each of the is substantial. However, the effect of mathematical literacy in PISA 2003, subscales. strong, positive attitudes and beliefs is Indigenous students performed 86 also significant, and can be encouraged However, in terms of attitudes and score points lower on average than through school programs. Also motivation amongst Indigenous

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 79 important is attendance at school – numeracy. Washington D.C.: National Indigenous students were found to be Academy Press. far more likely than non-Indigenous Thomson, S., Wernert, N., Underwood, students to be late to school on a C. & Nicholas, M. (2008). TIMSS 2007: regular basis, to miss consecutive Taking a closer look at mathematics and months of schooling and to change science in Australia. Camberwell: ACER. schools several times. In addition to lower levels of home educational resources and parental education experience, the gaps that appear at the beginning of primary school widen as a result of poor attendance at school.

Summary It is sometimes difficult for teachers and school leaders to see the purpose of PISA and TIMSS. However, the students we are educating today will compete in a global market, and we have to be sure that the education we are providing them with is one that will provide them with a strong base, both in knowledge and skills and in the ability to apply those skills to real-world problems. PISA and TIMSS provides us with that information, and much, much more. References Ministerial Council on Education, Employment, Training and Youth Affairs (MCEETYA) (1999). The Adelaide declaration on national goals for schooling in the twenty-first century. Available http://www.curriculum.edu. au/mceetya/nationalgoals/index.htm accessed May 2010 Measurement Framework for National Key Performance Measures (MCEETYA, 2008) Available http://www.mceecdya. edu.au/verve/_resources/PMRT_ Measurement_Framework_National_ KPMs.pdf accessed May 2010 De Bortoli, L & Thomson, S. (2009). The achievement of Australia’s Indigenous students in PISA 2000- 2006. Camberwell: ACER. Steen, L. A. (Ed). (1990). On the shoulders of giants: New approaches to

Research Conference 2010 80 Poster presentations

1 Ken Lountain, 2 Paul Waddell, 3 Alex Neill Barbara Reinfeld, Patrick Murray and New Zealand Council for Educational Phil Kimber and Stephen Murray Research Vivienne McQuade Mathematics.com.au NSW Processes surpass products: Department of Education and Children’s Services South Australia. Learning Online Maths Resources – Mapping multiplicative strategies Inclusion Team Creating deep mathematical to student ability thinking or lazy teachers When making judgements about Maths for Learning Inclusion – dispensing ‘busy work’? student understanding, the strategies that they use are far more revealing of action research into pedagogical With a plethora of online maths change their level of thinking than the answers programs available to teachers, students they produce. The poster will display Maths for Learning Inclusion is an and parents, how do we as educators a range of student responses to some initiative focussed on improving the distinguish between those that were multiplication problems, and explore teaching and learning of mathematics in created to entertain and occupy the relationship between students’ 28 primary schools in 6 clusters serving students from those that encourage and overall ability and the strategies that low socio-economic communities. develop deep mathematical thinking? they employ. An effective digital mathematics The aims of the project are: resource will be designed with student • all students achieving learning as the key goal. It should clearly demonstrate strategies to develop the • challenging and engaging curriculum building blocks of numeracy, provide • sustainable professional learning opportunities to discover better and communities varied ways of solving problems, and focus on the steps on the journey of • improvement informed by evidence discovery as well as the destination of and research improved student learning. Professional learning is composed This poster presentation will provide of maths knowledge and pedagogy, advice on strategies to evaluate the learning inclusion principles and purpose and place of digital resources practices. Teachers are supported in the teaching and learning of to establish and maintain a focus on mathematics. Insights drawn from over narrowing the achievement gap for 12 years of practice in the evaluation Aboriginal learners and students from and use of digital resources to support low socio-economic backgrounds effective student learning will inform through developing an action research this poster presentation. question. Learning is shared, analysed, critiqued and sustained as appropriate across schools and clusters by teachers’ and leaders’ participation in communities of practice. The program is supported by a concurrent and rigorous evaluation composed of multiple data sets including teacher narratives reflecting on pedagogical change. These narratives will be presented at the conference.

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 83 4 Cathryn Morris 5 Sonia White and 6 Michael Jennings Australian Association of Mathematics Dénes Sz cs The University of Queensland Teachers The Queensland University of Technology and The University of Cambridge First-year university students’ Make it count – Numeracy, mathematical understanding mathematics and Indigenous Number line estimation In recent years there has been a learners behaviours: Influence of noticeable increase in the diversity of The Australian Association of strategy? backgrounds, abilities and aspirations of Mathematics Teachers (AAMT) inc. The purpose of this study was to students entering bridging and first-year has established this national four year investigate number line estimation mathematics courses at The University project to develop an evidence base behaviours of children in Years 1-3 of Queensland. Much research has of practices that improve Indigenous and explore the potential influence of been undertaken into primary and students’ learning in mathematics and strategy during such tasks. Children secondary mathematics education but numeracy. The poster will provide: were asked to position target digits little in comparison has been done into tertiary mathematics and students’ • Information about the project on a series of 0-20 number lines transition from secondary to tertiary and its eight clusters of schools and their responses were analysed. mathematics. With the number of frameworks for intersecting Existing cognitive research has typically students entering Australian universities community with classroom and the modelled the development of number increasing, it is important to know what development of culturally responsive estimation as being a progression from level of mathematical understanding mathematics education logarithmic to linear representations. This trend was confirmed in this they bring with them. • Stories from the clusters involved study with children in Years 2 and 3 Diagnostic testing of first-year directly in the project demonstrating a significant preference engineering and science students at • Professional development, for a linear model; a result not The University of Queensland has communication and collaboration evident in the Year 1 participants. This been conducted at the beginning of through an online learning modelling approach had limitations first semester for the past four years. community (network ring) when attempting to understand the The data from the competency tests influence of strategy in number line was analysed to decide the best way • Examples of research/inquiry and estimation. To ascertain strategy, to improve students’ mathematical data collection we analysed estimation accuracy for knowledge and understanding. Results • Partnerships/friendships between individual target digits. These findings from the tests and subsequent community, school and universities point to a link between developmental outcomes will be presented. that support improved learning progression and strategy application for outcomes of Indigenous students certain target digits. It was concluded that further explorations into the types • A resource for others wanting of strategies children employ when to help their Indigenous students performing number estimation tasks better reach their potential in would be of great value, particularly mathematics and numeracy when referenced to classroom practice This project is funded by the Australian and the overt teaching of strategy in Government under the Closing the mathematics education. Gap Initiative.

Research Conference 2010 84 Conference program

Sunday 15 August 6.00-7.30 PM Cocktails with the Presenters – Crown Conference Centre – Entertainment by Fly Right Trio

Monday 16 August 7.30 AM Conference Registration Level 2 – Crown Conference Centre Hall 8.30 AM Welcome to Country Ian Hunter 8.45 AM Conference Opening Professor Geoff Masters, Chief Executive Officer, ACER 9.00 AM Keynote Address 1 Speaking in and about mathematics classrooms internationally: The technical vocabulary of students and teachers. Professor David Clarke, University of Melbourne Crown Conference Centre Hall Chair: Dr. John Ainley, ACER 10.15 AM Morning tea and poster presentations 10.45 AM Concurrent Sessions Block 1 Session A Session B Session C Session D Session E Issues of social equity in Primary students’ decoding Promoting the acquisition Mathematics assessment in Conversation with a Keynote access and success in mathematics tasks: The role of of higher order skills and primary classrooms: Making Professor Paul Ernest, mathematics learning for spatial reasoning understandings in primary it count University of Exeter Indigenous students and secondary mathematics Professor Tom Lowrie, Associate Professor Restricted to designated Professor Robyn Jorgenson, Charles Sturt University Professor John Pegg, Rosemary Callingham, delegates only. Griffith University University of New England University of Tasmania M11 M14 M 12 &13 Chair: Cath Pearn, ACER Crown Conference Centre Crown Conference Centre Chair: Kerry-Anne Hoad, Hall 1 Hall 2&3 ACER Chair: Dr Lawrence Ingvarson Chair: Dr Hilary Hollingsworth, ACER ACER 12.00 PM Lunch and poster presentations 12.15 PM Lunchtime talkback Mathematics or Numeracy – what are we actually talking about here? Does it matter? Talkback led by Mr Will Morony, Executive Officer, AAMT. Open to all delegates – bring your lunch and your views. M 15 &16 1.00 PM Keynote Address 2 Standards, what’s the difference?: A view from inside the development of the Common Core State Standards in the occasionally United States Mr Phil Daro, University of California Crown Conference Centre Hall Chair: Dr. John Ainley, ACER 2.15 PM Afternoon tea and poster presentations 2.45 PM Concurrent Sessions Block 2 Session F Session G Session H Session I Session J The case of technology in Reconceptualising early Learning about selecting Identifying cognitive processes Conversation with a Keynote senior secondary mathematics: mathematics learning classroom tasks and important to mathematics Professor Kaye Stacey, Curriculum and assessment Associate Professor Joanne structuring mathematics learning but often overlooked University of Melbourne congruence? lessons from students Mulligan, Mr. Ross Turner, ACER Restricted to designated Dr David Leigh-Lancaster, Macquarie University Professor Peter Sullivan, delegates only. Victorian Curriculum and Monash University M11 Assessment Authority Crown Conference Centre Chair: Marion Meiers, ACER M14 Hall 2 &3 Crown Conference Centre M 12 & 13 Chair: Kerry-Anne Hoad, Hall 1 Chair: Ray Peck, ACER ACER Chair: Dr. Lawrence Ingvarson, ACER 4.00 PM Close of Day 1 6.45 PM Pre dinner drinks Crown Conference Centre Hall Entertainment by Regent Strings 7.00 PM Conference dinner Crown Conference Centre Hall Entertainment by Pot Pourri

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 87 Tuesday 17 August 9.00 AM Keynote Address 3 Mathematics teaching and learning to reach beyond the basics Professor Kaye Stacey, University of Melbourne Crown Conference Centre Hall Chair: Dr. John Ainley, ACER 10.15 AM Morning tea and poster presentations 10.45 AM Concurrent Sessions Block 3 Session K Session L Session M Session N Session O Using mental representations Using technology to support Making connections to the Mathematics learning: What Conversation with a Keynote of space when words are effective mathematics big ideas in mathematics: TIMSS and PISA can tell us Mr Phil Daro, University of unavailable: Studies of teaching and learning: What Promoting proportional about what counts for all California, Berkley enumeration and arithmetic in counts? reasoning Australian students Indigenous Australia Restricted to designated Professor Merrilyn Goos, Dr Shelley Dole, Dr Sue Thomson, ACER delegates only. Associate Professor University of Queensland University of Queensland Robert Reeve, Crown Conference Centre M 14 University of Melbourne Crown Conference Centre Hall M 11 Hall 1 2 & 3 Chair: Marion Meiers, ACER Chair: Dr. Hilary Hollingsworth, M 12 & 13 Chair: Kerry-Anne Hoad, ACER Chair: Cath Pearn, ACER ACER 12.00 PM Lunch and poster presentations 12.15 PM Lunchtime talkback Mathematics or Numeracy – what are we actually talking about here? Does it matter? (repeat) Talkback led by Mr Will Morony, Executive Officer, AAMT. Open to all delegates – bring your lunch and your views. M 15 &16 1.00 PM Keynote Address 4 The social outcomes of school mathematics: Standard, unintended or visionary? Professor Paul Ernest, University of Exeter Crown Conference Centre Hall Chair: Dr. John Ainley, ACER 2.15 PM Closing Address Professor Geoff Masters, Chief Executive Officer, ACER

Research Conference 2010 88 Crown Conference Centre map and floorplan

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 91 Research Conference 2010 92 Conference delegates

Dinner table no. Delegate Name Delegate Organisation

13 Mr Ross Abbott Marist College, Canberra, ACT Head of Mathematics Ms Belinda Adams Lockleys North Primary School, SA Deputy Principal 2 Dr John Ainley ACER, VIC Deputy CEO (Research) and Research Director Mr Ronald Alderman DECS, SA Numeracy Coach Ms Jules Aldous Shelford Girls’ Grammar, VIC Ms Rosanna Algeri Casimir Catholic College, NSW Maths Teacher Ms Maria Alice CEO, Inner Western Region, NSW Project Officer: Primary and Numeracy 15 Ms Judith Allen Brighton Primary School, SA Principal Mr Nicholas Ambrozy St Anthony’s Catholic College, QLD Maths HOD 8 Dr Judy Anderson The University of Sydney, NSW Assoc. Prof. Mathematics Education Mrs Kay Anderson The Glennie School, QLD Maths Teacher Mr Lorne Anderson Taylors Lakes Secondary College, VIC Maths Coordinator Mrs Noxia Angelides Caulfield Junior Campus, VIC Curriculum Director Ms Janine Angove HOTmaths, NSW Manager Content Development Mrs Tania Angrove Catholic College Bendigo, VIC Maths Coordinator 9 Ms Gayle Appleby ACER INSTITUTE, VIC Administration Coordinator Mrs Sueanne Aquilina St Andrew’s Primary School, NSW Teacher Mrs Rebecca Armistead Killara Primary School, VIC Teacher Mrs Mary Asikas Seaford 6-12 School, SA Principal Ms Cynthia Athayde St John Bosco College Engadine, NSW Maths Coordinator Mrs Catherine Attard University of Western Sydney, NSW Lecturer Mr Brian Aulsebrook Sacred Heart School, NSW Principal Ms Vivienne Awad Loreto Kirribilli, NSW Deputy Principal Mrs Jessie Aziz Jacaranda, John Wiley & Sons Australia Ltd, Sales Coordinator, VIC Miss Veronica Azzopardi St Andrew’s Primary School, NSW Classroom Teacher

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 95 Dinner table no. Delegate Name Delegate Organisation

Mrs Marija Baggio Lefevre Primary School, SA Deputy Principal 14 Ms Jill Bain Wilderness School, SA Teacher Mathematics Mr Andrew Baker St Jerome’s Primary School, NSW Teacher Ms Julie Baker St Mary’s, Toukley, NSW Coordinator Mrs Ruth Bakogianis St Mary of the Angels Sec. College, VIC Teacher 19 Ms Maria Ball All Hallows’ School, QLD Head of Maths Mr Michael Barra Brisbane Catholic Education Office, QLD Education Officer Mathematics Ms Sue Barrington St Therese’s Primary, NSW Assistant Principal Mr Travis Bartlett Allenby Gardens Primary School, SA Deputy Principal Mrs Kim Bastock Presbyterian Ladies College, NSW Maths Coordinator Mr Mark Bateman OLGC Catholic School, NSW Principal Ms Jane Battrick Middle Park Primary School, VIC Leading Teacher - Numeracy Mr Kevin Bauer Holy Family Catholic Primary School, NSW Principal 17 Ms Geraldina Baxter Irymple Secondary School, VIC Teacher Mrs Donna Beauchamp-Whylie Carwatha College P-12, VIC Teacher Ms Naomi Belgrade Woodcroft College, SA Head of Mathematics 10 Ms Anne Bellert Catholic Education Office, NSW Additional Needs Officer Mr Richard Bennetts Malvern Primary School, VIC Principal Mr Steve Bentley The Friends’ School, TAS Teacher 7 Ms Dagmar Bevan DECS, SA Regional Curriculum Consultant 11 Ms Suzanne Bevan St Philip Neri, Northbridge, NSW Principal Mr Chris Biefeld St Martin’s School, NSW Assistant Principal 3 Ms Margaret Bigelow ACARA, NSW SPO Mathematics 20 Mrs Michelle Binney Whitsunday Anglican School, QLD Teacher 20 Mr Graham Bishop UWS College Pty Ltd, NSW Assistant Coordinator - Mathematics

Research Conference 2010 96 Dinner table no. Delegate Name Delegate Organisation

Mr Andrew Blackwood Claremont College, TAS Teacher Mr John Bleckly DECS, SA Numeracy Coach 8 Mr Christopher Blood Brisbane Boys’ College, QLD Head of Mathematics 11 Ms Janet Bohan St Mary’s Primary School, VIC Deputy Principal Mrs Elizabeth Bortolot Western Metropolitan Region, VIC Regional Numeracy Coach 10 Ms Trish Boschetti Primary Mathematics Association, SA Maths for Learning Inclusion Co-ordinator Mrs Caroline Boulis St Joseph’s Primary, Belmore, NSW Maths Coordinator Ms Mary Boutros Wooranna Park Primary School, VIC Teacher 20 Mr Robert Bowden West Beach Primary School, SA Deputy Principal Ms Benita Bowles Our Lady of Mercy College, VIC Head of Learning Support Mr Tony Boyd Our Lady of Fatima Primary, NSW Mr Russell Boyle Ruyton Girls’ School, VIC Dean of Mathematics 19 Ms Deborah Brassington Torrensville Primary School, SA Principal Mrs Natalie Bratby Holy Family School, NSW Teacher Mrs Karen Bredenhann Heights College, QLD Maths Coordinator Mr Bernard Bree Stuart Park Primary School, NT Principal Mr Christopher Brennan St Aidan’s Anglican Girls’ School, QLD Maths Teacher Miss Kellie Brennan Kingston State School, QLD Teacher Mrs Julie Bridgen Mary MacKillop, NSW Teacher 7 Mrs Fiona Brimmer Education Queensland, QLD PPO - Mathematics Mr Phil Brockbank All Saints’ College, WA Head of Mathematics Mr David Brooks The Friends’ School, TAS Maths Teacher Mrs Caroline Brown Sacre Coeur, VIC Teacher 10 Mr Garry Brown Qld Academy for Health Sciences, QLD Deputy Principal Mr Greg Brown Seaford Rise Primary School, SA Teacher

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 97 Dinner table no. Delegate Name Delegate Organisation

Mrs Julie Brown Catherine McAuley, Westmead, NSW Head of Mathematics 15 Mrs Safia Brown St Clares Catholic College, ACT Teacher Ms Julie Broz Steps Professional Development, WA Consultant Mr Steven Bruce Middle Park Primary School, VIC Teacher Mr Stuart Brunsdon Mary MacKillop for Girls, NSW Teacher Mrs Emily Buckley Canterbury Primary School, VIC Teacher Mrs Suzanne Budd All Saints Primary School, SA Numeracy Leader Ms Fiona Buining Orana Steiner School, ACT Teacher Ms Joan Burfitt Catholic Education Office, WA Consultant Ms Toni Burford Littlehampton Primary School, SA Coordinator, Maths Mr Paul Burke St Mary’s Primary School, VIC Teacher 21 Mrs Michele Burns Genazzano FCJ College, VIC Curriculum Leader Mathematics Miss Fiona Bylsma Christ the King Primary School, NSW Assistant Principal Mrs Dale Cain Catholic Schools Office, NSW Literacy/Numeracy Consultant Mrs Jacqueline Cain DECS, SA Numeracy Coach Mrs Kate Callea St Martin of Tours Primary School, VIC Numeracy Coordinator 3 Prof Rosemary Callingham University of Tasmania, TAS Ms Hilary Cameron St Gerard’s Primary School, NSW Assistant Principal 20 Ms Anne Cannizzaro West Lakes Shore Schools, SA Principal Mr David Carey St Andrew’s College, NSW Mathematics Coordinator Mr Peter Carmichael Education Queensland, QLD Project Officer - Mathematics Ms Beverley Carr The Friends’ School, TAS Teacher Mrs Beth Carroll St Joseph’s College, VIC Maths Domain Leader Mrs Cristi Carroll St Francis College, NSW Maths Coordinator Mr Shaun Carroll Concordia International School, CHINA Maths Facilitator

Research Conference 2010 98 Dinner table no. Delegate Name Delegate Organisation

Ms Amanda Carter Damascus College, VIC Head of Mathematics Mrs Louise Caruana St Mary’s Primary School, NSW Mr Greg Cashman Monte Sant’ Angelo Mercy College, NSW Teacher, Mathematics Mr Daryl Castellino Patrician Brothers’ College Fairfield, NSW Maths Coordinator 13 Mrs Marianne Castor St Dominic’s College, NSW Mr Steve Cauchi Mary MacKillop, NSW Coordinator Ms Melissa Chabran Bill and Melinda Gates Foundation, USA Program Officer Ms Cate Charles-Edwards Westbourne Grammar School, VIC Director of Maths 6 Mr Graeme Charlton Woodville Primary School and CHI, SA Principal Mr Seng Chong International Education Services Ltd, QLD Mathematics Coordinator 6 Ms Meredith Christie-Ling Woodville Primary School and CHI, SA Assistant Principal Prof David Clarke The University of Melbourne, VIC Director Miss Ruth Clarke Wycliffe Christian School, NSW Acting Head of Mathematics Ms Nicole Claxton Taylors Lakes Secondary College, VIC Numeracy Coach Ms Kathryn Cleary St Peter Chanel Primary, NSW Mr Grant Clifton Aitken College, VIC Head of Mathematics Mr Lance Coad St Michael’s Collegiate, TAS Teacher Mr Frank Cohen St John the Baptist Catholic Primary, NSW Principal Mr Ian Coleman St Augustine’s College, QLD Head of Department Mrs Lee Collie Macmillan Professional Learning, VIC Director Ms Carol Collins Braybrook Secondary College, VIC Teacher Mrs Pat Conheady North Shore Primary School, VIC Primary Maths Specialist 10 Mr Vince Connor Catholic Education Office, NSW Schools Consultant Ms Melanie Cook Good Samaritan Catholic College, NSW Maths Coordinator Mrs Bianca Cooke Good Shepherd School, NSW Teacher Mrs Merilyn Costa Malvern Primary School, VIC Maths Coordinator

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 99 Dinner table no. Delegate Name Delegate Organisation

Ms Julie Costelloe OLR The Entrance, NSW Teacher Mrs Sandra Cottam Department of Education of WA Curriculum Officer - Numeracy 12 Mr Noel Covill St Josephs College, QLD Head of Mathematics Mr Ian Cowan Terra Sancta College, NSW Teacher Mrs Melissa Cowan St Mary’s Primary School, VIC Reading Recovery Teacher Mr Peter Cranney St Joseph’s Primary School, NSW Assistant Principal Mr David Crees Flinders Christian Com. College, VIC Head of Mathematics Mrs Kimberley Crompton Leslie Barker College, NSW Academic Enrichment Coordinator 21 Mrs Shelley Cross A.B. Paterson College, QLD Teacher Mentor Maths Ms Susan Crouch Browns Plains SHS, QLD Maths Teacher Mrs Jacinta Crowe Our Lady of Rosary School, NSW Principal 8 Mrs Karen Crowley Trinity Lutheran College, QLD Head of Maths Mr Tom Crowley St Michael’s Primary School, NSW Maths Coordinator Mr Greg Cumming St Brendan’s School, NSW Deputy Principal Mrs Nicole Cumming St Patrick’s Primary School, NSW Principal Mrs Deborah Curkpatrick Presbyterian Ladies College, NSW Director, Student Learning Support & Extension Mrs Robin Curley Landsdale Primary School, WA Teacher Mr Chris Daly MacGregor Primary School, QLD Teacher Mrs Angela D’Angelo Catholic Education Office Sydney, NSW Adviser Mr Michael Darcy Assisi Catholic College, QLD Head of Mathematics 1 Mr Phil Daro University of California, USA Ms Andrea Dart Overnewton College, VIC Head of Curriculum Ms Maureen Davidson DECS, SA Numeracy Coach Mrs Beverley Davies Wycliffe Christian School, NSW Primary Teacher 15 Mr Gary Davies Newington College, NSW Head of Mathematics

Research Conference 2010 100 Dinner table no. Delegate Name Delegate Organisation

Mrs Helen Elizabeth Davies Gin Gin State School, QLD Principal Ms Tracey Davies Kidman Park Primary School, SA Deputy Principal Ms Patricia Davis Wenona School, NSW Head of Maths 12 Dr Alexandre Davyskib St. Aloysius College, NSW Senior Teacher Miss Susan Dawson Campbelltown P.A. High School, NSW Head Teacher Aboriginal Education Ms Fiona de St Germain St Rose Catholic School, NSW Year 5 Teacher Ms Eva De Vries Australian Catholic Education, Qld Principal Project Officer Ms Sandy Deam Kilkenny Primary School, SA Assistant Principal 15 Mr Michael Delean Brighton Primary School, SA Assistant Principal Mrs Thea Delfos St John’s Regional College, VIC Teacher 4 Mr Dean Dell’oro Geelong Grammar School, VIC Head of Mathematics Mrs Tracey D’elton Lowther Hall AGS, VIC Coordinator 16 Ms Jo Denton Daramalan College, ACT 10 Mr Chris Derwin Catholic Education Office, NSW Schools Consultant 5 Mr Lance Deveson ACER, VIC Library and Information Manager Mrs Elizabeth Devlin St Oliver’s Primary School, NSW Assistant Principal Mrs Elizabeth Devlin Mary MacKillop, NSW Teacher Mrs Margaret Devlin Stuartholme School, QLD Teacher Ms Jennie Dew Lloyd Street School, VIC Maths Coordinator 15 Mrs Tina Di Sano Saint Ignatius College, SA Teacher Ms Louise Dick Ascham School, NSW Maths Teacher Miss Alison Dickson St Thomas the Apostle, VIC Maths Coordinator Mrs Sue Dietrich MacKillop Catholic College, NSW Principal 17 Miss Claire Dillmann Kingston College, QLD Mathematics Teacher Mr Richard Dipane Georgiana Molloy Anglican School, WA Head of Mathematics

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 101 Dinner table no. Delegate Name Delegate Organisation

Mrs Louisa Doherty Calvin Secondary School, TAS Head of Mathematics 3 Dr Shelley Dole University of Queensland, QLD Ms Lyn Donaghue Learning Services North- West, TAS Numeracy Coordinator Mr Philip Donato Our Lady of the Sacred Heart College, SA Deputy Principal Mr Paul Dooley St Ursula’s College, QLD Teacher Mr Michael Dooner Clancy Catholic College, NSW Maths Coordinator 14 Ms Helen Douvartzidis Wilderness School, SA Head of Mathematics Mr John Dovey Melbourne High School, VIC Head of Mathematics Ms Amanda Dowdell St Peter Chanel Primary, NSW Teacher 15 Mr Graeme Downward Newington College, NSW Teacher Ms Melanie Doyle DECS, SA Numeracy Coach 17 Mr Glenn Dudley Pymble Ladies’ College, NSW Head of Mathematics Ms Jeanne Dudley All Saints Catholic Girls College, NSW Maths Coordinator 9 Mrs Mary-Ann Dudley Mt St Benedict College, NSW Maths Teacher/Pastoral Coordinator Miss Anne Duncan St John the Apostle Primary School, NSW Principal Mr Bruce Duncan Woodbridge School, TAS Numeracy Coordinator Miss Kerry Dundas Shelford Girls’ Grammar, VIC Mr David Dunstan AISWA, WA Numeracy Consultant 11 Miss Dominique Dybala St Mary’s Primary School, VIC Teacher Mrs Trish Dykes St Mary’s Primary School, VIC Teacher Mrs Maria Dyne Queen of Peace Primary School, VIC Maths Coordinator Ms Sylvia Eadie Learning Services North- West, TAS Numeracy Support Teacher Mrs Cheryl Eather Loyola Senior High School, NSW Administration Coordinator Mrs Jo Edwards Berserker Street State School, QLD HOC Mr Gavin Edwards DEECD, VIC Senior Project Officer Mrs Heather Efraimsen DECS, SA Principal

Research Conference 2010 102 Dinner table no. Delegate Name Delegate Organisation

Mr Deb Eldridge Ballarat Grammar School, VIC Maths Coordinator Ms Helen Elliott St Michael’s Primary, NSW Assistant Principal 7 Ms Ann-Marie Ellis DECS, SA Maths for All Facilitator Ms Sue Ellis Overnewton College, VIC Teacher Ms Cate Elshaug LLoyd Street School, VIC Assistant Principal Mr Andrew Emanuel Chisholm Catholic Primary School, NSW Assistant Principal Mrs Natalie Emberton All Saints Primary School, SA Teacher 1 Prof Paul Ernest The University of Exeter, UK Ms Gail Erskine St Jerome’s Primary, NSW Teacher Educator 19 Mrs Sue Evans Oberon High School, VIC Maths DBA Leader 9 Ms Frances Eveleigh ACER, NSW Research Fellow Mrs Caitlink Faiman Bialik College, VIC Head of Gifted Mrs Marilyn Faithfull Koonung Secondary College, VIC Senior Mathematics Administrator Mrs Wendy Falconer University of Waikato, NZ Numeracy Adviser Mrs Robyn Farnell Hampton Primary School, VIC Assistant Principal Ms Sally Farrell Palm Beach High School, QLD HOD Mr Antonio Fazzini Saint Ignatius’ College, SA Head of Mathematics Mr Luke Fensling McKinnon Primary School, VIC Numeracy Coordinator 15 Ms Candice Ferey Santa Sabina College, NSW Coordinator Learning Enrichment Mrs Margaret Ferguson Holy Family Primary School, NSW Teacher/Leadership Team Mr Bruce Ferrington Radford College, ACT Teacher Mrs Anita Fewster St Mary’s Primary School, VIC Teacher 18 Ms Jocelyn Field Penrhos College, WA Teacher 16 Mrs Joanne Findlay Bundaberg S.H.S., QLD Teacher Ms Anne Finlay De La Salle College Ashfield, NSW Mathematics Coordinator

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 103 Dinner table no. Delegate Name Delegate Organisation

Mr Simon Finniecome Domremy College, NSW Admin/Teacher 13 Mrs Megan Finnigan Marist College, Canberra, ACT Teacher Mrs Lauren Fitzhenry St Kevin’s Catholic Primary School, NSW Assistant Principal Mrs Lana Fleiszig Mt Scopus Memorial College, VIC Maths Coordinator Mrs Krishna Fleming Aquinas College, VIC Coordinator Mrs Sharon Fleming Loreto College, SA Teacher Mr Ken Fletcher Emmanual College, QLD Year 1 Teacher Ms Jacky Foley Nagle College, NSW Maths Coordinator Mrs Margaret Ford Seaford Rise Primary School, SA Teacher Mrs Robyn Ford Barker College, NSW Teacher Ms Michelle Fothergill Cambridge University Press, VIC Education Sales Coordinator Mrs Jo Fox St Peter Chanel Primary, NSW Principal Ms Kathryn Fox Catholic Schools Office, NSW Head of Teaching & Learning Services Mrs Elizabeth Fragopoulos St Joseph’s Primary, Belmore, NSW Teacher Educator Mr David Francis Citipointe Christian College, QLD Head of Mathematics Mrs Beaulah Frankson Good Shepherd School, NSW Teacher Miss Kyla Frazer Carwatha College P-12, VIC Teacher Ms Danielle Freeman Everton Park State School, QLD Mr Phil Freeman Craigslea Senior High School, QLD HOD - Mathematics Mrs Danielle Gagliardi Seaford 6-12 School, SA Teacher Ms Amanda Gahan St Peter Chanel Primary, NSW Teacher Mrs Susan Gahan Olsos Primary School, NSW Stage One Co-ordinator 15 Mrs Donielle Gale St Ignatius College, NSW Teacher Mr Todd Gallacher Carey Baptist Grammar, VIC Senior School Maths Mr Michael Gallagher St Joseph’s Catholic Primary School, NSW Assistant Principal

Research Conference 2010 104 Dinner table no. Delegate Name Delegate Organisation

Ms Gina Galluzzo Catholic Education Office, ACT Curriculum Officer Mr Craig Gannon Clarkson Community High School, WA Deputy Principal Ms Nicole Gardner Good Shepherd School, NSW Teacher Ms Martha Garkel Sacred Heart Girls’ College, VIC Head of Mathematics Ms Robyn Garnett Overnewton College, VIC Teacher Mrs Judy Gastin St Michael’s Primary, NSW Principal Mrs Elizabeth Gauld St Margaret Mary’s College, QLD Mathematics Coordinator Miss Michelle Gawronski North Shore Primary School, VIC Primary Maths Specialist Mr Andrew Gear Cedars Christian College, NSW Leading Teacher Mrs Katherine Gee Maria Regina Catholic Primary School, NSW Principal 5 Ms Katie Geerings Lorne Airey’s Inlet P12 College, VIC Teacher Ms Linda Gelati Catholic Education Office, SA Numeracy Consultant Mr Greg Georgiou Good Samaritan Catholic College, NSW Assistant Maths Coordinator Mrs Deborah Gibbs Massey University College of Education, NZ Mathematics Adviser Miss Melissa Gibbs Mount Gambier High School, SA Teacher 6 Mrs Bernadette Gibson Catholic Schools Office, NSW Education Officer 14 Ms Rhiannon Giles Wilderness School, SA Mathematics Teacher Ms Karen Gillespie Craigburn Primary School, SA Assistant Principal Mrs Trish Gleeson CSO Maitland-Newcastle, NSW Education Officer 2 Prof Merrilyn Goos The University of Queensland, QLD Director Mrs Johanna Gordon Brisbane Grammar School, QLD Teacher Ms Haley Graham Ballarat Clarendon College, VIC Co Head of Middle School Maths Mr Richard Grech Delany College, NSW Assistant Principal Mr David Green Sydney Grammar School, NSW Teacher 19 Mr James Green Trinity Catholic College, NSW Head of Mathematics

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 105 Dinner table no. Delegate Name Delegate Organisation

Mrs Denise Greenberg Wenona School, NSW Assistant Head of Mathematics Mr Martin Gregory Xavier College, VIC Teacher Mr William Grieve St Brendans College, QLD Head of Maths Ms Linda Grof St Kilda Primary School, VIC Instructional Practice Coach Ms Jacky Gruszka Taylors College Waterloo, NSW Maths Teacher Mrs Susan Guilfoyle Holy Family School, NSW Principal Mr Peter Hackett Corpus Christi College, WA Hola Mrs Robyn Hadfield Presbyterian Ladies College, NSW Maths Teacher Mrs Sue Hage Seaford Rise Primary School, SA Teacher Ms Belinda Haley Lockleys North Primary School, SA Teacher Mr Michael Hall St Andrew’s College, NSW Assistant Principal Mrs Lyn Hamilton DECS, SA Mrs Julie Hancock Catholic Education Office, SA Ms Judith Hanke DEECD, VIC Manager, Secretariat 11 Mrs Cynthia Harbor St Mary’s Primary School, VIC Teacher 4 Ms Christine Hardie Univ of Auckland, FoEd, Team Solutions, NZ Team Leader Miss Marina Hardy Mary MacKillop, NSW Assistant Principal Mr Matt Hardy Padua College, QLD Teacher Ms Joanna Harrisson Australind Senior High School, WA Teacher 14 Mr Bede Hart St Anne’s Primary School, NSW Principal Ms Jan Harte Catholic Education Office Sydney, NSW Curriculum Adviser 12 Mr Dave Hartley Merrimac State School, QLD Numeracy Coach Ms Jodie Hartmann Toormina High School, NSW 5 Ms Judy Hartnett Queensland University of Technology, QLD Lecturer Ms Liberty Hatzidimitriou Lowther Hall AGS, VIC Teacher Mrs Kerrin Hazard CSO Broken Bay, NSW Numeracy Project Officer

Research Conference 2010 106 Dinner table no. Delegate Name Delegate Organisation

Ms Carmel Healey Sacred Heart Catholic School, NSW Principal Ms Tracy Healy Lowther Hall AGS, VIC Coordinator 14 Mrs Christine Heath Pembroke School, SA Head of Middle School Mathematics Ms Jayne Heath Aust. Science & Math School, SA Assistant Principal Miss Karley Erin Hefferan DECS, SA Numeracy Coach Ms Tracy Herft Strathcona BGGS, VIC 7 Mrs Jenni Hewett DECS, SA Maths Facilitator/Numeracy Coordinator Ms Ann Hewitt Gympie SHS, QLD Teacher 14 Mr Ian Hilditch Pembroke School, SA Head of Mathematics Ms Jacky Hiscock Seaford 6-12 School, SA Teacher 3 Ms Kerry-Anne Hoad ACER INSTITUTE, VIC Director Mrs Giannina Hoffman SACE Board of SA, SA Assessor Trainer 5 Mr John Hogan Redgum Consulting, WA Mrs Birgit Holley Stuartholme School, QLD Teacher 7 Dr Hilary Hollingsworth ACER INSTITUTE, VIC Teaching Fellow 10 Mrs Janette Holmes Dept of Education, NSW Quality Teaching Consultant 11 Mrs Mary Hor St Philip Neri, Northbridge, NSW Assistant Principal 7 Ms Rhonda Horne DET, QLD Principal Education Officer Mr Nicholas Houghton St Anthony’s Primary School, NSW Teacher Mr Rodney Howard Bede Polding College, NSW Assistant Principal Mrs Rebecca Huddy DECS - Western Adelaide Region, SA Curriculum Coordinator Mr Cameron Hudson The Hutchins School, TAS Head of Mathematics Ms Judith Hunt DECS, SA Numeracy Coordinator Ms Janet Hunter Ascham School, NSW Teacher

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 107 Dinner table no. Delegate Name Delegate Organisation

Mrs Kylie Hyde Holy Family Catholic Primary, NSW Teacher 9 Mr Malcolm Hyland Ministry of Education, NZ Manager Mrs Diann Hynes Catholic Schools Office, NSW Schools Consultant Prof Lawrence Ingvarson ACER, VIC Principal Research Fellow 6 Mrs Bernadette Irvin Catholic Schools Office, NSW Education Officer Mrs Jane Irvin Morayfield State High School, QLD Head of Department Mathematics Miss Kim Irvine Italian Bilingual School, NSW Coordinator Mrs Terry Jacka St Hilda’s Schools, QLD Head of Faculty - Mathematics Mrs Ann Jackson MacKillop Catholic College, NSW Executive Team Ms Deirdre Jackson ACER, VIC Director, Assessment Services 10 Mrs Lorraine Jacob Murdoch University, WA Senior Lecturer Ms Kylie Jago Queechy High School, TAS Teacher Ms Jacinta James Simonds Catholic College, VIC Teacher 18 Miss Lauren James North Sydney Girls High School, NSW High School Teacher 11 Mrs Sheryl Jamieson Nuriootpa Primary School, SA Coordinator Mr Michael Jennings The University of Queensland, QLD Lecturer 21 Mr Paul Johansen St Paul’s School, QLD Head of Department Ms Janet Johnson Ocean View College, SA Teacher 18 Mrs Nicole Johnson Penrhos College, WA Teacher Mr Kevin Jones Bede Polding College, NSW Principal Ms Maureen Jones Christ the King Primary, NSW Principal Miss Brianna Jordan DECS, SA Numeracy Coach 1 Prof Robyn Jorgenson Griffith University, QLD Ms Fran Kane OLR The Entrance, NSW Assistant Principal

Research Conference 2010 108 Dinner table no. Delegate Name Delegate Organisation

Miss Pauline Kaszubowski Holy Family School, NSW Teacher Mrs Clare Kavanagh St Patrick’s College, VIC Head of Mathematics Mrs Robyn Kay MacGregor Primary School, QLD Deputy Principal 10 Mr Alexander Keech Dept. of Education, QLD Classroom Teacher Ms Jo Kellaway Aust. Science & Math School, SA Coordinator Dennis Kelly St Mary’s Primary School, VIC Teacher Ms Mary Kelly Holy Family Primary School, NSW Assistant Principal Mr Paul Kelly Catholic Ladies’ College, VIC Head of Mathematics Mr Tim Kelly Lismore C.E.O. QLD Education Officer Sr Brenda Kennedy Holy Family Primary School, NSW Principal Ms Jennifer Kerby Our Lady of Sion College, VIC Maths Teacher Miss Suzanne Khatib McKinnon Primary School, VIC Dr Siek Toon Khoo ACER, VIC Research Director 14 Mrs Diane Kibble St Catherine’s Catholic College, NSW Mathematics Co-ordinator Ms Katherine Kilburn Shore School, NSW Teacher Miss Linda Kloeden North Haven Primary School, SA Teacher Mrs Jacqui Klowss Marist College Ashgrove, QLD HOD Maths 15 Ms Margaret Knight St Columba College, SA Assistant Head of Primary Mr Michael Knight Terra Sancta College, NSW Teacher Ms Pat Knight ACER, VIC Senior Librarian Mrs Rebecca Knight DECS, SA Numeracy Coach Ms Carol Knox Lindisfarne Anglican Grammar School, NSW Maths Director Ms Karen Knox DECS, SA Numeracy Coach Mr Kimon Kousparis Casimir Catholic College, NSW Maths Coordinator 20 Ms Miriam Krakovska UWS College Pty Ltd, NSW Academic Teacher

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 109 Dinner table no. Delegate Name Delegate Organisation

Mr Andre Kristovskis The Riverina Anglican College, NSW Teacher Ms Jan Ladhams Steps Professional Development, WA Mathematics Consultant 18 Mr Greg Ladner Presbyterian Ladies College, WA Head Maths Ms Anni Lahdesluoma Retired Teacher 18 Ms Tania Lamble North Sydney Girls High School, NSW Teacher 16 Mrs Siobhan Lanskey Bundaberg S.H.S., QLD Teacher Ms Felisa Lapuz Marian College, VIC Head of Mathematics Mrs Jenny Lawrence Overnewton College, VIC Teacher Mrs Mary Leask Nagle College, NSW Principal 3 Dr David Leigh-Lancaster Victorian Curriculum & Assess Authority, VIC Curriculum Manager Mathematics Ms Elisabeth Lenders Carey Baptist Grammar, VIC Deputy Principal 17 Ms Dianne Ley Gilroy Catholic College, NSW Teacher Mr John Ley Xavier College, NSW Acting Principal Mr Cameron Lievore Our Lady of the Nativity School, NSW Principal Mrs Deborah Lilly Lowther Hall AGS, VIC Teacher Mr Julian Lindsay Runcorn State High School, QLD Head of Department - Mathematics Mrs Heather Lines Westminster School, SA Head of Mathematics Miss Charlotte Lipnicki St Mary’s Primary School, VIC Year 2 Teacher Mrs Jeanette Little Loreto College, QLD Head of Mathematics Mrs Carole Livesey Catholic Education Office, VIC Education Officer Mrs Sharyn Livy MAV, VIC Professional Officer Ms Shayne Llanda St Monica’s College, VIC Teacher Mr Peter Lorenti Reservoir District Sec. College, VIC Dr Ian Lowe MAV, VIC Professional Officer 2 Prof Tom Lowrie Charles Sturt University, NSW 11 Ms Donna Ludvigsen Grampians DEECD, VIC Network Improvement

Research Conference 2010 110 Dinner table no. Delegate Name Delegate Organisation

Mr Chris Lynagh St Luke’s Anglican School, QLD Teacher Mrs Carol Lynch Holy Family Catholic Primary, NSW Teacher Mr Des Lyristis Hunting Tower School, VIC Maths Department Mrs Ann MacMillan DECS, SA Coordinator Maths For Learning inclusion Mr Michael MacNeill St. Josephs College, VIC Learning Development Ms Robyn Macready-Bryan Carey Baptist Grammar, VIC Head of Maths/IT-Senior School Ms Christine Mae St Aloysius’ Primary , NSW Coordinator Dr Bryan Maher St Joseph’s High School, NSW Assistant Principal 11 Miss Danielle Mahony St Mary’s Primary School, VIC Teacher 8 Mr Chicri Maksoud Brisbane Boys’ College, QLD Coordinator Mathematics Mr Chris Malberg Taylors Lakes Secondary College, VIC Assistant Principal Ms Nita Maloney DECS, SA Numeracy Coach Miss Amanda Mamo Domremy College, NSW Teacher 18 Miss Alice Manning Penrhos College, WA Teacher Mrs Katrina Mansfield Craigslea Senior High School, QLD Teacher Mr Paul Mansfield Padua College, QLD Head of Curriculum - Mathematics 21 Mr Gareth Manson AB Paterson College, QLD Classroom Teacher Ms Juvy Marcellano Nagle College, NSW Maths Teacher 4 Mrs Anne Martin Geelong Grammar School, VIC Maths Teacher Mr David Martin St Peter’s College, SA Teacher 1 Prof Geoff Masters ACER, VIC CEO Ms Stamatiki Matheos North Haven Primary School, SA Teacher Ms Catherine Mathews Catholic Education Office, NSW Teacher Mr Lukas Matysek Cedars Christian College, NSW Dean Mr Richard Maynard Seaford 6-12 School, SA Program Manager Mrs Caroline Mazurkiewicz DEECD, VIC Teaching and Learning Coach WMR

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 111 Dinner table no. Delegate Name Delegate Organisation

19 Ms Fiona McAlister Aquinas College, WA Mathematics Teacher Ms Cara McCarthy DEECD, VIC Project Officer Mrs Sheila McCarthy Norlane West Primary School, VIC Maths Specialist 16 Ms Margaret McCaskie Daramalan College, ACT Teacher 11 Mr Terence McClelland Mareeba State High School, QLD Head of Department Ms Catherine McCluskey Catholic Education Office, SA Numeracy Consultant 2 Dr Barry McCrae ACER, VIC Principal Research Fellow Ms Kim McDonald St Andrew’s Primary School, NSW Assistant Principal Ms Michele McDonald Catholic Education Office, NSW Teaching & Learning Devl. Consultant Mrs Yvonne McGarry Canberra Girls’ Grammar, ACT Teacher Ms Bernadette McGill Our Lady of the Sacred Heart College, VIC Maths Domain Leader Mrs Patricia McGregor St Paul’s Manly, NSW Teacher Mrs Kim McHugh Steps Professional Development, WA Numeracy Consultant Mr Jesse McInnes Wesley College, VIC Teacher 17 Ms Narelle McKay Jamison High School, NSW

Mrs Jennifer McKeown St Thomas School, NSW Principal Ms Nicola McKinnon ACER, VIC Research Fellow Mrs Ellen McLagan Our Lady of Lourdes Catholic School, TAS Teacher Mrs Lorraine McLaren Reservoir District Sec. College, VIC Maths Coordinator 12 Ms Jillian McNamara St Mary’s Primary School, VIC Teacher Mr Colin McNeil Macmillan Education Australia, VIC Publisher Mrs Frances McPhee Caulfield Junior Campus, VIC Assistant Principal Ms Vivienne McQuade DECS, SA Curriculum Manager 19 Mr Peter Mee Mercedes College, WA Head of Mathematics Mrs Anita Meehan Bede Polding College, NSW Administration Coordinator

Research Conference 2010 112 Dinner table no. Delegate Name Delegate Organisation

Mrs Margaret Meehan Mary MacKillop, NSW Teacher Ms Jenny Meibusch Canberra Girls’ Grammar, ACT Teacher 8 Mrs Marion Meiers ACER, VIC Senior Research Fellow Mrs Silva Mekerdichian Covenant Christian School, NSW Mathematics Teacher Mr Paul Menday Catholic Education Office, NSW Head of School Services 6 Mrs Carey Menz-Dowling Catholic Schools Office, NSW Education Officer 7 Mrs Jenny Merrett Yarra Valley Grammar, VIC Head of Mathematics Mrs Chris Miethke DECS, SA Maths & Science Facilitator Mr Christopher Mills Richmond River High School, NSW Head Teacher 5 Mrs Dianne Mills Schools Industry Partnership, NSW Partnership Broker Mrs Leonie Mitchell Mary MacKillop, NSW Teacher Mr Brett Molloy Qld Studies Authority, QLD Manager 9 Mr Nick Moloney Marcellin College, VIC Learning Coordinator Ms Samantha Monteiro Education QLD Senior Teacher 9 Mr David Moran Marcellin College, VIC Teacher 3 Mr Will Morony Aust. Assoc. of Mathematics Teachers, SA Executive Officer Mrs Caty Morris Aust Assoc of Mathematic Teachers, SA National Manger: Indigenous Programs Mr Andrew Morrison Mossfiel Primary School, VIC Maths Leader 19 Mr Rodney Morrison Aquinas College, WA Assistant Head of Mathematics 19 Mrs Sally Morse Belmont High School, VIC Maths Domain Leader Ms Rachael Mowe Queenwood School for Girls, NSW Teacher 3 Assoc Prof Joanne Mulligan CRiMSE Macquarie University, NSW Ms Kerry Mulvogue Our Lady of Mercy College, VIC Ms Catherine Murray Catholic Schools Office, NSW Education Officer Mrs Vanessa Murray Holy Family Catholic Primary School, NSW Teacher Mr Bruce Murrie DECS, SA Teacher

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 113 Dinner table no. Delegate Name Delegate Organisation

Mr Robert Muscatello Catholic Education Office, NSW Education Officer 20 Mrs Anne Myhill William Carey Christian School, NSW Assistant Head of Mathematics Mrs Debra Needham Corpus Christi College, WA Assistant Hola 4 Mr Alex Neill NZCER, NZ Researcher Mr Michael Nekvapil Orana Steiner School, ACT Teacher Mr Mark Newhouse Association of Independent Schools of WA Manager of Curriculum 9 Mrs Kathy Nolan Catholic Education Office, VIC Project Officer Maths Ms Olivia Norris St Jerome’s Primary, NSW Numeracy Coordinator Ms Rosalie Nott Catholic Education Commission, NSW Assistant Director Mrs Debbie Oates Sydney Grammar School, NSW Maths Coordinator 8 Ms Gayl O’Connor Education Services Australia, VIC Assessment Advisor 10 Ms Lisa-Jane O’Connor Primary Mathematics Association, SA Educational Consultant Mrs Wendy Ogilvie DECS, SA Numeracy Coach Mr Michael O’Halloran Aquinas College, VIC KLAC Ms Delwyn Oliver Ballarat High School, VIC Head of Maths 18 Mrs Jennifer Olma Perth College, WA Mathematics Co-ordinator Ms Patricia Olsen Chisholm Institute, VIC Teacher 16 Mrs Sharon Olsen Bundaberg S.H.S., QLD Teacher Ms Joanne O’Malley St Kilda Primary School, VIC Acting Assistant Principal Mr Frank O’Mara Downlands College, QLD Teacher Ms Effie Orlando Mary MacKillop College, NSW Assistant Maths Coordinator 17 Mrs Carol Osborne Loreto Normanhurst, NSW Head of Mathematics Mr Peter Osland Board of Studies, NSW Maths Inspector Mrs Yvette Owens St John the Baptist Catholic Primary, NSW Assistant Principal Miss Attilia Pagano Sacred Heart School, NSW Teacher Educator

Research Conference 2010 114 Dinner table no. Delegate Name Delegate Organisation

13 Mr Chris Page Marist, Eastwood, NSW Maths Teacher Mr Michael Palme Brigidine College, NSW Head of Mathematics Mrs Deborah Palmer CEO, Inner Western Region, NSW Mrs Kathryn Palmer Western Metro. Region, VIC Regional Coach 14 Ms Katerina Papetros Seymour College, SA Maths Teacher 21 Mrs Larra Paron Genazzano FCJ College, VIC Mathematics Teacher Mrs Heather Parrington SACE Board of SA, SA Senior Curriculum Coordinator Ms Sheila Parsons MacGregor Primary School, QLD Teacher 20 Dr Anne Paterson Wesley College, WA Teacher Mrs Carol Patterson Haileybury, VIC Head of Mathematics Mr Jacob Pearce ACER, VIC Research Officer 7 Mrs Cath Pearn ACER INSTITUTE, VIC Teaching Fellow Ms Melinda Pearson Australian Assoc of Math Teachers, SA Project Officer Mrs Suzanne Pearson DET, WA Senior Curriculum Officer Mr Ray Peck ACER, VIC Senior Research Fellow 2 Prof John Pegg University of New England, NSW Mr Geoff Pell Taylors Lakes Secondary College, VIC Principal Ms Teresa Peluso Cheltenham Secondary College, VIC Maths Coordinator Mr Brett Perkins St. Cecilia’s Catholic School, NSW Classroom Teacher Ms Michelle Perry St Patrick’s Catholic Primary School, NSW Assistant Principal Mr Gregory Petherick DECS - Western Adelaide Region, SA Assistant Regional Director Mr Joemon Philip Mount Annan Christian College, NSW Coordinator Mr Ray Philpot ACER, VIC Research Fellow Ms Sue Pickup Our Lady of Mt Carmel, NSW Coordinator Mrs Samantha Pinkerton Guilford Young College, TAS Teacher Ms Meredith Plaisted Carey Baptist Grammar, VIC Head of Maths/IT Senior School

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 115 Dinner table no. Delegate Name Delegate Organisation

Mrs Pauline Pollock St Thomas Catholic School, NSW Teacher Mrs Karen Post DECS, SA Numeracy Coach Ms Maureen Price Mossfiel Primary School, VIC Principal Mr Rob Proffitt-White Education Queensland, QLD Numeracy Coach Ms Susanne Prosenica Copperfield College, VIC Teacher Ms Yianna Pullen Wooranna Park Primary School, VIC Assistant Principal Ms Robyn Purcell Marist Sister’s College, NSW Maths Coordinator 4 Mr Brendan Pye ACER INSTITUTE, VIC Project Officer Ms Mary Quill Holy Spirit College, NSW Mathematics Coordinator Mrs Kylie Quin Overnewton College, VIC Teacher Ms Mary Quinane Catholic Education Office, ACT Primary Numeracy Officer Mr Jeremy Rackham The Friends’ School, TAS Teacher Ms Jane Ralston-Palmer Carey Baptist Grammar, VIC Senior Teacher Ms Christine Ratcliff DECS, SA Numeracy Coach Ms Dympna Reavey Nagle College, NSW Leader of Teaching & Learning Mr Mark Redington Seaford 6-12 School, SA Teacher 13 Mr Max Redmayne Marist, Eastwood, NSW Maths Coordinator 3 Assoc Prof Robert Reeve The University of Melbourne, VIC 10 Miss Deborah Reeves Waikato University, NZ Numeracy Adviser 6 Ms Glenys Reid Department of Education, WA Principal Consultant Mrs Jenny Rendall Middle Park Primary School, VIC Principal Ms Anna Rerakis DEECD, VIC Project Officer Mrs Frances Reynolds Catholic Schools Office, NSW Schools Consultant Ms Louise Reynolds ACER, VIC Corporate Publicity & Comm. Manager Ms Mary Reynolds Eltham College of Education, VIC Numeracy Leader

Research Conference 2010 116 Dinner table no. Delegate Name Delegate Organisation

Mrs Penelope Reynolds Department of Education, WA Curriculum Officer - Numeracy Ms Elisabeth Rhodes Lowther Hall AGS, VIC Deputy Principal Mr Joshua Richmond Ballarat Grammar, VIC Maths Teacher Ms Joanne Riddell Catholic Education Office Sydney, NSW Mathematics Adviser (Primary) Mrs Janet Ridley Landsdale Primary School, WA Teacher Mr Paul Rijken Cardijn College, SA Principal Ms Nicole Riles St Laurence’s College, QLD Head of Mathematics Ms Sue Riquelme Lowther Hall AGS, VIC Coordinator Miss Karen Roberts Sandringham East Primary School, VIC Lead Teacher 12 Ms Trish Roberts St Mary’s Primary School, VIC Support Teacher Mr Andrew Robertson Kingswood College, VIC Head Faculty 8 Ms Leanne Robertson Education Services Australia, VIC Senior Project Manager Mr Greg Robinson Education Queensland, QLD Project Officer Ms Karen Robson St Peter’s Primary School, NSW Assistant Principal Mrs Kathleen Roffey Trinity Catholic College, NSW Mathematics Coordinator 5 Ms Honor Ronowicz University of Waikato, NZ Numeracy Adviser Mrs Sarah Rosenweg Shelford Girls’ Grammar, VIC Head of Faculty 4 Ms Lynda Rosman ACER INSTITUTE, VIC Manager Programs and Projects Mrs Jennifer Rowland DECS, SA Numeracy Coach Mr Peter Rundle Barker College, NSW Head of Mathematics Mrs Irene Ruscigno Epping Views Primary School, VIC Numeracy Coordinator 12 Mr Bradley Ryall St John’s College, NSW Maths Coordinator Ms Sophie Ryan Catholic Education Office, NSW Head of School Service Mrs Nicole Sadler St Mary’s Primary School, VIC Year 6 Teacher Mr John Sagner Browns Plains High School, QLD Head of Department Mathematics

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 117 Dinner table no. Delegate Name Delegate Organisation

19 Mr Darius Samojlowicz The Hills Grammar School, NSW Head of Stage Two Mr Jared Sanders Canterbury Primary School, VIC Teacher Mr Peter Sanders La Trobe University, VIC Lecturer Mrs Susan Sanders Our Lady of Mercy College, VIC Head of Maths Miss Alicia Sandersan Holy Family School, NSW Teacher Ms Emily Sangster Queensland Studies Authority, QLD Acting Manager Mrs Rosa Santopietro Our Lady of the Sacred Heart College, SA Maths Coordinator 6 Mr Ralph Saubern ACER, VIC General Manager, Schools Program Mr Keat Saw Australind Senior High School, WA Teacher Mrs Fiona Scannell Palm Beach High School, QLD HOD Mrs Ronelle Scheepers St Teresa’s College, QLD Learning Coordinator - Maths 11 Mr Bruce Schmidt Grampians DEECD, VIC Project Officer Ms Cathy Scott Chisholm Catholic Primary School, NSW Principal Mrs Lynda Secombe Assoc. of Independent Schools of SA Adviser Ms Judith Selby Cowra High School, NSW HT Mathematics Mrs Emma Sellars St Mary’s, Toukley, NSW Coordinator Mr Mark Sellen Shore School, NSW HOD Mrs Yvette Semler Queenwood School for Girls, NSW Teacher Mrs Katherine Serbin Nagle College, NSW Maths Teacher 21 Mr Ferruccio Servello Genazzano FCJ College, VIC Mathematics Teacher 12 Ms Michelle Sexton St Mary’s Primary School, VIC Teacher Mr Barry Shanley St John Fisher School, NSW Principal Ms Linda Shardlow Methodist Ladies College, VIC Head of Mathematics 16 Mrs Amy Shaw Bunbury Cathedral Grammar, WA Teacher Mrs Margaret Sheahan St Oliver’s Primary School, NSW Coordinator

Research Conference 2010 118 Dinner table no. Delegate Name Delegate Organisation

12 Mr James Sheedy St Mary’s Primary School, VIC Principal Mrs Debra Sheehan Overnewton College, VIC Teacher Mrs Kylie Shelton Berserker Street State School, QLD Teacher Ms Debra Shephard Killara Primary School, VIC Teacher Mr Ian Sheppard Wesley College, WA Head of Mathematics Mrs Joy Short Catholic Education Office, Parramatta, NSW Head of School Service Miss Jodie Sibbald Holy Family Primary School, NSW Teacher/Leadership Team Mr Michael Siciliano St Michael’s Primary School, NSW Assistant Principal Mrs Wendy Silvestri DECS, SA Numeracy Coach Miss Vanessa Simiele St Mary’s Primary School, VIC Teacher Miss Megan Skinner Wooranna Park Primary School, VIC Maths Specialist Miss Amy Skuthorp St Mary’s Primary School, VIC Prep Teacher Ms Christine Slattery CEO, SA Consultant Mrs Judy Slattery St John the Baptist, NSW Principal 20 Mr Roy Smalley Chisholm Institute, VIC Teacher 6 Mrs Barbara Smith ACER, VIC Sales Manager Ms Catherine Smith Marist Sister’s College, NSW Maths Teacher 4 Ms Denise Smith Univ of Auckland, FoEd, Team Solutions, NZ Team Leader 21 Mr Glen Smith St Paul’s School, QLD Head of Studies, Senior School Ms Jacqui Smith Western Port Secondary College, VIC Numeracy Co-ordinator 16 Ms Julie Smith Bunbury Cathedral Grammar, WA Teacher 6 Mrs Michelle Smith Holy Family Primary School, NSW Teacher/Leadership Team Miss Michelle Smith Catholic Schools Office, NSW Schools Consultant 8 Mr Vaughan Smith Caulfield Grammar School, VIC Head of Research Ms Gabriella Spadaro Marymount International School, ITALY Special Needs Teacher

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 119 Dinner table no. Delegate Name Delegate Organisation

4 Mr Ken Spanks Gin Gin State High School, QLD Teacher Mrs Susan Spencer Spencer Education, VIC Special Education Consultant Miss Dominique Spindler Routledge, UK Exhibitions Administrator 18 Mr Peter Sprent North Sydney Girls High School, NSW Teacher Mrs Lois Staatz Gatton State School, QLD Principal 1 Prof Kaye Stacey The University of Melbourne, VIC Miss Ellie Stanford Ascham School, NSW Teacher Mr Mitchell Staples Canterbury College, QLD Teacher Miss Liz Starling MacKillop Catholic College, NSW Executive Team Mr David Steele Wesley College, VIC Dept. Head of Campus Mr Greg Steele Norlane West Primary School, VIC Maths Specialist Ms Marie Stenning MacGregor Primary School, QLD Teacher Dr Andrew Stephanou ACER, VIC Senior Research Fellow Mrs Robyn Stephens Croydon Primary School, VIC Maths Coach 21 Mr David Stephenson Grovedale College, VIC Maths Coordinator 16 Mr Michael Stjepcevic Ipswich Grammar School, QLD HOD Junior Maths 12 Ms Melinda Stockwell Trinity Anglican College, QLD Teacher Mr Peter Stone Unley High School, SA Maths Coordinator Mr Max Stowe Ballarat Grammar School, VIC Maths Teacher Mr Dirk Strasser Pearson Australia, VIC Publishing Manager Mrs Susannah Stredwick Ravenswood School for Girls, NSW Learning Enrichment Coordinator 2 Prof Peter Sullivan Monash University, VIC Mrs Michele Sunnucks OLMC Primary School, NSW Assistant Principal Ms Nancy Surace MacKillop College, VIC Teacher Ms Jenny Sutton St Mary’s Primary School, VIC Classroom Teacher Miss Carla Sweeting Ascham School, NSW Teacher

Research Conference 2010 120 Dinner table no. Delegate Name Delegate Organisation

Ms Carmel Tapley Catholic Schools Office, NSW Education Officer Mrs Bernadette Taylor St. Cecilia’s Catholic School, NSW Classroom Teacher Ms Christine Taylor Board of Studies, NSW Inspector, Primary Ms Debbie Taylor The Friends’ School, TAS Teacher 9 Ms Margaret Taylor ACER INSTITUTE, VIC Administration Officer 5 Mrs Gaynor Terrill University of Waikato, NZ Numeracy Adviser 18 Mr Ken Terry North Sydney Girls High School, NSW Teacher 20 Mr Greg Thackeray William Carey Christian School, NSW Head Teacher Mr Greg Thomas St Martin of Tours Primary School, VIC Deputy Principal Ms Julie Thompson St Francis Xavier’s Primary, NSW Assistant Principal Mr Lincoln Thompson Queenwood School for Girls, NSW Teacher 2 Dr Sue Thomson ACER, VIC Principal Research Fellow 4 Mr Gregory Thrupp Gin Gin State High School, QLD Teacher Mr Gregory Tier Brisbane Grammar School, QLD Teacher Mrs Lorna Tobin St John the Apostle Primary School, NSW Assistant Principal Miss Melissa Tomaszewski Sacred Heart School, NSW Teacher Educator Ms Dianne Tomazos Department of Education, WA Principal Curriculum Officer - Numeracy Mr Leigh Toomey Aquinas College, VIC Teacher Mr Paul Toomey St. Cecilia’s Catholic School, NSW Principal 9 Mr George Toth Catholic Education Office, VIC Senior Project Officer Maths Ms Diane Touzell Good Shepherd School, NSW Teacher 16 Mr Ian Tranent Bundaberg S.H.S., QLD Teacher 13 Mrs Tanya Travers St Mary’s Primary School, VIC R.E.C. 7 Mr Bruce Trenerry DET, QLD Principal Education Officer Ms Jennifer Trevitt ACER, VIC Library Information Dissemination

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 121 Dinner table no. Delegate Name Delegate Organisation

Ms Liesa Trinder Mary MacKillop College, NSW Maths Coordinator 13 Mrs Gail Tull St Mary’s Primary School, VIC Special Needs Mr Geoff Tunnecliffe Eltham High School, VIC Maths Teacher Ms Susan Turnbull Australian Intern. School, HONG KONG Head of Mathematics Mr Mark Turkington Catholic Education Office Sydney, NSW Regional Director 16 Mrs Ann Marie Turner Ipswich Grammar School, QLD HOD Senior Maths Mr John Turner Kingston State School, QLD Teacher 6 Mr Kevin Turner OLHC Primary School, NSW Principal 1 Mr Ross Turner ACER, VIC Principal Research Fellow Ms Stacey Van der Velders Taylors Lakes Secondary College, VIC Numeracy Coordinator Ms Anja van Hooydonk St Mary’s Catholic College, QLD Teacher Ms Christine Van Ryswyk Catherine McAuley, Westmead, NSW Assistant Mathematics Coordinator 18 Miss Nikky Vanderhout North Sydney Girls High School, NSW Head Teacher Mrs Sapna Vats Wooranna Park Primary School, VIC Teacher Ms Jackie Vella Catholic Education Office Sydney, NSW Mathematics Adviser (Primary) Ms Rosemary Vellar CEO Sydney, NSW Head Educational Measurement 17 Mrs Veronica Verdi Gilroy Catholic College, NSW Teacher Mrs Josie Vescio St Rose Catholic School, NSW Principal Mrs Lorraine Vincent OLGC Catholic School, NSW Assistant Principal Miss Jacinta Vistoli Loreto College, SA Year 7 Teacher Ms Catherine Volpe Kew High School, VIC Teacher 9 Mr Paul Waddell Mathematics.com.au NSW Director & General Manager Mr Greg Wagner Moriah College, NSW Teacher Mr Jeff Wait Craigburn Primary School, SA Principal Mrs Angela Waite Loreto College, QLD Teacher

Research Conference 2010 122 Dinner table no. Delegate Name Delegate Organisation

Ms Julie Walker Loreto Kirribilli, NSW Mathematics Coordinator Mr Doug Wallace Wesley College, VIC Curriculum Coordinator Mrs Sue Walpole Carey Baptist Grammar, VIC Head of House 13 Mrs Lyn Walsh St Mary’s Primary School, VIC Teacher Mrs Sue Walsh Catholic Education Office, NSW Head of School Service Mr Graeme Walters Kinross Wolaroi School, NSW Head of Mathematics Mrs Renae Wan Italian Bilingual School, NSW Teacher 13 Ms Kerri Ward St Mary’s Primary School, VIC Grade 5 Teacher Mrs Rosemary Ward Xavier College, VIC Teacher Ms Dora Warlond DEECD, VIC Project Officer 17 Mr Paul Waters Mackay North SHS, QLD HOD Maths 13 Mr Robert Watt St Mary’s Primary School, VIC Teacher 5 Mr John Watters Schools Industry Partnership, NSW Executive Officer 8 Dr Jennifer Way The University of Sydney, NSW Associate Dean Undergraduate Programs Mrs Nerida Way Helensvale State School, QLD Numeracy Coach Ms Mignon Weckert Intern. Baccalaureate Org., SINGAPORE Regional Manager Mr Robert Wellham Erina High School, NSW Head Teacher Mathematics Mr Pieter Wepener Citipointe Christian College, QLD Maths Teacher Ms Helen Weston Sandringham East Primary School, VIC Instructional Coach 19 Mrs Valerie Westwell Bridgewater Primary School, SA Maths Teacher (yrs 3 to 7) Mr Jonathan Wever Mentone Grammar, VIC Faculty Head 14 Mr Glen Whiffen Pembroke School, SA Asst. Head of Mathematics 5 Dr Sonia White Qld University of Technology, QLD Lecturer Ms Ann Whitmore Marist, Burnie, TAS Teacher 17 Mrs Rosemary Wiffen Kingston College, QLD Head of Department - Mathematics

Teaching Mathematics? Make it count: What research tells us about effective teaching and learning of mathematics 123 Dinner table no. Delegate Name Delegate Organisation

20 Ms Damith Wijeratne UWS College Pty Ltd, NSW Academic Teacher 17 Mrs Kylie Wiles Loreto Normanhurst, NSW Teacher Mrs Desley Williams Tara Anglican School, NSW HOD - Mathematics 14 Mr John Williams St Aloysius’ College, NSW Mr Kevin Williams St Mary’s, Toukley, NSW Principal Mr David Willmott LaSalle Catholic College Bankstown, NSW Maths Coordinator Ms Robin Willmott MacGregor Primary School, QLD Teacher 15 Ms Debra Wilson Roseville College, NSW Head of Mathematics Mrs Jenny Wood The Friends’ School, TAS Teacher Mrs Kathryn Wood Holy Family Catholic Primary, NSW Teacher Ms Rosemary Wood Balaklava Primary School, SA Teacher Mrs Jennifer Woods MLC School, NSW Teacher Mr Peter Woolfe Waverley Christian College, VIC Mathematics Coordinator Mr Geoff Wright De La Salle College Cronulla, NSW Maths Coordinator Mr Lachie Wright Scotch Oakburn College, TAS Head of Junior School Mrs Noelene Wright Lindisfarne Anglican Grammar School, NSW Assistant Principal Mr Jason Yates DECS, SA Numeracy Coordinator Ms Joanna Yaxley John Paul College, QLD Head of Learning (Maths & IT) 21 Mr Alec Young ITE, TAS CEO 7 Ms Camille Young Trinity College Senior, SA Teacher Mr John Young Clarkson Community High School, WA Principal Mrs Mirella Zalakos Overnewton College, VIC Teacher Mr Gregory Zerounian St Leo’s College, NSW Maths Coordinator Ms Stavroula Zoumboulis ACER, VIC Research Fellow Mrs Dihna Zuvela Education Department, WA Teacher-in Charge for Mathematics

767 delegates listed as of Friday, 16 July 2010.

Research Conference 2010 124