Proceedings of the 31 st Conference of the International Group for the Psychology of Mathematics Education Volume 2 Editors Jeong-Ho Woo, Hee-Chan Lew Kyo-Sik Park, Dong-Yeop Seo The Korea Society of Educational Studies in Mathematics The Republic of Korea The Proceedings are also available on CD-ROM Copyright © 2007 left to the authors All rights reserved ISSN 0771-100X Cover Design: Hyun-Young Kang 2-ii PME31―2007 TABLE OF CONTENTS VOLUME 2 Research Reports Alcock, Lara 2-1 How Do Your Students Think about Proof? A DVD Resource for Mathematicians Applebaum, Mark & Leikin, Roza 2-9 Teachers’ Conceptions of Mathematical Challenge in School Mathematics Arzarello, Ferdinando & Paola, Domingo 2-17 Semiotic Games: The Role of the Teacher Asghari, Amir.H. 2-25 Examples, a Missing Link Askew, Mike 2-33 Scaffolding Revisited: From Tool for Result to Tool-and-Result Barmby, Patrick & Harries, Tony & Higgins, Steve & Suggate, Jennifer 2-41 How Can We Assess Mathematical Understanding? Barwell, Richard 2-49 The Discursive Construction of Mathematical Thinking: The Role of Researchers’ Descriptions Baturo, Annette R. & Cooper, Tom J. & Doyle, Katherine 2-57 Authority and Esteem Effects of Enhancing Remote Indigenous Teacher-Assistants’ Mathematics-Education Knowledge and Skills Bogomolny, Marianna 2-65 Raising Students’ Understanding: Linear Algebra Canada, Daniel L. 2-73 Informal Conceptions of Distribution Held by Elementary Preservice Teachers PME31―2007 2-iii Cayton, Gabrielle A. & Brizuela, Bárbara M. 2-81 First Graders’ Strategies for Numerical Notation, Number Reading and the Number Concept Chang, Y. L. & Wu, S. C. 2-89 An Exploratory Study of Elementary Beginning Mathematics Teacher Efficacy Chapman, Olive 2-97 Preservice Secondary Mathematics Teachers’ Knowledge and Inquiry Teaching Approaches Charalambous, Charalambos Y. 2-105 Developing and Testing a Scale for Measuring Students’ Understanding of Fractions Cheng, Ying-Hao & Lin, Fou-Lai 2-113 The Effectiveness and Limitation of Reading and Coloring Strategy in Learning Geometry Proof Chick, Helen L. & Harris, Kiri 2-121 Grade 5/6 Teachers’ Perceptions of Algebra in the Primary School Curriculum Chin, Erh-Tsung & Lin, Yung-Chi & Chuang, Chih-Wei & Tuan, Hsiao-Lin 2-129 The Influence of Inquiry-Based Mathematics Teaching on 11 th Grade High Achievers: Focusing on Metacognition Chino, Kimiho & Morozumi, Tatsuo & Arai, Hitoshi & Ogihara, Fumihiro & Oguchi, Yuichi & Miyazaki, Mikio 2-137 The Effects of “Spatial Geometry Curriculm with 3D DGS” in Lower Secondary School Mathematics Chiu, Mei-Shiu 2-145 Mathematics as Mother/Basis of Science in Affect: Analysis of TIMSS 2003 Data Cho, Han Hyuk & Song, Min Ho & Kim, Hwa Kyung 2-153 Mediating Model between Logo and DGS for Planar Curves Chung, Insook & Lew, Hee-Chan 2-161 Comparing Korean and U.S. Third Grade Elementary Student Conceptual Understanding of Basic Multiplication Facts 2-iv PME31―2007 Collet, Christina & Bruder, Regina & Komorek, Evelyn 2-169 Self-monitoring by Lesson Reports from Teachers in Problem-Solving Maths Lessons Cooper, Tom J. & Baturo, Annette R. & Ewing, Bronwyn & Duus, Elizabeth & Moore, Kaitlin 2-177 Mathematics Education and Torres Strait Islander Blocklaying Students: The Power of Vocational Context and Structural Understanding Dawn, Ng Kit Ee & Stillman, Gloria & Stacey, Kaye 2-185 Interdisciplinary Learning and Perceptions of Interconnectedness of Mathematics Delaney, Seán & Charalambous, Charalambos Y. & Hsu, Hui-Yu & Mesa, Vilma 2-193 The Treatment of Addition and Subtraction of Fractions in Cypriot, Irish, and Taiwanese Textbooks Diezmann, Carmel & Lowrie, Tom 2-201 The Development of Primary Students’ Knowledge of the Structured Number Line Dogan-Dunlap, Hamide 2-209 Reasoning with Metaphors and Constructing an Understanding of the Mathematical Function Concept Essien, Anthony & Setati, Mamokgethi 2-217 Exploring the English Proficiency-Mathematical Proficiency Relationship in Learners: An Investigation Using Instructional English Computer Software Ewing, Bronwyn & Baturo, Annette & Cooper, Tom & Duus, Elizabeth & Moore, Kaitlin 2-225 Vet in the Middle: Catering for Motivational Differences in Vocational Access Courses in Numeracy Forgasz, Helen J. & Mittelberg, David 2-233 The Gendering of Mathematics in Israel and Australia Fox, Jillian L. 2-241 21 st Century Children, Numeracy and Technology: An Analysis of Peer-reviewed Literature PME31―2007 2-v Fuglestad, Anne Berit 2-249 Teaching and Teacher’s Competence with ICT in Mathematics in a Community of Inquiry García-Alonso, Israel & García-Cruz, Juan Antonio 2-257 Statistical Inference in Textbooks: Mathematical and Everyday Contexts Gholamazad, Soheila 2-265 Pre-service Elementary School Teachers’ Experences with the Process of Creating Proofs Greenes, Carole & Chang, Kyung Yoon & Ben-Chaim, David 2-273 International Survey of High School Students’ Understanding of Key Concepts of Linearity Halverscheid, Stefan & Rolka, Katrin 2-281 Mathematical Beliefs in Pictures and Words Seen through “Multiple Eyes” 2-vi PME31―2007 HOW DO YOUR STUDENTS THINK ABOUT PROOF? A DVD RESOURCE FOR MATHEMATICIANS Lara Alcock University of Essex, UK This paper is about the construction and initial testing of a DVD resource designed to help mathematicians learn more about their students’ reasoning and engage with issues from mathematics education. The DVD uses specially annotated video data, together with screens of prompts for reflection, to encourage discussion among mathematicians about students’ difficulties and successes. In the paper I describe part of the content in detail, showing how a short episode can raise many major issues in the learning of proof. I also report on design issues in the structure of the DVD, and on mathematicians’ responses to an early version of the content. INTRODUCTION: DVD PURPOSE AND STRUCTURE Few mathematicians have time to read education research papers, so if we want them to be able to benefit from the ideas and results of mathematics education, we need to find ways to make these more accessible. This paper reports on a project that aims to do this by providing a DVD resource in which mathematicians can watch individual students engage with proof-based mathematics problems 1. Video episodes have been edited so that half of the screen shows the student working and the other half shows subtitles at the bottom in blue, and the student’s written work in large print in black at the top (a screen shot is provided in Figure 1). Both the subtitles and written work change in real time, enabling the viewer to follow the student’s thinking as easily as possible. The content of each episode is divided into 2-3-minute segments, after each of which the viewer sees a screen of questions designed to prompt reflection on what has been seen. In this way, mathematicians are invited to analyse this data for themselves, attempting to accurately characterise what they see and debating the learning and teaching issues that arise. They are thus able to engage with the process of mathematics education, rather than to passively consume its products. For another project with a similar ethos, see Nardi & Iannone (2004). In this paper I describe one video episode, which has now been shown in presentations to four groups of mathematicians in the UK and the USA. In this episode, Nick attempts a problem about upper bounds. In this paper I demonstrate how a single short episode can raise numerous issues in mathematics education at the transition-to-proof level, including: 1This projectisaHigher EducationAcademyMaths,Stats&OR Networkfundedactivity. 2007. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31 st Conference of the International Group for the Psychology of Mathematics Education , Vol. 2, pp. 1-8. Seoul: PME . 2-1 Alcock • Students’ understanding of abstract mathematical concepts and the way in which these must relate to definitions ( cf. Vinner, 1991; Moore, 1994). • Students’ use of examples in their mathematical reasoning ( cf . Bills et. al., 2006; Dahlberg & Housman, 1997). • Students’ strategies for proving and the extent to which they apply these effectively (cf . Alcock & Weber, in press; Weber, 2001). • Students’ understanding of what constitutes a proof, and the relationship between counterexamples and proof ( cf . Recio & Godino, 2001; Iannone & Nardi, 2004). Figure 1: Screen shot from the DVD. DATA: DVD CONTENT The data used for the DVD was gathered at a large state university in the USA. The participants were all enrolled in a transition-to-proof course called “Introduction to mathematical reasoning,” in which they studied methods of proof in the context of abstract topics such as sets, functions and number theory. Each participant was interviewed individually, and during the interview was asked to attempt three tasks. 2-2 PME31―2007 Alcock Each task was given on a piece of paper; the student was told they could write on the paper and was asked to describe their thinking out loud. The interviewer remained silent except to remind the participant of this request and to clarify the nature of the task. In this paper we see Nick working on the third of the three tasks: Let A ⊆ R be a nonempty set. U is an upper bound for A if andonly if ∀a ∈ A, a ≤ u . Task: Suppose that A, B ⊆ R are nonempty sets, u is anupper boundfor A and v is an upper boundfor B.Proveordisprove: 1. u + v isanupper boundfor A ∪ B . 2. uv isanupper boundfor A ∪ B . 3. u − v isanupper boundfor A ∩ B . The reader will note that all three statements are false, so that in each case a “disproof” is required and hence a counterexample is needed. This makes the task different from those that most students will encounter in such courses. This was a deliberate research design choice, since I wanted to avoid questions that could be answered using a standard procedure. The data was originally intended for research purposes only, so the content that could be used for creating the DVD was constrained by the limited number of students who agreed to allow their interviews to be used for this purpose.