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We illustrate an analytic framework for teachers' mathematics discourse in instruction (MDI). MDI is built on three interacting components of a mathematics lesson: a sequence of examples and related tasks; accompanying talk; patterns of interaction. Together these illuminate what is made available to learn. MDI is grounded empirically in mathematics teaching practices in South Africa, and theoretically in socio-cultural theoretical resources. The framework is responsive to the goals of a particular research and professional development project with potential for wider use. [For the complete proceedings, see ED597799.]

This document contains the second volume of the proceedings of the 30th Annual Conference of the International Group for the Psychology of Mathematics Education. Conference presentations are centered around the theme "Mathematics at the Centre." This volume features 60 research reports by presenters with last names beginning between Abr and Dri: (1) The Odds of Understanding the Law of Large Numbers: A Design for Grounding Intuitive Probability in Combinatorial Analysis (Dor Abrahamson and Rose M. Cendak); (2) Imaginary-Symbolic Relations, Pedagogic Resources and the Constitution of Mathematics for Teaching in In-Service Mathematics Teacher Education (Jill Adler and Zain Davis); (3) Relationship between Pre-Service Mathematics Teachers' Teaching and Learning Beliefs and Their Practices (Hatice Akkoc and Feral Ogan-Bekiroglu); (4) Teachers' Awareness of Dimensions of Variation: A Mathematics Intervention Project (Thabit Al-Murani); (5) The Student Teacher and the Others: Multimembership on the Process of Introducing Technology in the Classroom (Nelia Amado and Susana Carreira); (6) Improving Student Teachers' Understanding of Fractions (Solange Amorim Amato); (7) Autodidactic Learning of Probabilistic Concepts through Games (Miriam Amit and Irma Jan); (8) Graduate Students' Processes in Generating Examples of Mathematical Objects (Samuele Antonini); (9) Reasoning in an Absurd World: Difficulties with Proof by Contradiction (Samuele Antonini and Maria Alessandra Mariotti); (10) Will Penelope Choose Another Bridegroom? Looking for an Answer through Signs (Ferdinando Arzarello, Luciana Bazzini, Francesca Ferrara, Ornella Robutti, Cristina Sabena, and Bruna Villa); (11) Motivation and Perceptions of Classroom Culture in Mathematics of Students across Grades 5 to 7 (Chryso Athanasiou and George N. Philippou); (12) Deductive Reasoning: Different Conceptions and Approaches (Michal Ayalon and Ruhama Even); (13) The Tendency to Use Intuitive Rules among Students with Different Piagetian Cognitive Levels (Reuven Babai); (14) Coming to Appreciate the Pedagogical Uses of CAS (Lynda Ball and Kaye Stacey); (15) Students' Conceptions of "m" and "c": How to Tune a Linear Function (Caroline Bardini and Kaye Stacey); (16) A Contradiction between Pedagogical Content Knowledge and Teaching Indications (Ibrahim Bayazit and Eddie Gray); (17) Identifying and Supporting Mathematical Conjectures through the Use of Dynamic Software (David Benitez Mojica and Manuel Santos Trigo); (18) Students Constructing Representations for Outcomes of Experiments (Palma Benko and Carolyn A. Maher); (19) Logarithms: Snapshots from Two Tasks (Tanya Berezovski and Rina Zazkis); (20) Trying to Reach the Limit--The Role of Algebra in Mathematical Reasoning (Christer Bergsten); (21) Semiotic Sequence Analysis--Constructing Epistemic Types Empirically (Angelika Bikner-Ahsbahs); (22) Service Teaching: Mathematical Education of Students of Client Departments (Erhan Bingolbali, John Monaghan, and Tom Roper); (23) Students' Thinking about the Tangent Line (Irene Biza, Constantinos Christou, and Theodossios Zachariades); (24) Habermas' Theory of Rationality as a Comprehensive Frame for Conjecturing and Proving in School (Paulo Boero); (25) Extending Students' Understanding of Decimal Numbers via Realistic Mathematical Modeling and Problem Posing (Cinzia Bonotto); (26) Different Media, Different Types of Collective Work in Online Continuing Teacher Education: Would You Pass the Pen, Please? (Marcelo C. Borba and Rubia B. A. Zulatto); (27) Reformulating "Mathematical Modelling" in the Framework of the Anthropological Theory of Didactics (Marianna Bosch, Fco. Javier Garcia, Josep Gascon, and Luisa Ruiz Higueras); (28) Students' Impressions of the Value of Games for the Learning of Mathematics (Leicha A. Bragg); (29) The Transition from Arithmetic to Algebra: To Reason, Explain, Argue, Generalize and Justify (Trygve Breiteig and Barbro Grevholm); (30) Resisting Reform Pedagogy: Teacher and Learner Contributions (Karin Brodie); (31) Manifestations of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE) (Jill P. Brown); (32) Types of Representations of the Number Line in Textbooks (Alicia Bruno and Noemi Cabrera); (33) Educational Neuroscience: New Horizons for Research in Mathematics Education (Stephen R. Campbell); (34) Variability in a Probability Context: Developing Pre-Service Teachers' Understanding (Daniel L. Canada); (35) Implementing a Reform-Oriented Mathematics Syllabus: A Survey of Secondary Teachers (Michael Cavanagh); (36) Student's Modelling with a Lattice of Conceptions in the Domain of Linear Equations and Inequations (Hamid Chaachoua, Marilena Bittar, and Jean-Francois Nicaud); (37) Using Reading and Coloring to Enhance Incomplete Prover's Performance in Geometry Proof (Ying-Hao Cheng and Fou-Lai Lin); (38) Aspects of Teachers' Pedagogical Content Knowledge for Decimals (Helen Chick, Monica Baker, Thuy Pham, and Hui Cheng); (39) Collaborative Action Research on Implementing Inquiry-Based Instruction in an Eighth Grade Mathematics Class: An Alternative Mode for Mathematics Teacher Professional Development (Erh-Tsung Chin, Yung-Chi Lin, Yann-Tyng Ko, Chi-Tung Chien, and Hsiao-Lin Tuan); (40) Routine and Novel Mathematical Solutions: Central-Cognitive or Peripheral-Affective Participation in Mathematics Learning (Mei-Shiu Chiu); (41) The Role of Self-Generated Problem Posing in Mathematics Exploration (Victor V. Cifarelli and Jinfa Cai); (42) A Longitudinal Study of Children's Mental Computation Strategies (Barbara Clarke, Doug M. Clarke, and Marj Horne); (43) Assessing Fraction Understanding Using Task-Based Interviews (Doug M. Clarke, Michal Sukenik, Anne Roche, and Annie Mitchell); (44) Evaluation of a Teaching Concept for the Development of Problem Solving Competences in Connection with Self-Regulation (Christina Collet and Regina Bruder); (45) Developing Probability Thinking in Primary School: A Case Study on the Constructive Role of Natural Language in Classroom Discussions (Valeria Consogno, Teresa Gazzolo, and Paulo Boero); (46) Collaboration with Teachers to Improve Mathematics Learning: Pedagogy at Three Levels (Tom J. Cooper, Annette R. Baturo, and Edlyn J. Grant); (47) "Aim High--Beat Yourself": Effective Mathematics Teaching in a Remote Indigenous Community (Tom J. Cooper, Annette R. Baturo, Elizabeth Warren, and Edlyn J. Grant); (48) Development of Children's Understanding of Length, Area, and Volume Measurement Principles (Margaret Curry, Michael Mitchelmore, and Lynne Outhred; (49) Mathematics-for-Teaching: The Cases of Multiplication and Division (Brent Davis, Elaine Simmt, and Dennis Sumara); (50) Generative Concept Images (Gary E. Davis and Catherine A. Pearn); (51) Developmental Assessment of Data Handling Performance Age 7-14 (Pauline Davis, Maria Pampaka, Julian Williams, and Lawrence Wo); (52) The Effect of Different Teaching Tools in Overcoming the Impact of the Intuitive Rules (Eleni Deliyianni, Eleni Michael, and Demetra Pitta-Pantazi); (53) Investigating Social and Individual Aspects in Teacher's Approaches to Problem Solving (Fien Depaepe, Erik De Corte, and Lieven Verschaffel); (54) Maths Avoidance and the Choice of University (Pietro Di Martino and Francesca Morselli); (55) Primary Students' Reasoning about Diagrams: The Building Blocks of Matrix Knowledge (Carmel M. Diezmann); (56) Integrating Errors into Developmental Assessment: "Time" for Ages 8-13 (Brian Doig, Julian Williams, Lawrence Wo, and Maria Pampaka); (57) Vygotsky's Everyday Concepts/Scientific Concepts Dialectics in School Context: A Case Study (Nadia Douek); (58) Creating Mathematical Models with Structures (Katherine Doyle); (59) Mechanisms for Consolidating Knowledge Constructs (Tommy Dreyfus, Nurit Hadas, Rina Hershkowitz, and Baruch Schwarz); and (60) Reconciling Factorizations Made with CAS and with Paper-and-Pencil: The Power of Confronting Two Media (Paul Drijvers, Carolyn Kieran, Andre Boileau, Fernando Hitt, Denis Tanguay, Luis Saldanha, and Jose Guzman). (Individual papers contain references.)

The first volume of the 29th annual conference of the International Group for the Psychology of Mathematics Education contains plenary lecture and research forum papers as listed below. Short oral communications papers, poster presentations, brief summaries of discussion groups, and working sessions are also included in the volume. The plenary lectures include: (1) Modelling Students' Learning in Argumentation and Mathematics Proof (Fou-Lai Lin); (2) Travelling the Road to Expertise (Stacey); (3) Telling Identities (Sfar and Prusak); and (4) Co-Constructing Artefacts and Knowledge in Net-Based Teams (Reimann). The plenary panel section includes: (1) What Do Studies Like PISA Mean to the Mathematics Education Community? (Jones); (2) From a Profile to the Scrutiny of Student Performance (Yoshinori); (3) The PISA-Study (Neubrand); (4) Some Results from the PISA 2003 International Assessment of Mathematics Learning (Kieran); and (5) The Foundation and Spectacle of [the Leaning Tower of] PISA (Williams). The first research forum (RF01) includes: (1) Not Everything Is Proportional (De Brock, Van Dooren, and Verschaffel); (2) Designing Instruction on Proportional Reasoning with Average Speed (Gravemeijer, van Galen, and Keijzer); (3) Folding Perimeters (Friedlander and Arcavi); and (4) The Dolls' House Classroom (Ainley and Pratt). The second research forum (RF02) includes: (1) Shaping a Multi-Dimensional Analysis of Signs (Arzarello, Ferrara, Robutti, Paola, and Sabena); (2) Working with Artefacts (Bussi and Maschietto); (3) The Role of Gestures in Mathematical Discourse (Edwards); (4) Connecting Talk, Gesture, and Eye Motion for the Microanalysis of Mathematics Learning (Ferrara and Nemirovsky); (5) Why Do Gestures Matter? Gestures as Semiotic Means of Objectification (Radford); (6) Gestures, Signs and Mathematisation (Williams); and (7) Building Intellectual Infrastructure to Expose and Understand Ever-Increasing Complexity (Kaput). The third research forum (RF03) includes: (1) Using Growth Points to Describe Pathways for Young Children's Number Learning (Gervasoni); (2) Number Attainment in Sri Lankan Primary Schools (Hart); and (3) Mathematics Recovery (Pearn). The fourth research forum (RF04) includes: (1) The Place of Theory in Mathematics Education Research (Lester, Jr.); (2) Theories of Mathematics Education (Lerman); (3) The Articulation of Symbol and Mediation in Mathematics Education (Armella); (4) Using Theory to Advance Our Understandings of Student Cognitive Development (Pegg and Tall); (5) Trends in the Evolution of Models and Modeling Perspectives on Mathematical Learning and Problem Solving (Lesh and English); and (6) Issues and Tendencies in German Mathematics-Didactics (Torner and Sriraman). (Individual papers contain references.)

This volume of the 27th International Group for the Psychology of Mathematics Education Conference presents papers from: plenary panels; research forums; working sessions; discussion groups; short oral communications; and poster sessions from the meeting. Plenary lectures included: (1) Studying and Capturing the Complexity of Practice: The Case of the Dance of Agency (Jo Boaler); (2) Voyaging from Theory to Practice in Teaching and Learning: A View from Hawaii (Barbara Dougherty and Joseph Zilliox); (3) The Dialectics between Theory and Practice: Theoretical Issues and Practice Aspects from an Early Algebra Project (Nicolina Malara); and (4) Probing Students' Understanding of Variables through Cognitive Conflict: Is the Concept of a Variable So Difficult for Students to Understand (Toshiakaira Fujii). Plenary Panel, Teachers Who Navigate between Their Research and Their Practice, included: (1) The Teacher/Researcher Roles (Jarmila Novotna); (2) A Moment in Time, Twisted between History and Practice (Agatha Lebethe); (3) Navigating between Research and Practice: Finding My Own Way (Vicki Zack); and (4) With Less Do More (Gershon Rosen). The first research forum, "Perceptuo-Motor Activity and Imagination in Mathematics Learning," included: (1) Three Conjectures Concerning the Relationship between Body Activity and Understanding Mathematics (Ricardo Nemiroskvy); (2) Approaching Algebra through a Motion Experiences (Ornella Robutti and Fernando Arzarello); (3) Incorporating Experiences of Motion into a Calculus Classroom (Marty Schnepp and Daniel Chazan); (4) Sensors, Body, Technology and Multiple Representations (Marcelo Borba and Nilce Scheffer); and (5) Becoming Friends with Acceleration: The Role of Tools and Bodily Activity in Mathematical Learning (Chris Rasmussen and Ricardo Nemiroskvy). The second research forum, "Equity, Mathematics Learning and Technology," included: (1) Equity, Mathematics Learning and Computers: Who Gets a Fair Deal in Australian SecondarySchools? (Helen J. Forgasz); (2) Mathematical Literacy in Higher Education: Attracting Women to Engineering Professions by Using ICT (Christine Keitel); (3) Availability and (Non-) Use of Technology in and for Mathematics Education in Poor Schools in South Africa (Mamokgethi Setati); (4) Equity, Mathematics Learning and Technology--Introduction (Gilah C. Leder); (5) Feminist Frameworks for Researching Mathematics (Gabriele Kaiser); (6) Windows on Practice: Investigating Equity in Technology Based Mathematics Classrooms (Colleen Vale); and (7) Some Issues Facing Work for Equity in Mathematics Education (Walter G. Secada). Research Report, Facilitating the Teaching of Space Mathematics: An Evaluation (Kay Diane Owens) concludes this volume of the 27th proceedings. Information relating to: working sessions; discussion groups; short oral communications; and poster sessions is also included. (Individual papers contain references.) [For 2002 Proceedings Volumes 1-4, see ED476065.]

The first volume of this proceedings contains an introduction and the plenary, research forum, working group, and discussion group papers. Papers include: (1) "Resources as a Verb: Recontextualizing Resources in and for School Mathematics" (Jill Adler); (2) "Markets and Standards: The Politics of Education in a Conservative Age" (Michael Apple); (3) "Analyzing the Mathematical Learning of the Classroom Community: The Case of Statistical Data Analysis" (Paul Cobb); (4) "The Production of Artifacts as Goal for School Mathematics?" (Cyril Julie); (5) "A Moment in the Zoom of a Lens: Towards a Discursive Psychology of Mathematics Teaching and Learning" (Stephen Lerman); (6) "Diversity and Change in Mathematics Teacher Education" (Chris Breen); (7) "Pilot In-Service Mathematics Teacher Project" (Barbara Jaworski); (8) "Talking Math: Proposal for School Change" (Lena Licon Khisty); (9) "The CAPE Project--Constructing Activities for Peer-Education" (Konrad Krainer); (10) "Reflecting on Mathematics Learning In and Out-of-School from a Cultural Psychology Perspective" (Guida de Abreu); (11) "Reflections on Mathematics Learning and Teaching: Implications of Cultural Perspectives" (Philip C. Clarkson); (12) "A Semiotic Analysis of Students' Own Cultural Mathematics" (Norma Presmeg); (13) "Linking Students' Own Cultural Mathematics and Academic Mathematics" (Marta Civil); (14) "Developing Chains of Signifiers: A New Tool for Teachers?" (Kathryn C. Irwin); (15) "Learning through Problem Solving" (Hanlie Murray, Alwyn Olivier, and Piet Human); (16) "Learning through Problem Solving" (Anne Reynolds); (17) "Supporting Students' Construction of Increasingly Sophisticated Ways of Reasoning through Problem Solving" (Koeno Gravemejier, Kay McClain, and Michelle Stephan); (18) "Supporting Students' Reasoning through Problem Solving--Implications for Classroom Practice" (Susie Groves); (19) "Building the Meaning of Statistical Association through Data Analysis Activities" (Carmen Batanero, Juan D. Godino, and Antonio Estepa); (20) "Building the Meaning of Statistical Association through Data Analysis Activities" (Michael Glencross); (21) "Graphing as a Computer-Mediated Tool" (Janet Ainley, Elena Nardi, and Dave Pratt); (22) "Symbol-Use, Fusion, and Logical Necessity: On the Significance of Children's Graphing" (Ricardo Nemirovsky); (23) "Advanced Mathematical Thinking" (David Reid); (24) "Algebra: Epistemology, Cognition and New Technologies" (Jean-Philippe Drouhard, Alan Bell, and Sonia Ursini); (25) "Classroom Research" (Donald Cudmore and Lyn English); (26) "Cultural Aspects in the Learning of Mathematics" (Nomma Presmeg, Marta Civil, Judit Moschkovich, and Phil Clarkson); (27) "Geometry" (Maria Alessandra Mariotti and Ana Mesquita); (28) "Research on the Psychology of Mathematics Teacher Development" (Andrea Peter-Koop and Vania Santos-Wagner); (29) "Research in the Social Aspects of Mathematics Education" (Jo Boaler and Paola Valero); (30) "The Teaching and Learning of Stochastics" (John Truran, Kath Truran, and Carmen Batanero); (31) "Understanding of Multiplicative Concepts" (Tom Cooper and Tad Watanabe); (32) "Embodiment, Enactivism and Mathematics Education" (Rafael Nunes, Lawrie Edwards, Joao Filipe Matos, and Stephen Campbell); (33) "Learning and Teaching Number Theory" (Stephen Campbell and Rina Zazkis); (34) "Under-Represented Countries in PME: National Mathematics Education Research Communities and Priorities" (Paola Valero, Bernadette Denys, Renuka Vithal, and Pedro Gomez); (35) "Frameworks for Research on Open-ended Mathematical Tasks" (Peter Sullivan and Shukkan Susan Leung); and (36) "Exploring Different Ways of Working with Videotape in Research and Inservice Work" (Susan Pirie and Chris Breen). (ASK)

The theme of the 38th meeting of the International Group for the Psychology of Mathematics Education (PME 38) and the 36th meeting of the North American Chapter of the Psychology of Mathematics Education (PME-NA 36) was "Mathematics Education at the Edge." Academically, the theme provides opportunities to highlight and examine mathematics education research that is: (1) breaking new ground or on the cutting edge of innovative research and research methodologies; and (2) exploring issues with groups that are often positioned at the edge or periphery of educational research, such as social justice, peace education, equity, and Indigenous education. Geographically, the theme "Mathematics Education at the Edge" describes the very place of the conference setting, Vancouver, a city situated at the edge of Canada on the Pacific Ocean and Coast Mountain Range. The papers in the six volumes of these proceedings are organized according to the type of presentation. Volume 1 contains the presentations of the plenary speakers, Research Forum activities, Discussion Group activities, Working Session activities and the National Presentation of mathematics education in Canada. Volumes 2-5 contain the Research Reports of the conference, while Volume 6 consists of the Short Oral and Poster Presentations. The organization of PME 2014 is a collaborative effort involving teams of colleagues at the University of British Columbia and Simon Fraser University.

This volume of the 27th International Group for the Psychology of Mathematics Education Conference includes the following research reports: (1) Improving Decimal Number Conception by Transfer from Fractions to Decimals (Irita Peled and Juhaina Awawdy Shahbari); (2) The Development of Student Teachers' Efficacy Beliefs in Mathematics during Practicum (George Philippou, Charalambos Charalambous and Leonidas Kyriakides); (3) Recognising Equivalent Algebraic Expressions: An Important Component of Algebraic Expectation for Working with Cas (Robyn Pierce, Lynda Ball and Kaye Stacey); (4) Heuristics of Twelfth-Graders Building Isomorphisms (Arthur B. Powell and Carolyn A. Maher); (5) Interactive Whiteboards and the Construction of Definitions for the Kite (Dave Pratt and Ian Davison); (6) Development of Personal Constructs about Mathematical Tasks--A Qualitative Study using Repertory Grid Methodology (Susanne Prediger and Katja Lengnink); (7) Lesson Study Characterized as a Multi-Tiered Teaching Experiment (Norma Presmeg and Jeff Barrett); (8) Calculators, Graphs, Gestures and the Production of Meaning (Luis Radford, Serge Demers, Jose Guzman and Michele Cerulli); (9) The Effects of Numerical and Figural Cues on the Induction Processes of Preservice Elementary Teachers (Ferdinand D. Rivera and Joanne Rossi Becker); (10) Five Key Considerations for Networking in a Handheld-Based Mathematics (Jeremy Roschelle, Phillip Vahey, Deborah Tatar, Stephen Hegedus and Jim Kaput); (11) An Interpreting Game in a Third Grade Classroom (Adalira Saenz-Ludlow); (12) Thinking in Images and its Role in Learning Mathematics (Ildar S. Safuanov and Valery A. Gusev); (13) Primary Teachers' Conceptions about the Concept of Volume: The Case of Volume--Measurable Objects (Mariana Saiz); (14) Assessment Practices in School Mathematics: Acting and Debating (Haralambos Sakonidis and Anna Klothou); (15) Influential Aspects of Dynamic Geometry Activities in the Construction of Proofs (Ernesto Sanchez and Ana Isabel Sacristan); (16) Students' Use of Technology in Mathematical Problem Solving: Transforming Technological Artifacts into Mathematical Tools (Manuel Santos, Evelyn Aguero, Alexander Borbon and Cristhian Paez); (17) Algebra in Elementary School (Analucia Schliemann, David Carraher, Barbara Brizuela, Darrell Earnest, Anne Goodrow, Susanna Lara-Roth and Irit Peled); (18) The Impact of Preparing for the Test on Classroom Practice (Roberta Y. Schorr and Sylvia Bulgar); (19) An Analysis of Mental Space Construction in Teaching Linear Equation Word Problems (Yasuhiro Sekiguchi); (20) Language Use in a Multilingual Mathematics Classroom In South Africa: A Different Perspective (Mamokgethi Setati); (21) Middle School Students' Thinking about Variability in Repeated Trials: A Cross-Task Comparison (J. Michael Shaughnessy, Dan Canada and Matt Ciancetta); (22) Students' Understanding Of Z(Subscript)n (Daniel Siebert and Steven R. Williams); (23) Teachers' Mathematics: Curious Obligations (Elaine Simmt, Brent Davis, Lynn Gordon and Jo Towers); (24) Logico-Mathematical Activity versus Empirical Activity: Examining a Pedagogical Distinction (Martin A. Simon); (25) The Provision of Accurate Images with Dynamic Geometry (Margaret P. Sinclair,); (26) Aesthetic Values in Mathematics: A Value-Oriented Epistemology (Nathalie Sinclair); (27) From Cognitive Science to School Practice: Building the Bridge (Florence Mihaela Singer); (28) Connecting Theory and Reflective Practice through the Use of Personal Theories (Tracey Smith); (29) Two Meanings of the "Equal" Sign and Senses of Comparison and Substitution Methods (Armando Solares, Eugenio Filloy and Teresa Rojano); (30) Goal Sketches in Fraction Learning (Catherine Sophian and Samara Madrid); (31) Using an Empowerment Professional Development Model to Support Beginning Primary Mathematics Teachers (Len Sparrow and Sandra Frid); (32) Getting at the Mathematics: Sara's Journal (Bob Speiser and Chuck Walter); (33) Emergence ofMath Knowledge Structures. Introspection (Nad'a Stehlikova); (34) Grade-Related Trends in the Prevalence and Persistence of Decimal Misconceptions (Vicki Steinle and Kaye Stacey); (35) Being Explicit about Aspects of Mathematics Pedagogy (Peter Sullivan, Robyn Turner Harrison, Judy Mousley and Robyn Zevenbergen); (36) Self-Efficacy in Mathematics and Students' Use of Self-Regulated Learning Strategies during Assessment Events (Howard Tanner and Sonia Jones); (37) The Assessment of Mathematical Logic: Abstract Patterns and Familiar Contexts (Anne R. Teppo, Warren W. Esty and Kay Kirkpatrick); (38) The Role of Representation in Teacher Understanding of Function (Mike Thomas); (39) Effective Teaching with Virtual Material: Years Six and Seven Case Studies (Kylie Thompson, Annette Baturo and Tom Cooper); (40) Obstacles for Mental Representations of Real Numbers: Observations from a Case Study (Gunter Torner); (41) Teacher and Students' Joint Production of a Reversible Fraction Conception (Ron Tzur); (42) On the Search for Gender-Related Differences in Dutch Primary Mathematics Classrooms (Marja Van Den Heuvel-Panhuizen); (43) Remedying Secondary School Students' Illusion of Linearity: A Teaching Experiment (Wim Van Dooren, Dirk De Bock, An Hessels, Dirk Janssens and Lieven Verschaffel); (44) The Value of Wenger's Concepts of Modes of Participation and Regimes of Accountability in Understanding Teacher Learning (Laura R. Van Zoest and Jeffrey V. Bohl); (45) Computers in the Primary Classroom: Barriers to Effective Use (Carolyn Vela, presented by Michael O.J. Thomas); (46) Students and Teachers Listening to Themselves: Language Awareness in the Mathematics Classroom (David Wagner); (47) The Context Sensitivity of Mathematical Generalizations (Joseph F. Wagner); (48) How Does Flexible Mathematical Thinking Contribute to the Growth of Understanding? (Lisa Warner, Lara J. Alcock, Joseph Coppolo, Jr. and Gary E. Davis); (49) Young Children's Understanding of Equals: A LongalStudy (Elizabeth Warren); (50) Statistical Variation in a Chance Setting (Jane M. Watson, and Ben A. Kelly); (51) A Procedural Route toward Understanding the Concept of Proof (Keith Weber); (52) Teaching Angles by Abstraction from Physical Activities with Concrete Materials (Paul White and Michael Mitchelmore); (53) Undergraduate Students' Mental Operations in Systems of Differential Equations (Karen Whitehead and Chris Rasmussen); (54) Empirical Generalisation as an Inadequate Cognitive Scaffold to Theoretical Generalisation of a More Complex Concept (Gaye Williams); (55) Comparing Competence in Transformational and Generational Algebraic Activities (Kirsty Wilson, Janet Ainley and Liz Bills); (56) Complexity in Teaching and Children's Mathematical Thinking (Terry Wood and Betsy Mcneal); (57) Students' Understanding of Proof by Contradiction (Jya-Yi Wu Yu, Fou-Lai Lin and Yuan-Shun Lee); (58) What Does "Positive" Attitude Really Mean? (Rosetta Zan and Pietro Di Martino); and (59) Translation of a Function: Coping with Perceived Inconsistency (Rina Zazkis, Peter Liljedahl and Karen Gadowsky). An author index is included. (Individual papers contain references.) [For Volume 3 of the 2003 Proceedings, see ED500858.]

The second volume of this proceedings contains the first portion of the research reports. Papers include: (1) "Learning Algebraic Strategies Using a Computerized Balance Model" (James Aczel); (2) "Children's Perception of Multiplicative Structure in Diagrams" (Bjornar Alseth); (3) "A Discussion of Different Approaches to Arithmetic Teaching" (Julia Anghileri); (4) "A Model for Analyzing the Transition to Formal Proofs in Geometry" (Ferdinando Arzarello, Chiara Micheletti, Federica Olivero, Ornella Robutti, and Domingo Paola); (5) "Dragging in Cabri and Modalities of Transition from Conjectures to Proofs in Geometry" (Ferdinando Arzarello, Chiara Micheletti, Federica Olivero, Ornella Robutti, Domingo Paola, and Gemma Gallino); (6) "The Co-Construction of Mathematical Knowledge: The Effect of an Intervention Program on Primary Pupils' Attainment" (Mike Askew, Tamara Bibby, and Margaret Brown); (7) "Dialectical Proof: Should We Teach It to Physics Students?" (Roberto Ribeiro Baldino); (8) "Lacan and the School's Credit System" (Roberto Ribeiro Baldino and Tania Cristina Baptista Cabral); (9) "Which Is the Shape of an Ellipse? A Cognitive Analysis of an Historical Debate" (Maria G. Bartolini Bussi and Maria Alessandra Mariotti); (10) "Children's Understanding of the Decimal Numbers through the Use of the Ruler" (Milena Basso, Cinzia Bonotto, and Paolo Sozio); (11) "Construction of Multiplicative Abstract Schema for Decimal-Number Numeration" (Annette R. Baturo and Tom J. Cooper); (12) "Classroom-Based Research to Evaluate a Model Staff Development Project in Mathematics" (Joanne Rossi Becker and Barbara J. Pence); (13) "Some Misconceptions Underlying First-Year Students' Understanding of 'Average Rate' and of 'Average Value'" (Jan Bezuidenhout, Piet Human, and Alwyn Olivier); (14) "Operable Definitions in Advanced Mathematics: The Case of the Least Upper Bound" (Liz Bills and David Tall); (15) "Beyond 'Street' Mathematics: The Challenge of Situated Cognition" (Jo Boaler); (16) "The 'Voices and Echoes Game' and the Interiorization of Crucial Aspects of Theoretical Knowledge in a Vygotskian Perspective: Ongoing Research" (Paolo Boero, Bettina Pedemonte, Elisabetta Robotti, and Giampaolo Chiappini); (17) "Children's Construction of Initial Fraction Concepts" (George Booker); (18) "Graphing Calculators and Reorganization of Thinking: The Transition from Functions to Derivative" (Marcelo C. Borba and Monica E. Villarreal); (19) "Pre-Algebra: A Cognitive Perspective" (G.M. Boulton-Lewis, T. Cooper, B. Atweh, H. Pillay, and L. Wilss); (20) "The Right Baggage?" (Mary Briggs); (21) "Learner-Centered Teaching and Possibilities for Learning in South African Mathematics Classrooms" (Karin Brodie); (22) "Researching Transition in Mathematical Learning" (Tony Brown, Frank Eade, and Dave Wilson); (23) "Metaphor as Tool in Facilitating Preservice Teacher Development in Mathematical Problem Solving" (Olive Chapman); (24) "Restructuring Conceptual and Procedural Knowledge for Problem Representation" (Mohan Chinnappan); (25) "The Structure of Students' Beliefs towards the Teaching of Mathematics: Proposing and Testing a Structural Model" (Constantinos Christou and George N. Philippou); (26) "Abstract Schema versus Computational Proficiency in Percent Problem Solving" (Tom J. Cooper, Annette R. Baturo, and Shelley Dole); (27) "Implicit Cognitive Work in Putting Word Problems into Equation Form" (Anibal Cortes); (28) "Three Sides Equal Means It Is Not Isosceles" (Penelope Currie and John Pegg); (29) "Making Sense of Sine and Cosine Functions through Alternative Approaches: Computer and 'Experimental World' Contexts" (Nielce Lobo Da Costa and Sandra Magina); (30) "Teacher and Students' Flexible Thinking in Mathematics: Some Relations" (Maria Manuela M.S. David and Maria da Penha Lopes); (31) "The Influence of Metacognitive and Visual Scaffolds on the Predominance of the Linear Model" (Dirk De Bock, Lieven Verschaffel, and Dirk Janssens); (32) "To Teach Definitions in Geometry or Teach To Define?" (Michael De Villiers); (33) "Student thinking about Models of Growth and Decay" (Helen M. Doerr); (34) "Analysis of a Long Term Construction of the Angle Concept in the Field of Experience of Sunshadows" (Nadia Douek); (35) "On Verbal Addition and Subtraction in Mozambican Bantu Languages" (Jan Draisma); (36) "Teachers' Beliefs and the 'Problem' of the Social" (Paula Ensor); (37) "From Number Patterns to Algebra: A Cognitive Reflection on a Cape Flats Experience" (Clyde B.A. Felix); (38) "Affective Dimensions and Tertiary Mathematics Students" (Helen J. Forgasz and Gilah C. Leder); (39) "Social Class Inequalities in Mathematics Achievement: A Multilevel Analysis of TlMSS South Africa Data" (George Frempong); (40) "Context Influence on Mathematical Reasoning" (Fulvia Furinghetti and Domingo Paola); (41) "What Do They Really Think? What Students Think about the Median and Bisector of an Angle in the Triangle, What They Say and What Their Teachers Know about It" (Hagar Gal); (42) "Levels of Generalization in Linear Patterns" (Juan Antonio Garcia-Cruz and Antonio Martinon); (43) "The Evolution of Pupils' Ideas of Construction and Proof Using Hand-Held Dynamic Geometry Technology" (John Gardiner and Brian Hudson); and (44) "Cognitive Unity of Theorems and Difficulty of Proof" (Rosella Garuti, Paolo Boero, and Enrica Lemut). (ASK)