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This third volume of the 31st annual proceedings of the International Group for the Psychology of Mathematics Education conference presents research reports for author surnames beginning Han- through Miy-. Reports include: (1) Elementary Education Students' Memories of Mathematics in Family Context (Markku S. Hannula, Raimo Kaasila, Erkki Pehkonen, and Anu Laine); (2) Mistake-Handling Activities in the Mathematics Classroom: Effects of an In-Service Teacher Training on Students' Performance in Geometry (Aiso Heinze and Kristina Reiss); (3) Gender Similarities instead of Gender Differences: Students' Competences in Reasoning and Proof (Aiso Heinze, Stefan Ufer, and Kristina Reiss); (4) Studying Lesson Structure from the Perspective of Students' Meaning Construction: The Case of Two Japanese Mathematics Classrooms (Keiko Hino); (5) A Framework for Creating or Analyzing Japanese Lessons from the Viewpoint of Mathematical Activities: A Fraction Lesson (Kenji Hiraoka and Kaori Yoshida-Miyauchi); (6) Revisiting Discourse as an Instructional Resource: Practices that Create Spaces for Learning and Student Contributions (Lynn Liao Hodge, Qing Zhao, Jana Visnovska, and Paul Cobb); (7) An Illustration of Students' Engagement with Mathematical Software Using Remote Observation (Anesa Hosein, James Aczel, Doug Clow, and John T. E. Richardson); (8) Geometric Calculations are More than Just the Application of Procedural Knowledge (Hui-Yu Hsu); (9) Constructing Pedagogical Representations to Teach Linear Relations in Chinese and U.S. Classrooms (Rongjin Huang and Jinfa Cai); (10) Teachers as Researchers: Putting Mathematics at the Core (Danielle Huillet); (11) Can You Convince Me: Learning to Use Mathematical Argumentation (Roberta Hunter); (12) On the Mathematical Knowledge under Construction in the Classroom: A Comparative Study (M. Kaldrimidou, H. Sakonidis, and M. Tzekaki); (13) Students' Beliefs and Attitudes about Studying and Learning Mathematics (Eleftherios Kapetanas and Theodosios Zachariades); (14) "How Can We Describe the Relation between the Factored Form and the Expanded Form of These Trinomials? We Don't even Know If Our Paper-and-Pencil Factorizations are Right": The Case for Computer Algebra Systems (CAS) with Weaker Algebra Students (Carolyn Kieran and Caroline Damboise); (15) What Is a Beautiful Problem? An Undergraduate Students' Perspective (Boris Koichu, Efim Katz, and Abraham Berman); (16) Can Lessons Be Replicated? (Angelika Kullberg); (17) Problem Posing as a Means for Developing Mathematical Knowledge of Prospective Teachers (Ilana Lavy and Atara Shriki); (18) Activity-Based Class: Dilemma and Compromise (KyungHwa Lee); (19) Induction, Analogy, and Imagery in Geometric Reasoning (KyungHwa Lee; MinJung Kim; GwiSoo Na, DaeHee Han, and SangHun Song); (20) The Analysis of Activity That Gifted Students Construct Definition of Regular Polyhedra (KyungHwa Lee, EunSung Ko, and SangHun Song); (21) Multiple Solution Tasks as a Magnifying Glass for Observation of Mathematical Creativity (Roza Leikin and Miri Lev); (22) Interactive Whiteboards as Mediating Tools for Teaching Mathematics: Rhetoric or Reality? (Steve Lerman and Robyn Zevenbergen); (23) From Construction to Proof: Explanations in Dynamic Geometry Environment (Allen Leung and Chi Ming Or); (24) Prospective Middle School Teachers' Knowledge in Mathematics and Pedagogy for Teaching--The Case of Fraction Division (Yeping Li and Dennie Smith); (25) Improving Students' Algebraic Thinking: The Case of Talia (Kien Lim); (26) The Effect of a Mentoring Development Program on Mentors' Conceptualizing Mathematics Teaching and Mentoring (Pi-Jen Lin); (27) Uses of Examples in Geometric Conjecturing (Miao-Ling Lin and Chao-Jung Wu); (28) Algebrification of Arithmetic: Developing Algebraic Structure Sense in the Context of Arithmetic (Drora Livneh and Liora Linchevski); (29) The Potential of Patterning Activities to Generalization (Hsiu-Lan Ma); (30) Infinite Magnitude vs. Infinite Representation: The Story of [Pi] (Ami Mamolo); (31) The Ability of Sixth Grade Students in Korea and Israel to Cope with Number Sense Tasks (Zvia Markovits and JeongSuk Pang); (32) Creating Your Own Symbols: Beginning Algebraic Thinking with Indigenous Students (Chris Matthews, Tom J. Cooper, and Annette R. Baturo); (33) Exploring Students' Mathematics-Related Self Image as Learners (Silvana Martins Melo and Marcia Maria Fusaro Pinto); (34) Difficulties on Understanding the Indefinite Integral (N. Metaxas); (35) Detecting the Emergence and Development of Mathematical Discourse: A Novel Approach (Christina Misailidou); and (36) The Nature and Role of Proof When Installing Theorems: The Perspective of Geometry Teachers (Takeshi Miyakawa and Patricio Herbst). (Individual papers contain references.) [For other volumes in the series, see ED499417, ED499418, and ED499419.]

This document contains the second volume of the proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education. Conference papers are centered around the theme of "Learners and Learning Environments." This volume features 43 research reports by presenters with last names beginning between Adl and Fre: (1) Working with Learners' Mathematics: Exploring a Key Element of Mathematical Knowledge for Teaching (Jill Adler, Zain Davis, Mercy Kazima, Diane Parker, and Lyn Webb); (2) A Comparison between Teachers' and Pupils' Tendency to Use a Representativeness Heuristic (Thekla Afantiti-Lamprianou, Julian S. Williams, and Iasonas Lamprianou); (3) Purposeful Task Design and the Emergence of Transparency (Janet G. Ainley, Liz Bills, and Kirsty Wilson); (4) A Developmental Model for Proportional Reasoning in Ratio Comparison Tasks (Silvia Alatorre and Olimpia Figueras); (5) Referential and Syntactic Approaches to Proof: Case Studies from a Transition Course (Lara Alcock and Keith Weber); (6) Teachers' Beliefs about Students' Development of the Pre-Algebraic Concept of Equation (Vassiliki Alexandrou-Leonidou and George N. Philippou); (7) Developing Students' Understanding of the Concept of Fractions as Numbers (Solange Amorim Amato); (8) Multiple Representations in 8th Grade Algebra Lessons: Are Learners Really Getting It? (Miriam Amit and Michael N. Fried); (9) Reform-Oriented Teaching Practices: A Survey of Primary School Teachers (Judy Anderson and Janette Bobis); (10) The Genesis of Signs by Gestures: The Case of Gustavo (Ferdinando Arzarello, Francesca Ferrara, Ornella Robutti, and Domingo Paola); (11) Students' Experience of Equivalence Relations: A Phenomenological Approach (Amir H. Asghari and David Tall); (12) How Series Problems Integrating Geometric and Arithmetic Schemes Influence Prospective Secondary Teachers' Pedagogical Understanding (Leslie Aspinwall, Kenneth L. Shaw, and Hasan Unal); (13) Dealing with Learning in Practice: Tools for Managing the Complexity of Teaching and Learning (Sikunder Ali Baber and Bettina Dahl); (14) Situations of Psychological Cognitive No-Growth (Roberto R. Baldino and Tania C. B. Cabral); (15) Good CAS Written Records: Insight from Teachers (Lynda Ball and Kaye Stacey); (16) Developing Procedure and Structure Sense of Arithmetic Expressions (Rakhi Banerjee and K. Subramaniam); (17) Struggling with Variables, Parameters, and Indeterminate Objects, or How to Go Insane in Mathematics (Caroline Bardini, Luis Radford, and Cristina Sabena); (18) Exploring How Power is Enacted in Small Groups (Mary Barnes); (19) A Framework for the Comparison of PME Research into Multilingual Mathematics Education in Different Sociolinguistic Settings (Richard Barwell); (20) Vygotsky's Theory of Concept Formation and Mathematics Education (Margot Berger); (21) Preservice Teachers' Understandings of Relational and Instrumental Understanding (Kim Beswick); (22) The Transformation of Mathematics in On-Line Courses (Marcelo C. Borba); (23) Using Cognitive and Situated Perspectives to Understand Teacher Interactions with Learner Errors (Karin Brodie); (24) Identification of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE) (Jill P. Brown); (25) The "A4-Project": Statistical World Views Expressed through Pictures (Michael Bulmer and Katrin Rolka); (26) A Whole-School Approach to Developing Mental Computation Strategies (Rosemary Callingham); (27) A Comparison of Perceived Parental Influence on Mathematics Learning among Students in China and Australia (Zhongjun Cao, Helen Forgasz, and Alan Bishop); (28) Using Word Problems in Malaysian Mathematics Education: Looking beneath the Surface (Kah Yein Chan and Judith Mousley); (29) Constructing Pedagogical Knowledge of Problem Solving: Preservice Mathematics Teachers (Olive Chapman); (30) Revisiting a Theoretical Model on Fractions: Implications for Teaching and Research (Charalambos Y. Charalambous and Demetra Pitta-Pantazi); (31) Students' Reflection on Their Sociomathematical Small-Group Interaction: A Case Study (Petros Chaviaris and Sonia Kafoussi); (32) Investigating Teachers' Responses to Student Misconceptions (Helen L. Chick and Monica K. Baker); (33) Studying the Distribution of Responsibility for the Generation of Knowledge in Mathematics Classrooms in Hong Kong, Melbourne, San Diego and Shanghai (David Clarke and Lay Hoon Seah); (34) Indigenous and Non-Indigenous Teaching Relationships in Three Mathematics Classrooms in Remote Queensland (Tom J. Cooper, Annette R. Baturo, and Elizabeth Warren); (35) Exploring the Strategies Used by Grade 1 to 3 Children through Visual Prompts, Symbols and Worded Problems: A Case for a Learning Pathway for Number (Ty Corvell Cranfield, Cally Kuhne, and Gary Powell); (36) Primary Students' Knowledge of the Properties of Spatially-Oriented Diagrams (Carmel Diezmann); (37) A Conceptual Framework for Studying Teacher Preparation: The Pirie-Kieren Model, Collective Understanding, and Metaphor (Maria A. Droujkova, Sarah B. Berenson, Kelli Slaten, and Sue Tombes); (38) Mathematical Modelling with 9-Year-Olds (Lyn English and James Watters); (39) Exploring "Lesson Study" in Teacher Preparation (Maria L. Fernandez); (40) Child-Initiated Mathematical Patterning in the Pre-Compulsory Years (Jillian Fox); (41) The Tacit-Explicit Nature of Students' Knowledge: A Case Study on Area Measurement (Cristina Frade); (42) Teachers as Interns in Informal Mathematics Research (John M. Francisco and Carolyn A. Maher); and (43) Exploring Excellence and Equity within Canadian Mathematics Classrooms (George Frempong). (Individual papers contain references.)

This document is the fourth volume of the proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education. Conference papers are centered around the theme of "Learners and Learning Environments." This volume features 42 research reports by presenters with last names beginning between Mul and Wu: (1) Case Studies of Children's Development of Structure in Early Mathematics: A Two-Year Longitudinal Study (Joanne Mulligan, Michael Mitchelmore, and Anne Prescott); (2) A Case Study of How Kinesthetic Experiences Can Participate in and Transfer to Work with Equations (Ricardo Nemirovsky and Chris Rasmussen); (3) The Construction of Proportional Reasoning (Stephen Norton); (4) The Transition of a Secondary Mathematics Teacher: From a Reform Listener into a Believer (Jo Clay Olson and Karmen Kirtley); (5) Substantive Communication of Space Mathematics in Upper Primary School (Kay Owens); (6) Transforming Korean Elementary Mathematics Classrooms to Student-Centered Instruction (JeongSuk Pang); (7) The Effect of Improved Automaticity and Retrieval of Basic Number Skills on Persistently Low-Achieving Students (John Pegg, Lorraine Graham, and Anne Bellert); (8) Degrees of Freedom in Modeling: Taking Certainty out of Proportion (Irit Peled and Ronit Bassan-Cincinatus); (9) "I Know That You Don't Have to Work Hard": Mathematics Learning in the First Year of Primary School (Bob Perry and Sue Dockett); (10) Disentangling Mentors' Role in the Development of Prospective Teachers' Efficacy Beliefs in Teaching Mathematics (George Philippou and Charalambos Y. Charalambous); (11) Linear Functions and a Triple Influence of Teaching on the Development of Students' Algebraic Expectation (Robyn Pierce); (12) Engaging the Learner's Voice? Catechetics and Oral Involvement in Reform Strategy Lessons (Adrian J. Pinel); (13) Teaching Projectile Motion to Eliminate Misconceptions (Anne Prescott and Michael Mitchelmore); (14) An Investigation of a Preservice Teacher's Use of Representations in Solving Algebraic Problems Involving Exponential Relationships (Norma Presmeg and Rajeev Nenduradu); (15) On Embodiment, Artifacts, and Signs: A Semiotic-Cultural Perspective on Mathematical Thinking (Luis Radford, Caroline Bardini, Cristina Sabena, Pounthioun Diallo, and Athanase Simbagoye); (16) Generalization Strategies of Beginning High School Algebra Students (Joanne Rossi Becker and Ferdinand Rivera); (17) Synchronizing Gestures, Words and Actions in Pattern Generalizations (Cristina Sabena, Luis Radford, and Caroline Bardini); (18) Analyzing Student Modeling Cycles in the Context of a "Real World" Problem (Roberta Y. Schorr and Miriam Amit); (19) Negotiating about Perceived Value Differences in Mathematics Teaching: The Case of Immigrant Teachers in Australia (Wee Tiong Seah); (20) Development of Mathematical Norms in an Eighth-Grade Japanese Classroom (Yasuhiro Sekiguchi); (21) Solving Additive Problems at Pre-Elementary School Level with the Support of Graphical Representation (Ana Coelho Vieira Selva, Jorge Tarcisio da Rocha Falcao, and Terezinha Nunes); (22) From the Everyday, through the Authentic, to Mathematics: Reflecting on the Process of Teaching Mathematics through the Everyday (Godfrey Sethole); (23) Personal Experiences and Beliefs in Early Probabilistic Reasoning: Implications for Research (Sashi Sharma); (24) Assimilating Innovative Learning/Teaching Approaches into Teacher Education: Why Is It so Difficult? (Atara Shriki and Ilana Lavy); (25) Student Thinking Strategies in Reconstructing Theorems (Tatag Yuli Eko Siswono); (26) A Comparison of How Textbooks Teach Multiplication of Fractions and Division of Fractions in Korea and in U.S. (Ji-Won Son); (27) Mathematical Knowledge of Pre-Service Primary Teachers (Beth Southwell and Marina Penglase); (28) Analysing Longitudinal Data on Students' Decimal Understanding Using Relative Risk and Odds Ratios (Vicki Steinle and Kaye Stacey); (29) Girls Journey toward Proportional Reasoning (Olof Bjorg Steinthorsdottir); (30) University Student Perceptions of CAS Use in Mathematics Learning (Sepideh Stewart and Michael O. J. Thomas); (31) Prospective Teachers' Understanding of Proof: What if the Truth Set of an Open Sentence Is Broader than that Covered by the Proof? (Andreas J. Stylianides, Gabriel J. Stylianides, and George Philippou); (32) Planning and Teaching Mathematics Lessons as a Dynamic, Interactive Process (Peter Sullivan, Robyn Zevenbergen, and Judy Mousley); (33) Teacher Factors in Integration of Graphic Calculators into Mathematics Learning (Michael O. J. Thomas and Ye Yoon Hong); (34) Students' Overreliance on Linearity: An Effect of School-Like Word Problems (Wim Van Dooren, Dirk De Bock, Dirk Janssens, and Lieven Verschaffel); (35) A Process of Abstraction by Representations of Concepts (N. C. Verhoef and H. G. B. Broekman); (36) Argumentation Profile Charts as Tools for Analysing Students' Argumentations (Jill Vincent, Helen Chick, and Barry McCrae); (37) Characterizing Middle School Students' Thinking in Estimation (Tanya N. Volkova); (38) Reviewing and Thinking the Affect/Cognition Relation (Margaret Walshaw and Tania Cabral); (39) Young Children's Ability to Generalise the Pattern Rule for Growing Patterns (Elizabeth Warren); (40) Consolidating One Novel Structure whilst Constructing Two More (Gaye Williams); (41) Spreadsheets, Pedagogic Strategies and the Evolution of Meaning for Variable (Kirsty Wilson, Janet Ainley, and Liz Bills); and (42) A Study of the Geometric Concepts of the Elementary School Students Who Are Assigned to the van Hiele Level One (Der-bang Wu and Hsiu-Lan Ma). (Individual papers contain references.)

This fourth and final volume of the 31st annual proceedings of the International Group for the Psychology of Mathematics Education conference presents research reports for author surnames beginning Na- through Zod-. Reports include: (1) Mathematically Gifted Students' Problem Solving Approaches on Conditional Probability (GwiSoo Na, DaeHee Han, KyungHwa Lee, and SangHun Song); (2) Students Engaged in Proving--Participants in an Inquiry Process or Executers of a Predetermined Script? (Talli Nachlieli and Patricio Herbst); (3) Exploring the Idea of Curriculum Materials Supporting Teacher Knowledge (Jihwa Noh and Ok-Ki Kang); (4) What Is the Price of Topaze? (Jarmila Novotna and Alena Hospesova); (5) Designing Unit for Teaching Proportion Based on Cultural-Historical Activity Theory: Process of Symbolizing through Collective Discourse (Minoru Ohtani); (6) Prototype Phenomena and Common Cognitive Paths in the Understanding of the Inclusion Relations between Quadrilaterals in Japan and Scotland (Masakazu Okazaki and Taro Fujita); (7) A Study of Gender Differences in Language Used by Parents and Children Working on Mathematical Tasks (Melfried Olson, Judith Olson, and Claire Okazaki); (8) Students' Motivation and Achievement and Teachers' Practices in the Classroom (Marilena Pantziara and George Philippou); (9) Using a Multiplicative Approach to Construct Decimal Structure (Irit Peled, Ruth Meron, and Shelly Rota); (10) If You Don't Listen to the Teacher, You Won't Know What to Do: Voices of Pasifika Learners (Pamela Perger); (11) Fostering Generalization in Connecting Registers of Dynamic Geometry and Euclidean Constructions (Norma Presmeg, Jeff Barrett, and Sharon McCrone); (12) Addressing the Issue of the Mathematical Knowledge of Secondary Mathematics Teachers (Jerome Proulx); (13) Abduction in Pattern Generalization (F. D. Rivera and Joanne Rossi Becker); (14) An Activity for Development of the Understanding of the Concept of Limit (Kyeong Hah Roh); (15) Early Access to Algebraic Ideas: The Role of Representations and of the Mathematics of Variation (Teresa Ceballos Rojano and Elvia Maximo Perrusquia); (16) The Role of Cognitive Conflict in Belief Changes (Katrin Rolka, Bettina Rosken, and Peter Liljedahl); (17) Factors Affecting Seventh Graders' Cognitive Perceptions of Patterns Involving Constructive and Deconstructive Generalizations (Joanne Rossi Becker and F. D. Rivera); (18) Mathematically Gifted Students' Spatial Visualization Ability of Solid Figures (HyunAh Ryu, YeongOk Chong, and SangHun Song); (19) Genetic Approach to Teaching Geometry (Ildar S. Safuanov); (20) On Primary Teachers' Assessment of Pupils' Written Work in Mathematics (H. Sakonidis and A. Klothou); (21) Qualities Co-Valued in Effective Mathematics Lessons in Australia: Preliminary Findings (Wee Tiong Seah); (22) Resolving Cognitive Conflict with Peers--Is There a Difference between Two and Four? (Hagit Sela and Orit Zaslavsky); (23) Explicit Linking in the Sequence of Consecutive Lessons in Mathematics Classrooms in Japan (Yoshinori Shimizu); (24) On the Teaching Situation of Conceptual Change: Epistemological Considerations of Irrational Numbers (Yusuke Shinno); (25) Posing Problems with Use the "What If Not?" Strategy in NIM Game (SangHun Song, JaeHoon Yim, EunJu Shin, and HyangHoon Lee); (26) Embodied, Symbolic and Formal Aspects of Basic Linear Algebra Concepts (Sepideh Stewart and Michael O. J. Thomas); (27) The Application of Dual Coding Theory in Multi-Representational Virtual Mathematics Environments (Jennifer M. Suh and Patricia S. Moyer-Packenham); (28) Interactions between Teaching Norms of Teacher's Professional Community and Learning Norms of Classroom Communities (Wen-Huan Tsai); (29) Analysis of a Learning Case: Jasmine (Zelha Tunc-Pekkan); (30) Elementary Teachers' Linguistic Inventions and Semantic Warrants for Mathematical Inferences (Janet G. Walter and Christine Johnson); (31) How Do Mentors Decide: Intervening in Practice Teachers' Teaching of Mathematics or Not (Chih-Yeuan Wang and Chien Chin); (32) Exploring an Understanding of Equals as Quantitative Sameness with 5 Year Old Students (Elizabeth Warren); (33) Classroom Teaching Experiment: Eliciting Creative Mathematical Thinking (Gaye Williams); and (34) Is a Visual Example in Geometry Always Helpful? (Iris Zodik and Orit Zaslavsky). (Individual papers contain references.) [For other volumes in the series, see ED499416, ED499417, and ED499419.]

The first volume of the 31st annual proceedings of the International Group for the Psychology of Mathematics Education conference presents plenary lectures; research forums; discussion groups; working sessions; short oral communications; and posters from the meeting. Plenary lecture papers include: (1) On Humanistic Mathematics Education: A Personal Coming of Age? (Chris Breen); (2) Certainty, Explanation and Creativity in Mathematics (Michael Otte); (3) I Need the Teacher to Tell Me If I Am Right or Wrong (Anna Sierpinska); and (4) School Mathematics and Cultivation of Mind (Jeong-Ho Woo). Plenary panel papers include: (1) Introduction to the PME Plenary Panel, "School Mathematics for Humanity Education" (Koeno Gravemeijer); (2) Humanizing the Theoretical and the Practical for Mathematics Education (Cristina Frade); (3) Making Mathematics More Mundane--A Semiotic Approach (Willibald Dorfler); (4) Mathematics: A Human Potential (Martin A. Simon); and (4) Need for Humanising Mathematics Education (Masataka Koyama). The first research forum, Learning through Teaching: Development of Teachers' Knowledge in Practice (RF01) includes: (1) A View on the Teachers' Opportunities to Learn Mathematics through Teaching (Roza Leikin and Rina Zazkis); (2) Integrating Virtual and Face-to-Face Practice: A Model for Continuing Teacher Education (Marcelo C. Borba); (3) Teachers' Learning Reified: The Professional Growth of Inservice Teachers through Numeracy Task Design (Peter Liljedahl); (4) Constraints on What Teachers Can Learn from their Practice: Teachers' Assimilatory Schemes (Martin A. Simon); and (5) What and How Might Teachers Learn via Teaching: Contributions to Closing an Unspoken Gap (Ron Tzur). The second research forum, Researching Change in Early Career Teachers (RF02), includes: (1) Introduction (Peter Sullivan); (2) Researching Relief of Mathematics Anxiety among Pre-Service Elementary School Teachers (Markku S. Hannula, Peter Liljedahl, Raimo Kaasila, and Bettina Rosken);(3) Teachers' Learning from Learning Studies: An Example of Teaching and Learning Fractions in Primary Four (Lo Mun Ling and Ulla Runesson); (4) Tracking Teachers' Learning in Professional Development Centered on Classroom Artifacts (Lynn T. Goldsmith and Nanette Seago); (5) Teacher Change in the Context of Addressing Students' Special Needs in Mathematics (Orit Zaslavsky and Liora Linchevski); (6) Researching Change in Prospective and Beginning Teachers (Laurinda Brown and Alf Coles); and (7) Summary and Conclusions (Markku S. Hannula). Information relating to discussion groups, working sessions, short oral communications, and poster presentations conclude this volume of the 31st proceedings. (Individual papers contain references.)

This volume of the 30th annual proceedings of the International Group for the Psychology of Mathematics Education conference presents: plenary panel papers; research forum papers; short oral communication papers; and poster presentation papers from the meeting. Information relating to discussion groups and working sessions is also provided. Plenary lecture papers include: (1) Mathematics, didactical engineering and observation (G. Brousseau); (2) A Semiotic View of the Role of Imagery and Inscriptions in Mathematics Teaching and Learning (N. Presmeg); (3) School Mathematics as a Developmental Activity (S. Stech); and (4) PME 1 to 30--Summing Up And Looking Ahead (P. Tsamir and D. Tirosh). Plenary panel papers include: (1) The Necessity of Collaborations between Mathematicians and Mathematics Educators (Z. Gooya); (2) Generic versus Subject Specific Pedagogy (S. Groves); (3) How Can Schools Put Mathematics in Their Centre? (K. Krainer); and (4) Mathematics in the Centre (T. Rojano). The first research forum (RF01) includes: (1) Seeing More and Differently (L. Brown and A. Coles); (2) Joint Reflection as a Way to Cooperation between Researchers and Teachers (A. Hospesova, J. Machackova, and M. Ticha); (3) Opening the Space of Possibilities (A. Lebethe, N. Eddy, and K. Bennie); (4) Diverse Roles, Shared Responsibility (J. Novotna and A. Pelantova); (5) Research with Teachers (L. Poirier); (6) Developing a Voice (G. Rosen); and (7) Learning about Mathematics and about Mathematics Learning through and in Collaboration (V. Zack and D. Reid). The second research forum (RF02) includes: Exemplification in Mathematics Education (L. Bills, T. Dreyfus, J. Mason, P. Tsamir, A. Watson, and O. Zaslavsky). The third research forum (RF03) includes: (1) Conceptual Change in Mathematics Learning (D. Tirosh and P. Tsamir); (2) Aspects of Students' Understanding of Rational Numbers (X. Vamvakoussi and S. Vosniadou); (3) Conceptual Change in the Number Concept (K. Merenluoto and E. Lehtinen); (4) The Linear Imperative (W. Van Dooren, D. De Bock, and L. Verschaffel); (5) Conceptual Change in Advanced Mathematical Thinking (I. Biza and T. Zachariades); (6) Students' Interpretation of the Use of Literal Symbols in Algebra (K. P. Christou and S. Vosniadou); (7) The Dilemma of Mathematical Intuition in Learning (L. B. Resnick); and (8) Designing for Conceptual Change (B. Greer). Short oral communications papers include: (1) The Mathematics Teachers' Conceptions about the Possible Uses of Learning Objects from RIVED-Brazil Project (C. A. A. P. Abar and L. S. de Assis); (2) On the Way to Understanding Integration (S. Abdul-Rahman); (3) The Impact of Graphic-Calculator Use on Bedouin Students' Learning Functions (M. Abu-Naja and M. Amit); (4) How to Put It All Together? (M. Alagic); (5) Reading Mathematics Textbook as a Storybook (S. E. Anku); (6) Naming and Referring to Quantities When Solving Word Problems in a Spreadsheet Environment (D. Arnau and L. Puig); (7) If It Divides by 4, It Must Divide by 8 (J. Back); (8) Universal and Existential Mathematical Statements (R. Barkai, T. Dreyfus, D. Tirosh, and P. Tsamir); (9) Situtations, Linear Functions and the "Real World" (G. Benke); (10) The Relationship between High School Mathematics and Career Choices among High Achieving Young Women (S. B. Berenson, J. J. Michael, and M. Vouk); (11) Students' Understanding of Ambiguity in Symbols (K. P. Blair); (12) Reform-Oriented Teaching Practices and the Influence of School Context (J. Bobis and J. Anderson); (13) Approaching Linear (In)Dependence with Example-Generation (M. Bogomolny); (14) Metaphors in Teacher's Discourse (J. Bolite Frant, V. Font, and J. Acevedo); (15) Designing Instructional Programs that Facilitate Increased Reflection (J. Bowers and S. Nickerson); (16) Conformism in Teaching Mathematics (A. Braverman, P. Samovol, and M. Applebaum); (17) Constructing Multiplication (J. Brocardo, L. Serrazina, and I. Rocha); (18) The Trigonometric Connection (S. A. Brown); (19) Student Beliefs and Attitudes from Poetry Writing in Statistics (M. Bulmer, B. Lea, and K. Rolka); (20) The Teaching of Proof in Textbooks (R. Cabassut); (21) Classroom: A Learning Context for Teachers (A. P. Canavarro); (22) An Investigation of Differences in Performance in Mathematics between Parallel Students and Normal Entry Students at the Polytechnic--University of Malawi (P. C. Chamdimba); (23) A Study on Eliciting the Formula for the Area of Triangle from Students' Structuring of Tile Arrays and Figure Reconstructions (J.-H. Chen and S.-K. S. Leung); (24) Decision Making at Uncertainty (E. Chernoff and R. Zazkis); (25) A Study on Implementating Inquiry-Based Teaching to Facilitate Secondary School Students' Learning in the Retaking Mathematics Course (E.-T. Chin, C.-Y. Chen, C.-Y. Liu, and C.-P. Lin); (26) Girls Excelled Boys in Learning Geometric Transformation Using Tessellations (S. Choi-Koh and H. Ko); (27) A National Survey of Young Children's Understanding of Basic Time Concepts (J. Chung and C.-C. Yang); (28) An Exploration of the Mathematical Literacy of Irish Student Primary School Teachers (D. Corcoran); (29) Mathematics Teachers' Knowledge and Practice (J. P. da Ponte and O. Chapman); (30) Psychological Aspects of Students Thinking at the Stage of Graphical Representation in the Process of Investigation of Functions (M. Dagan); (31) Formative Feedback and Mindful Teaching of Undergraduate Mathematics (G. E. Davis and M. A. McGowen); (32) Mathematics Education in the South and Western Pacific (A. J. Dawson); (33) Teacher Meditation of Technology-Supported Graphing Activity (R. Deaney, S. Hennessy, and K. Ruthven); (34) A Categorization of Difficulties Encountered by 13-to-15-Year-Olds while Selecting Inverse Algebraic Operation (A. Demby); (35) "It's Infinity" (T. Dooley); (36) What Is to Be Known? (J.-P. Drouhard); (37) "The Most Normal Path" (M. Droujkova, S. Berenson, G. Mojica, K. Slaten, and H. Wilson); (38) Exploratory Mathematics Talk in Friendship Groups (J.-A. Edwards); (39) Conceptual Basis of Proof (L. D. Edwards); (40) In-Service Education under Market Conditions (L. R. Ejersbo); (41) Statements of Problems and Students' Choices between Linear or Non-Linear Models (C. Esteley, M. Villarreal, and H. Alagia); (42) Equity and Quality Mathematics Education (G. Frempong); (43) Is Sensitivity for the Complexity of Mathematics Teaching Measurable? (T. Fritzlar); (44) By Using the Outcome-Based Approach to Strengthen Students' Learning Capabilities (A. T.-F. Fung and K.-M. Leung); (45) Patterns of Students' Interactions while Doing Geometric Proofs in Groups (F. L. Gallos); (46) Dialogue: A Tool for Creating Mathematical Proof (S. Gholamazad); (47) Beginning Teachers in Mathematical Inquiry (B. Graves and C. Suurtamm); (48) Learning Trajectory of Fraction in Elementary Education Mathematics (S. Hadi); (49) Elementary Education Students' Affect towards and Advancement in Mathematics (M. S. Hannula, R. Kaasila, E. Pehkonen, and A. Laine); (50) Pre-Service Mathematics Teachers (B. Hartter and J. Olson); (51) Effectiveness of Video-Case Based Elementary Mathematics Teacher Training (R. Huang and J. Bao); (52) Improving Students' Level of Geometrical Thinking through Teacher's Regulating Roles (M. Imprasitha); (53) What's the Connection between Ears and Dice (I. Jan and M. Amit); (54) Tactile Perception in 3D Geometry (D. Jirotkova and G. Littler); (55) High Achieving Students' Conceptions of Limits (K. Juter); (56) Reading Visual Representations of Data with Kindergarten Children (S. Kafoussi); (57) Students' Use of Gestures to Support Mathematical Understandings in Geometry (L. H. Kahn); (58) Mathematical Abilities for Developing Understanding of Formal Proof (E. Kapetanas and T. Zachariades); (59) An Analysis of Connections between Errors and Prior Knowledge in Decimal Calculation (J. Kim, J. Pang, and K. Song); (60) Insights into Primary Teachers' Interpretations of Students' Written Answers in Mathematics (A. Klothou and H. Sakonidis); (61) The Role of Proof (S. Kmetic); (62) A Comparison of Mathematically Gifted and Non-Gifted Students in Intuitively Based, Probabilistic Misconception (E. S. Ko, B. H. Choi, and E. H. Lee); (63) Exploring Teaching and Learning of Letters in Algebra (A. Kullberg and U. Runesson); (64) Teaching Mathematics to Indigenous Students and Pupils from Multicultural Backgrounds (E. K. Lam); (65) Limitations of a Partitive Fraction Scheme in Developing Multiplicative Reasoning about Fractions (H. S. Lee); (66) Teachers' Reflection and Self-Assessment through the Use of a Videotape of Their Own Mathematics Instruction (S. Lee and J. Pang); (67) A Case Study on the Introducing Methods of the Irrational Numbers Based on the Freudenthal's Mathematising Instruction (Y. R. Lee); (68) A Case Study of an Elementary School Teacher's Professional Development on Mathematics Teaching in Context (Y.-C. Leu, C.-H. Hsu, and W.-L. Huang); (69) "But after All, We'll Need This for School" (N. Leufer and S. Prediger); (70) Developing Primary Students' Cognitive Skills through Interactive Mathematics Lessons (K.-M. Leung); (71) A Study on the Effects of Multiple Representation Curriculum on Fraction Number Learning Schemes for Fourth Grade Children (S.-K. S. Leung and I.-J. Wang); (72) Teachers' Knowledge about Definitions (E. Levenson and T. Dreyfus); (73) Supporting Teachers on Maintaining High-Level Instructional Tasks in Classroom by Using Research-Based Cases (P.-J. Lin); (74) Towards an Anti-Essentialist View of Technology in Mathematics Education (B. Lins and C. H. de Jesus Costa); (75) Comparing Teaching of Common Mathematical Tasks in Different Countries (G. Littler and M. Tzekaki); (76) New Approach of Neurocognition in Mathematical Education Research and further Implications (C. Liu, F.-L. Lin, and C.-N. Dai); (77) Reasoning and Generalizing about Functional Relationship in a Grade 2 Classroom (S. London McNab); (78) The MathematicalPerformances in Solving the Norming Problem (H.-L. Ma); (79) The Education of Reasoning (E. Macmillan); (80) The Effect of Rephrasing Word Problems on the Achievements of Arab Students in Mathematics (A. Mahajne and M. Amit); (81) An Approach to Eary Algebra Using Technology as an Enhancement (C. A. Maher and G. Gjone); (82) Teachers' Beliefs and Competencies of Creative Mathematical Activities (B. Maj); (83) Manipulative Representation (N. Mark-Zigdon and D. Tirosh); (84) Language, Power and Mathematics Learning (M. Mathye and M. Setati); (85) Children Learning as Participation in Web-Based Communities of Practice (J. F. Matos and M. Santos); (86) Recognizing Mathematical Competences (J. F. Matos, M. Santos, and M. Mesquita); (87) Mathematics Teachers' Preparation Program (A. S. Md. Yunus, R. Hamzah, H. Ismail, S. K. S. Hussain, and M. R. Ismail); (88) Mathematics Register Acquisition (T. Meaney); (89) Development of Spatial Abilities (H. Meissner); (90) An Encounter between Queer Theory and Mathematics Education (H. Mendick); (91) Establishing a Mathematics Learning Community in the Study of Mathematics for Teaching (J. Mgombelo and C. Buteau); (92) Talking Mathematics in a Second Language (H. Miranda); (93) Objects in Motion (I. Miranda, L. Radford, and J. G. Hernandez); (94) Researching the Appearance of Mathematical Argumentation (C. Misailidou); (95) Teachers' Pedagogical Content Knowledge in the Teaching of Quadrilaterals (I. A. C. Mok and M. Y. H. Park); (96) Out-of-School Experts in Mathematics Classes (J. Monaghan); (97) A Sequel to Trends in International Mathematics and Science Study (TIMSS), 2003, in Botswana (S. M. Montsho); (98) The Pattern and Structure Mathematics Awareness Project (PASMAP) (J. Mulligan and M. Mitchelmore); (99) Primary Pupils' Mathematics Achievement (C. Opolot-Okurut); (100) Images of Functions Defined in Pieces (R. Ovodenko and P. Tsamir); (101) A Comparative Analysis of Elementary Mathematics Textbooks of Korea and Singapore (J. Pang and H. Hwang); (102) Substitutions on Algebraic Statements, Based on Associations in Natural Reasoning (M. Panizza); (103) Virtual Learning Environments and Primary Teachers' Professional Development (M. C. Penalva-Martinez and C. Rey-Mas); (104) Cypriot Preservice Primary School Teachers' Subject-Matter Knowledge of Mathematics (M. Petrou); (105) Phenomenological Mathematics Teaching (P. Portaankorva-Koivisto); (106) Using the Debate to Educate Future Teachers of Mathematics (J. Proulx); (107) Surprise on the Way from Change of Length to Change of Area (N. Prusak, N. Hada, and R. Hershkowitz); (108) Discovering of Regularity (by 11-Years-Old Children) (M. Pytlak); (109) Using Manipulatives to Teach Students in College Developmental Math Classes about Fractions (S. L. Reynolds and E. B. Uptegrove); (110) Sixth Graders' Ability to Generalize Patterns in Algebra (F. Rivera and J. Rossi Becker); (111) Variety of Representational Environments in Early Geometry (F. Roubicek); (112) From Research on Using Problems Related to Functional Equations as Multifunctional Tools for Revealing Subject Mater Knowledge of Functions in Future Mathematics Teachers (M. Sajka); (113) Expert and Novice Primary Teachers' Intervening in Students' Mathematical Activity (H. Sakonidis, M. Kaldrimidou, and M. Tzekaki); (114) Examining Teachers' Reflections about Mathematics Teaching, Learning, and Assessment (V. M. Santos-Wagner); (115) Analyzing Students' Thought Process in Revealing Correspondence between Formulas and Geometrical Objects (P. Satianov and M. Dagan); (116) "No Need to Explain, We Had the Same" (K. Schreiber); (117) Characteristics of Malaysian Students' Understanding about Functions (S. A. Sh. Abdullah); (118) Mathematical Induction via Conceptual Representation (A. Sharif-Rasslan); (119) Exploring the Meanings of Events in Mathematics Classroom from Learners' Perspective (Y. Shimizu); (120) A Study on the Law of Large Numbers Instruction through Computer Simulation (B.-M. Shin and K.-H. Lee); (121) Mathematics Learning Quality for Gifted Junior High School Students in Taiwan (H.-Y. Shy, C.-H. Liang, and W.-M. Liang); (122) A Preservice Teacher's Growth in Subject Matter Knowledge while Planning a Trigonometry Lesson (K. M. Slaten); (123) Comparing Numbers: Counting-Based and Unit-Based Approaches (H. Slovin); (124) Mathematically Gifted 6th Grade Korean Students' Proof Level for a Geometric Problem (S Song, Y. Chong, J. Yim, and H. Chang); (125) Probability Reasoning Level of Gifted Students in Mathematics (S. Song, K. Lee, G. Na, and D. Han); (126) Analysis of Mathematically Gifted 5th and 6th Grade Students' Process of Solving "Straight Line Peg Puzzle" (S. Song, J. Yim, Y. Chong, and J. Kim); (127) Standard Mathematics Discourses of Developmental Algebra Undergraduates (S. K. Staats); (128) Novice Students, Experienced Mathematicians, and Advanced Mathematical Thinking Processes (E. Stadler); (129) The "Soil" of Extended Problems: The Cultural Background of the Chinese Mathematics Teaching Practice (X. Sun and N.-Y. Wong); (130) Immersion in Mathematical Inquiry: The Experiences of Beginning Teachers (C. Suurtamm and B. Graves); (131) The Whole Idea (S. Tobias); (132) The Teaching Modes (R. A. Tomas Ferreira); (133) Didactic Decisions (J. Trgalova and I. Lima); (134) Evaluating a Large-Scale National Program for Incorporating Computational Technologies to Mathematics Classrooms (M. Trigueros and A. I. Sacristan); (135) Symmetry: Equality or a Dynamic Transformation? (K. Tselepidis and C. Markopoulos); (136) Teaching Children to Count (F. Turner); (137) Student Conceptions and Textbook Messages (B. Ubuz); (138) Students' Errors in Transforming Terms and Equations (A. Ulovec and A. Tollay); (139) Mathematics with Technology (S. Ursini, G. Sanchez, and D. Santos); (140) Francisca Uses Decimal Numbers (M. E. Valedmoros Alvarez and E. F. Ledesma Ruiz); (141) Development of Numerical Estimation in Grade 1 to 3 (M. van Galen and P. Reitsma); (142) Mathematics Education and Neurosciences (MENS) (F. van Nex and T. Gebuis); (143) Symbolizing and Modeling to Promote a Flexible Use of the Minus Sign in Algebraic Operations (J. Vlassis); (144) An Analysis of Preservice Teachers' Estimation Strategies within the Context of Whole Numbers, Fractions, Decimals, and Percents (T. N. Volkova); (145) What Does It Mean to Interpret Students' Talk and Actions? (T. Wallach and R. Even); (146) The Research of Co-Teaching Math between Experienced and Preservice Teachers in Elementary School (J.-H. Wang); (147) The Influence of Teaching on Transforming Math Thinking (T.-Y. Wang and F.-J. Hsieh); (148) Searching for Common Ground (J. Watson, L. Webb, L. King, and P. Webb); (149) Are Beliefs and Practices Congruent or Disjoint? (L. Webb and P. Webb); (150) Working Memory and Children's Mathematics (M. Witt and S. Pickering); (151) Mathematics Education Reform in the United States (T. Wood); (152) A Modeling Perspective on Problem Solving in Students' Mathematics Project (F.-M. Yen and C.-K. Chang); and (153) Development of a Questionnaire to Measure Teachers' Mathematics-Related Beliefs (S.-Y. Yu and C.-K. Chang). Poster presentations include: (1) A Comparative Analysis of Mathematics Achievement and Attitudes of Male and Female Students in Botswana Secondary Schools (A. A. Adeyinka); (2) Logical-Mathematical Learning for Student with Down's Syndrome (R. M. Aguilar, A. Bruno, C. S. Gonzalez, V. Munoz, A. Noda, and L. Moreno); (3) The Math Fair as a Bridge between Mathematics and Mathematics Education, the University and Elementary or Junior High School (M. Beisiegel); (4) One Teaching Episode from a Learner's, an Observer's and a Teacher's Point of View (H. Binterova and J. Novotna); (5) A Framework for Studying Curricular Effects on Students' Learning (J. Cai and J. C. Moyer); (6) Preservice Elementary Teachers' Conceptual Understanding of Word Problems (O. Chapman); (7) Mathematics Education and School Failure (P.Chaviaris and S. Kafoussi); (8) Enhancing the Seventh Graders' Learning on Equality Axiom and Linear Equation through Inquiry-Oriented Teaching and Integrated Mathematics and Science Curriculum (K.-J. Chen, S.-Y. Yu, E.-T. Chin, and H.-L. Tuan); (9) To Conjecture the Staff Development Model of Mathematical Teacher According to Spark's Theory (Y.-T. Chen and S. Leou); (10) Discovery of Implementing Teaching by Discussion in Mathematics Classrooms (J. Chung); (11) Modeling Teachers' Questions in High School Mathematics Classes (S. Dalton, G. Davis, and S. Hegedus); (12) My Assistant, a Didactic Tool of Mathematics for Primary School Teachers (N. de Bengoechea-Olguin); (13) A Model to Interpret Teacher's Practices in Technology-Based Environment (N. C. Dedeoglu); (14) The Gnomon (P. Delikanlis); (15) The Teacher's Proactive Role in the Context of Word Problem Solving by Young Beginners in Algebra (I. Demonty); (16) Students' Geometrical Thinking Development at Grade 8 in Shanghai (L. Ding and K. Jones); (17) An Interdisciplinary Perspective on Learning to Teach Mathematical Writing (H. M. Doerr, K. Chandler-Olcott, and J. O. Masingila); (18) Multiplication Models (D. Droujkov and M. Droujkova); (19) Quantitative Grids and Cyclic Patterns (D. Droujkov and M. Droujkova); (20) Learners' Influence in Computer Environments (M. Droujkova nd D. Droujkov); (21) Alleviating Obstructions to Learning (D. Easdown); (22) Knowledge and Interpretation of Teachers to the School Content of Proportionality (H. Enriquez Ramirez and E. Jimenez de la Rosa Barrios); (23) Mathematical Flexibility in the Domain of School Trigonometry (C. Fi); (24) "Moving Fluidly among Worlds" (S. Gerofsky); (25) Making Practice Studyable (H. Ghousseini and L. Sleep); (26) Cognitive Roots for the Concept of Asymptote (V. Giraldo, M. Chaves, and E. Belfort); (27) Flemish and Spanish High School Students' Mathematics-Related Beliefs Systems (I. M. Gomez-Chacon, P. Op't Eynde, and E. De Corte); (28) Cube Nets (M.Hejny and D. Jirotkova); (29) From Word Notation of Relations between Constants and Unknown to Algebraic Notation (Pretest) (J. Herman); (30) Mathematics and Community Capacity Building (P. Howard and B. Perry); (31) A Study on the Mathematical Thinking in Learning Process (C.J.-Hsieh and F.-J. Hsieh); (32) A Case Study on Pre-Service Teachers Making Mathematical Model of Voronoi-Diagram (C.-T. Hu and T.-Y. Tso); (33) A Fast-Track Approach to Algebra for Adults (R. Hubbard); (34) The Validity of On-Screen Assessment of Mathematics (S. Hughes); (35) After Using Computer Algebra System, Change of Students' Rationales and Writing (I. K. Kim); (36) Using a Socrates' Method in a Course of Mathematics Education for Future Mathematics Teachers (N. H. Kim); (37) Beyond Visual Level (G. Kospentaris and T. Spirou); (38) Videopapers and Professional Development (T. Lima Costa and H. Nascentes Coelho); (39) Concurrent Calibration Design for Mathematics Learning Progress Investigation (C.-J. Lin, P.-H. Hung, and S. Lin); (40) Further Insights into the Proportion Reasoning and the Ratio Concept (C. Liu, F.-L. Lin, W. Kuo, and I.-L. Hou); (41) Coursework Patterns between Mathematics and Science among Secondary Students (X. Ma); (42) Beginning the Lesson (C. Mesiti and D. Clarke); (43) Development of Web Environment for Lower Secondary School Mathematics Teachers with 3D Dynamic Geometry Software (M. Miyazaki, H. Arai, K. Chino, F. Ogihara, Y. Oguchi, and T. Morozumi); (44) The Effect of the Teacher's Mode of Instruction inside Math Classrooms with a Computer (S. Mochon and M. E. F. Olvera); (45) Mathematics Education in Rural Schools (J. Mousley and G. Marks); (46) Mathematically Gifted Students' Conception of Infinity (G. Na and E. Lee); (47) A Good Moment in Time to Stop "Shying Away from the Nature of Our Subject"? (E. Nardi); (48) Toward Real Change through Virtual Communities (K. T. Nolan); (49) Powerful Ideas, Learning Stories and Early Childhood Mathematics (B. Perry, E.Harley, and S. Dockett); (50) Tracing the Development of Knowledge about Mathematics Teaching (C. Rey-Mas and C. Penalva-Martinez); (51) Preschool Children's Number Sense (L. de C. Ribeiro and A. G. Spinillo); (52) A First Approach to Students' Learning of Mathematical Contents (G. Sanchez-Matamoros and I. Escudero); (53) Young Children's Mathematics Education within a Philosophical Community of Inquiry (A. Sawyer); (54) Learning Mathematics in Austria (H. Schwetz and G. Benke); (55) The Influence of a Mathematician on His Students' Perceptions (A. Sharif-Rasslan); (56) Analysis on the Algebraic Generalization of Some Korean Mathematically Promising Elementary Students (S. Song, J. Yim, Y. Chong, and E. Park); (57) Students' Linguistic Strategies for Shared Authority in Undergraduate Algebra Discussions (S. Staats); (58) Contrasting Decimal Conceptions of Adult and School Students (K. Stacey and V. Steinle); (59) Mathematical Writing and the Development of Understanding (N. Stehlikova); (60) Enhancing Teachers' Professional Development through Developing Teaching Norms Based on Developing Classroom Learning Norms (W.-H. Tsai); (61) The Features in the Process of Mathematical Modeling with Dynamic Geometric Software (T.-Y. Tso); (62) Finding Instructive Characteristics of Picture Books that Support the Learning of Mathematics (S. van den Boogaard and M. van den Heuvel-Panhuizen); (63) High School Course Pathways of High Achieving Girls (P. H. Wilson, G. F. Mojica, K. M. Slaten, and S. B. Berenson); (64) The Developmental Stages of Representations of Simple Regular Space Figures of Elementary School Students (D.-B. Wu, J.-L. Ma, and D.-C. Chen); (65) How to Assess Mathematical Thinking? (S. Yesildere and E. B. Turnuklu); and (66) Lasting Effects of a Professional Development Initiative (S. Zehetmeier). (Individual papers contain references.)

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.)

The third volume of the 29th annual conference of the International Group for the Psychology of Mathematics Education contains full research report papers. Papers include: (1) Students' Use of ICT Tools: Choices and Reasons (Anne Berit Fuglestad); (2) Interaction of Modalities in Cabri: A Case Study (Fulvia Furinghetti, Francesca Morselli, and Domingo Paola); (3) The Duality of Zero in the Transition from Arithmetic to Algebra (Aurora Gallardo and Abraham Hernandez); (4) Conflicts in Offshore Learning Environments of a University Preparatory Mathematics Course (Linda Galligan); (5) The Diverse Learning Needs of Young Children who were Selected for an Intervention Program (Ann Gervasoni); (6) Using Contradictions in a Teaching and Learning Development Project (Simon Goodchild and Barbara Jaworski); (7) A Sociocultural Analysis of Learning to Teach (Merrilyn Goos); (8) Three Case Studies on the Role of Memorizing in Learning and Teaching Mathematics (Sue Gordon and Jackie Nicholas); (9) Reconstructing Norms (Nuria Gorgorio and Nuria Planas); (10) Forming Teachers as Resonance Mediators (Paolo Guidoni, Donatella Iannece, and Roberto Tortora); (11) Investigating the Problem Solving Competency of Pre Service Teachers in Dynamic Geometry Environment (Shajahan Haja); (12) Structure and Typical Profiles of Elementary Teacher Students' View of Mathematics (Markku Hannula, Raimo Kaasila, Erkki Pehkonen, and Anu Laine); (13) Preservice Teachers' View on y= x+5 and y = [pi] x [superscript 2] Expressed through the Utilization of Concept Maps: A Study of the Concept of Function (Orjan Hansson); (14) Mistake-Handling Activities in the Mathematics Classroom (Aiso Heinze); (15) One Teacher's Role in Promoting Understanding in Mental Computation (Ann Heirdsfield); (16) In the Middle of Nowhere: How a Textbook Can Position the Mathematics Learner (Beth Herbel-Eisenmann and David Wagner); (17) Chinese Whispers--Algebra Style: Grammatical, Notational, Mathematical and Activity Tensions (Dave Hewitt); (18) Pedagogy of Facilitation: How do we Best Help Teachers of Mathematics with New Practices? (Joanna Higgins); (19) Students' Difficulties with Applying a Familiar Formula in an Unfamiliar Context (Maureen Hoch and Tommy Dreyfus); (20) Learning Mathematics: Perspectives of Australian Aboriginal Children and Their Teachers (Peter Howard and Bob Perry); (21) Verification or Proof: Justification of Pythagoras' Theorem in Chinese Mathematics Classrooms (Rongjin Huang); (22) Mozambican Teachers' Professional Knowledge about Limits of Functions (Danielle Huillet); (23) Heuristic Biases in Mathematical Reasoning (Matthew Inglis and Adrian Simpson); (24) Three Utilities for the Equal Sign (Ian Jones and Dave Pratt); (25) Computer Algebra Systems (CAS) as a Tool for Coaxing the Emergence of Reasoning about Equivalence of Algebraic Expressions (Carolyn Kieran and Luis Saldanha); (26) Students' Colloquial and Mathematical Discourses on Infinity And Limit (Dong Joong Kim, Anna Sfard, and Joan Ferrini-Mundy); (27) The Effect of Mathmagic on the Algebraic Knowledge and Skills of Low-Performing High School Students (Hari P. Koirala); (28) Learning and Teaching Early Number: Teachers' Perceptions (Cally Kuhne, Marja Van Den Heuvel-Panhuizen, and Paula Ensor); (29) Situation-Specific and Generalized Components Of Professional Knowledge of Mathematics Teachers: Research on a Video-Based In-Service Teacher Learning Program (Sebastian Kuntze and Kristina Reiss); (30) Assessing Professional Growth of Pre-Service Teachers Using Comparison between Theoretical And Practical Image of the "Good Teacher" (Ilana Lavy and Atara Shriki); (31) Mathematically Gifted Students' Geometrical Reasoning and Informal Proof (Kyung Hwa Lee); (32) Investigation on an Elementary Teacher's Mathematics Pedagogical Values through Her Approach to Students' Errors (Yuh-Chyn Leu and Chao-Jung Wu); (33) How the Calculator-Assisted Instruction Enhances Two Fifth Grade Students' Learning Number Sense in Taiwan (Yung-Chi Lin and Erh-Tsung Chin); (34) Fourth-Grade Students' Performance on Graphical Languages in Mathematics (Tom Lowrie and Carmel Diezmann); (35) Preservice Teachers' Conceptions of Variation (Katie Makar and Dan Canada); (36) The Equivalence and Ordering of Fractions in Part-Whole and Quotient Situations (Ema Mamede, Terezinha Nunes, and Peter Bryant); (37) Growth of Mathematical Understanding in a Bilingual Context: Analysis and Implications (Sitaniselao Stan Manu); (38) Motivational Beliefs, Self-Regulated Learning and Mathematical Problem Solving (Andri Marcou and George Philippou); (39) Fractions in the Workplace: Folding Back and the Growth of Mathematical Understanding (Lyndon Martin, Lionel LaCroix, and Lynda Fownes); (40) Meaning Construction through Semiotic Means: The Case of the Visual Pyramid (Michela Maschietto and Maria G. Bartolini Bussi); (41) Students' Motivational Beliefs, Self-Regulation Strategies and Mathematics Achievement (Nikos Mousoulides and George Philippou); and (42) The Transition to Postgraduate Study in Mathematics: A Thinking Styles Perspective (Andreas Moutsios-Rentzos and Adrian Simpson). (Individual papers contain references.) [For Volume 2 of this Proceedings, see ED496859.]

In a longitudinal study of children in grades 0-4, gender differences are developing very early in some number domains, but there do not appear to be similar differences showing in the measurement and geometry domains studied. The Early Numeracy Research Project collected data at the beginning and end of each year from over 11,000 children in Grades 0-2 from 1999-2001. Of these, 1237 children (672 boys, 565 girls) formed a longitudinal cohort who started Grade 0. A subset of 572 (295 boys, 277 girls) of these children were also assessed at the end of Grade 3 in 2002 and Grade 4 in 2003. The students were assessed individually through a task based interview of about 30-40 minutes duration near the beginning and end of each of their Grade 0-2 years and at the end of the Grade 3-4 years. (Contains 1 table and 6 figures.) (Paper presented at the 28th Conference of the International Group for the Psychology of Mathematics Education, 2004.) [For complete proceedings, see ED489538.]

This document contains the second volume of the proceedings of the 28th annual conference of the International Group for the Psychology of Mathematics Education. Conference presentations are centered around the theme "Inclusion and Diversity". This volume features 65 research report papers: (1) Constructing Meanings and Utilities within Algebraic Tasks (Janet Ainley, Liz Bills, and Kirsty Wilson); (2) Proportional Reasoning of Quasi-Illiterate Adults (Silvia Alatorre and Olimpia Figueras); (3) Uses of Example Objects in Proving (Lara Alcock); (4) Improving Student Teachers' Attitudes to Mathematics (Solange Amorim Amato); (5) Time and Flow as Parameters in International Comparisons: A View from an Eighth Grade Algebra Lesson (Miriam Amit and Michael N. Fried); (6) Disciplined Calculators or Flexible Problem Solvers? (Julia Anghileri); (7) A Statement, the Contrapositive and the Inverse: Intuition and Argumentation (Samuele Antonini); (8) Solution--What Does It Mean? Helping Linear Algebra Students Develop the Concept While Improving Research Tools (Ilana Arnon and David DeVries); (9) Organizing with a Focus on Defining a Phenomenographic Approach (Amir H. Asghari); (10) Mediation and Interpretation: Exploring the Interpersonal and the Intrapersonal in Primary Mathematics Lessons (Mike Askew); (11) Being Sensitive to Students' Mathematical Needs: What Does It Take? (Cettina Axiak); (12) A New Practice Evolving in Learning Mathematics: Differences in Students' Written Records with CAS (Lynda Ball and Kaye Stacey); (13) Empowering Andrea to Help Year 5 Students Construct Fraction Understanding (Annette R. Baturo); (14) Understanding Inverse Functions: The Relationship between Teaching Practice and Student Learning (Ibrahim Bayazit and Eddie Gray); (15) The Impact of Teachers' Perceptions of Student Characteristics on the Enactment of Their Beliefs (Kim Beswick); (16) Towards the Emergence of Constructing Mathematical Meanings (Angelika Bikner-Ahsbahs); (17) Identity, Knowledge and Departmental Practices: Mathematics of Engineers and Mathematicians (Erhan Bingolbali and John Monaghan); (18) Elementary Grades Students' Capacity for Functional Thinking (Maria L. Blanton and James J. Kaput); (19) For the Sake of the Children: Maintaining the Momentum of Professional Development (Janette Bobis); (20) From Formal to Semi-Informal Algorithms: The Passage of a Classroom into a New Mathematical Reality (Ada Boufi and Frosso Skaftourou); (21) A Measure of Rulers--The Importance of Units in a Measure (Philippa Bragg and Lynne Outhred); (22) In the Serpent's Den: Contrasting Scripts Relating to Fear of Mathematics (Chris Breen); (23) Formal Inclusion and Real Diversity in an Engineering Program of a New Public University (Tania Cristina Baptista Cabral and Roberto Ribeiro Baldino); (24) Primary Students' Understanding of Tessellation: An Initial Exploration (Rosemary Callingham); (25) Facilitating Peer Interactions in Learning Mathematics: Teachers' Practical Knowledge (Olive Chapman); (26) Towards a Unified Model on Teachers' Concerns and Efficacy Beliefs Related to a Mathematics Reform (Charalambos Charambous, George Philippou, and Leonidas Kyriakides); (27) What Is Unusual? The Case of a Media Graph (Helen L. Chick and Jane M. Watson); (28) Proofs through Exploration in Dynamic Geometry Environments (C. Christou, N. Mousoulides, M. Pittalis, and D. Pitta-Pantazi); (29) Establishing a Professional Learning Community among Middle School Mathematics Teachers (Karen Koellner Clark and Hilda Borko); (30) Patterns of Participation in the Mathematics Classroom (David Clarke); (31) Young "White" Teachers' Perceptions of Mathematics Learning of Aboriginal and Non-Aboriginal Students in Remote Communities (Tom J. Cooper, Annette R. Baturo, Elizabeth Warren, and Shani M. Doig); (32) Two Important Invariant Tasks in Solving Equations: Analyzing the Equation and Checking the Validity of Transformations (Anibal Cortes and Nelly Kavafian); (33) Certainty during the Early Stages of Problem Solving (Maria de Hoyos, Eddie Gray, and Adrian Simpson); (34) Studying the Mathematical Concept of Implication through a Problem on Written Proofs (Virginie Deloustal-Jorrand); (35) From Single Beliefs to Belief Systems: A New Observational Tool (Pietro Di Martino); (36) Assessment as a Strategic Tool for Enhancing Learning in Teacher Education: A Case Study (Brian Doig and Susie Groves); (37) Patterns of Reasoning in Classroom (Paola Domingo, Marco Cartiglia, Fulvia Furinghetti); (38) Generalized Diagrams as a Tool for Young Children's Problem Solving (Barbara J. Dougherty and Hannah Slovin); (39) Correspondences, Functions and Assignation Rules (M. Downs and J. Mamona-Downs); (40) Unjustified Assumptions Based on Diagrams in Geometry (Tali Dvora and Tommy Dreyfus); (41) The Impact of Individual Curricula on Teaching Stochastics (Andreas Eichler); (42) The Functions of Pictures in Problem Solving (Iliada Elia and George Philippou); (43) Mathematical Modelling with Young Children (Lyn D. English and James J. Watters); (44) Extending Linear Models to Non-Linear Contexts: An In-Depth Study about Two University Students' Mathematical Productions (Cristina Esteleyi, Monica Villarreal, and Humberto Alagia); (45) University Students' Conceptions of Function (Anastasia Evangelidou, Panayiotis Spyrou, Iliada Elia, and Athanasios Gagatsis); (46) Practical Constraints upon Teacher Development in Pakistani Schools (Razia Fakir-Mohammad); (47) Towards a Definition of Function (Rossana Falcade, Maria Alessandra Mariotti, and Colette Laborde); (48) "Why Doesn't It Start From the Origin?": Hearing the Cognitive Voice of Signs (Francesca Ferrara); (49) Mathematical Language and Advanced Mathematics Learning (Pier Luigi Ferrari); (50) Arithmetic/Algebraic Problem-Solving and the Representation of Two Unknown Quantities (Eugenio Filloy, Teresa Rojano, and Armando Solares); (51) Equity and Computers for Mathematics Learning: Access and Attitudes (Helen J. Forgasz); (52) The Tacit-Explicit Dynamic in Learning Processes (Cristina Frade); (53) Tracking Primary Students' Understanding of the Equality Sign (Viktor Freiman and Lesley Lee); (54) Levels of Student Responses in a Spreadsheet-Based Environment (Alex Friedlander and Michal Tabach); (55) Sensitivity for the Complexity of Problem Oriented Mathematics Instruction--A Challenge to Teacher Education (Torsten Fritzlar); (56) ICT Tools and Students' Competence Development (Anne Berit Fuglestad); (57) The Effects of Different Modes of Representation on Mathematical Problem Solving (Athanasios Gagatsis and Iliada Elia); (58) Descriptions and Conflicts in Dynamic Geometry (Victor Giraldo, Elizabeth Belfort, and Luiz Mariano Carvalho); (59) Students Problem Solving and Justification (Barbara Glass and Carolyn A. Maher); (60) Didactical Knowledge Development of Pre-Service Secondary Mathematics Teachers (Pedro Gomez and Luis Rico); (61) Legitimization of the Graphic Register in Problem Solving at the Undergraduate Level: The Case of the Improper Integral (Alejandro S. Gonzalez-Martin and Matias Camacho); (62) Will "The Way They Teach" Be "The Way They Have Learned"? Pre-Service Teachers' Beliefs Concerning Computer Embedding in Math Teaching (D. Gorev, I. Gurevich, and M. Barabash); (63) Progressive Discourse in Mathematics Classes--The Task of the Teacher (Susie Groves and Brian Doig); (64) Teachers' Practices and Dynamic Geometry (Luiz Carlos Guimaraes and Elizabeth Belfort); and (65) Characterization of Students' Reasoning and Proof Abilities in 3-Dimensional Geometry (Angel Gutierrez, John Pegg, and Christine Lawrie). (Individual papers contain references.)

This document contains the fourth volume of the proceedings of the 28th annual conference of the International Group for the Psychology of Mathematics Education. Conference presentations are centered around the theme "Inclusion and Diversity". This volume features 64 research report papers: (1) Situated or Abstract: The Effect of Combining Context and Structure on Constructing an Additive (Part-Part-Whole) Schema (Irit Peled and Ruth Meron); (2) Using Graphical Profiles to Study the Learning and Teaching of Mathematics (Dvora Peretz); (3) The Role of Number in Proportional Reasoning: A Prospective Teacher's Understanding (Axelle C. Person, Sarah B. Berenson, and Paula J. Greenspon); (4) Learning to Use CAS: Voices from a Classroom (Robyn Pierce and Kaye Stacey); (5) Technical School Students' Conceptions of Tangent Lines (Marcia Maria Fusaro Pinto and Valeria Guimaraes Moreira); (6) Elementary School Students' Mental Representations of Fractions (Demetra Pitta-Pantazi, Eddie M. Gray, and Constantinos Christou); (7) A Structural Model for Problem Posing (M. Pittalis, C. Christou, N. Mousoulides, and D. Pitta-Pantazi); (8) Some Undergraduates' Experiences of Learning Mathematics (Hilary Povey and Corinne Angier); (9) Normalising Geometrical Constructions: A Context for the Generation of Meanings for Ratio and Proportion (Georgos Psycharis and Chronis Kynigos); (10) The Sensual and the Conceptual: Artefact-Mediated Kinesthetic Actions and Semiotic Activity (Luis Radford, Serge Demers, Jose Guzman, and Michele Cerulli); (11) A Sociocultural Account of Students' Collective Mathematical Understanding of Polynomial Inequalities in Instrumented Activity (Ferdinand Rivera and Joanne Rossi Becker); (12) Infinity as a Multi-Faceted Concept in History and in the Mathematics Classroom (Ornella Robutti, Ferdinando Arzarello, and Maria G. Bartolini Bussi); (13) Successful Undergraduate Mathematicians: A Study of Students in Two Universities (Melissa Rodd and Margaret Brown); (14) Bilingual Lessons and Mathematical World Views--A German Perspective (Katrin Rolka); (15) The Use of Models in Teaching Proof by Mathematical Induction (Gila Ron and Tommy Dreyfus); (16) Reflecting on Prospective Elementary Teachers' Mathematics Content Knowledge: The Case of Laura (Tim Rowland, Peter Huckstep, and Anne Thwaites); (17) The Competent Use of the Analytic Method in the Solution of Algebraic Word Problems. A Didactical Model Based on a Numerical Approach with Junior High Students (Guillermo Rubio and Rafael Del Valle); (18) Language and Concept Development in Geometry (Frode Ronning); (19) The Role of Gestures in Conceptualisation: An Exploratory Study on the Integral Function (Cristina Sabena); (20) Psychological Aspects of Genetic Approach to Teaching Mathematics (Ildar S. Safuanov); (21) Routines in Non-Routine Problem Solving Processes (Wolfgang Schloeglmann); (22) Teacher Guidance of Knowledge Construction (Baruch Schwarz, Tommy Dreyfus, Nurit Hadas, and Rina Hershkowitz); (23) Types of Student Reasoning on Sampling Tasks (J. Michael Shaughnessy, Matt Ciancetta, and Dan Canada); (24) Didactic Model--Bridging a Concept with Phenomena (Beba Shternberg and Michal Yerushalmy); (25) Elaborating the Teacher's Role--Towards a Professional Language (Dianne Siemon, Jo Virgona, Maria Lasso, Vanessa Parsons, and Juli Cathcart); (26) Making the Connection: Procedural and Conceptual Students' Use of Linking Words in Solving Problems (Adrian Simpson and Nora Zakaria); (27) Children's Conceptual Understanding of Counting (Hannah Slovin and Barbara J. Dougherty); (28) Adding Fractions Using "Half" as an Anchor for Reasoning (Alina Galvao Spinillo and Maria Soraia Silva Cruz); (29) Persistence of Decimal Misconceptions and Readiness to Move to Expertise (Vicki Steinle and Kaye Stacey); (30) Molly and Equations in A2: A Case Study of Apprehending Structure (Nada Stehlikova and Adrian Simpson); (31) Image--Metaphor--Diagram: Visualisation in Learning Mathematics (Rudolf Straber and Gert Kadunz); (32) Images of Fractions "as" Processes and Images of Fractions "in" Processes (Marek Sulista, Jan Herman, Lucia Ilucova, Veronika Kremsova, Jiri Pribyl, Janka Ruppeldtova, Adrian Simpson, Nada Stehlikova, and Michaela Ulrychova); (33) Describing Elements of Mathematics Lessons that Accommodate Diversity in Student Background (Peter Sullivan, Judy Mousley, and Robyn Zevenbergen); (34) Orchestrating Mathematical Proof through the Use of Digital Tools (Rosamund Sutherland, Federica Olivero, and Marnie Weeden); (35) School-Based Community of Teachers and Outcomes for Students (Paola Sztajn, Martha Alexsaht-Snider, Dorothy Y. White, and Amy Hackenberg); (36) Thinking through Three Worlds of Mathematics (David Tall); (37) The Effect of Students' Roles on the Establishment of Shared Meanings During Problem Solving (Konstantinos Tatsis and Eugenia Koleza); (38) Integrating CAS Calculators into Mathematics Learning: Partnership Issues (Michael O. J. Thomas and Ye Yoon Hong); (39) The Development of Structure in the Number System (Noel Thomas); (40) Orientations to Numeracy: Teachers' Confidence and Disposition to Use Mathematics Across the Curriculum (Steve Thornton and John Hogan); (41) Efficiency and Adaptiveness of Multiple School-Taught Strategies in the Domain of Simple Addition (Joke Torbeyns, Lieven Verschaffel, and Pol Ghesquiere); (42) Supporting Teachers on Developing Teaching Norms Based on Children's Learning Mathematics (Wen-Huan Tsai); (43) Prospective Teachers' Images and Definitions: The Case of Inflection Points (Pessia Tsamir and Regina Ovodenko); (44) Fostering Mathematical Meaning via Scientific Inquiry: A Case Study (Ron Tzur, Rita A. Hagevik, and Mary E. Watson); (45) Students Building Isomorphisms (Elizabeth B. Uptegrove and Carolyn A. Maher); (46) How Do High School Students Interpret Parameters in Algebra? (Sonia Ursini and Maria Trigueros); (47) Causes Underlying Pre-Service Teachers' Negative Beliefs and Anxieties about Mathematics (Liisa Uusimaki and Rod Nason); (48) Fractions in Adult's Elementary School: The Lucina (Marta Elena Valdemoros-Alvarez); (49) Students' Overreliance on Proportionality: Evidence from Primary School Pupils Solving Arithmetic Word Problems (Wim Van Dooren, Dirk De Bock, An Hessels; Dirk Janssens, and Lieven Verschaffel); (50) From Functions to Equations: Introduction of Algebraic Thinking to 13 Year-Old Students (Petros Verikios, Vasiliki Farmaki, and Nikos Nikoudatos); (51) Critical Awareness of Voice in Mathematics Classroom Discourse: Learning the Steps in the "Dance of Agency" (David Wagner); (52) Metaphors and Cultural Models Afford Communication Repairs of Breakdowns between Mathematical Discourses (G. D. Wake and J. S. Williams); (53) Generalising Arithmetic: Supporting the Process in the Early Years (Elizabeth Warren); (54) A Framework for Describing the Processes that Undergraduates Use to Construct Proofs (Keith Weber); (55) The Nature of Spontaneity in High Quality Mathematics Learning Experiences (Gaye Williams); (56) Spreadsheet Generalising and Paper and Pencil Generalising (Kirsty Wilson, Janet Ainley, and Liz Bills); (57) A Comparison of a Visual-Spatial Approach and a Verbal Approach to Teaching Mathematics (Pamela Woolner); (58) Differentiation of Students' Reasoning on Linear and Quadratic Geometric Number Patterns (Kai-Lin Yang and Fou-Lai Lin); (59) Map Construction as a Context for Studying the Notion of Variable Scale (Nikoleta Yiannoutsou and Chronis Kynigos); (60) Understanding How the Concept of Fractions Develops: A Vygotskian Perspective (Kaori Yoshida); (61) Secondary Mathematics Teachers' Knowledge Concerning the Concept of Limit and Continuity (T. Zachariades and E. Mastorides); (62) Characteristics of Mathematical Problem Solving Tutoring in an Informal Setting (Orit Zaslavsky and Iris Zodik); (63) Making Sense of Irrational Numbers: Focusing on Representation (Rina Zazkis and Natasa Sirotic); and (64) Numeracy Practices of Young Workers (Robyn Zevenbergen and Kelly Zevenbergen). (Individual papers contain references.)

The aim of the study (Johnsen Hoines 2002) reported here was to shed light on the process through which student teachers express and interpret their understandings about mathematical notions and thereby gain insight into them. This paper focuses on how students cooperate and move between different ways of understandings. It emphases in particular on how differences in the ways of explaining a mathematical content make different conditions for the dialogue and learning processes that develop. [For complete proceedings, see ED 489538].

This paper presents the early results of an experiment involving a class of elementary student teachers within the context of their mathematics preparation. The motivation of the exercise centered on giving them an experience with mathematical research at their own level and ascertaining its impact on their attitudes and beliefs. The students spent the first month working on open-ended geometrical topics. In the second month, working alone or in groups of up to four, they chose one or more of these topics then worked on a problem of their own design. The students spent the class time developing their ideas using strategies such as generating examples and non-examples, generalising, etc. Reference to books was not accepted as a research tool, but the instruction team monitored student progress and was available for questions. [For complete proceedings, see ED489538.]

This document contains the first volume of the proceedings of the 28th Annual Conference of the International Group for the Psychology of Mathematics Education. Conference presentations are centered around the theme "Inclusion and Diversity". In total, 147 presentations centered around the vision of mathematics for all. This volume features eight plenary and five research forum papers, and papers from four groups of discussions groups, working sessions, and short orals and poster presentations. Plenary Lectures; Plenary Panel compilations; and Research Forum compilations: (1) Classroom Business Same as Usual? (What) Do Researchers and Policy Makers Learn from Classroom Studies? (Kirsti Klette); (2) Grappling with Complexity: Co-Learning in Inquiry Communities in Mathematics Teaching Development (Barbara Jaworski); (3) Class, Diversity, and Race in Research on the Psychology of Mathematics Learning (Arthur B. Powell); (4) From Diversity to Inclusion and Back: A Lens on Learning (Rina Hershkowitz); (5) Lives, learning and Liberty: The Impact and Responsibilities of Mathematics Education (Peter Gates); (6) Do We Welcome Children's Mathematics? Who are We? (Marit Johnsen Hoines); (7) Learning (and Researching) as Participation in Communities of Practice (Madalena Pinto dos Santos); (8) Researching and Learning Mathematics at the Margin: From "Shelter" to School (Renuka Vithal); (9) Affect in Mathematics Education--Exploring Theoretical Frameworks (Markku Hannula, Jeff Evans, George Philippou, and Rosetta Zan); (10) Algebraic Equations and Inequalities: Issues for Research and Teaching (Luciana Bazzini and Pessia Tsamir); (11) International Perspectives on the Nature of Mathematical Knowledge for Secondary Teaching: Progress and Dilemmas (Helen Doerr and Terry Wood); (12) Contrasting Comparative Research on Teaching and Learning in Mathematics (Jonas Emanuelsson and David Clarke); and (13) Researching Mathematics Education in Multilingual Contexts: Theory, Methodology and Teaching Mathematics. (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.]

This second volume of the 31st annual proceedings of the International Group for the Psychology of Mathematics Education conference presents research reports for author surnames beginning Alc- through Hal-. Reports include: (1) How Do Your Students Think about Proof? A DVD Resource for Mathematicians (Lara Alcock); (2) Teachers' Conceptions of Mathematical Challenge in School Mathematics (Mark Applebaum and Roza Leikin); (3) Semiotic Games: The Role of the Teacher (Ferdinando Arzarello and Domingo Paola); (4) Examples, a Missing Link (Amir H. Asghari); (5) Scaffolding Revisited: From Tool for Result to Tool-and-Result (Mike Askew); (6) How Can We Assess Mathematical Understanding? (Patrick Barmby, Tony Harries, Steve Higgins, and Jennifer Suggate); (7) The Discursive Construction of Mathematical Thinking: The Role of Researchers' Descriptions (Richard Barwell); (8) Authority and Esteem Effects of Enhancing Remote Indigenous Teacher-Assistants' Mathematics-Education Knowledge and Skills (Annette R. Baturo, Tom J. Cooper, and Katherine Doyle); (9) Raising Students' Understanding: Linear Algebra (Marianna Bogomolny); (10) Informal Conceptions of Distribution Held by Elementary Preservice Teachers (Daniel L. Canada); (11) First Graders' Strategies for Numerical Notation, Number Reading and the Number Concept (Gabrielle A. Cayton and Barbara M. Brizuela); (12) An Exploratory Study of Elementary Beginning Mathematics Teacher Efficacy (Y. L. Chang and S. C. Wu); (13) Preservice Secondary Mathematics Teachers' Knowledge and Inquiry Teaching Approaches (Olive Chapman); (14) Developing and Testing a Scale for Measuring Students' Understanding of Fractions (Charalambos Y. Charalambous); (15) The Effectiveness and Limitation of Reading and Coloring Strategy in Learning Geometry Proof (Ying-Hao Cheng and Fou-Lai Lin); (16) Grade 5/6 Teachers' Perceptions of Algebra in the Primary School Curriculum (Helen L. Chick and Kiri Harris); (17) The Influence of Inquiry-Based MathematicsTeaching on 11th Grade High Achievers: Focusing on Metacognition (Erh-Tsung Chin, Yung-Chi Lin, Chih-Wei Chuang, and Hsiao-Lin Tuan); (18) The Effects of "Spatial Geometry Curriculum with 3D DGS" in Lower Secondary School Mathematics (Kimiho Chino, Tatsuo Morozumi, Hitoshi Arai, Fumihiro Ogihara, Yuichi Oguchi, and Mikio Miyazaki); (19) Mathematics as Mother/Basis of Science in Affect: Analysis of TIMSS 2003 Data (Mei-Shiu Chiu); (20) Mediating Model between Logo and DGS for Planar Curves (Han Hyuk Cho, Min Ho Song, and Hwa Kyung Kim); (21) Comparing Korean and U.S. Third Grade Elementary Student Conceptual Understanding of Basic Multiplication Facts (Insook Chung and Hee-Chan Lew); (22) Self-Monitoring by Lesson Reports from Teachers in Problem-Solving Maths Lessons (Christina Collet, Regina Bruder, and Evelyn Komorek); (23) Mathematics Education and Torres Strait Islander Blocklaying Students: The Power of Vocational Context and Structural Understanding (Tom J. Cooper, Annette R. Baturo, Bronwyn Ewing, Elizabeth Duus, and Kaitlin Moore); (24) Interdisciplinary Learning and Perceptions of Interconnectedness of Mathematics (Ng Kit Ee Dawn, Gloria Stillman, and Kaye Stacey); (25) The Treatment of Addition and Subtraction of Fractions in Cypriot, Irish, and Taiwanese Textbooks (Sean Delaney, Charalambos Y. Charalambous, Hui-Yu Hsu, and Vilma Mesa); (26) The Development of Primary Students' Knowledge of the Structured Number Line (Carmel Diezmann and Tom Lowrie); (27) Reasoning with Metaphors and Constructing an Understanding of the Mathematical Function Concept (Hamide Dogan-Dunlap); (28) Exploring the English Proficiency-Mathematical Proficiency Relationship in Learners: An Investigation Using Instructional English Computer Software (Anthony Essien and Mamokgethi Setati); (29) Vet in the Middle: Catering for Motivational Differences in Vocational Access Courses in Numeracy (Bronwyn Ewing, Annette Baturo, Tom Cooper, Elizabeth Duus, and Kaitlin Moore); (30) The Gendering of Mathematics in Israel and Australia (Helen J. Forgasz and David Mittelberg); (31) 21st Century Children, Numeracy and Technology: An Analysis of Peer-Reviewed Literature (Jillian L. Fox); (32) Teaching and Teacher's Competence with ICT in Mathematics in a Community of Inquiry (Anne Berit Fuglestad); (33) Statistical Inference in Textbooks: Mathematical and Everyday Contexts (Israel Garcia-Alonso and Juan Antonio Garcia-Cruz); (34) Pre-Service Elementary School Teachers' Experiences with the Process of Creating Proofs (Soheila Gholamazad); (35) International Survey of High School Students' Understanding of Key Concepts of Linearity (Carole Greenes, Kyung Yoon Chang, and David Ben-Chaim); and (36) Mathematical Beliefs in Pictures and Words Seen through "Multiple Eyes" (Stefan Halverscheid and Katrin Rolka). (Individual papers contain references.) [For other volumes in the series, see ED499416, ED499418, and ED499419.]

This document contains the fourth volume of the proceedings of the 30th 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 59 research reports by presenters with last names beginning between Kun and Ros: (1) Evaluational Research on a Video-Based In-Service Mathematics Teacher Training Project--Reported Instructional Practice and Judgements on Instructional Quality (Sebastian Kuntze and Kristina Reiss); (2) Constructing a Sinusoidal Periodic Covariation (Chronis Kynigos and Kostas Gavrilis; (3) Modelling Fractions with Area: The Salience of Vertical Partitioning (Andreas O. Kyriakides); (4) Dynamic Visualization and the Case of "Stars in Cage" (Ilana Lavy); (5) Mathematics, Gender, and Large Scale Data: New Directions or More of the Same? (Gilah C. Leder, Helen J. Forgasz, and Peter J. Taylor); (6) Evaluation and Reconstruction of Mathematics Textbooks by Prospective Teachers (Kyung Hwa Lee); (7) Maths, ICT and Pedagogy: An Examination of Equitable Practice in Diverse Contexts (Stephen Lerman and Robyn Zevenbergen); (8) Solving Problems in Different Ways: Teachers' Knowledge Situated in Practice (Anat Levav-Waynberg and Roza Leikin); (9) Persona-Based Journaling: Aligning the Product with the Process (Peter Liljedahl); (10) The Power of Goal Orientation in Predicting Student Mathematics Achievement (Chuan-Ju Lin, Pi-Hsia Hung, and Suwei Lin); (11) Why Do Student Teachers Teach or Not Teach the Professed Values? (Fang-Chi Lin, Chih-Yeuan Wang, Chien Chin, and Gi-Yuan Chang); (12) Investigating Mathematics Learning with the Use of Computer Programmes in Primary Schools (Maria Dolores Lozano, Ivonne Twiggy Sandoval, and Maria Trigueros); (13) Is Subject Matter Knowledge Affected by Experience? The Case of Composition of Functions (Calin A. Lucus); (14) Classroom Factors Supporting Progress in Mathematics (Irene F. Mackay); (15) A Remedial Intervention in Algebra (Laura Maffei and Maria Alessandra Mariotti); (16) The Problem-Solving Element in Young Students' Work Related to the Concept of Area (Joanna Mamona-Downs and Ioannis Papadopoulos); (17) On Students' Conceptions in Vector Space Theory (Mirko Maracci); (18) Towards the Development of a Self-Regulated Mathematical Problem Solving Model (Andri Marcou and Stephen Lerman); (19) Visual Cognition: Content Knowledge and Beliefs of Preschool Teachers (Zvia Markovits, Sherman Rosenfeld, and Bat-Sheva Eylon); (20) An Unexpected Way of Thinking about Linear Function Tables (Mara Martinez and Barbara Brizuela); (21) Levels of Understanding of Patterns in Multiple Representations (Stalo Michael, Iliada Elia, Athanasios Gagatsis, Athina Theoklitou, and Andreas Savva); (22) Can the Spontaneous and Uncritical Application of the Linear Model be Questioned? (Modestina Modestou and Athanasios Gagatsis); (23) Primary Students' Reasoning in Problem Solving and Teachers' Evaluation of Their Arguments (Annita Monoyiou, Xenia Xistouri, and George Philippou); (24) Use of Examples in Conjecturing and Proving: An Exploratory Study (Francesca Morselli); (25) Knowledge Building and Knowledge Forum: Grade 4 Students Collaborate to Solve Linear Generalizing Problems (Joan Moss and Ruth Beatty); (26) Improving Mathematical Knowledge through Modeling in Elementary Schools (Nicholas Mousoulides, Marios Pittalis, and Constantinos Christou); (27) Integrating Concepts and Processes in Early Mathematics: The Australian Pattern and Structure Mathematics Awareness Project (PASMAP) (Joanne Mulligan, Michael Mitchelmore, and Anne Prescott); (28) Embodiment and Reasoning in Children's Invented Calculation Strategies (Carol Murphy); (29) When the Wrong Answer is the "Good" Answer: Problem-Solving as a Means for Identifying Mathematical Promise (Dorit Neria and Miriam Amit); (30) The Concept of Equations: What Have Students Met Before? (Rosana Nogueira de Lima and David Tall); (31)Teaching Becomes You: The Challenges of Placing Identity Formation at the Centre of Mathematics Pre-Service Teacher Education (Kathleen T. Nolan); (32) Structure Sense for University Algebra (Jarmila Novotna, Nad'a Stehlikova and Maureen Hoch); (33) Semiotic Chaining in an Expression Constructing Activity Aimed at the Transition from Arithmetic to Algebra (Masakazu Okazaki); (34) Kikan-Shido: Through the Lens of Guiding Student Activity (Catherine A. O'Keefe, Xu Li Hua, and David J. Clarke); (35) Hiding and Showing Construction Elements in a Dynamic Geometry Software: A Focusing Process (Federica Olivero); (36) Prompting Growth for Prospective Teachers Using Cognitive Dissonance (Jo Clay Olson, Meghan Colasanti, and Karmin Trujillo); (37) Metacognition and Reading--Criteria for Comprehension of Mathematics Texts (Magnus Osterholm); (38) Classification Activities and Definition Construction at the Elementary Level (Cecile Ouvrier-Buffet); (39) Abstraction, Scaffolding and Emergent Goals (Mehmet Fatih Ozmantar and John Monaghan); (40) Cognitive and Metacognitive Performance on Mathematics (Areti Panaoura and Georgia Panaoura); (41) Measuring and Relating Primary Students' Motives, Goals and Performance in Mathematics (Marilena Pantziara and George Philippou); (42) The Intuitive Rule More A-More B: The Impact of a Dynamic Geometry Software (Pandelitsa Papageorgiou, Annita Monoyiou, and Demetra Pitta-Pantazi); (43) The Reflective Abstraction in the Construction of the Concept of the Definite Integral: A Case Study (Theodorus Paschos and Vassiliki Farmaki); (44) Infinity of Numbers: How Students Understand It (Erkki Pehkonen, Markku S. Hannula, Hanna Maijala, and Riitta Soro); (45) The Assessment of Undergraduate Mathematicians: Recrafting Assessment of Learning to Provide Opportunities for Assessment as Learning (Hillary Povey and Corinne Angier); (46) Social Cognition Emerging from Student-to-Student Discursive Interactions during Mathematical Problem Solving (Arthur B. Powell); (47) Understanding Teachers' Mathematical Knowledge for Teaching: A Theoretical and Methodological Approach (Arthur B. Powell and Evelyn Hanna); (48) Continuities and Discontinuities for Fractions: A Proposal for Analysing in Different Levels (Susanne Prediger); (49) Dynamic Manipulation Schemes of Geometrical Constructions: Instrumental Genesis as an Abstraction Process (Giorgos Psycharis); (50) Rhythm and the Grasping of the General (Luis Radford, Caroline Bardini, and Cristina Sabena); (51) Developing Mathematical Initiative in Minority Students (Iuliana Radu, Barbara Tozzi, and Keith Weber); (52) A Model of Textbook Use (Sebastian Rezat); (53) Identical Twins' Perceptions of Two Different Instructional Approaches to Learning Mathematics (Candice L. Ridlon); (54) When Successful Comparison of Decimals Doesn't Tell the Full Story (Anne Roche and Doug M. Clarke); (55) Analysis of Proofs Produced by University Mathematics Students, and the Influence of Using Cabri Software (Felix Rodriguez and Angel Gutierrez); (56) Challenging the Mathematical Beliefs of Preservice Elementary School Teachers (Katrin Rolka, Bettina Rosken, and Peter Liljedahl); (57) Partial Knowledge Constructs for the Probability Area Model (Gila Ron, Tommy Dreyfus, and Rina Hershkowitz); (58) A Picture is Worth a Thousand Words--The Role of Visualization in Mathematics Learning (Bettina Rosken and Katrin Rolka); and (59) Establishing and Justifying Algebraic Generalization at the Sixth Grade Level (Joanne Rossi Becker and Ferdinand Rivera). (Individual papers contain references.)

This document contains the fifth volume of the proceedings of the 30th 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 59 research reports by presenters with last names beginning between Sac and Zaz: (1) Bryan's Story: Classroom Miscommunication about General Symbolic Notation and the Emergence of a Conjecture during a CAS-Based Algebra Activity (Ana Isabel Sacristan and Carolyn Kieran); (2) A Teacher's Method to Introduce Story-Problems: Student-Generated Problems (Adalira Saenz-Ludlow); (3) "There's More than Meets the Eye": Analysing Verbal Protocols, Gazes and Sketches on External Mathematical Representations (Jonathan P. San Diego, James Aczel, Barbara Hodgson, and Eileen Scanlon); (4) Notions of Variability in Chance Settings (Ernesto Sanchez and Miguel Mercado Martinez); (5) Elementary Preservice Teacher Learning Levels (Victoria Sanchez and Miguel Mercado Martinez); (6) Mathematics Achievement: Sex Differences vs. Gender Differences (Dora Santos, Sonia Ursini, Martha Patricia Ramirez, and Gabriel Sanchez); (7) Self Assessment and Appropriation of Assessment Criteria (Leonor Santos and Anabela Gomes); (8) Drawing as Problem-Solving: Young Children's Mathematical Reasoning through Pictures (Carole Saundry and Cynthia Nicol); (9) The Role of the Teacher in Turning Claims to Arguments (Baruch Schwarz, Rina Hershkowitz, and Shirly Azmon); (10) Learning Mathematics for Teaching (Nanette Seago and Lynn Goldsmith); (11) Coherence of Mathematics Lessons in Japanese Eighth-Grade Classrooms (Yasuhiro Sekiguchi); (12) Triangle Property Relationships: Making the Connections (Penelope Serow); (13) Access to Mathematics versus Access to the Language of Power (Mamokgethi Setati); (14) Compartmentalization of Representation in Tasks Related to Addition and Subtraction Using the Number Line (Myria Shiakalli and Athanasios Gagatsis); (15)The Derivation of a Learning Assessment Framework for Multiplicative Thinking (Dianne Siemon, John Izard, Margarita Breed, and Jo Virgona); (16) Sources of Students' Frustration in Bridging Mathematics Courses (Anna Sierpinska); (17) What Makes a Good Problem? An Aesthetic Lens (Nathalie Sinclair and Sandra Crespo); (18) Discovering a Rule and Its Mathematical Justification in Modeling Activities Using Spreadsheet (Hong-chan Son and Hee-chan Lew); (19) Investigating Preservice Teachers' Understanding and Strategies on a Student's Errors of Reflective Symmetry (Ji-Won Son); (20) Exploring the Role Played by the Remainder in the Solution of Division Problems (Alina Galvao Spinillo and Sintria Labres Lautert); (21) Incomplete or Incorrect Understanding of Decimals: An Important Deficit for Student Nurses (Vicki Steinle and Robyn Pierce); (22) Proportional Reasoning: Variable Influencing the Problems Difficulty Level and One's Use of Problem Solving Strategies (Olof Bjorg Steinthorsdottir); (23) The Impact of the Intuitive Rule "If A Then B, If Not A Then Not B" in Perimeter and Area Tasks (Lambros Stephanou and Demetra Pitta-Pantazi); (24) Process-Object Difficulties in Linear Algebra: Eigenvalues and Eigenvectors (Sepideh Stewart and Michael O. J. Thomas); (25) Conceptual Changes as Dialectical Transformation (Nadia Stoyanova Kennedy); (26) Content Knowledge for Mathematics Teaching: The Case of Reasoning and Proving (Andreas J. Stylianides and Gabriel J. Stylianides); (27) "Making Proof Central to Pre-High School Mathematics Is an Appropriate Instructional Goal": Provable, Refutable, or Undecidable Proposition? (Gabriel J. Stylianides and Andreas J. Stylianides); (28) Power and Poverty--Whose, Where, and Why?: School Mathematics, Context and the Social Construction of "Disadvantage" (Dalene M. Swanson); (29) Documenting Learning in School-Based Mathematics Communities of Teachers (Paola Sztajn); (30) Research and Teaching--Can One Person Do Both? A Case Study (MichalTabach); (31) Computer "Knowledge" and Student's Images of Figures: The Case of Dragging (Varda Talmon and Michal Yerushalmy); (32) Graphics Calculators for Mathematics Learning in Singapore and Victoria (Australia): Teachers' Views (Hazel Tan and Helen J. Forgasz); (33) Vague Language in Greek and English Mathematical Talk: A Variation Study in Face-Work (Konstantinos Tatsis and Tim Rowland); (34) Teachers Using Computers in Mathematics: A Longitudinal Study (Michael O. J. Thomas); (35) Analysing Classroom Interactions Using Critical Discourse Analysis (Steve Thornton and Noemi Reynolds); (36) Pre-Service and In-Service Mathematics Teachers' Concept Images of Radian (Tahsin Topcu, Mahmut Kertil, Hatice Akkoc, Kamil Yilmaz, and Osman Onder); (37) Adaptive Expertise in the Number Domain 20-100 (Joke Torbeyns, Lien Vanderveken, Lieven Verschaffel, and Pol Ghesquiere); (38) Mathematical Activity in a Technological Workplace: Results from an Ethnographic Study (Chrissavgi Triantafillou and Despina Potari); (39) Pupils' Over-Use of Proportionality on Missing-Value Problems: How Numbers May Change Solutions (Wim Van Dooren, Dirk De Bock, Marleen Evers, and Lieven Verschaffel); (40) Arithmetical Procedures in the Solution of a Problem Involving Velocity (Veronica Vargas and Jose Guzman); (41) Introducing Algebraic Thinking to 13 Year-Old Students: The Case of the Inequality (Petros Verikios and Vassiliki Farmaki); (42) Why Is a Discontinuous Function Differentiable? (Antti Viholainen); (43) Overgeneralization of Linear Models: Presence and Characteristics among University Students (Monica Villareal, Cristina Esteley, and Humberto Alagia); (44) Educational Design Research in Mozambique: Starting Mathematics from Authentic Resources (Pauline Vos, Tiago G. Devesse, and Assane Rassul); (45) "The Big Test": A School Community Experiences Standardized Mathematics Assessment (Fiona Walls); (46) Numeracy Reform in New Zealand: Factors that Influence Classroom Enactment (Margaret Walshaw and Glenda Anthony); (47) An Investigation of Factors Influencing Teachers' Scoring Student Responses to Mathematics Constructed-Response Assessment Tasks (Ning Wang and Jinfa Cai); (48) Teacher Actions that Assist Young Students to Write Generalizations in Words and in Symbols (Elizabeth Warren); (49) Promoting Pre-Service Teachers' Understanding of Decimal Notation and Its Teaching (Wanty Widjaja and Kaye Stacey); (50) Impetus to Explore: Approaching Operational Deficiency Optimistically (Gaye Williams); (51) Generating and Evaluating Geometry Conjectures with Self-Directed Experiments (Chao-Jung Wu, Wing-Kwong Wong, Ying-Hao Cheng, and Yunn-Wen Lien); (52) The Distributions of van Hiele Levels of Geometric Thinking among 1st and 6th Graders (Der-bang Wu and Hsiu-Ian Ma); (53) Elementary Teacher Education and Teacher Efficacy toward Mathematics and Science (Su-Chiao Wu and Yu Liang Chang); (54) Spatial Rotation and Perspective Taking Abilities in Relation to Performance in Reflective Symmetry Tasks (Xenia Xistouri and Demetra Pitta-Pantazi); (55) Elementary Teachers' Mathematics Beliefs and Teaching Practices after a Curriculum Reform (Shirley M. Yates); (56) Inquiry Activities in a Classroom: Extra-Logical Processes of Illumination vs. Logical Process of Deductive and Inductive Reasoning. A Case Study (Oleksiy Yevdokimov); (57) Developments of a Child's Fraction Concepts with the Help of Psychological Tools: A Vygotsky's Cultural-Historical Perspective (Kaori Yoshida); (58) A Teacher's Treatment of Examples as Reflection of Her Knowledge-Base (Orit Zaslavsky, Guershon Harel, and Alfred Manaster); and (59) Cognitive Conflict and its Resolution via Pivotal/Bridging Example (Rina Zazkis and Egan Chernoff). (Individual papers contain references.)

This document contains the third volume of the proceedings of the 30th 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 Ead and Kou: (1) Exploring Realistic Mathematics Education in English Schools (Frank Eade and Paul Dickinson); (2) Reasoning about Non-Linearity in 6- to 9-Year-Olds: The Role of Task Presentation (Mirjam Ebersbach and Wilma C. M. Resing); (3) Teachers, Classroom, Students--A Holistic View on Beliefs Concerning the Educational Practice (Andreas Eichler); (4) The Effects of Different Modes of Representation on Problem Solving: Two Experimental Programs (Iliada Elia and Athanasios Gagatsis); (5) Can Moderate Hermeneutics Help Us to Understand Learning and Teaching in the Mathematics Classroom? (Fiona Ell); (6) Sociocultural Approaches to Emotions in Mathematics Education: Initial Comparisons (Jeff Evans and Rosetta Zan); (7) Moral Education in the Teaching of Mathematics (Thomas Falkenberg); (8) The Abductive System (Elisabetta Ferrando); (9) Remembering and Imagining: Moving Back and Forth between Motion and Its Representation (Francesca Ferrara); (10) From Verbal Texts to Symbolic Expressions: A Semiotic Approach to Early Algebra (Pier Luigi Ferrari); (11) Verbal Memory Span Limitations as a Factor in Early Mathematical Learning Difficulties (Maureen Finnane); (12) Connecting Algebraic Development to Mathematical Patterning in Early Childhood (Jillian Fox); (13) Aline's and Julia's Stories: Reconceptualizing Transfer from a Situated Point of View (Cristina Frade, Peter Winbourne, and Selma Moura Braga); (14) Insights into Students' Algebraic Reasoning (John Francisco and Markus Hahkioniemi); (15) Proof, Authority, and Agency: Intimations from an 8th Grade Classroom (Michael N. Fried and Miriam Amit); (16) Students' Thoughts about ICT in School Mathematics (Anne Berit Fuglestad); (17) Primary Trainee Teachers' Understanding of Basic Geometrical Figures in Scotland (Taro Fujita and Keith Jones); (18) An Empirical Four-Dimensional Model for the Understanding of Function (Athanasios Gagatsis, Iliada Elia, Areti Panaoura, Katerina Gravvani, and Panayotis Spyrou); (19) The Hidden Side in Taiwanese Classrooms--Through the Lens of PLS in Geometry (Hagar Gal, Fou-Lai Lin, and Jia-Ming Ying); (20) The Zero and Negativity among Secondary School Students (Aurora Gallardo and Abraham Hernandez); (21) Students' Actions in Open and Multiple-Choice Questions Regarding Understanding of Averages (Juan Antonio Garcia Cruz and Alexandre Joaquim Garrett); (22) Paradoxes: The Interplay of Gender, Social Class and Mathematics in the Caribbean (Patricia George); (23) Insights about the Addition Strategies Used by Grade 1 and Grade 2 Children Who Are Vulnerable in Number Learning (Ann Gervasoni); (24) A Generic Organizer for the Enrichment of the Concept Image of Derivative (Victor Giraldo and Luiz Mariano Carvalho); (25) Pupils' Attitudes to Integrated Learning Systems in Mathematics (Chrysanthi Gkolia and Alan Jervis); (26) The Discourse of Logical Necessity: Rules for Action in Pre-School Mathematics (Geraldo Gomes and Luciano Meira); (27) An Online Community of Practice for Pre-Service and Beginning Teachers of Secondary Mathematics (Merrilyn Goos and Anne Bennison); (28) Development of Abstract Mathematical Thinking through Artistic Patterns (Ivona Grzegorczyk and Despina A. Stylianou); (29) Rational Numbers after Elementary School: Realizing Models for Fractions on the Real Line (Stefan Halverscheid, Melanie Henseleit, and Klaus Lies); (30) Student Beliefs about Mathematics Encoded in Pictures and Words (Stefan Halverscheid and Katrin Rolka); (31) Algebra Teachers' Ways of Thinking Characterizing the Mental Act of Problem Posing (Guershon Harel, Boris Koichu, and Alfred Manaster); (32) Mathematical Impairment among Epileptic Children (Izabel Hazin, Jorge T. da Rocha Falcao, and Selma Leitao); (33) Patterns of Participation in Networked Classrooms (Stephen Hegedus, Sara Dalton, Laura Cambridge, and Gary Davis); (34) Teacher Training Students' Probabilistic Reasoning about Compound Stochastic Events in an ICT Environment (Tore Heggem and Kjaerand Iversen); (35) Learning to Prove with Heuristic Worked-out Examples (Aiso Heinze, Kristina Reiss, and Christian GroB); (36) Teacher Actions: Enhancing the Learning of Mental Computation in Year 2 (Ann Heirdsfield and Janeen Lamb); (37) Early Conceptual Thinking (Milan Hejny, Darina Jirotkova, and Jana Kratochvilova); (38) Diversity in the Construction of a Group's Shared Knowledge (Rina Hershkowitz, Nurit Hadas, and Tommy Dreyfus); (39) Structure Sense versus Manipulation Skills: An Unexpected Result (Maureen Hoch and Tommy Dreyfus); (40) Young Children Developing Place Value Understandings (Marj Horne and Sharyn Livy); (41) New Insights into Learning Processes from Some Neuroscience Issues (Donatella Iannece, Maria Mellone, and Roberto Tortora); (42) Fostering Conceptual Mathematical Thinking in the Early Years: A Case Study (Paola Iannone and Anne D. Cockburn); (43) The Role of Mathematical Context in Evaluating Conditional Statements (Matthew Inglis and Adrian Simpson); (44) Knowledge for Teaching Fraction Arithmetic: Partitioning Drawn Representations (Andrew Izsak); (45) Inquiry Community in an Activity Theory Frame (Barbara Jaworski and Simon Goodchild); (46) Gender Differences in Patterns of Strategy Use amongst Secondary School Mathematics Students (Tim Jay); (47) Positioning of a Subject Based and Investigative Dialogue in Practice Teaching (Marit Johnsen Hoines and Beate Lode); (48) Everyday Computer-Based Maths Teaching: The Predominance of Practical Activities (Helga Jungwirth); (49) Facilitators for Change of Elementary Teacher Student's View of Mathematics (Raimo Kaasila, Markku S. Hannula, Anu Laine, and Erkki Pehkonen); (50) The Mathematical Beliefs of Teachers about Applications and Modelling--Results of an Empirical Study (Gabriele Kaiser); (51) An Analysis of Solving Groups of Problems (toward the Study of Problem Solving Instruction) (Alexander Karp); (52) Advancing Low Achievers in Secondary Schools by Using Non-Professional Mathematics Tutoring (Ronnie Karsenty); (53) Mathematical Knowledge for Teaching: Adding to the Description through a Study of Probability in Practice (Mercy Kazima and Jill Adler); (54) Measurements with a Physical and a Virtual Quadrant: Students' Understanding of Trigonometric Tangent (Stefanos Keisoglou and Chronis Kynigos); (55) Language and Mathematics: Towards Social Justice for Linguistically Diverse Students (Lena Licon Khisty); (56) Concept Definition, Concept Image and the Discrete-Continuous Interplay (Ivy Kidron and Thierry Dana Picard); (57) Reconceiving Strategic Knowledge in Proving from the Student's Perspective (Jessica Knapp and Keith Weber); (58) Patterns of Middle School Students' Heuristic Behaviors in Solving Seemingly Familiar Problems (Boris Koichu, Abraham Berman, and Michael Moore); (59) Improper Proportional Reasoning: A Comparative Study in High School (Katerina Kontoyianni, Modestina Modestou, Maria Erodotou, Polina Ioannou, Athinos Constantinides, Marinos Parisinos, and Athanasios Gagatsis); and (60) Integer Instruction: A Semiotic Analysis of the "Compensation Strategy" (Andreas Koukkoufis and Julian Williams). (Individual papers contain references.)

In extra-curriculum activities, the nature of the instructor-learner relationship differs from that in class. This constellation, if accompanied by a smaller gap between the mathematical skills of tutor and learner, has an impact on motivational aspects and on the choice of contents in these activities: The use of problem questions of competition type often plays a dominant role as a means to include many individuals in these activities. A survey is presented which indicates two risks of that approach: On the long term, it seems to have a negative effect on the motivation to work on open or more complex problems. Furthermore, it tends to discourage those with individual reference norm and motivation to participate that is intrinsic and directed on the mathematical content. (Contains 2 tables.) [For complete proceedings, see ED 489538.]

This research report describes the findings of a study on teachers' ways of interpreting student responses to tasks involving equivalent expressions. The teachers in this study were engaged in model-eliciting activities designed to promote the development of their knowledge and reveal their models (or interpretations) of their students' algebraic thinking about equivalent expressions by creating a library of their students' work. This report focuses on one teacher's model of his algebraic practice. Results showed that this teacher devoted a significant amount of time to the implementation of the algebraic unit. The teacher employed visual strategies for the first time and began to perceive their usefulness in demonstrating the equivalency of two expressions. (Contains 2 figures.) [For complete proceedings, see ED489178.]

In 1943 Jacques Hadamard gave a series of lectures on mathematical invention at the Ecole Libre des Hautes Etudes in New York City. These talks were subsequently published as The Psychology of Mathematical Invention in the Mathematical Field (Hadamard, 1945). In this article I present a study that mirrors the work of Hadamard. Results both confirm and extend the work of Hadamard on the inventive process. In addition, the results also speak to the larger context of "doing" and learning mathematics. [For complete proceedings, see ED489538.]

We present two vignettes of classroom episodes that exemplify new activity structures for introducing core algebra ideas such as linear functions, slope as rate and parametric variation within a new educational technology environment that combines two kinds of classroom technology affordances, one based in dynamic representation and the other based in connectivity. These descriptions of how mathematical and social structures interact in the classroom help account for significant algebra learning gains in recent SimCalc teaching experiments among 13-16 year old students. (Contains 1 table and 2 figures.) [For complete proceedings, see ED 489538.]

Student learning depends on the teacher's actions, which are, in turn, dependent on the teacher's knowledge base--defined here by three components: knowledge of mathematics content, knowledge of student epistemology, and knowledge of pedagogy. The purpose of this study is to construct models for teachers' knowledge base and for their development in an on-site professional development project. [For complete proceedings, see ED489538.]

This document contains the third volume of the proceedings of the 28th Annual Conference of the International Group for the Psychology of Mathematics. Conference presentations are centered around the theme "Inclusion and Diversity". A total of 65 research reports are presented here: (1) A Teacher's Model of Students Algebraic Thinking About Equivalent Expressions (Jean E. Hallagan); (2) On Motivational Aspects of Instructor-Learner Interactions in Extra-Curriculum Activities (Stefan Halverscheid); (3) Development of Understanding and Self-Confidence in Mathematics, Grades 5-8 (Markku S. Hannula, Hanna Maijala, and Erkki Pehkonen); (4) Mathematics Teachers' Knowledge Base: Preliminary Results (Guershon Harel and Kien H. Lim); (5) The Role of Tool and Teacher Mediations in the Construction of Meanings for Reflection (Lulu Healy); (6) The Proving Process in Mathematics Classroom--Method and Results of a Video Study (Aiso Heinze); (7) Structure Sense in High School Algebra: The Effect of Brackets (Maureen Hoch and Tommy Dreyfus); (8) Explaining Variability in Retrieval Times for Addition Produced by Students with Mathematical Learning Difficulties (Sarah L. Hopkins and Michael J. Lawson); (9) Early Gender Differences (Marj Horne); (10) Perceptual and Symbolic Representations as a Starting Point of the Acquisition of the Derivative (Markus Hahkioniemi); (11) Implementation of a Model Using Authentic Investigative Activities for Teaching Ratio & Proportion in Pre-Service Teacher Education (Bat-Sheva Ilany, Yaffa Keret, and David Ben-Chaim); (12) Mathematicians and the Selection Task (Matthew Inglis and Adrian Simpson); (13) Insight into Pupils' Understanding of Infinity in a Geometrical Context (Darina Jirotkova and Graham Littler); (14) Textual Differences as Conditions for Learning Processes (Marit Johnsen Hoines); (15) Students' Gender Attitudes Towards the Use of Calculators in Mathematics Instruction (L. M. Kaino and E. B. Salani); (16) Teaching Arithmetic and Algebraic Expressions (K. Subramakhi Banerjee); (17) An Introduction to the Profound Potential of Connected Algebra Activities: Issues of Representation, Engagement and Pedagogy (James J. Kaput and Stephen J. Hegedus); (18) The Tension between Teacher Beliefs and Teacher Practice: The Impact of the Work Setting (M. Kerem Karaagac and John Threlfall); (19) What a Simple Task Can Show: Teachers Explore the Complexity of Children's Thinking (Heather Kelleher, Cynthia Nicol, and Carole Saundry); (20) Constructing Knowledge about the Bifurcation Diagram: Epistemic Actions and Parallel Constructions (Ivy Kidron and Tommy Dreyfus); (21) Experiencing Research Practice in Pure Mathematics in a Teacher Training Context (Eva Knoll, Paul Ernest, and Simon Morgan); (22) Enhancing Mathematical Literacy with the Use of Metacognitive Guidance in Forum Discussion (Bracha Kramarski); (23) The Impact of Developing Teacher Conceptual Knowledge on Students' Knowledge of Division (Janeen Lamb and George Booker); (24) Kinds of Arguments Emerging While Exploring in a Computerized Environment (Ilana Lavy); (25) Using Solo to Analyse Group Responses (Christine Lawrie and Chris Reading); (26) Connections between Qualitative and Quantitative Thinking about Proportion: The Case of Paulina (Elena Fabiola Ruiz Ledesma and Marta Elena Valdemoros Alvarez); (27) Towards High Quality Geometrical Tasks: Reformulation of a Proof Problem (Roza Leikin); (28) Mathematical Thinking & Human Nature: Consonance & Conflict (Uri Leron); (29) The Mathematics Pedagogical Values Delivered by an Elementary Teacher in Her Mathematics Instruction: Attainment of Higher Education and Achievement (Yuh-Chyn Leu and Chao-Jung Wu); (30) Visual Reasoning in Computational Environment: A Case of Graph Sketching (Allen Leung and King Wah Chan); (31) Elementary School Students' Use of Mathematically-Based and Practically-Based Explanations: The Case of Multiplication (Esther Levenson, Dina Tirosh, and Pessia Tsamir); (32) Mathematical Discovery: Hadamard Resurrected (Peter Liljedahl); (33)Supporting Teachers on Designing Problem-Posing Tasks as a Tool of Assessment to Understand Students' Mathematical Learning (Pi-Jen Lin); (34) Prospective Elementary School Teachers' Solution Strategies and Reasoning for a Missing Value Proportion Task (Jane Lo); (35) Introduction of Calculus in 12th Grade: The Role of Artefacts (Michela Maschietto); (36) The Critical Role of Institutional Context in Teacher Development (Kay McClain and Paul Cobb); (37) Students' Perceptions of Factors Contributing to Successful Participation in Mathematics (Andrea McDonough, Peter Sullivan, and Robyn Turner Harrison); (38) The Cognitive--Motivational Profiles of Students Dealing with Decimal Numbers and Fractions (Kaarina Merenluoto); (39) The Number Line as a Representation of Decimal Numbers: A Research with Sixth Grade Students (Niki Michaelidou, Athanasios Gagatsis, and Demetra Pitta-Pantazi); (40) Preservice Teachers Conceptions of Mathematics-Based Software (James Middleton, Terri Kurz, and H. Bahadir Yanik); (41) Helping Children to Model Proportionally in Group Argumentation: Overcoming the "Constant Sum" Error (Christina Misailidou and Julian Williams); (42) Abstraction in Mathematics and Mathematics Learning (Michael Mitchelmore and Paul White); (43) Reflective Symmetry in Construction and Proving (Takeshi Miyakawa); (44) Students' Improper Proportional Reasoning: The Case of Area and Volume of Rectangular Figures (Modestina Modestou, Athanasios Gagatsis, and Demetra Pitta-Pantazi); (45) Abstraction and Consolidation (John Monaghan and Mehmet Fatih Ozmantar); (46) Critical Sense in Interpretations of Media Graphs (Carlos Monteiro and Janet Ainley); (47) Between Affect and Cognition: Proving at University Level (Francesca Morselli and Fulvia Furinghetti); (48) An Aspect of Mathematical Understanding: The Notion of "Connected Knowing" (Judith Mousley); (49) Algebraic and Geometric Approach in Function Problem Solving (Nikos Mousoulides and Athanasios Gagatsis); (50) Children's Development of Structure in Early Mathematics (Joanne Mulligam, Anne Prescott, and Michael Mitchelmore); (51) On the Fragile, Yet Crucial, Relationship between Mathematicians and Researchers in Mathematics Education (Elena Nardi and Paola Iannone); (52) Students' Preference of Non-Algebraic Representations in Mathematical Communication (Dorit Neria and Miriam Amit); (53) Learning to See in Mathematics Classrooms (Cynthia Nicol and Sandra Crespo); (54) Students' Ways of Interpreting Aspects of Chance Embedded in a Dice Game (Per Nilsson); (55) The Impact of State-Wide Numeracy Testing on the Teaching of Mathematics in Primary Schools (Steven Nisbet); (56) "The Poetry of the Universe": New Mathematics Teachers' Metaphoric Meaning-Making (Andrew Noyes); (57) What Students Do When Hearing Others Explaining (Kazuhiko Nunokawa and Satoshi Ohzeki); (58) Changes in Teachers' Practices While Assuming New Leadership Roles (Jo Clay Olson); (59) Students' Structuring of Rectangular Arrays (Lynne Outhred and Michael Mitchelmore); (60) Construction of Mathematical Definitions: An Epistemological and Didactical Study (Cecile Ouvrier-Buffet); (61) Mathematical Abstraction through Scaffolding (Mehmet Fatih Ozmantar and Tom Roper); (62) The Use of Diagrams in Solving Non Routine Problems (Marilena Pantziara, Athanasios Gagatsis, and Demetra Pitta-Pantazi); (63) Designing for Local and Global Meanings of Randomness (Efi Paparistodemou and Richard Noss); (64) Integrating the History of Mathematics in Educational Praxis: An Euclidean Geometry Approach to the Solution of Motion Problems (Theodorus Paschos, Vassilike Farmaki, and Nikos Klaudatos); and (65) The Magic Circle of the Textbook--An Option or an Obstacle for Teacher Change (Leila Pehkonen). (Individual papers contain references.)

The project's focus was to learn more about the ways to best assess and support the development of numeracy in the early grades. Project goals include involving teachers and researchers in: (1) the creation and use of performance-based tasks most appropriate for assessing numeracy in young learners; (2) the development and refinement of instructional strategies to support numeracy in school and at home; and (3) the development of reference standards on key assessment items that provide a portrait of young students' mathematical thinking. [For complete proceedings, see ED489538.]

The purpose of the present study was to investigate the effects of forum discussion embedded within metacognitive guidance on mathematical literacy. In particular the study compares two learning environments: (a) Forum discussion with metacognitive guidance (FORUM+META); and (b) Forum discussion without metacognitive guidance (FORUM). Participants were 43 seventh-grade students (boys and girls) who practiced online problem solving in two classes. It was found that students who were exposed to FORUM+META discussion outperformed students that were not exposed to metacognitive guidance (FORUM discussion) on mathematical literacy. The effects were observed on various aspects of solving real-life tasks: (a) Understanding the task; (b) Using mathematical strategies; (c)Processing information; and (d) Using mathematical reasoning. A section entitled The Pizza Task is appended. [For complete proceedings, see ED489538.]

I have different approaches to this contribution. My concerns as a teacher and a teacher educator are to discuss how we manage to organise for inclusion of the variety of children's mathematics. How do we organise for the mathematics to be included into their mathematics? A message from children about what they expect mathematics to be has impact on this complexity. Their voice is a voice of the culture and affects the position of "we". What impact has the authoritarian nature of the mathematics for our discussions about diversity and inclusion? [For complete proceedings, see ED489178]

This volume of the 27th International Group for the Psychology of Mathematics Education Conference includes the following research reports: (1) The Affective Views of Primary School Children (Peter Grootenboer); (2) Theoretical Model of Analysis of Rate Problems in Algebra (Jose Guzman, Nadine Bednarz and Fernando Hitt); (3) Locating Fractions on A Number Line (Markku S. Hannula); (4) Preservice Teachers' Conceptions about Y=X+5: Do They See a Function? (Orjan Hansson and Barbro Grevholm); (5) Daring to Ask the Hard Questions: The Effect of Clinical Interview Training upon Teachers Classroom Questioning (Hanna Haydar); (6) Reducing Abstraction: The Case of Elementary Mathematics (Orit Hazzan and Rina Zazkis); (7) The Effect of a Simcalc Connected Classroom on Students' Algebraic Thinking (Stephen J. Hegedus and James J. Kaput); (8) "Spontaneous" Mental Computation Strategies (Ann Heirdsfield); (9) Notation Issues: Visual Effects and Ordering Operations (Dave Hewitt); (10) Attitudes of Mathematics and Language Teachers Towards New Educational Trends (Marie Hofmannova, Jarmila Novotna and Zuzana Hadj-Moussova); (11) Gender Differences in the Early Years in Addition and Subtraction ( Marj Horne); (12) High Achieving Girls in Mathematics: What's Wrong with Working Hard? (Ann C. Howe and Sarah B. Berenson); (13) Mental Functioning of Instruments in the Learning of Geometrical Transformations (Veronica Hoyos); (14) A Perspective for Examining the Link between Problem Solving and Problem Posing (Stephen Hwang and Jinfa Cai): (15) Multiplicative Strategies of New Zealand Secondary School Authors (Kathryn Irwin); (16) Effective Vs. Efficient: Teaching Methods of Solving Linear Equations (Kathy M. C. Ivey); (17) Student's Concept of Infinity in the Context if a Simple Geometrical Construct (Darina Jirotkova and Graham H. Littler); (18) Beyond Discourse: A Multimodal Perspective of Learning Mathematics in a Multilingual Context ( Lena Licon Khisty, Hector Morales and Kathryn Chval); (19) Spontaneous Emergence of Elementary Number-Theoretic Concepts and Techniques in Interaction with Computing Technology (Carolyn Kieran and Jose Guzman); (20) Secondary School Mathematics Preservice Teachers' Probabilistic Reasoning in Individual and Pair Settings (Hari P. Koirala); (21) Social Transformation of Students' Conceptual Model: Analysis of Students' Use of Metaphor for Differential Equations (Oh-Nam Kwon, Kyoung Hee Cho, Kyung Hee Shin and Jeong Sook Park); (22) 13 Year-Olds' Meanings around Intrinsic Curves with a Medium for Symbolic Expression and Dynamic Manipulation (Kynigos Chronis, and Georgos Psycharis); (23) The Probabilistic Thinking of Primary School Pupils in Cyprus: The Case of Tree Diagrams (Iasonas Lamprianou and Thekla Afantiti Lamprianou); (24) Pre-Service Teachers' Transition from "Knowing That" to "Knowing Why" via Computerized Project-Based-Learning (Ilana Lavy and Atara Shriki): (25) Mathematics Competitions, Gender, and Grade Level: Does Time Make a Difference (Gilah C. Leder and David G. Pederson, Graham H. Pollard); (26) Dynamic Geometry and the Theory of Variation (Allen Leung); (27) Enhancing Teachers' Understanding of Students' Learning by using Assessment Tasks (Pi-Jen Lin); (28) Early Mathematics Teaching: The Relationship between Teachers' Beliefs and Classroom Practices (Francis Lopez-Real and Nirmala Rao); (29) Factors Motivating Reform: Learning from Teachers' Stories (Azita Manouchehri); (30) Difficulties in Vector Space Theory: A Compared Analysis in Terms of Conceptions and Tacit Models (Mirko Maracci); (31) Function and Graph in Dgs Environment (Rossana Falcade, Maria Alessandra Mariotti and Colette Laborde); (32) Collective Mathematical Understanding as an Improvisational Process (Lyndon Martin and Jo Towers); (33) Supporting Teacher Change: A Case from Statistics (Kay McClain); (34) Describing the Practice of Effective Teachers of Mathematics in the Early Years (Andrea McDonough and Doug Clarke); (35) Mathematics Professional Development as the Development of Communities of Practice (Rebecca McGraw, Fran Arbaugh, Kathleen Lynch and Catherine A. Brown); (36) The Emergence of Mathematical Goals in a Recreational Practice (Luciano Meira and Monica Correira); (37) Abstracting the Density of Numbers on the Number Line--A Quasi-Experimental Study (Kaarina Merenluoto); (38) Measuring Children's Proportional Reasoning, The "Tendency" for an Additive Strategy and the Effect of Models (Christina Misailidou and Julian Williams); (39) Mathematicians' Writing (Morten Misfeldt); (40) A Co-Learning Partnership in Mathematics Lower Secondary Classroom in Pakistan: Theory into Practice (Razia Fakir Mohammad); (41) Prospective Elementary Teachers' Misunderstandings in Solving Ratio and Proportion Problems (Cecilia Monteiro); (42) What Counts as Mathematical Discourse? (Judit Moschkovich); (43) Mathematical and Pedagogical Understanding as Situated Cognition (Judith A. Mousley); (44) The Relative Influence of the Teacher in Third Grade Mathematics Classrooms (Hanlie Murray); (45) The Activity of Defining (Talli Nachlieli and Anna Sfard); (46) Collective Learning Structures: Complexity Science Metaphors for Teaching (Immaculate Namukasa); (47) Mathematicians on Concept Image Construction: "Single Landscape" vs "Your Own Tailor-Made Brain Version" (Elena Nardi and Paola Iannone); (48) Learning in and from Practice: Pre-Service Teachers Investigate their Mathematics Teaching (Cynthia Nicol and Sandra Crespo); (49) Mentoring Teaching of Mathematics in Teacher Education (Vivi Nilssen); (50) Getting Organised: The Role of Data Organisation in Students' Representation of Numerical Data (Steven Nisbet); (51) Strong and Weak Metaphors for Limits (Michael C. F. Oehrtman); (52) Applying Theory of Planned Behavior Model on Studying Teachers' Change in Mathematics Instruction (Youngyoul Oh); (53) Characteristics of 5th Graders' Logical Development through Learning Division with Decimals (Masakazu Okazaki); (54) Nathan's Strategies for Simplifying and Adding Fractions in Third Grade (John Olive); (55) Cabri as a Shared Workspace within the Proving Process (Federica Olivero); (56) The Construct Validity of an Inventory for the Measurement of Young Pupils' Metacognitive Abilities in Mathematics (Areti Panaoura and George Philippou); (57) Student-Centered Teaching Practices in Korean Elementary Mathematics Classrooms (Jeong Suk Pang); and (58) On Pupils' Self-Confidence in Mathematics: Gender Comparisons (Erkki Pehkonen, Anu Nurmi, Markku Hannula and Hanna Maijala). An author index is included. (Individual papers contain references.) [For Volume 2 of the 2003 Proceedings, see ED500859. For Volume 4, see ED500860.]

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.]

This volume of the 27th International Group for the Psychology of Mathematics Education Conference presents the following research reports: (1) Text Talk, Body Talk, Table Talk: A Design of Ratio and Proportion as Classroom Parallel Events (Dor Abrahamson); (2) Generalizing the Context and Generalising the Calculation (Janet Ainley); (3) Interview Design for Ratio Comparison Tasks (Silvia Alatorre and Olimpia Figueras); (4) Modeling Outcomes from Probability Tasks: Sixth Graders Reasoning Together (Alice Alston and Carolyn A. Maher); (5) A Web-Based Survey to Assess Prospective Elementary School Teachers' Beliefs about Mathematics and Mathematics Learning: An Alternative to Likert Scales (Rebecca C., Ambrose, Randolph Philipp, Jennifer Chauvot, and Lisa Clement); (6) "Sensing": Supporting Student Understanding of Decimal Knowledge (Glenda Anthony); (7) Non-Examples and Proof By Contradiction (Samuele Antonini); (8) Thematization of the Calculus Graphing (Bernadette Baker, Maria Trigueros, and Laurel Cooley); (9) Attention to Mathematical Structure during Participation in a Mathematics Classroom Task By Learners of English as an Additional Language (EAL) (Richard Barwell); (10) Levels of Sophistication in Elementary Students' Reasoning about Length (Michael T. Battista); (11) Australian Indigenous Students' Knowledge of Two-Digit Numeration: Adding One Ten (Annette Baturo); (12) Using Instructional Representations of Ratio as an Assessment Tool ff Subject Matter Knowledge (Sarah B. Berenson and Rod Nason); (13) A Social Extension of a Psychological Interest Theory (Angelika Bikner-Ahsbahs); (14) Particular and General in Early Symbolic Manipulation (Liz Bills, Janet Ainley and Kirsty Wilson); (15) The Nature of Scaffolding in Undergraduate Students' Transition to Mathematical Proof (Maria Blanton, Despina A. Stylianou, and Manuela David); (16) Children's Conceptions of Infinity of Numbers in a Fifth Grade Classroom Discussion Context (Paolo Boero, Nadia Douek, and Rossella Garuti); (17) Investigating the Mathematics Incorporated in the Real World as a Starting Point for Mathematics Classroom Activities (Cinzia Bonotto); (18) Fourth Graders Solving Equations (Barbara M. Brizuela and Analucia Schliemann); (19) An Examination of How People with Diverse Background Talk about Mathematics Teaching and Learning Both Face-To-Face and On-Line (Catherine A. Brown, and Yusuf Koc); (20) Mathematical Identity in Initial Teacher Training (Tony Brown); (21) Using Research to Inform Practice: Children Make Sense of Division of Fractions (Sylvia Bulgar); (22) Developing and Connecting Calculus Students' Notions of Rate-of-Change and Accumulation: The Fundamental Theorem of Calculus (Marilyn Carson, Nanci Smith, and Joni Persson); (23) Guess My Rule Revisited (David W. Carraher and Darrell S. Earnest); (24) Building Theories: Working In A Microworld and Writing the Mathematical Notebook (Michele Cerulli and Maria Alessandra Mariotti); (25) Teacher Investigations of Students' Work: Meeting the Challenge of Attending to Students' Thinking (Michelle T. Chamberlin); (26) Teachers' Conceptions of Mathematical Word Problems: A Basis for Professional Development (Olive Chapman); (27) Testing A Comprehensive Model for Measuring Problem Solving and Problem Posing Skills of Primary Pupils (Charalambos Charalambous, Leonidas Kyriakides, and George Philippou); (28) Mathematical Proof as Formal Procept in Advanced Mathematical Thinking (Erh-Tsung Chin); (29) A Methodological Approach for Developing Rubrics for Research Purposes (Lisa Clement, Jennifer Chauvot, Randolph Philipp, and Rebecca Ambrose); (30) Curved Solids Nets (Nitsa Cohen); (31) Preservice Elementary Teachers' Solution Processes to Problematic Addition and Subtraction Word Problems Involving Ordinal Numbers and Their Interpretations of Solutions (Jose N. Contreras and Armando M. Martinez-Cruz); (32) Open-Ended Realistic Division Problems, Generalisation and Early Algebra (Tom J. Cooper and Elizabeth Warren); (33) A Cognitive Model of Experts' Algebraic Solving Methods (Anibal Cortes); (34) Learning to Investigate Students' Mathematical Thinking: The Role of Student Interviews (Sandra Crespo and Cynthia Nicol); (35) Affective Aspects on Mathematics Conceptualization: From Dichotomies to an Integrated Approach (Falcao Da Rocha, Jorge Tarcisio, Claudia Roberta De Araujo, Fernanda Andrade, Izabel Hazin, Jorge Costa Do Nascimento, and Monica Maria Lins Lessa); (36) What Can We Learn about Cognitive Learning Processes by Asking the Pupils? (Bettina Dahl); (37) Knowledge Sharing Systems: Advantages of Public Anonymity and Private Accountability (Sarah M. Davis); (38) Secondary School Students' Improper Proportional Reasoning: The Role of Direct Versus Indirect Measures (Dirk De Bock, Wim Van Dooren, Elke De Bolle, Dirk Janssens, and Lieven Verschaffel); (39) Thinking in Ordinary Lessons: What Happened When Nine Teachers Believed Their Failing Students Could Think Mathematically (Els De Geest, Anne Watson, and Steph Prestage); (40) Towards a Redefinition of the Mathematics Culture in the Classroom (Inaqui De Olaizola and Manuel Santos-Trigo); (41) Scaling Up Strategies for Change (Truus Dekker and Els Feijs); (42) Representational Ability and Understanding of Derivative (Delos Santos, Alan Gil and Michael O. J. Thomas); (43) Using Students' Ways of Thinking to Re-Cast the Tasks of Teaching about Functions (Helen M. Doerr); (44) From Oral to Written Texts in Grade 1 and the Approach to Mathematical Argumentation (Nadia Douek and Michel Pichat); (45) Young Children's Understanding of Geometric Shapes: The Role of Geometric Models (Iliada Elia, Athanasios Gagatsis, and Leonidas Kyriakides); (46) Perspective-Taking in Middle-School Mathematical Modelling: A Teacher Case Study (Lyn D. English and Helen M. Doerr); (47) Bridging Mathematical Knowledge from Different Cultures: Proposals for an Intercultural and Interdisciplinary Curriculum (Franco, M. Favilli, Luisa Oliveras, and Margarida Cesar); (48) Metaphors as Vehicles of Knowledge: An Exploratory Analysis (Francesca Ferrara); (49) Equity and Beliefs about the Efficacy of Computers for Mathematics Learning (Helen J. Forgasz); (50) Identifying a Research Agenda: The Interaction of Technology with the Teaching and Learning of Data Analysis and Statistics (Susan N. Friel); (51) To Produce Conjectures and to Prove Them within a Dynamic Geometry Environment: A Case Study (Fulvia Furinghetti and Domingo Paola); (52) "It Is Possible to Die before Being Born". Negative Integers Subtraction: A Case Study (Aurora Gallardo); (53) Evolution of Forms of Representation In a Modelling Activity: A Case Study (Rossella Garuti, Carlo Dapueto, and Paolo Boero); (54) Key Transitions in Counting Development for Young Children Who Experience Difficulty (Ann Gervasoni); (55) One Line Proof: What Can Go Wrong? (Soheila Gholamazad, Peter Liljedahl, and Rina Zazkis); (56) On Line Professional Community Development and Collaborative Discourse In Geometry (Joaquin Gimenez and Marcelo Barrial); and (57) Descriptions and Definitions in the Teaching of Elementary Calculus (Victor Giraldo, Luiz Mariano Carvalho, and David Tall). (Individual papers contain references, tables, and figures.)[For volume 1 of these proceedings, see ED500857.]