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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 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.]

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.)