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.]