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.]