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.]