The fourth volume of this proceedings contains 29 full research reports continuing on from Volume 3, 84 short oral communications (abstracts only) and 34 poster presentations (abstracts only). The full papers include: (1) "Beliefs, Teacher Education and the History of Mathematics" (George N. Philippou and Constantinos Christou); (2) "Working Class Students and the Culture of Mathematics Classrooms in the UK" (Hilary Povey and Mark Boylan); (3) "The Co-ordination of Meanings for Randomness" (Dave Pratt and Richard Noss); (4) "Learning Obstacles in Differential Equations" (Chris L. Rasmussen); (5) "Images and Definitions for the Concept of Increasing/Decreasing Function" (Shakre Rasslan and Shlomo Vinner); (6) "Why Is Proof by Contradiction Difficult?" (David A. Reid and Joann Dobbin); (7) "Self-Efficacy Beliefs as Mediators in Math Learning: A Structural Model" (Martin Risnes); (8) "Teachers' Pedagogical Content Knowledge of Geometry" (Lynn Rossouw and Eddie Smith); (9) "Conviction, Explanation and Generic Examples" (Tim Rowland); (10) "Children's Multiplicative Problem-Solving Strategies in Real-World Situations" (Silke Ruwisch); (11) "National Tests: Educating Teachers about Their Children's Mathematical Thinking" (J.T. Ryan, J.S. Williams, and B.A. Doig); (12) "Patterns of Mathematical Misunderstanding Exhibited by Calculus Students in a Problem Solving Course" (Manuel Santos-Trigo); (13) "The Evolution of Mathematical Practices: How One First-Grade Classroom learned to measure." (Michelle Stephan and Paul Cobb); (14) "Conceptualizing Mathematics Teaching: The Role of Autonomy in Stimulating Teacher Reflection" (Peter Sullivan and Judith Mousley); (15) "Learning through Reflection with Mature, Low Attaining Students" (Malcolm Swan); (16) "Using Research into Children's Understanding of the Symmetry of Dice in Order to Develop a Model of How They Perceive the Concept of a Random ZGenerator" (John M. Truran); (17) "Students' Solutions of Inequalities" (Pessia Tsamir, Nava Almog, and Dina Tirosh); (18) "Do Equilateral Polygons Have Equal Angles?" (Pessia Tsamir, Dina Tirosh, and Ruth Stavy); (19) "Meaningfully Assembling Mathematical Pieces: An Account of a Teacher in Transition" (Ron Tzur, Martin A. Simon, Karen Heinz, and Margaret Kinzel); (20) "Research Methods of the 'North' Revisited from the 'South'" (Paola Valero and Renuka Vithal); (21) "An Experiment in Developing Proof through Pattern" (Sue Waring, Anthony Orton, and Tom Roper); (22) "What Makes a Mathematical Performance Noteworthy in Informal Teacher Assessment?" (Anne Watson); (23) "Participating in Learning Mathematics through Shared Local Practices" (Peter Winbourne and Anne Watson); (24) "Intuitive Counting Strategies of 5-6 year old Children within a Transformational Arithmetic Framework" (David Womack and Julian Williams); (25) "Differences in Teaching for Conceptual Understanding of Mathematics" (Terry Wood); (26) "Children's Beginning Knowledge of Numerals and Its Relationship to Their Knowledge of Number Words: An Exploratory, Observational Study" (Bob Wright); (27) "A Study of Argumentation in a Second-Grade Mathematics Classroom" (Erna Yackel); (28)"Teacher Perceptions, Learned Helplessness and Mathematics Achievement" (Shirley M. Yates); (29) "Students' Understandings of the Role of Counter-Examples" (Orit Zaslavsky and Gila Ron). (ASK)