This book contains volume 1 of the proceedings of the 23rd annual meeting of the International Group for the Psychology of Mathematics Education (PME) held October, 2001 in Snowbird, Utah. Papers include: (1) "Opening the Dimensions of Mathematical Capability: The Development of Knowledge, Practice and Identity in Mathematics Classrooms" (Jo Boaler); (2) "Learning Mathematics as Developing a Discourse" (Anna Sfard); (3) "Gender and Mathematics: Themes Within Reflective Voices" (Diana B. Erchick and Linda Condron); (4) "Representations and Mathematics Visualization" (Fernando Hitt); (5) "Models and Modeling" (Richard Lesh, Helen Doerr, Thomas Post, Judith S. Zawojewski, and Guadalupe Carmona); (6) "Geometry and Technology" (Douglas McDougall and Jean J. McGehee); (7) "The Complexity of Learning to Reason Probabilistically" (Carolyn A. Maher and Robert Speiser); and (8) "Problem Posing Research: Answered and Unanswered Questions" (Lyn D. English). Volume 2 of the proceedings of the 23rd annual meeting of the International Group for the Psychology of Mathematics Education (PME) held October, 2001 in Snowbird, Utah is also included. Papers include: (1) "Growth in Student Mathematical Understanding Through Precalculus Student and Teacher Interactions" (Daniel R. Ilaria and Carolyn A. Maher); (2) "Investigating the Teaching and Learning of Proof: First Year Results" (Tami S. Martin and Sharon Soucy McCrone); (3) "Beliefs About Proof in Collegiate Calculus" (Manya Raman); (4) "Metaphors of the Novice: Emergent Aspects of Calculus Students' Reasoning" (Mike Oehrtman); (5) "An Examination of the Interaction Patterns of a Single- Gender Mathematics Class" (Amy Burns Ellis); (6) "The Role of Intellectual Culture of Mathematics in Doctoral Student Attrition" (Abbe H. Herzig); (7) "Group Tests as a Context for Learning How to Better Facilitate Small Group Discussion" (Judith Kysh); and (8) "Learning Paths to 5- and 10-Structured Understanding of Quantity: Addition and Subtraction Solution Strategies of Japanese Children" (Aki Murata and Karen Fuson). (MM)