This two volume collection of proceedings contains working group reports, research reports, oral reports, poster session reports and discussion group reports presented at PME-NA 21. Only the plenary and research reports are full reports; the others are brief abstracts. Full reports include: (1) "Representation, Vision and Visualization: Cognitive Functions in Mathematical Thinking. Basic Issues for Learning" (Raymond Duval); (2) "On the Development of Human Representational Competence from an Evolutionary Point of View: From Episodic to Virtual Culture" (James J. Kaput); (3) "Representation and Evolution: A Discussion of Duval's and Kaput's Papers" (Patrick Thompson); (4) "The Role of Visual Representations in the Learning of Mathematics" (Abraham Arcavi); (5) "Concept and Representation in the Research on Probability Education" (Ana Marza Ojeda Salazar); (6) "On Representations and Situated Tools" (Luis Moreno Armella); (7) "Language Use in Embodied Action and Interaction in Knowing Fractions" (Thomas Kieren); (8) "The Use of Technology as a Means to Explore Mathematics Qualities in Proposed Problems" (Manuel Santos Trigo); (9) "Rethinking Representations" (Luis Radford); (10) "On Visualization and Generalization in Mathematics" (Norma C. Presmeg); (11) "School Algebra: Theory and Practice" (Eugenio Filloy, Teresa Rojano, James Kaput, Carolyn Kieran, Luis Puig, and Tenoch Cedillo); (12) "Social and Cultural Contexts in Mathematics Pedagogy (Yolanda de la Cruz); (13) "Determining Linearity: The Use of Visualization in Problem Solving" (Despina A. Stylianou and Ed Dubinsky); (14) "Mathematical Representation of Money Interest" (Youngyul Oh); (15) "The Notation Students Develop to Express a Rule" (Carol W. Bellisio and Alice S. Alston); (16) "Professionals Read Graphs (Imperfectly?)" (Wolff-Michael Roth); (17) "Spreadsheet and Composition of Functions" (J. Armando Landa H. and Sonia Ursini); (18) "Investigating Students' Developing Concept of Rate" (Susan Nickerson and Janet Bowers); (19) "A Conceptual Network for the Teaching-Learning Process of the Concept of Volume" (Mariana Saiz and Olympia Figueras); (20) "The Teacher's Role in Supporting Students' Development of Statistical Reasoning" (Kay McClain); (21) "Cross-National Comparisons of Representational Skill for Problem Solving" (Mary E. Brenner); (22) "Variations in Preference for Visualization among Mathematics Students and Teachers" (Norma C. Presmeg); (23) "The Little Teacher: A Student Role in Mathematics Small Groups" (Kathy M.C. Ivey and Sharon B. Walen); (24) "Narration as a Tool for Analyzing Beliefs on Calculus -- A Case Study" (Guenter Toerner); (25) "Writing about Mathematics in a Secondary Methods Course (Kate Masarik); (26) "A Web-Based Database of Problems and Practices and Real Communities of ESP Teachers and Students" (Eric Hsu). (AA)