Paul Cobb - Symbolizing and Communicating in Mathematics Classrooms: Perspectives on Discourse, Tools, and Instructional Design

In 1993, we, in collaboration with Koeno Gravemeijer of the Fruedenthal Institute, Utrecht, The Netherlands, began working on a National Science Foundation-funded project titled Mathematizing, Modeling, and Communicating in Reform Classrooms. A major purpose of the project was to investigate the role of models and symbols in mathematical learning, thereby addressing a previously underdeveloped area of learning theory. Our intention was to build on and extend previous research by coordinating a cognitive analysis of the role of modeling in mathematical development with an analysis of the fundamentally social nature of model and symbol use in innovative classrooms. To that end, we conducted a series of classroom teaching experiments in first-, second-, and third-grade mathematics classes where mathematics instruction followed an inquiry tradition. These teaching experiments differed significantly from those that we had conducted in our previous work in that, in these, we were guided not only by our understanding of individual students mathematical conceptual development, but also by the instructional design theory of Realistic Mathematics Education (RME). Of particular importance in each case was developing one or more realistic (in the sense of RME) scenarios that could serve as starting points for students initial informal activity and that had the potential to facilitate students development of more formal mathematical reasoning. In the process we were able to investigate the role that student-generated models can play in supporting their transition from informal, situated problem solving to more formal yet personally-meaningful mathematical activity.

As the project progressed it became apparent to us that it would be helpful to engage in in-depth discussions with others who were also grappling with issues of communicating, symbolizing, modeling, and mathematizing, particularly as these issues relate to learning in the classroom. It was for this purpose that we assembled a small international group at Vanderbilt University in the fall of 1995 for a symposium on symbolizing, communicating, and mathematizing. This volume is a result of that symposium.

This book is intended for those who are seeking to expand their understanding of the complexity of learning as we are, so that we can enhance the learning experiences students have in schools.

We wish to acknowledge the assistance of a number of people who contributed significantly to the successful production of this book. We are grateful to the National Science Foundation, and especially to our program officers, Richard Lesh and Barbara Lovitts, for supporting the symposium which led to preparation of the book itself. We also thank the Office of Educational Research and Improvement and especially Thomas Romberg, Director of the National Research and Development Center on Achievement in School Mathematics and Science, for supporting the development of the book. We are grateful to Andy Anderson, for providing a detailed review that contributed significantly to the improvement of this volume, and to Naomi Silverman of Lawrence Erlbaum Associates for her assistance with the preparation and publication of this book. We are also indebted to Cheryl Burkey for formatting the manuscript, and Pamela Hunter, for her invaluable assistance with the editing process. Finally, we thank members of our families for their support throughout this entire project.

Paul Cobb

Erna Yackel

Kay McClain