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'An excellent state-of-the-art sourcebook.' - Anna O. Graeber, University of Maryland, College Park

Two of the most important concepts children develop progressively throughout their mathematics education years are additivity and multiplicativity. Additivity is associated with situations that involve adding, joining, affixing, subtracting, separating and removing. Multiplicativity is associated with situations that involve duplicating, shrinking, stressing, sharing equally, multiplying, dividing, and exponentiating.

This book presents multiplicativity in terms of a multiplicative conceptual field (MCF), not as individual concepts. It is presented in terms of interrelations and dependencies within, between, and among multiplicative concepts. The authors share the view that research on the mathematical, cognitive, and instructional aspects of multiplicative concepts must be situated in an MCF framework.

Guershon Harel is Associate Professor of Mathematics at Purdue University, and Jere Confrey is Associate Professor of Education at Cornell University.

Table of Contents

Introduction

Part I: Theoretical Approaches

1. Children's Multiplying Schemes

Leslie Steffe

2. Multiplicative Conceptual Field: What and Why?

Gerard Vergnaud

3. Extending the Meaning of Multiplication and Division

Brian Greer

Part II: The Role of The Unit

4. Ratio and Proportion: Cognitive Foundations in Unitizing and Norming

Susan Lamon

5. Units of Quantity: A Conceptual Basis Common to Additive and Multiplicative Structures

Merlyn Behr, Guershon Harel, Thomas Post, and Richard Lesh

Part III: Ratio and Rate

6. The Development of the Concept of Speed and Its Relationship to Concepts of Rate

Patrick Thompson

7. Missing-Value Proportional Reasoning Problems: Factors Affecting Informal Reasoning Patterns

James Kaput and Mary Maxwell West

Part IV: Multiplicative Worlds

8. Splitting, Similarity, and Rate of Change: A New Approach to Multiplication and Exponential Functions

Jere Confrey

9. Multiplicative Structures and the Development of Logarithms: What Was Lost by the Invention of Function?

Erick Smith and Jere Confrey

Part V: Intuitive Models

10. The Impact of the Number Type on the Solution of Multiplication and Division Problems: Further Investigations

Guershon Harel, Merlyn Behr, Thomas Post, and Richard Lesh

Summary

11. Multiple Views of Multiplicative Structures

Thomas Kieren

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