Theories of Mathematical Learning

Chemists, working with only mortars and pestles, could not get very far unless they had mathematical models to explain what was happening "inside" of their elements of experience -- an example of what could be termed mathematical learning. This volume contains the proceedings of Work Group 4: Theories of Mathematics, a subgroup of the Seventh International Congress on Mathematical Education held at Universite Laval in Quebec. Bringing together multiple perspectives on mathematical thinking, this volume presents elaborations on principles reflecting the progress made in the field over the past 20 years and represents starting points for understanding mathematical learning today. This volume will be of importance to educational researchers, math educators, graduate students of mathematical learning, and anyone interested in the enterprise of improving mathematical learning worldwide.

Preface	p. ix
Sociological and Anthropological Perspectives on Mathematics Learning	p. 1
Emergent and Sociocultural Views of Mathematical Activity	p. 3
Negotiation of Mathematical Meaning in Classroom Processes: Social Interaction and Learning Mathematics	p. 21
Emergent Mathematical Environments in Children's Games	p. 51
Negotiating the Negotiation of Meaning: Comments on Voigt (1992) and Saxe and Bermudez (1992)	p. 69
Negotiating Mathematical Meanings in and out of School	p. 77
Social Interaction and Individual Cognition	p. 85
Learning Mathematics as a Meaningful Activity	p. 91
Learning Mathematics as Participation in Classroom Practice: Implications of Sociocultural Theory for Educational Reform	p. 115
Cultural Processes and Learning: Expectations, Actions, and Outcomes	p. 131
Traditions of School Mathematics in Japanese and American Elementary Classrooms	p. 149
Cognitive Science Theories and Their Contributions to the Learning of Mathematics	p. 177
Theories of Mathematics Education: the Role of Cognitive Analyses	p. 179
A Conception of Knowledge Acquisition and Its Implications for Mathematics Education	p. 197
The Theory of Conceptual Fields	p. 219
Learning about Fractions	p. 241
Imagery and the Development of Mathematical Reasoning	p. 267
Cognition, Mathematics, and Education	p. 285
The Contribution of Constructivism to the Learning of Mathematics	p. 301
Theory of Mathematics Education: the Contributions of Constructivism	p. 303
Aspects of Radical Constructivism and Its Educational Recommendations	p. 307
Phenomenography and Children's Experience of Division	p. 315
Varieties of Constructivism: a Framework for Comparison	p. 335
The Construction of Conceptual Schemes in Mathematics	p. 351
Constructing Mathematical Conventions Formed by the Abstraction and Generalization of Earlier Ideas: the Development of Initial Fraction Ideas	p. 381
A Joint Perspective on the Idea of Representation in Learning and Doing Mathematics	p. 397
Young Children Invent Methods of Proof: the Gang of Four	p. 431
Constructivism and Its Consequences for Training Teachers	p. 449
Perspectives on the Nature of Mathematical Learning	p. 465
Is the Metaphor of Mental Object Appropriate for a Theory of Learning Mathematics?	p. 467
On the Nature of a Model of Mathematical Learning	p. 477
Author Index	p. 499
Subject Index	p. 507
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Theories of Mathematical Learning

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