Teaching and learning proof across the grades: a K-16 perspective

409 Pages·2009·2.41 MB·English

by  Despina A. Stylianou

Teaching and Learning Proof
Across the Grades
In recent years there has been increased interest in the nature and role of proof in
mathematics education, with many mathematics educators advocating that proof
should be a central part of the mathematics education of students at all grade
levels. This important new collection provides that much needed forum for
mathematics educators to articulate a connected K–16 “story” of proof. Such a
story includes understanding how the forms of proof, including the nature of
argumentation and justification as well as what counts as proof, evolve chrono-
logically and cognitively, and how curricula and instruction can support the
development of students’ understanding of proof. Collectively, these studies
inform educators and researchers at different grade levels about the teaching and
learning of proof at each level and, thus, help advance the design of further
empirical and theoretical work in this area. By building and expanding on exist-
ing research and by allowing a variety of voices from the field to be heard,
Teaching and Learning Proof Across the Grades not only highlights the main ideas
that have recently emerged on proof research, but also defines an agenda for
future study.
Despina A. Stylianou is an Associate Professor of Mathematics Education at the
City College of the City University of New York.
Maria L. Blanton is a Senior Executive Research Associate at the James J. Kaput
Center for Research and Innovation in Mathematics Education and an Associate
Professor of Mathematics Education at the University of Massachusetts,
Dartmouth.
Eric J. Knuth is an Associate Professor of Mathematics Education at the University
of Wisconsin-Madison.
STUDIES IN MATHEMATICAL THINKING AND LEARNING
Alan H. Schoenfeld, Series Editor
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Guide for Observation and Self-Assessment, Second Edition
Baroody/Dowker (Eds.) The Development of Arithmetic Concepts and Skills:
Constructing Adaptive Expertise
Boaler Experiencing School Mathematics: Traditional and Reform Approaches to
Teaching and their Impact on Student Learning
Carpenter/Fennema/Romberg (Eds.) Rational Numbers: An Integration of Research
Chazan/Callis/Lehman (Eds.) Embracing Reason: Egalitarian Ideals and the Teach-
ing of High School Mathematics
Cobb/Bauersfeld (Eds.) The Emergence of Mathematical Meaning: Interaction in
Classroom Cultures
Cohen Teachers’ Professional Development and the Elementary Mathematics Class-
room: Bringing Understandings to Light
Clements/Sarama/DiBiase (Eds.) Engaging Young Children in Mathematics:
Standards for Early Childhood Mathematics Education
English (Ed.) Mathematical and Analogical Reasoning of Young Learners
English (Ed.) Mathematical Reasoning: Analogies, Metaphors, and Images
Fennema/Nelson (Eds.) Mathematics Teachers in Transition
Fennema/Romberg (Eds.) Mathematics Classrooms that Promote Understanding
Fernandez/Yoshida Lesson Study: A Japanese Approach to Improving Mathematics
Teaching and Learning
Kaput/Carraher/Blanton (Eds.) Algebra in the Early Grades
Lajoie Reflections on Statistics: Learning, Teaching, and Assessment in Grades K–12
Lehrer/Chazan (Eds.) Designing Learning Environments for Developing Under-
standing of Geometry and Space
Ma Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of
Fundamental Mathematics in China and the United States
Martin Mathematics Success and Failure among African-American Youth: The Roles
of Sociohistorical Context, Community Forces, School Influence, and Individual
Agency
Reed Word Problems: Research and Curriculum Reform
Romberg/Carpenter/Dremock (Eds.) Understanding Mathematics and Science
Matters
Romberg/Fennema/Carpenter (Eds.) Integrating Research on the Graphical Repre-
sentation of Functions
Romberg/Shafer The Impact of Reform Instruction on Student Mathematics
Achievement: An Example of a Summative Evaluation of a Standards-Based
Curriculum
Sarama/Clements Early Childhood Mathematics Education Research: Learning Tra-
jectories for Young Children
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Arithmetic: From Children’s Ideas to Classroom Practice
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Senk/Thompson (Eds.) Standards-Based School Mathematics Curricula: What are
They? What do Students Learn?
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Sophian The Origins of Mathematical Knowledge in Childhood
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Stylianou/Blanton/Knuth (Eds.) Teaching and Learning Proof Across the Grades:
A K–16 Perspective
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Watson/Mason Mathematics as a Constructive Activity: Learners Generating
Examples
Wilcox/Lanier (Eds.) Using Assessment to Reshape Mathematics Teaching: A Case-
book for Teachers and Teacher Educators, Curriculum and Staff Development
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Wood/Nelson/Warfield (Eds.) Beyond Classical Pedagogy: Teaching Elementary
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and Prospects

Teaching and Learning Proof
Across the Grades
A K–16 Perspective
Edited by
Despina A. Stylianou, Maria L. Blanton,
and Eric J. Knuth
First published 2009
by Routledge
270 Madison Ave, New York, NY 10016
Simultaneously published in the UK
by Routledge
2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN
Routledge is an imprint of the Taylor & Francis Group, an informa business
This edition published in the Taylor & Francis e-Library, 2009.
“To purchase your own copy of this or any of Taylor & Francis or Routledge’s
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Library of Congress Cataloging-in-Publication Data
Stylianou, Despina A.
Teaching and learning proof across the grades : a K–16 perspective/
Despina A. Stylianou, Maria L. Blanton, Eric J. Knuth.
p. cm.—(Studies in mathematical thinking and learning)
Includes bibliographical references and index.
1. Logic, Symbolic and mathematical—Study and teaching. 2. Proof
theory—Study and teaching. 3. Mathematical analysis—Study and teaching.
4. Problem solving—Study and teaching. I. Blanton, Maria L. II. Knuth,
Eric J. III. Title.
QA8.7.S79 2008
511.3′6071—dc22 2008031609
ISBN 0-203-88200-8 Master e-book ISBN
ISBN 10: 0–415–98984–1 (hbk)
ISBN 10: 0–203–88200–8 (ebk)
ISBN 13: 978–0–415–98984–8 (hbk)
ISBN 13: 978–0–203–88200–9 (ebk)
NCTM Stock Number: 13356
We wish to thank our families as this book is
proof of their patience and support

Contents
Series Editor’s Foreword: The Soul of Mathematics
alan h. schoenfeld xii
Preface xvii
List of Contributors xix
Introduction 1
Section I
Theoretical Considerations on the Teaching and Learning of Proof 13
1 What I Would Like My Students to Already Know About Proof 17
reuben hersh
2 Exploring Relationships Between Disciplinary Knowledge and School
Mathematics: Implications for Understanding the Place of Reasoning
and Proof in School Mathematics 21
daniel chazan and h. michael lueke
3 Proving and Knowing in Public: The Nature of Proof in a Classroom 40
patricio herbst and nicolas balacheff
Section II
Teaching and Learning of Proof in the Elementary Grades 65
4 Representation-based Proof in the Elementary Grades 71
deborah schifter
5 Representations that Enable Children to Engage in
Deductive Argument 87
anne k. morris
6 Young Mathematicians at Work: The Role of Contexts and Models
in the Emergence of Proof 102
catherine twomey fosnot and bill jacob
7 Children’s Reasoning: Discovering the Idea of Mathematical Proof 120
carolyn a. maher
ix