Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction

216 Pages·2001·3.06 MB·English

by  Catherine Twomey Fosnot

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n our efforts to reform mathematics education, we?ve learned a tremendous lksn YO U N G
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amount about young students? strategies and the ways they cons-truct knowl t
edge, without fully understanding how to support such development over time. MATHEMATICIANS
The Dutch do. So, funded by the NSF and Exxon, Mathematics in the City was
begun, a collaborative inservice project that pooled the best thinking from both
countries. YIonung Mathematicians at WorCka,therine Fosnot and Maarten Dolk AT W O R K
reveal what they learned after several years of intensive study in numerous urban
classrooms.
The first in a three-volumYeousnegt,Mathematicians at Work C
focuses on young children between the ages of four and eight o
n Constructing Number Sense,
as they construct a deep understanding of number and the
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operations of addition and subtraction. Rather than- offer unre t
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lated activities, Fosnot and Dolk provide a concerted, unified uY
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description of development, with a focus on big-ideas, pro t Addition, and Subtraction
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gressive strategies, and emerging models. Drawing from the
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work of the Dutch mathematician, Hans Freudenthal, they define mathematics asNG
mathematizing the activity of structuring, modeling, and interpreting one?su
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lived world mathematically. And they describe teachers who use-rich problemm
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atic situations to promote inquiry, problem solving, and constru-ction, and chibelT
dren who raise and pursue their own mathematical ideas. rH
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In contrast to other books on math rYeofuonrgm,Mathematicians at Woprrko- eM
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vides a new look at the teaching of computation. It moves beyond the current sA
debate about algorithms to argue for deep number sense and the development eT
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of a repertoire of strategies based on landmark numbers and operations. SampleAIC
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minilessons on the use of the open number line model are provided to dI
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show you how to support the development of efficient computation. itN
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Catherine Twomey Fosnoits Professor of Education at the City College nS
of New York and Director of Mathematics in the City, the i-nservice proj,A
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ect describedYionung Mathematicians at Wo.rkShe has twice received n
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the best writing awards from AERA?s Constructivist SIG as well as the
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young scholar award fromEtdhuecational Communication and uO
Technology Journ.alHer previous books inclEundqeu:iring Teachers, Enquiring bR
Learners; Constructivism: Theory, Perspectives, anadndPractice; trK
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Reconstructing Mathematics Educa(tciooanuthored with Deborah c
t
Schifter). i
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Maarten Dolkis a researcher and developer of mathematics education at
the Freudenthal Institute in the Netherlands, where he has been involved
in the development of inservice materials for teachers and of multimedia
learning environments for student teachers. He has also directed an
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YOUNG MATHEMATICIANS
AT WORK
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00-H1709-FM 3/1/01 3:24 PM Page iii
YOUNG MATHEMATICIANS
AT WORK
CONSTRUCTING NUMBER SENSE,
ADDITION, AND SUBTRACTION
CATHERINE TWOMEY FOSNOT
MAARTEN DOLK
HEINEMANN • Portsmouth, NH
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Heinemann
A division of Reed Elsevier Inc.
361 Hanover Street
Portsmouth, NH, 03801-3912
www.heinemann.com
Offices and agents throughout the world
Copyright © 2001 by Catherine Twomey Fosnot and Maarten Dolk. All rights reserved. No part of this
book may be reproduced in any form or by any electronic or mechanical means, including information
storage and retrieval systems, except by a reviewer, who may quote brief passages in a review.
Table 7.1 is reprinted with permission from “The Harmful Effects of Algorithms in Grades 1–4” by
Constance Kamii and Ann Dominick in The Teaching and Learning of Algorithms in School Mathematics
by the National Council of Teachers of Mathematics.
The photo of the tapestry “L’artithmetique” is used courtesy of the Musée national du Moyen Age, Paris,
Thermas at hotel de Cluny.
This material is supported in part by the National Science Foundation under Grant No. 9550080 and
Grant No. 9911841. Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect the views of the National Science
Foundation.
Library of Congress Cataloging-in-Publication Data
Fosnot, Catherine Twomey.
Young mathematicians at work : constructing number sense, addition, and subtraction /
Catherine Twomey Fosnot, Maarten Dolk.
p. cm.
Includes bibliographical references and index.
ISBN 0-325-00353-X (alk. paper)
1. Mathematics—Study and teaching (Elementary) I. Dolk, Maarten Ludovicus
Antonius Marie, 1952– II. Title.
QA135.5 .F6318 2001
372.7—dc21
2001016587
Editors: Victoria Merecki and Leigh Peake
Cover design: Darci Mehall/Aureo Design
Cover photograph: Haynes Images
Text photographs: Herbert Seignoret
Manufacturing: Louise Richardson
Printed in the United States of America on acid-free paper
04 03 02 RRD 3 4 5
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To the teachers with whom we have worked
and from whom we have learned so much,
particularly Karen During,
who died of leukemia in the third year of the project
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CONTENTS
Acknowledgments xiii
Preface xv
About This Book xix
CHAPTER ONE: “MATHEMATICS” OR “MATHEMATIZING”?
Learning and Teaching in the Classroom 2
What Is Revealed 3
Back to the Classroom 5
Teaching and Learning as Development 9
Strategies, Big Ideas, and Models in a Teaching/Learning Framework 10
Strategies as Schemes 10
Big Ideas as Structures 11
Models as Tools for Thought 11
Walking the Edge 12
Summing Up... 12
CHAPTER TWO: THE LANDSCAPE OF LEARNING
Describing the Journey 15
Linear Frameworks 15
vii
Learning Trajectories 16
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viii CONTENTS The Role of Context 19
Word Problems vs. Truly Problematic Situations 19
Finding Situations for Mathematizing 22
Building in Constraints 23
Open vs. Closed Situations 23
Turning Classrooms into Mathematical Communities 25
The Edge Between the Individual and the Community 25
Facilitating Dialogue 26
Structuring Math Workshop 27
Summing Up... 28
CHAPTER THREE: NUMBER SENSE ON THE HORIZON
Emerging Strategies and Big Ideas 31
Struggling with Counting 32
Struggling to Understand “How Many?” 33
Identifying Landmarks 36
Designing Contexts 37
Using Games 37
Using Routines 41
Designing and Using Investigations 46
Summing Up... 49
CHAPTER FOUR: PLACE VALUE ON THE HORIZON
Additive Numeration Systems 51
Sticks, Stones, and Bones 51
The Invention of Numerals 52
Multiplicative Numeration 53
The Invention of Place Value 53
Why Did the Development of Place Value Take So Long? 53
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Helping Children Develop Mathematical Notation 54 Contents ix
A Window into a Classroom 54
The Development of Organization and a System of Tens 64
Investigations to Facilitate Place Value Development 67
Summing Up... 75
CHAPTER FIVE: DEVELOPING MATHEMATICAL MODELS
What Are Mathematical Models? 77
Modeling Actions and Situations 78
Facilitating the Development of Models 81
How Many People Are on the Bus? 81
Generalizing the Situation, Generalizing the Model 85
A Walk in the City 86
The Role of Context 89
Subtraction Contexts 89
Connecting Addition and Subtraction; Connecting the Models 90
Summing Up... 95
CHAPTER SIX: ADDITION AND SUBTRACTION
FACTS ON THE HORIZON
Teaching Basic Facts 97
To Memorize, or Not to Memorize? 97
Common Addition Strategies 98
Memorization or Automaticity? 98
Focusing on Relationships 99
Doubles 99
Combinations That Make Tens 102