Elementary and Middle School Mathematics Teaching Developmentally
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S
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John A. Van de Walle
Late of Virginia Commonwealth University
Karen S. Karp
University of Louisville
Jennifer M. Bay-Williams
University of Louisville
Allyn & Bacon
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Library of Congress Cataloging-in-Publication Data
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Van de Walle, John A.
Elementary and middle school mathematics: teaching developmentally. —
7th ed. / John A. Van de Walle, Karen S. Karp, Jennifer M. Bay-Williams.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-0-205-57352-3
ISBN-10: 0-205-57352-5
1. Mathematics—Study and teaching (Elementary) 2. Mathematics—
Study and teaching (Middle school) I. Karp, Karen. II. Bay-Williams, Jennifer M.
QA135.6.V36 2008
510.71'2—dc22
III. Title.
2008040112
Printed in the United States of America
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WEB
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Allyn & Bacon
is an imprint of
www.pearsonhighered.com
ISBN-10: 0-205-57352-5
ISBN-13: 978-0-205-57352-3
In Memoriam
“Do you think anyone will ever read it?” our father
asked with equal parts hope and terror as the first
complete version of the first manuscript of this
book ground slowly off the dot matrix printer.
Dad envisioned his book as one that teachers would
not just read but use as a toolkit and guide in helping
students discover math. With that vision in mind, he
had spent nearly two years pouring his heart, soul,
and everything he knew about teaching mathematics
into “the book.” In the two decades since that first
manuscript rolled off the printer, “the book” became a
part of our family—sort of a child in need of constant
love and care, even as it grew and matured and made
us all enormously proud.
Many in the field of math education referred to
our father as a “rock star,” a description that utterly
baffled him and about which we mercilessly teased
him. To us, he was just our dad. If we needed any
“Believe in kids!”
proof that Dad was in fact a rock star, it came in the
—John A. Van de Walle
stories that poured in when he died—from countless
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teachers, colleagues, and most importantly from
elementary school students about how our father had
taught them to actually do math. Through this book, millions of children all over the world will be able to
use math as a tool that they understand, rather than as a set of meaningless procedures to be memorized
and quickly forgotten. Dad could not have imagined a better legacy.
Our deepest wish on our father’s behalf is that with the guidance of “the book,” teachers will continue
to show their students how to discover and to own for themselves the joy of doing math. Nothing would
honor our dad more than that.
—Gretchen Van de Walle and Bridget Phipps
(daughters of John A. Van de Walle)
Dedication
As many of you may know, John Van de Walle passed away suddenly after the release of the sixth edition.
It was during the development of the previous edition that we (Karen and Jennifer) first started writing
for this book, working toward becoming coauthors for the seventh edition. Through that experience, we
appreciate more fully John’s commitment to excellence—thoroughly considering recent research, feedback
from others, and quality resources that had emerged. His loss was difficult for all who knew him and we
miss him greatly.
We believe that our work on this edition reflects our understanding and strong belief in John’s
philosophy of teaching and his deep commitment to children and prospective and practicing teachers.
John’s enthusiasm as an advocate for meaningful mathematics instruction is something we keep in the
forefront of our teaching, thinking, and writing. In recognition of his contributions to the field and his
lasting legacy in mathematics teacher education, we dedicate this book to John A. Van de Walle.
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Over the past 20 years, many of us at Pearson Allyn & Bacon and Longman have had the privilege to
work with John Van de Walle, as well as the pleasure to get to know him. Undoubtedly, Elementary
and Middle School Mathematics: Teaching Developmentally has become the gold standard for elementary
mathematics methods courses. John set the bar high for math education. He became an exemplar of
what a textbook author should be: dedicated to the field, committed to helping all children make sense
of mathematics, focused on helping educators everywhere improve math teaching and learning, diligent
in gathering resources and references and keeping up with the latest research and trends, and meticulous
in the preparation of every detail of the textbook and supplements. We have all been fortunate for the
opportunity to have known the man behind “the book”—the devoted family man and the quintessential
teacher educator. He is sorely missed and will not be forgotten.
—Pearson Allyn & Bacon
About the Authors
John A. Van de Walle was a professor emeritus at Virginia Commonwealth
University. He was a mathematics education consultant who regularly
gave professional development workshops for K–8 teachers in the United
States and Canada. He visited and taught in elementary school classrooms
and worked with teachers to implement student-centered math lessons.
He co-authored the Scott Foresman-Addison Wesley Mathematics K–6
series and contributed to the new Pearson School mathematics program,
enVisionMATH. Additionally, he wrote numerous chapters and articles for
the National Council of Teachers of Mathematics (NCTM) books and journals and was very active in NCTM. He served as chair of the Educational
Materials Committee and program chair for a regional conference. He was
a frequent speaker at national and regional meetings, and was a member of
the board of directors from 1998–2001.
S. KarpEnhancer
is a professor of mathematics education at the University
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of Louisville (Kentucky). Prior to entering the field of teacher education
she was an elementary school teacher in New York. Karen is a coauthor
of Feisty Females: Inspiring Girls to Think Mathematically, which is aligned
with her research interests on teaching mathematics to diverse populations.
With Jennifer, Karen co-edited Growing Professionally: Readings from NCTM
Publications for Grades K–8. She is a member of the board of directors of the
NCTM and a former president of the Association of Mathematics Teacher
Educators (AMTE).
Jennifer M. Bay-Williams is an associate professor of mathematics education at the University of Louisville (Kentucky). Jennifer has published
many articles on teaching and learning in NCTM journals. She has also
coauthored the following books: Math and Literature: Grades 6–8, Math and
Nonfiction: Grades 6–8, and Navigating Through Connections in Grades 6–8.
Jennifer taught elementary, middle, and high school in Missouri and in Peru,
and continues to work in classrooms at all levels with students and with
teachers. Jennifer serves as the president of the Association of Mathematics
Teacher Educators (AMTE) and chair of the NCTM Emerging Issues
Committee.
v
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Brief Contents
SECTION I
Teaching Mathematics: Foundations and Perspectives
CHAPTER 1 Teaching Mathematics
in the Era of the NCTM Standards
1
CHAPTER 5 Building Assessment
into Instruction
76
93
CHAPTER 2 Exploring What It Means
to Know and Do Mathematics
13
CHAPTER 6 Teaching Mathematics
Equitably to All Children
CHAPTER 3 Teaching Through
Problem Solving
32
CHAPTER 7 Using Technology
to Teach Mathematics
CHAPTER 4 Planning in the ProblemBased Classroom
58
111
SECTION II
Development of Mathematical Concepts and Procedures
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CHAPTER 8 Developing Early Number
Concepts and Number Sense
125
CHAPTER 18 Proportional Reasoning
348
CHAPTER 9 Developing Meanings for
the Operations
CHAPTER 19 Developing Measurement
Concepts
145
369
CHAPTER 10 Helping Children
Master the Basic Facts
CHAPTER 20 Geometric Thinking
and Geometric Concepts
167
399
CHAPTER 11 Developing WholeNumber Place-Value Concepts
CHAPTER 21 Developing Concepts
of Data Analysis
187
436
CHAPTER 12 Developing Strategies
for Whole-Number Computation
CHAPTER 22 Exploring Concepts
of Probability
213
456
CHAPTER 13 Using Computational
Estimation with Whole Numbers
CHAPTER 23 Developing Concepts of
Exponents, Integers, and Real Numbers
240
473
CHAPTER 14 Algebraic Thinking:
Generalizations, Patterns, and Functions
254
APPENDIX A Principles and Standards
for School Mathematics: Content Standards
and Grade Level Expectations
A-1
CHAPTER 15 Developing Fraction
Concepts
286
APPENDIX B Standards for Teaching
Mathematics
B-1
CHAPTER 16 Developing Strategies
for Fraction Computation
309
APPENDIX C Guide to Blackline
Masters
C-1
CHAPTER 17 Developing Concepts
of Decimals and Percents
328
vii
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Contents
Preface
xix
SECTION I
Teaching Mathematics: Foundations and Perspectives
The fundamental core of effective teaching of mathematics combines an understanding of how children learn, how to
promote that learning by teaching through problem solving, and how to plan for and assess that learning on a daily
basis. Introductory chapters in this section provide perspectives on trends in mathematics education and the process of
doing mathematics. These chapters develop the core ideas of learning, teaching, planning, and assessment. Additional
perspectives on mathematics for children with diverse backgrounds and the role of technology are also discussed.
CHAPTER 1
CHAPTER 2
Teaching Mathematics in the
Era of the NCTM Standards
Exploring What It Means to
Know and Do Mathematics
The National Standards-Based Movement
1
What Does It Mean to Do Mathematics?
1
13
13
Mathematics Is the Science of Pattern and Order 13
A Classroom Environment for Doing Mathematics 14
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Principles and Standards for School Mathematics
The Six Principles 2
The Five Content Standards
The Five Process Standards
2
An Invitation to Do Mathematics
3
Let’s Do Some Mathematics! 15
Where Are the Answers? 19
3
Curriculum Focal Points: A Quest for Coherence
5
The Professional Standards for Teaching
Mathematics and Mathematics Teaching Today
Shifts in the Classroom Environment
The Teaching Standards 5
What Does It Mean to Learn Mathematics?
5
5
Influences and Pressures on Mathematics Teaching
National and International Studies
State Standards 7
Curriculum 7
A Changing World Economy 8
6
An Invitation to Learn and Grow
9
Becoming a Teacher of Mathematics 9
6
Constructivist Theory 20
Sociocultural Theory 21
Implications for Teaching Mathematics
20
21
What Does It Mean to Understand Mathematics?
23
Mathematics Proficiency 24
Implications for Teaching Mathematics 25
Benefits of a Relational Understanding 26
Multiple Representations to Support Relational
Understanding 27
Connecting the Dots
29
REFLECTIONS ON CHAPTER 1
REFLECTIONS ON CHAPTER 2
Writing to Learn 11
For Discussion and Exploration
Writing to Learn 30
For Discussion and Exploration
11
15
30
RESOURCES FOR CHAPTER 1
RESOURCES FOR CHAPTER 2
Recommended Readings 11
Standards-Based Curricula 12
Online Resources 12
Field Experience Guide Connections
Recommended Readings 30
Online Resources 31
Field Experience Guide Connections
31
12
ix
x
Contents
Planning for All Learners
CHAPTER 3
Teaching Through Problem
Solving
Teaching Through Problem Solving
32
32
Drill or Practice?
Problems and Tasks for Learning Mathematics 33
A Shift in the Role of Problems 33
The Value of Teaching Through Problem Solving 33
Examples of Problem-Based Tasks 34
Homework
Multiple Entry Points 36
Creating Meaningful and Engaging Contexts 37
How to Find Quality Tasks and Problem-Based
Lessons 38
Writing to Learn 72
For Discussion and Exploration 72
43
RESOURCES FOR CHAPTER 4
Let Students Do the Talking 43
How Much to Tell and Not to Tell 44
The Importance of Student Writing 44
Metacognition 46
Disposition 47
Attitudinal Goals 47
Recommended Readings 73
Online Resources 73
Field Experience Guide Connections 73
EXPANDED LESSON
Fixed Areas
74
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47
The Before Phase of a Lesson 48
Teacher Actions in the Before Phase 48
The During Phase of a Lesson 51
Teacher Actions in the During Phase 51
The After Phase of a Lesson 52
Teacher Actions in the After Phase 53
CHAPTER 5
Building Assessment into
Instruction
Integrating Assessment into Instruction
55
What Is Assessment? 76
The Assessment Standards 76
Why Do We Assess? 77
What Should Be Assessed? 78
REFLECTIONS ON CHAPTER 3
Writing to Learn 56
For Discussion and Exploration 56
RESOURCES FOR CHAPTER 3
Performance-Based Assessments
78
Examples of Performance-Based Tasks 79
Thoughts about Assessment Tasks 80
Recommended Readings 56
Online Resources 57
Field Experience Guide Connections 57
Rubrics and Performance Indicators
Simple Rubrics 80
Performance Indicators 81
Student Involvement with Rubrics 82
CHAPTER 4
Planning in the ProblemBased Classroom
Planning a Problem-Based Lesson
72
REFLECTIONS ON CHAPTER 4
42
Teaching in a Problem-Based Classroom
Frequently Asked Questions
71
Practice as Homework 71
Drill as Homework 71
Provide Homework Support
Four-Step Problem-Solving Process 42
Problem-Solving Strategies 43
A Three-Phase Lesson Format
69
New Definitions of Drill and Practice 69
What Drill Provides 69
What Practice Provides 70
When Is Drill Appropriate? 70
Students Who Don’t Get It 71
Selecting or Designing Problem-Based Tasks and
Lessons 36
Teaching about Problem Solving
64
Make Accommodations and Modifications 65
Differentiating Instruction 65
Flexible Groupings 67
Example of Accommodating a Lesson: ELLs 67
58
Planning Process for Developing a Lesson 58
Applying the Planning Process 62
Variations of the Three-Phase Lesson 63
Textbooks as Resources 64
Observation Tools
58
82
Anecdotal Notes 83
Observation Rubric 83
Checklists for Individual Students 83
Checklists for Full Classes 84
Writing and Journals
84
The Value of Writing 84
Journals 85
80
76
76
Contents
REFLECTIONS ON CHAPTER 6
Writing Prompts and Ideas 85
Journals for Early Learners 86
Student Self-Assessment 87
Diagnostic Interviews
Tests
Writing to Learn 109
For Discussion and Exploration 109
87
RESOURCES FOR CHAPTER 6
88
Improving Performance on High-Stakes Tests
Recommended Readings 110
Online Resources 110
Field Experience Guide Connections 110
89
Teach Fundamental Concepts and Processes 89
Test-Taking Strategies 89
Grading
xi
90
Grading Issues 90
REFLECTIONS ON CHAPTER 5
CHAPTER 7
Writing to Learn 91
For Discussion and Exploration 91
Using Technology to Teach
Mathematics
111
RESOURCES FOR CHAPTER 5
Calculators in Mathematics Instruction
Recommended Readings 91
Online Resources 92
Field Experience Guide Connections 92
112
When to Use a Calculator 112
Benefits of Calculator Use 112
Graphing Calculators 113
Data-Collection Devices
Computers in Mathematics Instruction
CHAPTER 6
Mathematics for All Children
Tools for Developing Geometry 116
93
Tools for Developing Algebraic Thinking
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Instructional Software 118
Problem Solving 118
Drill and Reinforcement 118
Guidelines for Selecting and Using Software
95
Guidelines for Using Software 119
Response to Intervention 95
Students with Mild Disabilities 96
Students with Significant Disabilities 100
Culturally and Linguistically Diverse Students
How to Select Software 119
Resources on the Internet
102
Windows and Mirrors 102
Culturally Relevant Mathematics Instruction 102
Ethnomathematics 103
English Language Learners (ELLs) 104
Strategies for Teaching Mathematics to ELLs 104
Working Toward Gender Equity
How to Select Internet Resources 120
Emerging Technologies 120
Writing to Learn 122
For Discussion and Exploration 123
RESOURCES FOR CHAPTER 7
106
Reducing Resistance and Building Resilience
120
REFLECTIONS ON CHAPTER 7
Possible Causes of Gender Inequity 106
What Can Be Done? 106
107
Providing for Students Who Are Mathematically
Gifted 107
109
118
Concept Instruction 118
94
Providing for Students with Special Needs
Final Thoughts
Tools for Developing Probability and Data Analysis 117
93
Diversity in Today’s Classroom 94
Tracking and Flexible Grouping 94
Instructional Principles for Diverse Learners 95
Strategies to Avoid 108
Strategies to Incorporate 108
115
Tools for Developing Numeration 115
Teaching Mathematics
Equitably to All Children
Creating Equitable Instruction
114
Recommended Readings 123
Online Resources 123
Field Experience Guide Connections 124
119
xii
Contents
SECTION II
Development of Mathematical Concepts and Procedures
This section serves as the application of the core ideas of Section I. Here you will find chapters on every major content area in the pre-K–8 mathematics curriculum. Numerous problem-based activities to engage students are interwoven with a discussion of the mathematical content and how children develop their understanding of that content.
At the outset of each chapter, you will find a listing of “Big Ideas,” the mathematical umbrella for the chapter. Also
included are ideas for incorporating children’s literature, technology, and assessment. These chapters are designed to
help you develop pedagogical strategies and to serve as a resource for your teaching now and in the future.
CHAPTER 8
CHAPTER 9
Developing Early Number
Concepts and Number Sense
Developing Meanings
for the Operations
Promoting Good Beginnings
125
Addition and Subtraction Problem Structures
125
Number Development in Pre-K and Kindergarten
126
Early Counting 127
145
Examples of the Four Problem Structures 146
Teaching Addition and Subtraction
The Relationships of More, Less, and Same 126
148
Contextual Problems 148
Numeral Writing and Recognition 128
INVESTIGATIONS IN NUMBER, DATA, AND SPACE
Counting On and Counting Back 128
Early Number Sense
145
Grade 2, Counting, Coins, and Combinations
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Problems 151
129
Relationships among Numbers 1 Through 10
130
Properties of Addition and Subtraction 153
Patterned Set Recognition 130
Multiplication and Division Problem Structures
One and Two More, One and Two Less 131
Examples of the Four Problem Structures 154
Anchoring Numbers to 5 and 10 132
Teaching Multiplication and Division
Part-Part-Whole Relationships 134
Dot Cards as a Model for Teaching Number
Relationships 137
Remainders 157
138
Pre-Place-Value Concepts 138
Extending More Than and Less Than Relationships 139
Doubles and Near-Doubles 139
Model-Based Problems 158
Properties of Multiplication and Division 160
Strategies for Solving Contextual Problems
Analyzing Context Problems 161
140
Two-Step Problems 163
Estimation and Measurement 140
REFLECTIONS ON CHAPTER 9
Data Collection and Analysis 141
Extensions to Early Mental Mathematics
142
Writing to Learn 164
For Discussion and Exploration 164
REFLECTIONS ON CHAPTER 8
RESOURCES FOR CHAPTER 9
Writing to Learn 143
For Discussion and Exploration 143
Literature Connections 165
Recommended Readings 165
Online Resources 166
Field Experience Guide Connections 166
RESOURCES FOR CHAPTER 8
Literature Connections 143
Recommended Readings 144
Online Resources 144
Field Experience Guide Connections 144
157
Contextual Problems 157
Relationships for Numbers 10 Through 20
Number Sense in Their World
150
161
154
Contents
Basic Ideas of Place Value
CHAPTER 10
188
Integration of Base-Ten Groupings with Count
by Ones 188
Helping Children Master
the Basic Facts
167
Developmental Nature of Basic Fact Mastery
Role of Counting 189
Integration of Groupings with Words 189
167
Integration of Groupings with Place-Value Notation 190
Approaches to Fact Mastery 168
Models for Place Value
Guiding Strategy Development 169
Reasoning Strategies for Addition Facts
191
Base-Ten Models and the Ten-Makes-One Relationship 191
170
Groupable Models 191
One More Than and Two More Than 170
Adding Zero 171
Using 5 as an Anchor 172
10 Facts 172
Up Over 10 172
Doubles 173
Near-Doubles 173
Reinforcing Reasoning Strategies 174
Reasoning Strategies for Subtraction Facts
Pregrouped or Trading Models 192
Nonproportional Models 192
Developing Base-Ten Concepts
193
Grouping Activities 193
The Strangeness of Ones, Tens, and Hundreds 195
Grouping Tens to Make 100 195
Equivalent Representations 195
Oral and Written Names for Numbers
175
Reasoning Strategies for Multiplication Facts
197
Two-Digit Number Names 197
Subtraction as Think-Addition 175
Down Over 10 176
Take from the 10 176
Three-Digit Number Names 198
Written Symbols 198
Patterns and Relationships with Multidigit
Numbers 200
177
Doubles 178
Fives 178
Zeros and Ones 178
Nifty Nines 179
Using Known Facts to Derive Other Facts 180
The Hundreds Chart 200
Relationships with Landmark Numbers 202
Number Relationships for Addition and Subtraction
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Connections to Real-World Ideas 207
Division Facts and “Near Facts”
Mastering the Basic Facts
Numbers Beyond 1000
181
204
207
Extending the Place-Value System 208
182
Conceptualizing Large Numbers 209
Effective Drill 182
Fact Remediation
xiii
REFLECTIONS ON CHAPTER 11
184
Writing to Learn 210
For Discussion and Exploration 210
REFLECTIONS ON CHAPTER 10
Writing to Learn 185
For Discussion and Exploration 185
RESOURCES FOR CHAPTER 11
Literature Connections 211
Recommended Readings 211
Online Resources 212
Field Experience Guide Connections 212
RESOURCES FOR CHAPTER 10
Literature Connections 185
Recommended Readings 186
Online Resources 186
Field Experience Guide Connections 186
CHAPTER 12
CHAPTER 11
Developing Whole-Number
Place-Value Concepts
Pre-Base-Ten Concepts
188
Developing Strategies for
Whole-Number Computation
187
Toward Computational Fluency
Direct Modeling 214
Children’s Pre-Base-Ten View of Numbers 188
Student-Invented Strategies 215
Count by Ones 188
Traditional Algorithms 217
214
213
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Contents
Development of Student-Invented Strategies
Computational Estimation Strategies
218
Creating an Environment for Inventing Strategies 218
Front-End Methods 245
Models to Support Invented Strategies 218
Rounding Methods 246
Student-Invented Strategies for Addition
and Subtraction 219
245
Compatible Numbers 247
Clustering 248
Adding and Subtracting Single-Digit Numbers 219
Use Tens and Hundreds 248
Adding Two-Digit Numbers 220
Estimation Experiences
Subtracting by Counting Up 220
249
Calculator Activities 249
Take-Away Subtraction 221
Using Whole Numbers to Estimate Rational
Numbers 251
Extensions and Challenges 222
Traditional Algorithms for Addition and Subtraction
223
Addition Algorithm 223
Subtraction Algorithm 225
Student-Invented Strategies for Multiplication
226
REFLECTIONS ON CHAPTER 13
Writing to Learn 252
For Discussion and Exploration 252
Useful Representations 226
RESOURCES FOR CHAPTER 13
Multiplication by a Single-Digit Multiplier 227
Literature Connections 252
Recommended Readings 252
Online Resources 252
Field Experience Guide Connections 253
Multiplication of Larger Numbers 228
Traditional Algorithm for Multiplication
230
One-Digit Multipliers 230
Two-Digit Multipliers 231
Student-Invented Strategies for Division
232
CHAPTER 14
Missing-Factor Strategies 232
Cluster Problems 233
Traditional Algorithm for Division
234
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One-Digit Divisors 234
Two-Digit Divisors 235
Algebraic Thinking:
Generalizations, Patterns,
Enhancer
and Functions
254
REFLECTIONS ON CHAPTER 12
Algebraic Thinking
Writing to Learn 237
For Discussion and Exploration 238
Generalization from Arithmetic and from Patterns
255
Generalization in the Hundreds Chart 256
RESOURCES FOR CHAPTER 12
Generalization Through Exploring a Pattern 257
Literature Connections 238
Recommended Readings 239
Online Resources 239
Field Experience Guide Connections 239
Meaningful Use of Symbols
257
The Meaning of the Equal Sign 258
The Meaning of Variables 262
Making Structure in the Number System Explicit
Making Conjectures about Properties 265
CHAPTER 13
Justifying Conjectures 266
Using Computational Estimation
with Whole Numbers
240
Introducing Computational Estimation
255
Generalization with Addition 255
240
Understanding Computational Estimation 241
Suggestions for Teaching Computational
Estimation 242
Computational Estimation from Invented
Strategies 244
Odd and Even Relationships 266
Study of Patterns and Functions
267
Repeating Patterns 267
Growing Patterns 269
Linear Functions 274
Mathematical Modeling
Teaching Considerations
276
277
Emphasize Appropriate Algebra Vocabulary 277
Multiple Representations 278
Stop Before the Details 244
Connect Representations 280
Use Related Problem Sets 244
Algebraic Thinking Across the Curriculum 280
265
Contents
CONNECTED MATHEMATICS
Grade 7, Variables and Patterns
xv
Online Resources 307
Field Experience Guide Connections 308
281
REFLECTIONS ON CHAPTER 14
Writing to Learn 283
For Discussion and Exploration 283
CHAPTER 16
Developing Strategies
for Fraction Computation
RESOURCES FOR CHAPTER 14
Literature Connections 283
Recommended Readings 284
Online Resources 284
Field Experience Guide Connections 285
Number Sense and Fraction Algorithms
309
310
Conceptual Development Takes Time 310
A Problem-Based Number Sense Approach 310
Computational Estimation 310
Addition and Subtraction
CHAPTER 15
Invented Strategies 312
Developing Fraction
Concepts
Meanings of Fractions
286
Developing an Algorithm 315
Mixed Numbers and Improper Fractions 317
Multiplication
287
317
Developing the Concept 317
Fraction Constructs 287
Developing the Algorithm 320
Building on Whole-Number Concepts 287
Models for Fractions
312
Factors Greater Than One 320
288
Division
Region or Area Models 288
321
Partitive Interpretation of Division 321
Length Models 289
Measurement Interpretation of Division 323
Set Models 290
Concept of Fractional Parts
Answers That Are Not Whole Numbers 324
291
Developing the Algorithms
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Sharing Tasks 291
324
Fraction Language 293
REFLECTIONS ON CHAPTER 16
Equivalent Size of Fraction Pieces 293
Writing to Learn 326
For Discussion and Exploration 326
Partitioning 294
Using Fraction Language and Symbols
294
RESOURCES FOR CHAPTER 16
Counting Fraction Parts: Iteration 294
Literature Connections 326
Recommended Readings 327
Online Resources 327
Field Experience Guide Connections 327
Fraction Notation 296
Fractions Greater Than 1 296
Assessing Understanding 297
Estimating with Fractions
298
Benchmarks of Zero, One-Half, and One 299
CHAPTER 17
Using Number Sense to Compare 299
Equivalent-Fraction Concepts
Developing Concepts
of Decimals and Percents
301
Conceptual Focus on Equivalence 301
Equivalent-Fraction Models 302
Connecting Fractions and Decimals
Developing an Equivalent-Fraction Algorithm 304
Teaching Considerations for Fraction Concepts
REFLECTIONS ON CHAPTER 15
Writing to Learn 306
For Discussion and Exploration 306
306
328
Base-Ten Fractions 329
Extending the Place-Value System 330
Fraction-Decimal Connection 332
Developing Decimal Number Sense
333
Familiar Fractions Connected to Decimals 334
RESOURCES FOR CHAPTER 15
Approximation with a Nice Fraction 335
Literature Connections 307
Recommended Readings 307
Ordering Decimal Numbers 336
Other Fraction-Decimal Equivalents 337
328
xvi
Contents
Introducing Percents
RESOURCES FOR CHAPTER 18
337
Models and Terminology 338
Literature Connections 366
Recommended Readings 367
Online Resources 368
Field Experience Guide Connections 368
Realistic Percent Problems 339
Estimation 341
Computation with Decimals
342
The Role of Estimation 342
Addition and Subtraction 342
Multiplication 343
CHAPTER 19
Division 344
Developing Measurement
Concepts
REFLECTIONS ON CHAPTER 17
Writing to Learn 345
For Discussion and Exploration 345
The Meaning and Process of Measuring
370
Concepts and Skills 371
Nonstandard Units and Standard Units:
Reasons for Using Each 372
RESOURCES FOR CHAPTER 17
Literature Connections 345
Recommended Readings 346
Online Resources 346
Field Experience Guide Connections 347
The Role of Estimation and Approximation 372
Length
373
Comparison Activities 373
Units of Length 374
Making and Using Rulers 375
CHAPTER 18
Area
Proportional
Reasoning
348
376
Comparison Activities 376
INVESTIGATIONS IN NUMBER, DATA, AND SPACE
Ratios
Grade 3, Perimeter, Angles, and Area
348
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Enhancer
of Area 378
Types of Ratios 349
Proportional Reasoning
The Relationship Between Area and Perimeter 380
350
Volume and Capacity
Additive Versus Multiplicative Situations 351
Units of Volume and Capacity 381
Equivalent Ratios 353
Weight and Mass
Different Ratios 354
Grade 7, Comparing and Scaling
Units of Weight or Mass 383
356
Time
Ratio Tables 356
Proportional Reasoning Across the Curriculum
Algebra 359
383
Duration 383
358
Clock Reading 383
Elapsed Time 384
Measurement and Geometry 359
Scale Drawings 360
Statistics 361
363
382
Comparison Activities 382
CONNECTED MATHEMATICS
Proportions
380
Comparison Activities 381
Indentifying Multiplicative Relationships 352
Number: Fractions and Percent
362
Money
385
Coin Recognition and Values 385
Counting Sets of Coins 385
Making Change 386
Angles
386
Within and Between Ratios 363
Comparison Activities 386
Reasoning Approaches 364
Units of Angular Measure 386
Cross-Product Approach 365
REFLECTIONS ON CHAPTER 18
Writing to Learn 366
For Discussion and Exploration 366
377
Using Protractors and Angle Rulers 386
Introducing Standard Units
Instructional Goals
387
387
Important Standard Units and Relationships 389
369
Contents
Estimating Measures
xvii
REFLECTIONS ON CHAPTER 20
389
Strategies for Estimating Measurements 390
Writing to Learn 433
For Discussion and Exploration 433
Tips for Teaching Estimation 390
Measurement Estimation Activities 391
RESOURCES FOR CHAPTER 20
Developing Formulas for Area and Volume
391
Students’ Misconceptions 391
Areas of Rectangles, Parallelograms, Triangles,
and Trapezoids 392
Literature Connections 433
Recommended Readings 434
Online Resources 434
Field Experience Guide Connections 435
Circumference and Area of Circles 394
Volumes of Common Solid Shapes 395
Connections among Formulas 396
CHAPTER 21
REFLECTIONS ON CHAPTER 19
Developing Concepts
of Data Analysis
Writing to Learn 397
For Discussion and Exploration 397
RESOURCES FOR CHAPTER 19
What Does It Mean to Do Statistics?
436
437
Is It Statistics or Is It Mathematics? 437
Variability 437
The Shape of Data 438
Process of Doing Statistics 439
Literature Connections 397
Recommended Readings 397
Online Resources 398
Field Experience Guide Connections 398
Formulating Questions
439
Ideas for Questions 439
CHAPTER 20
Data Collection
Geometric Thinking and Geometric
Concepts 399
440
Using Existing Data Sources 440
Data Analysis: Classification
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Attribute Materials
Geometry Goals for Students
399
441
441
Data Analysis: Graphical Representations
Spatial Sense and Geometric Reasoning 400
443
Bar Graphs and Tally Charts 443
Geometric Content 400
Circle Graphs 444
The Development of Geometric Thinking
400
The van Hiele Levels of Geometric Thought 400
Implications for Instruction 404
Continuous Data Graphs 445
Scatter Plots 447
Data Analysis: Measures of Center
Learning about Shapes and Properties
405
449
Averages 449
Shapes and Properties for Level-0 Thinkers 405
Understanding the Mean: Two Interpretations 449
Shapes and Properties for Level-1 Thinkers 410
Box-and-Whisker Plots 452
Shapes and Properties for Level-2 Thinkers 416
Learning about Transformations
419
Transformations for Level-0 Thinkers 419
Transformations for Level-1 Thinkers 422
Transformations for Level-2 Thinkers 424
Learning about Location
453
REFLECTIONS ON CHAPTER 21
Writing to Learn 454
For Discussion and Exploration 454
RESOURCES FOR CHAPTER 21
424
Location for Level-0 Thinkers 424
Location for Level-1 Thinkers 426
Location for Level-2 Thinkers 428
Learning about Visualization
Interpreting Results
429
Visualization for Level-0 Thinkers 429
Visualization for Level-1 Thinkers 430
Visualization for Level-2 Thinkers 432
Literature Connections 454
Recommended Readings 455
Online Resources 455
Field Experience Guide Connections 455
xviii
Contents
CHAPTER 22
CHAPTER 23
Exploring Concepts
of Probability
Introducing Probability
457
Exponents
Likely or Not Likely 457
The Probability Continuum 459
473
Exponents in Expressions and Equations 473
Theoretical Probability and Experiments
Negative Exponents 476
460
Scientific Notation 477
Theoretical Probability 461
Integers
Experiments 462
Implications for Instruction 464
479
Contexts for Exploring Integers 479
Use of Technology in Experiments 464
Meaning of Negative Numbers 481
Sample Spaces and Probability of Two Events
465
Independent Events 465
Two-Event Probabilities with an Area Model 466
Dependent Events 467
Simulations
Developing Concepts
of Exponents, Integers,
and Real Numbers
456
Two Models for Teaching Integers 481
Operations with Integers
482
Addition and Subtraction 482
Multiplication and Division 484
468
Real Numbers
486
REFLECTIONS ON CHAPTER 22
Rational Numbers 486
Writing to Learn 470
For Discussion and Exploration 470
Irrational Numbers 487
Density of the Real Numbers 489
RESOURCES FOR CHAPTER 22
REFLECTIONS ON CHAPTER 23
Literature Connections 471
Recommended Readings 471
Online Resources 472
Field Experience Guide Connections 472
Writing to Learn 489
For Discussion and Exploration 489
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Enhancer
FOR CHAPTER 23
Literature Connections 490
Recommended Readings 490
Online Resources 490
Field Experience Guide Connections 490
APPENDIX A
Principles and Standards for School Mathematics:
Content Standards and Grade Level Expectations A-1
APPENDIX B
Standards for Teaching Mathematics
APPENDIX C
Guide to Blackline Masters
References
Index I-1
R-1
C-1
B-1
473
Preface
WHAT YOU WILL FIND IN THIS BOOK
If you look at the table of contents, you will see that the chapters are separated into two distinct
sections. The first section, consisting of seven chapters, deals with important ideas that cross
the boundaries of specific areas of content. The second section, consisting of 16 chapters, offers
teaching suggestions for every major mathematics topic in the pre-K–8 curriculum. Chapters in
Section I offer perspective on the challenging task of helping children learn mathematics. The
evolution of mathematics education and underlying causes for those changes are important components of your professional knowledge as a mathematics teacher. Having a feel for the discipline
of mathematics—that is, to know what it means to “do mathematics”—is also a critical component
of your profession. The first two chapters address these issues.
Chapters 2 and 3 are core chapters in which you will learn about a constructivist view of learning, how that is applied to learning mathematics, and what it means to teach through problem
solving. Chapter 4 will help you translate these ideas of how children best learn mathematics into
the lessons you will be teaching. Here you will find practical perspectives on planning effective
lessons for all children, on the value of drill and practice, and other issues. A sample lesson plan is
found at the end of this chapter. Chapter 5 explores the integration of assessment with instruction
to best assist student learning.
In Chapter 6, you will read about the diverse student populations in today’s classrooms including students who are English language learners, are gifted, or have special needs. Chapter 7
provides perspectives on the issues related to using technology in the teaching of mathematics. A
strong case is made for the use of handheld technology at all grade levels. Guidance is offered for
the selection and use of computer software and resources on the Internet.
Each chapter of Section II provides a perspective of the mathematical content, how children
best learn that content, and numerous suggestions for problem-based activities to engage children
in the development of good mathematics. The problem-based tasks for students are integrated
within the text, not added on. Reflecting on the activities as you read can help you think about the
mathematics from the perspective of the student. Read them along with the text, not as an aside.
As often as possible, take out pencil and paper and try the problems so that you actively engage in
your learning about children learning mathematics.
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SOME SPECIAL FEATURES OF THIS TEXT
By flipping through the book, you will notice many section headings, a large number of figures,
and various special features. All are designed to make the book more useful as a textbook and as a
long-term resource. Here are a few things to look for.
NEW!
MyEducationLab 3
MyEd
New to this edition, you will find margin notes that connect chapter ideas to the
MyEducationLab website (www.myeducationlab.com). Every chapter in Section I
connects to new video clips of John Van de Walle presenting his ideas and activities
to groups of teachers. For a complete list of the new videos of John Van de Walle, see
the inside front cover of your text.
Think of MyEducationLab as an extension of the text. You will find practice test
questions, lists of children’s literature organized by topic, links to useful websites, classroom videos, and videos of John Van de Walle talking with students and teachers. Each
of the Blackline Masters mentioned in the book can be downloaded as a PDF file. You will
also find seven Expanded Lesson plans based on activities in the book. MyEducationLab is
easy to use! In the textbook, look for the MyEducationLab logo in the margins and follow links
to access the multimedia assignments in MyEducationLab that correspond with the chapter
content.
Go to the Activities and Application section of Chapter
3 of MyEducationLab. Click
on Videos and watch the
video entitled “John Van
de Walle on Teaching
Through Problem Solving”
to see him working on a
problem with teachers during a training workshop.
xix
xx
Preface
Big Ideas
Much of the research and literature espousing a student-centered approach suggests that
teachers plan their instruction around “big ideas” rather than isolated skills or concepts. At
the beginning of each chapter in Section II, you will find a list of the key mathematical ideas
associated with the chapter. Teachers find these lists helpful for quickly getting a picture of the
mathematics they are teaching.
Mathematics Content Connections
Following the Big Ideas lists are brief descriptions of other content areas in mathematics that
are related to the content of the current chapter. These lists are offered to help you be more
aware of the potential interaction of content as you plan lessons, diagnose students’ difficulties,
and learn more yourself about the mathematics you are teaching.
388
Z Activities
Chapter 19 Developing Measurement Concepts
of required precision. (Would you measure your lawn to
purchase grass seed with the same precision as you would
use in measuring a window to buy a pane of glass?) Students
need practice in using common sense in the selection of
appropriate standard units.
3. Knowledge of relationships between units. Students
should know those relationships that are commonly used,
such as inches, feet, and yards or milliliters and liters. Tedious conversion exercises do little to enhance measurement sense.
Developing Unit Familiarity. Two types of activities can
help develop familiarity with standard units: (1) comparisons that focus on a single unit and (2) activities that develop personal referents or benchmarks for single units or
easy multiples of units.
Activity 19.21
Developing Decimal Number Sense
333
About
One Unit
for students to learn from the beginning
that decimals
are
simply fractions.
Give students a model of a standard unit, and have
The calculator can also play a significant role in decithem search for objects that measure about the same
mal concept development.
as that one unit. For example, to develop familiarity
with the meter, give students a piece of rope 1 meter
long. Have them make lists of things that are about 1
meter. Keep separate lists for things that are a little
Recall how to make the calculator “count” by
less (or more) or twice as long (or half as long). Enpressing
1
. . . Now have students
students
to find familiar items in their daily
press
0.1
. . . When the courage
display shows
0.9,
stop and discuss what this means lives.
and what
Inthe
thedisplay
case of lengths, be sure to include curved
will look like with the next press. Many students will
or circular lengths. Later, students can try to predict
predict 0.10 (thinking that 10 comes after 9). This
a given
prediction is even more interestingwhether
if, with each
press, object is more than, less than, or
the students have been accumulating
base-ten
close
to 1strips
meter.
Activity 17.3
Calculator Decimal Counting
as models for tenths. One more press would mean
one more strip, or 10 strips. Why should the calculator
same
activity can be done with other unit lengths.
not show 0.10? When the tenth pressThe
produces
a display of 1 (calculators are not usually
set to display
Families
can be enlisted to help students find familiar distrailing zeros to the right of the decimal), the discustances that are about 1 mile or about 1 kilometer. Sugsion should revolve around trading 10 strips for a
a letter
square. Continue to count to 4 gest
or 5 byin
tenths.
How that they check the distances around the
many presses to get from one whole
number to the to the school or shopping center, or along
neighborhood,
next? Try counting by 0.01 or by 0.001. These counts
other frequently traveled paths. If possible, send home (or
illustrate dramatically how small one-hundredth and
use in10class)
one-thousandth really are. It requires
countsaby1-meter or 1-yard trundle wheel to measure
0.001 to get to 0.01 and 1000distances.
counts to reach 1.
For capacity units such as cup, quart, and liter, students
The fact that the calculator need
counts a0.8,
0.9, 1, 1.1 incontainer
that holds or has a marking for a single
stead of 0.8, 0.9, 0.10, 0.11 should give rise to the question
unit. They should then find other containers at home and at
“Does this make sense? If so, why?”
school
that hold
about as much, more, and less. Remember
Calculators that permit entry
of fractions
also have
a fraction-decimal conversion key.
On
some
calculators
that the shapes of acontainers can be very deceptive when
decimal such as 0.25 will convert to the base-ten fraction
estimating their capacity.
25
and allow for either manual or automatic simplification.
100
For
standard
Graphing calculators can be set so that
thethe
conversion
is weights of gram, kilogram, ounce, and
either with or without simplification.
Thestudents
ability of fracpound,
can compare objects on a two-pan balance
tion calculators to go back and forth between fractions and
with single copies of these units. It may be more effective to
decimals makes them a valuable tool as students begin to
work with 10 grams or 5 ounces. Students can be encourconnect fraction and decimal symbolism.
aged to bring in familiar objects from home to compare on
the classroom scale.
Standard area units are in terms of lengths such as
square inches or square feet, so familiarity with lengths is
important. Familiarity with a single degree is not as important as some idea of 30, 45, 60, and 90 degrees.
The second approach to unit familiarity is to begin with
very familiar items and use their measures as references or
benchmarks. A doorway is a bit more than 2 meters high
and a doorknob is about 1 meter from the floor. A bag of
flour is a good reference for 5 pounds. A paper clip weighs
about a gram and is about 1 centimeter wide. A gallon of
milk weighs a little less than 4 kilograms.
Activity 19.22
Familiar References
Use the book Measuring Penny (Leedy, 2000) to get
students interested in the variety of ways familiar
items can be measured. In this book, the author
bridges between nonstandard (e.g., dog biscuits) and
standard units to measure Penny the pet dog. Have
your students use the idea of measuring Penny to find
something at home (or in class) to measure in as many
ways as they can think using standard units. The measures should be rounded to whole numbers (unless
children suggest adding a fractional unit to be more
precise). Discuss in class the familiar items chosen and
their measures so that different ideas and benchmarks
are shared.
The numerous activities found in every chapter of Section
II have always been rated by readers as one of the most
valuable parts of the book. Some activity ideas are described
directly in the text and in the illustrations. Others are
presented in the numbered Activity boxes. Every activity
is a problem-based task (as described in Chapter 3) and is
designed to engage students in doing mathematics. Some
activities incorporate calculator use; these particular activities are marked with a calculator icon.
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Of special interest for length are benchmarks found
on our bodies. These become quite familiar over time and
can be used as approximate rulers in many situations. Even
though young children grow quite rapidly, it is useful for
them to know the approximate lengths that they carry
around with them.
Teaching Con
siderations
Activity 19.23
Personal Benchmarks
Measure your body. About how long is your foot, your
stride, your hand span (stretched and with fingers together), the width of your finger, your arm span (finger to finger and finger to nose), the distance around
your wrist and around your waist, and your height to
waist, to shoulder, and to head? Some may prove to
be useful benchmarks, and some may be excellent
models for single units. (The average child’s fingernail
width is about 1 cm, and most people can find a
10-cm length somewhere on their hands.)
To help remember these references, they must be used
in activities in which lengths, volumes, and so on are compared to the benchmarks to estimate measurements.
Investigations in Number,
Data, and Space and
Connected Mathematics 3
In Section II, four chapters include features that
describe an activity from the standards-based curriculum Investigations in Number, Data, and Space (an
elementary curriculum) or Connected Mathematics
Project (CMP II) (a middle school curriculum). These
features include a description of an activity in the
program as well as the context of the unit in which it is
found. The main purpose of this feature is to acquaint
you with these materials and to demonstrate how the
spirit of the NCTM Standards and the constructivist
theory espoused in this book have been translated into
existing commercial curricula.
Grade 7, Varia
Context
Much of this
unit is built on
the context of
a group of stud
ents who take
a multiday bike
trip from Phil
adelphia to Wil
liamsburg, Virginia, and who
then decide to
set up a bike
tour business
of their own.
Students expl
a variety of func
ore
tional relation
ships between
time, distance
, speed, expe
nses, profits,
so on. When
and
data are plot
ted as discrete
points, students
consider wha
t the graph
might look like
between poin
ts. For example,
what interpre
tations could
be given to each
of these five
graphs showing
speed change
from 0 to 15
mph in the first
10 minutes of
a trip?
Investigation
bles and Patte
rns
3: Analyzing Gra
phs and Tables
Area
377
Task Descrip
tion
In this investig
ation, the ficti
onal students
in the unit beg
an gathering
data in preparation for setti
ng up their tour
business. As
their first task
, they sought
data from two
ferent bike rent
difal companies
as shown here
given by one
s, company in the
,
le
g
n
A
form
eter,and by the other in the form of of a table
a graph. The
task is interest
, Perim
3
e
ing
d
beca
use of the first
Gra
way in which
han
a
d
stud
ents eas rien
ce the value
of one represen ent of idexpe
and Are
tation overisan
S
E
V
E
N
T
H
E
D
I
T
I
O
N
Apago PDF Enhancer
John A. Van de Walle
Late of Virginia Commonwealth University
Karen S. Karp
University of Louisville
Jennifer M. Bay-Williams
University of Louisville
Allyn & Bacon
Boston
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editions.
Library of Congress Cataloging-in-Publication Data
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Van de Walle, John A.
Elementary and middle school mathematics: teaching developmentally. —
7th ed. / John A. Van de Walle, Karen S. Karp, Jennifer M. Bay-Williams.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-0-205-57352-3
ISBN-10: 0-205-57352-5
1. Mathematics—Study and teaching (Elementary) 2. Mathematics—
Study and teaching (Middle school) I. Karp, Karen. II. Bay-Williams, Jennifer M.
QA135.6.V36 2008
510.71'2—dc22
III. Title.
2008040112
Printed in the United States of America
10
9
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6
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4
3
2
1
WEB
13
12
11
10
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08
Allyn & Bacon
is an imprint of
www.pearsonhighered.com
ISBN-10: 0-205-57352-5
ISBN-13: 978-0-205-57352-3
In Memoriam
“Do you think anyone will ever read it?” our father
asked with equal parts hope and terror as the first
complete version of the first manuscript of this
book ground slowly off the dot matrix printer.
Dad envisioned his book as one that teachers would
not just read but use as a toolkit and guide in helping
students discover math. With that vision in mind, he
had spent nearly two years pouring his heart, soul,
and everything he knew about teaching mathematics
into “the book.” In the two decades since that first
manuscript rolled off the printer, “the book” became a
part of our family—sort of a child in need of constant
love and care, even as it grew and matured and made
us all enormously proud.
Many in the field of math education referred to
our father as a “rock star,” a description that utterly
baffled him and about which we mercilessly teased
him. To us, he was just our dad. If we needed any
“Believe in kids!”
proof that Dad was in fact a rock star, it came in the
—John A. Van de Walle
stories that poured in when he died—from countless
Apago PDF Enhancer
teachers, colleagues, and most importantly from
elementary school students about how our father had
taught them to actually do math. Through this book, millions of children all over the world will be able to
use math as a tool that they understand, rather than as a set of meaningless procedures to be memorized
and quickly forgotten. Dad could not have imagined a better legacy.
Our deepest wish on our father’s behalf is that with the guidance of “the book,” teachers will continue
to show their students how to discover and to own for themselves the joy of doing math. Nothing would
honor our dad more than that.
—Gretchen Van de Walle and Bridget Phipps
(daughters of John A. Van de Walle)
Dedication
As many of you may know, John Van de Walle passed away suddenly after the release of the sixth edition.
It was during the development of the previous edition that we (Karen and Jennifer) first started writing
for this book, working toward becoming coauthors for the seventh edition. Through that experience, we
appreciate more fully John’s commitment to excellence—thoroughly considering recent research, feedback
from others, and quality resources that had emerged. His loss was difficult for all who knew him and we
miss him greatly.
We believe that our work on this edition reflects our understanding and strong belief in John’s
philosophy of teaching and his deep commitment to children and prospective and practicing teachers.
John’s enthusiasm as an advocate for meaningful mathematics instruction is something we keep in the
forefront of our teaching, thinking, and writing. In recognition of his contributions to the field and his
lasting legacy in mathematics teacher education, we dedicate this book to John A. Van de Walle.
Apago PDF Enhancer
Over the past 20 years, many of us at Pearson Allyn & Bacon and Longman have had the privilege to
work with John Van de Walle, as well as the pleasure to get to know him. Undoubtedly, Elementary
and Middle School Mathematics: Teaching Developmentally has become the gold standard for elementary
mathematics methods courses. John set the bar high for math education. He became an exemplar of
what a textbook author should be: dedicated to the field, committed to helping all children make sense
of mathematics, focused on helping educators everywhere improve math teaching and learning, diligent
in gathering resources and references and keeping up with the latest research and trends, and meticulous
in the preparation of every detail of the textbook and supplements. We have all been fortunate for the
opportunity to have known the man behind “the book”—the devoted family man and the quintessential
teacher educator. He is sorely missed and will not be forgotten.
—Pearson Allyn & Bacon
About the Authors
John A. Van de Walle was a professor emeritus at Virginia Commonwealth
University. He was a mathematics education consultant who regularly
gave professional development workshops for K–8 teachers in the United
States and Canada. He visited and taught in elementary school classrooms
and worked with teachers to implement student-centered math lessons.
He co-authored the Scott Foresman-Addison Wesley Mathematics K–6
series and contributed to the new Pearson School mathematics program,
enVisionMATH. Additionally, he wrote numerous chapters and articles for
the National Council of Teachers of Mathematics (NCTM) books and journals and was very active in NCTM. He served as chair of the Educational
Materials Committee and program chair for a regional conference. He was
a frequent speaker at national and regional meetings, and was a member of
the board of directors from 1998–2001.
S. KarpEnhancer
is a professor of mathematics education at the University
ApagoKaren
of Louisville (Kentucky). Prior to entering the field of teacher education
she was an elementary school teacher in New York. Karen is a coauthor
of Feisty Females: Inspiring Girls to Think Mathematically, which is aligned
with her research interests on teaching mathematics to diverse populations.
With Jennifer, Karen co-edited Growing Professionally: Readings from NCTM
Publications for Grades K–8. She is a member of the board of directors of the
NCTM and a former president of the Association of Mathematics Teacher
Educators (AMTE).
Jennifer M. Bay-Williams is an associate professor of mathematics education at the University of Louisville (Kentucky). Jennifer has published
many articles on teaching and learning in NCTM journals. She has also
coauthored the following books: Math and Literature: Grades 6–8, Math and
Nonfiction: Grades 6–8, and Navigating Through Connections in Grades 6–8.
Jennifer taught elementary, middle, and high school in Missouri and in Peru,
and continues to work in classrooms at all levels with students and with
teachers. Jennifer serves as the president of the Association of Mathematics
Teacher Educators (AMTE) and chair of the NCTM Emerging Issues
Committee.
v
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Brief Contents
SECTION I
Teaching Mathematics: Foundations and Perspectives
CHAPTER 1 Teaching Mathematics
in the Era of the NCTM Standards
1
CHAPTER 5 Building Assessment
into Instruction
76
93
CHAPTER 2 Exploring What It Means
to Know and Do Mathematics
13
CHAPTER 6 Teaching Mathematics
Equitably to All Children
CHAPTER 3 Teaching Through
Problem Solving
32
CHAPTER 7 Using Technology
to Teach Mathematics
CHAPTER 4 Planning in the ProblemBased Classroom
58
111
SECTION II
Development of Mathematical Concepts and Procedures
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CHAPTER 8 Developing Early Number
Concepts and Number Sense
125
CHAPTER 18 Proportional Reasoning
348
CHAPTER 9 Developing Meanings for
the Operations
CHAPTER 19 Developing Measurement
Concepts
145
369
CHAPTER 10 Helping Children
Master the Basic Facts
CHAPTER 20 Geometric Thinking
and Geometric Concepts
167
399
CHAPTER 11 Developing WholeNumber Place-Value Concepts
CHAPTER 21 Developing Concepts
of Data Analysis
187
436
CHAPTER 12 Developing Strategies
for Whole-Number Computation
CHAPTER 22 Exploring Concepts
of Probability
213
456
CHAPTER 13 Using Computational
Estimation with Whole Numbers
CHAPTER 23 Developing Concepts of
Exponents, Integers, and Real Numbers
240
473
CHAPTER 14 Algebraic Thinking:
Generalizations, Patterns, and Functions
254
APPENDIX A Principles and Standards
for School Mathematics: Content Standards
and Grade Level Expectations
A-1
CHAPTER 15 Developing Fraction
Concepts
286
APPENDIX B Standards for Teaching
Mathematics
B-1
CHAPTER 16 Developing Strategies
for Fraction Computation
309
APPENDIX C Guide to Blackline
Masters
C-1
CHAPTER 17 Developing Concepts
of Decimals and Percents
328
vii
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Contents
Preface
xix
SECTION I
Teaching Mathematics: Foundations and Perspectives
The fundamental core of effective teaching of mathematics combines an understanding of how children learn, how to
promote that learning by teaching through problem solving, and how to plan for and assess that learning on a daily
basis. Introductory chapters in this section provide perspectives on trends in mathematics education and the process of
doing mathematics. These chapters develop the core ideas of learning, teaching, planning, and assessment. Additional
perspectives on mathematics for children with diverse backgrounds and the role of technology are also discussed.
CHAPTER 1
CHAPTER 2
Teaching Mathematics in the
Era of the NCTM Standards
Exploring What It Means to
Know and Do Mathematics
The National Standards-Based Movement
1
What Does It Mean to Do Mathematics?
1
13
13
Mathematics Is the Science of Pattern and Order 13
A Classroom Environment for Doing Mathematics 14
Apago PDF Enhancer
Principles and Standards for School Mathematics
The Six Principles 2
The Five Content Standards
The Five Process Standards
2
An Invitation to Do Mathematics
3
Let’s Do Some Mathematics! 15
Where Are the Answers? 19
3
Curriculum Focal Points: A Quest for Coherence
5
The Professional Standards for Teaching
Mathematics and Mathematics Teaching Today
Shifts in the Classroom Environment
The Teaching Standards 5
What Does It Mean to Learn Mathematics?
5
5
Influences and Pressures on Mathematics Teaching
National and International Studies
State Standards 7
Curriculum 7
A Changing World Economy 8
6
An Invitation to Learn and Grow
9
Becoming a Teacher of Mathematics 9
6
Constructivist Theory 20
Sociocultural Theory 21
Implications for Teaching Mathematics
20
21
What Does It Mean to Understand Mathematics?
23
Mathematics Proficiency 24
Implications for Teaching Mathematics 25
Benefits of a Relational Understanding 26
Multiple Representations to Support Relational
Understanding 27
Connecting the Dots
29
REFLECTIONS ON CHAPTER 1
REFLECTIONS ON CHAPTER 2
Writing to Learn 11
For Discussion and Exploration
Writing to Learn 30
For Discussion and Exploration
11
15
30
RESOURCES FOR CHAPTER 1
RESOURCES FOR CHAPTER 2
Recommended Readings 11
Standards-Based Curricula 12
Online Resources 12
Field Experience Guide Connections
Recommended Readings 30
Online Resources 31
Field Experience Guide Connections
31
12
ix
x
Contents
Planning for All Learners
CHAPTER 3
Teaching Through Problem
Solving
Teaching Through Problem Solving
32
32
Drill or Practice?
Problems and Tasks for Learning Mathematics 33
A Shift in the Role of Problems 33
The Value of Teaching Through Problem Solving 33
Examples of Problem-Based Tasks 34
Homework
Multiple Entry Points 36
Creating Meaningful and Engaging Contexts 37
How to Find Quality Tasks and Problem-Based
Lessons 38
Writing to Learn 72
For Discussion and Exploration 72
43
RESOURCES FOR CHAPTER 4
Let Students Do the Talking 43
How Much to Tell and Not to Tell 44
The Importance of Student Writing 44
Metacognition 46
Disposition 47
Attitudinal Goals 47
Recommended Readings 73
Online Resources 73
Field Experience Guide Connections 73
EXPANDED LESSON
Fixed Areas
74
Apago PDF Enhancer
47
The Before Phase of a Lesson 48
Teacher Actions in the Before Phase 48
The During Phase of a Lesson 51
Teacher Actions in the During Phase 51
The After Phase of a Lesson 52
Teacher Actions in the After Phase 53
CHAPTER 5
Building Assessment into
Instruction
Integrating Assessment into Instruction
55
What Is Assessment? 76
The Assessment Standards 76
Why Do We Assess? 77
What Should Be Assessed? 78
REFLECTIONS ON CHAPTER 3
Writing to Learn 56
For Discussion and Exploration 56
RESOURCES FOR CHAPTER 3
Performance-Based Assessments
78
Examples of Performance-Based Tasks 79
Thoughts about Assessment Tasks 80
Recommended Readings 56
Online Resources 57
Field Experience Guide Connections 57
Rubrics and Performance Indicators
Simple Rubrics 80
Performance Indicators 81
Student Involvement with Rubrics 82
CHAPTER 4
Planning in the ProblemBased Classroom
Planning a Problem-Based Lesson
72
REFLECTIONS ON CHAPTER 4
42
Teaching in a Problem-Based Classroom
Frequently Asked Questions
71
Practice as Homework 71
Drill as Homework 71
Provide Homework Support
Four-Step Problem-Solving Process 42
Problem-Solving Strategies 43
A Three-Phase Lesson Format
69
New Definitions of Drill and Practice 69
What Drill Provides 69
What Practice Provides 70
When Is Drill Appropriate? 70
Students Who Don’t Get It 71
Selecting or Designing Problem-Based Tasks and
Lessons 36
Teaching about Problem Solving
64
Make Accommodations and Modifications 65
Differentiating Instruction 65
Flexible Groupings 67
Example of Accommodating a Lesson: ELLs 67
58
Planning Process for Developing a Lesson 58
Applying the Planning Process 62
Variations of the Three-Phase Lesson 63
Textbooks as Resources 64
Observation Tools
58
82
Anecdotal Notes 83
Observation Rubric 83
Checklists for Individual Students 83
Checklists for Full Classes 84
Writing and Journals
84
The Value of Writing 84
Journals 85
80
76
76
Contents
REFLECTIONS ON CHAPTER 6
Writing Prompts and Ideas 85
Journals for Early Learners 86
Student Self-Assessment 87
Diagnostic Interviews
Tests
Writing to Learn 109
For Discussion and Exploration 109
87
RESOURCES FOR CHAPTER 6
88
Improving Performance on High-Stakes Tests
Recommended Readings 110
Online Resources 110
Field Experience Guide Connections 110
89
Teach Fundamental Concepts and Processes 89
Test-Taking Strategies 89
Grading
xi
90
Grading Issues 90
REFLECTIONS ON CHAPTER 5
CHAPTER 7
Writing to Learn 91
For Discussion and Exploration 91
Using Technology to Teach
Mathematics
111
RESOURCES FOR CHAPTER 5
Calculators in Mathematics Instruction
Recommended Readings 91
Online Resources 92
Field Experience Guide Connections 92
112
When to Use a Calculator 112
Benefits of Calculator Use 112
Graphing Calculators 113
Data-Collection Devices
Computers in Mathematics Instruction
CHAPTER 6
Mathematics for All Children
Tools for Developing Geometry 116
93
Tools for Developing Algebraic Thinking
Apago PDF Enhancer
Instructional Software 118
Problem Solving 118
Drill and Reinforcement 118
Guidelines for Selecting and Using Software
95
Guidelines for Using Software 119
Response to Intervention 95
Students with Mild Disabilities 96
Students with Significant Disabilities 100
Culturally and Linguistically Diverse Students
How to Select Software 119
Resources on the Internet
102
Windows and Mirrors 102
Culturally Relevant Mathematics Instruction 102
Ethnomathematics 103
English Language Learners (ELLs) 104
Strategies for Teaching Mathematics to ELLs 104
Working Toward Gender Equity
How to Select Internet Resources 120
Emerging Technologies 120
Writing to Learn 122
For Discussion and Exploration 123
RESOURCES FOR CHAPTER 7
106
Reducing Resistance and Building Resilience
120
REFLECTIONS ON CHAPTER 7
Possible Causes of Gender Inequity 106
What Can Be Done? 106
107
Providing for Students Who Are Mathematically
Gifted 107
109
118
Concept Instruction 118
94
Providing for Students with Special Needs
Final Thoughts
Tools for Developing Probability and Data Analysis 117
93
Diversity in Today’s Classroom 94
Tracking and Flexible Grouping 94
Instructional Principles for Diverse Learners 95
Strategies to Avoid 108
Strategies to Incorporate 108
115
Tools for Developing Numeration 115
Teaching Mathematics
Equitably to All Children
Creating Equitable Instruction
114
Recommended Readings 123
Online Resources 123
Field Experience Guide Connections 124
119
xii
Contents
SECTION II
Development of Mathematical Concepts and Procedures
This section serves as the application of the core ideas of Section I. Here you will find chapters on every major content area in the pre-K–8 mathematics curriculum. Numerous problem-based activities to engage students are interwoven with a discussion of the mathematical content and how children develop their understanding of that content.
At the outset of each chapter, you will find a listing of “Big Ideas,” the mathematical umbrella for the chapter. Also
included are ideas for incorporating children’s literature, technology, and assessment. These chapters are designed to
help you develop pedagogical strategies and to serve as a resource for your teaching now and in the future.
CHAPTER 8
CHAPTER 9
Developing Early Number
Concepts and Number Sense
Developing Meanings
for the Operations
Promoting Good Beginnings
125
Addition and Subtraction Problem Structures
125
Number Development in Pre-K and Kindergarten
126
Early Counting 127
145
Examples of the Four Problem Structures 146
Teaching Addition and Subtraction
The Relationships of More, Less, and Same 126
148
Contextual Problems 148
Numeral Writing and Recognition 128
INVESTIGATIONS IN NUMBER, DATA, AND SPACE
Counting On and Counting Back 128
Early Number Sense
145
Grade 2, Counting, Coins, and Combinations
Apago PDF Model-Based
Enhancer
Problems 151
129
Relationships among Numbers 1 Through 10
130
Properties of Addition and Subtraction 153
Patterned Set Recognition 130
Multiplication and Division Problem Structures
One and Two More, One and Two Less 131
Examples of the Four Problem Structures 154
Anchoring Numbers to 5 and 10 132
Teaching Multiplication and Division
Part-Part-Whole Relationships 134
Dot Cards as a Model for Teaching Number
Relationships 137
Remainders 157
138
Pre-Place-Value Concepts 138
Extending More Than and Less Than Relationships 139
Doubles and Near-Doubles 139
Model-Based Problems 158
Properties of Multiplication and Division 160
Strategies for Solving Contextual Problems
Analyzing Context Problems 161
140
Two-Step Problems 163
Estimation and Measurement 140
REFLECTIONS ON CHAPTER 9
Data Collection and Analysis 141
Extensions to Early Mental Mathematics
142
Writing to Learn 164
For Discussion and Exploration 164
REFLECTIONS ON CHAPTER 8
RESOURCES FOR CHAPTER 9
Writing to Learn 143
For Discussion and Exploration 143
Literature Connections 165
Recommended Readings 165
Online Resources 166
Field Experience Guide Connections 166
RESOURCES FOR CHAPTER 8
Literature Connections 143
Recommended Readings 144
Online Resources 144
Field Experience Guide Connections 144
157
Contextual Problems 157
Relationships for Numbers 10 Through 20
Number Sense in Their World
150
161
154
Contents
Basic Ideas of Place Value
CHAPTER 10
188
Integration of Base-Ten Groupings with Count
by Ones 188
Helping Children Master
the Basic Facts
167
Developmental Nature of Basic Fact Mastery
Role of Counting 189
Integration of Groupings with Words 189
167
Integration of Groupings with Place-Value Notation 190
Approaches to Fact Mastery 168
Models for Place Value
Guiding Strategy Development 169
Reasoning Strategies for Addition Facts
191
Base-Ten Models and the Ten-Makes-One Relationship 191
170
Groupable Models 191
One More Than and Two More Than 170
Adding Zero 171
Using 5 as an Anchor 172
10 Facts 172
Up Over 10 172
Doubles 173
Near-Doubles 173
Reinforcing Reasoning Strategies 174
Reasoning Strategies for Subtraction Facts
Pregrouped or Trading Models 192
Nonproportional Models 192
Developing Base-Ten Concepts
193
Grouping Activities 193
The Strangeness of Ones, Tens, and Hundreds 195
Grouping Tens to Make 100 195
Equivalent Representations 195
Oral and Written Names for Numbers
175
Reasoning Strategies for Multiplication Facts
197
Two-Digit Number Names 197
Subtraction as Think-Addition 175
Down Over 10 176
Take from the 10 176
Three-Digit Number Names 198
Written Symbols 198
Patterns and Relationships with Multidigit
Numbers 200
177
Doubles 178
Fives 178
Zeros and Ones 178
Nifty Nines 179
Using Known Facts to Derive Other Facts 180
The Hundreds Chart 200
Relationships with Landmark Numbers 202
Number Relationships for Addition and Subtraction
Apago PDF Enhancer
Connections to Real-World Ideas 207
Division Facts and “Near Facts”
Mastering the Basic Facts
Numbers Beyond 1000
181
204
207
Extending the Place-Value System 208
182
Conceptualizing Large Numbers 209
Effective Drill 182
Fact Remediation
xiii
REFLECTIONS ON CHAPTER 11
184
Writing to Learn 210
For Discussion and Exploration 210
REFLECTIONS ON CHAPTER 10
Writing to Learn 185
For Discussion and Exploration 185
RESOURCES FOR CHAPTER 11
Literature Connections 211
Recommended Readings 211
Online Resources 212
Field Experience Guide Connections 212
RESOURCES FOR CHAPTER 10
Literature Connections 185
Recommended Readings 186
Online Resources 186
Field Experience Guide Connections 186
CHAPTER 12
CHAPTER 11
Developing Whole-Number
Place-Value Concepts
Pre-Base-Ten Concepts
188
Developing Strategies for
Whole-Number Computation
187
Toward Computational Fluency
Direct Modeling 214
Children’s Pre-Base-Ten View of Numbers 188
Student-Invented Strategies 215
Count by Ones 188
Traditional Algorithms 217
214
213
xiv
Contents
Development of Student-Invented Strategies
Computational Estimation Strategies
218
Creating an Environment for Inventing Strategies 218
Front-End Methods 245
Models to Support Invented Strategies 218
Rounding Methods 246
Student-Invented Strategies for Addition
and Subtraction 219
245
Compatible Numbers 247
Clustering 248
Adding and Subtracting Single-Digit Numbers 219
Use Tens and Hundreds 248
Adding Two-Digit Numbers 220
Estimation Experiences
Subtracting by Counting Up 220
249
Calculator Activities 249
Take-Away Subtraction 221
Using Whole Numbers to Estimate Rational
Numbers 251
Extensions and Challenges 222
Traditional Algorithms for Addition and Subtraction
223
Addition Algorithm 223
Subtraction Algorithm 225
Student-Invented Strategies for Multiplication
226
REFLECTIONS ON CHAPTER 13
Writing to Learn 252
For Discussion and Exploration 252
Useful Representations 226
RESOURCES FOR CHAPTER 13
Multiplication by a Single-Digit Multiplier 227
Literature Connections 252
Recommended Readings 252
Online Resources 252
Field Experience Guide Connections 253
Multiplication of Larger Numbers 228
Traditional Algorithm for Multiplication
230
One-Digit Multipliers 230
Two-Digit Multipliers 231
Student-Invented Strategies for Division
232
CHAPTER 14
Missing-Factor Strategies 232
Cluster Problems 233
Traditional Algorithm for Division
234
Apago PDF
One-Digit Divisors 234
Two-Digit Divisors 235
Algebraic Thinking:
Generalizations, Patterns,
Enhancer
and Functions
254
REFLECTIONS ON CHAPTER 12
Algebraic Thinking
Writing to Learn 237
For Discussion and Exploration 238
Generalization from Arithmetic and from Patterns
255
Generalization in the Hundreds Chart 256
RESOURCES FOR CHAPTER 12
Generalization Through Exploring a Pattern 257
Literature Connections 238
Recommended Readings 239
Online Resources 239
Field Experience Guide Connections 239
Meaningful Use of Symbols
257
The Meaning of the Equal Sign 258
The Meaning of Variables 262
Making Structure in the Number System Explicit
Making Conjectures about Properties 265
CHAPTER 13
Justifying Conjectures 266
Using Computational Estimation
with Whole Numbers
240
Introducing Computational Estimation
255
Generalization with Addition 255
240
Understanding Computational Estimation 241
Suggestions for Teaching Computational
Estimation 242
Computational Estimation from Invented
Strategies 244
Odd and Even Relationships 266
Study of Patterns and Functions
267
Repeating Patterns 267
Growing Patterns 269
Linear Functions 274
Mathematical Modeling
Teaching Considerations
276
277
Emphasize Appropriate Algebra Vocabulary 277
Multiple Representations 278
Stop Before the Details 244
Connect Representations 280
Use Related Problem Sets 244
Algebraic Thinking Across the Curriculum 280
265
Contents
CONNECTED MATHEMATICS
Grade 7, Variables and Patterns
xv
Online Resources 307
Field Experience Guide Connections 308
281
REFLECTIONS ON CHAPTER 14
Writing to Learn 283
For Discussion and Exploration 283
CHAPTER 16
Developing Strategies
for Fraction Computation
RESOURCES FOR CHAPTER 14
Literature Connections 283
Recommended Readings 284
Online Resources 284
Field Experience Guide Connections 285
Number Sense and Fraction Algorithms
309
310
Conceptual Development Takes Time 310
A Problem-Based Number Sense Approach 310
Computational Estimation 310
Addition and Subtraction
CHAPTER 15
Invented Strategies 312
Developing Fraction
Concepts
Meanings of Fractions
286
Developing an Algorithm 315
Mixed Numbers and Improper Fractions 317
Multiplication
287
317
Developing the Concept 317
Fraction Constructs 287
Developing the Algorithm 320
Building on Whole-Number Concepts 287
Models for Fractions
312
Factors Greater Than One 320
288
Division
Region or Area Models 288
321
Partitive Interpretation of Division 321
Length Models 289
Measurement Interpretation of Division 323
Set Models 290
Concept of Fractional Parts
Answers That Are Not Whole Numbers 324
291
Developing the Algorithms
Apago PDF Enhancer
Sharing Tasks 291
324
Fraction Language 293
REFLECTIONS ON CHAPTER 16
Equivalent Size of Fraction Pieces 293
Writing to Learn 326
For Discussion and Exploration 326
Partitioning 294
Using Fraction Language and Symbols
294
RESOURCES FOR CHAPTER 16
Counting Fraction Parts: Iteration 294
Literature Connections 326
Recommended Readings 327
Online Resources 327
Field Experience Guide Connections 327
Fraction Notation 296
Fractions Greater Than 1 296
Assessing Understanding 297
Estimating with Fractions
298
Benchmarks of Zero, One-Half, and One 299
CHAPTER 17
Using Number Sense to Compare 299
Equivalent-Fraction Concepts
Developing Concepts
of Decimals and Percents
301
Conceptual Focus on Equivalence 301
Equivalent-Fraction Models 302
Connecting Fractions and Decimals
Developing an Equivalent-Fraction Algorithm 304
Teaching Considerations for Fraction Concepts
REFLECTIONS ON CHAPTER 15
Writing to Learn 306
For Discussion and Exploration 306
306
328
Base-Ten Fractions 329
Extending the Place-Value System 330
Fraction-Decimal Connection 332
Developing Decimal Number Sense
333
Familiar Fractions Connected to Decimals 334
RESOURCES FOR CHAPTER 15
Approximation with a Nice Fraction 335
Literature Connections 307
Recommended Readings 307
Ordering Decimal Numbers 336
Other Fraction-Decimal Equivalents 337
328
xvi
Contents
Introducing Percents
RESOURCES FOR CHAPTER 18
337
Models and Terminology 338
Literature Connections 366
Recommended Readings 367
Online Resources 368
Field Experience Guide Connections 368
Realistic Percent Problems 339
Estimation 341
Computation with Decimals
342
The Role of Estimation 342
Addition and Subtraction 342
Multiplication 343
CHAPTER 19
Division 344
Developing Measurement
Concepts
REFLECTIONS ON CHAPTER 17
Writing to Learn 345
For Discussion and Exploration 345
The Meaning and Process of Measuring
370
Concepts and Skills 371
Nonstandard Units and Standard Units:
Reasons for Using Each 372
RESOURCES FOR CHAPTER 17
Literature Connections 345
Recommended Readings 346
Online Resources 346
Field Experience Guide Connections 347
The Role of Estimation and Approximation 372
Length
373
Comparison Activities 373
Units of Length 374
Making and Using Rulers 375
CHAPTER 18
Area
Proportional
Reasoning
348
376
Comparison Activities 376
INVESTIGATIONS IN NUMBER, DATA, AND SPACE
Ratios
Grade 3, Perimeter, Angles, and Area
348
Apago PDF Units
Enhancer
of Area 378
Types of Ratios 349
Proportional Reasoning
The Relationship Between Area and Perimeter 380
350
Volume and Capacity
Additive Versus Multiplicative Situations 351
Units of Volume and Capacity 381
Equivalent Ratios 353
Weight and Mass
Different Ratios 354
Grade 7, Comparing and Scaling
Units of Weight or Mass 383
356
Time
Ratio Tables 356
Proportional Reasoning Across the Curriculum
Algebra 359
383
Duration 383
358
Clock Reading 383
Elapsed Time 384
Measurement and Geometry 359
Scale Drawings 360
Statistics 361
363
382
Comparison Activities 382
CONNECTED MATHEMATICS
Proportions
380
Comparison Activities 381
Indentifying Multiplicative Relationships 352
Number: Fractions and Percent
362
Money
385
Coin Recognition and Values 385
Counting Sets of Coins 385
Making Change 386
Angles
386
Within and Between Ratios 363
Comparison Activities 386
Reasoning Approaches 364
Units of Angular Measure 386
Cross-Product Approach 365
REFLECTIONS ON CHAPTER 18
Writing to Learn 366
For Discussion and Exploration 366
377
Using Protractors and Angle Rulers 386
Introducing Standard Units
Instructional Goals
387
387
Important Standard Units and Relationships 389
369
Contents
Estimating Measures
xvii
REFLECTIONS ON CHAPTER 20
389
Strategies for Estimating Measurements 390
Writing to Learn 433
For Discussion and Exploration 433
Tips for Teaching Estimation 390
Measurement Estimation Activities 391
RESOURCES FOR CHAPTER 20
Developing Formulas for Area and Volume
391
Students’ Misconceptions 391
Areas of Rectangles, Parallelograms, Triangles,
and Trapezoids 392
Literature Connections 433
Recommended Readings 434
Online Resources 434
Field Experience Guide Connections 435
Circumference and Area of Circles 394
Volumes of Common Solid Shapes 395
Connections among Formulas 396
CHAPTER 21
REFLECTIONS ON CHAPTER 19
Developing Concepts
of Data Analysis
Writing to Learn 397
For Discussion and Exploration 397
RESOURCES FOR CHAPTER 19
What Does It Mean to Do Statistics?
436
437
Is It Statistics or Is It Mathematics? 437
Variability 437
The Shape of Data 438
Process of Doing Statistics 439
Literature Connections 397
Recommended Readings 397
Online Resources 398
Field Experience Guide Connections 398
Formulating Questions
439
Ideas for Questions 439
CHAPTER 20
Data Collection
Geometric Thinking and Geometric
Concepts 399
440
Using Existing Data Sources 440
Data Analysis: Classification
Apago PDF Enhancer
Attribute Materials
Geometry Goals for Students
399
441
441
Data Analysis: Graphical Representations
Spatial Sense and Geometric Reasoning 400
443
Bar Graphs and Tally Charts 443
Geometric Content 400
Circle Graphs 444
The Development of Geometric Thinking
400
The van Hiele Levels of Geometric Thought 400
Implications for Instruction 404
Continuous Data Graphs 445
Scatter Plots 447
Data Analysis: Measures of Center
Learning about Shapes and Properties
405
449
Averages 449
Shapes and Properties for Level-0 Thinkers 405
Understanding the Mean: Two Interpretations 449
Shapes and Properties for Level-1 Thinkers 410
Box-and-Whisker Plots 452
Shapes and Properties for Level-2 Thinkers 416
Learning about Transformations
419
Transformations for Level-0 Thinkers 419
Transformations for Level-1 Thinkers 422
Transformations for Level-2 Thinkers 424
Learning about Location
453
REFLECTIONS ON CHAPTER 21
Writing to Learn 454
For Discussion and Exploration 454
RESOURCES FOR CHAPTER 21
424
Location for Level-0 Thinkers 424
Location for Level-1 Thinkers 426
Location for Level-2 Thinkers 428
Learning about Visualization
Interpreting Results
429
Visualization for Level-0 Thinkers 429
Visualization for Level-1 Thinkers 430
Visualization for Level-2 Thinkers 432
Literature Connections 454
Recommended Readings 455
Online Resources 455
Field Experience Guide Connections 455
xviii
Contents
CHAPTER 22
CHAPTER 23
Exploring Concepts
of Probability
Introducing Probability
457
Exponents
Likely or Not Likely 457
The Probability Continuum 459
473
Exponents in Expressions and Equations 473
Theoretical Probability and Experiments
Negative Exponents 476
460
Scientific Notation 477
Theoretical Probability 461
Integers
Experiments 462
Implications for Instruction 464
479
Contexts for Exploring Integers 479
Use of Technology in Experiments 464
Meaning of Negative Numbers 481
Sample Spaces and Probability of Two Events
465
Independent Events 465
Two-Event Probabilities with an Area Model 466
Dependent Events 467
Simulations
Developing Concepts
of Exponents, Integers,
and Real Numbers
456
Two Models for Teaching Integers 481
Operations with Integers
482
Addition and Subtraction 482
Multiplication and Division 484
468
Real Numbers
486
REFLECTIONS ON CHAPTER 22
Rational Numbers 486
Writing to Learn 470
For Discussion and Exploration 470
Irrational Numbers 487
Density of the Real Numbers 489
RESOURCES FOR CHAPTER 22
REFLECTIONS ON CHAPTER 23
Literature Connections 471
Recommended Readings 471
Online Resources 472
Field Experience Guide Connections 472
Writing to Learn 489
For Discussion and Exploration 489
Apago PDFRESOURCES
Enhancer
FOR CHAPTER 23
Literature Connections 490
Recommended Readings 490
Online Resources 490
Field Experience Guide Connections 490
APPENDIX A
Principles and Standards for School Mathematics:
Content Standards and Grade Level Expectations A-1
APPENDIX B
Standards for Teaching Mathematics
APPENDIX C
Guide to Blackline Masters
References
Index I-1
R-1
C-1
B-1
473
Preface
WHAT YOU WILL FIND IN THIS BOOK
If you look at the table of contents, you will see that the chapters are separated into two distinct
sections. The first section, consisting of seven chapters, deals with important ideas that cross
the boundaries of specific areas of content. The second section, consisting of 16 chapters, offers
teaching suggestions for every major mathematics topic in the pre-K–8 curriculum. Chapters in
Section I offer perspective on the challenging task of helping children learn mathematics. The
evolution of mathematics education and underlying causes for those changes are important components of your professional knowledge as a mathematics teacher. Having a feel for the discipline
of mathematics—that is, to know what it means to “do mathematics”—is also a critical component
of your profession. The first two chapters address these issues.
Chapters 2 and 3 are core chapters in which you will learn about a constructivist view of learning, how that is applied to learning mathematics, and what it means to teach through problem
solving. Chapter 4 will help you translate these ideas of how children best learn mathematics into
the lessons you will be teaching. Here you will find practical perspectives on planning effective
lessons for all children, on the value of drill and practice, and other issues. A sample lesson plan is
found at the end of this chapter. Chapter 5 explores the integration of assessment with instruction
to best assist student learning.
In Chapter 6, you will read about the diverse student populations in today’s classrooms including students who are English language learners, are gifted, or have special needs. Chapter 7
provides perspectives on the issues related to using technology in the teaching of mathematics. A
strong case is made for the use of handheld technology at all grade levels. Guidance is offered for
the selection and use of computer software and resources on the Internet.
Each chapter of Section II provides a perspective of the mathematical content, how children
best learn that content, and numerous suggestions for problem-based activities to engage children
in the development of good mathematics. The problem-based tasks for students are integrated
within the text, not added on. Reflecting on the activities as you read can help you think about the
mathematics from the perspective of the student. Read them along with the text, not as an aside.
As often as possible, take out pencil and paper and try the problems so that you actively engage in
your learning about children learning mathematics.
Apago PDF Enhancer
SOME SPECIAL FEATURES OF THIS TEXT
By flipping through the book, you will notice many section headings, a large number of figures,
and various special features. All are designed to make the book more useful as a textbook and as a
long-term resource. Here are a few things to look for.
NEW!
MyEducationLab 3
MyEd
New to this edition, you will find margin notes that connect chapter ideas to the
MyEducationLab website (www.myeducationlab.com). Every chapter in Section I
connects to new video clips of John Van de Walle presenting his ideas and activities
to groups of teachers. For a complete list of the new videos of John Van de Walle, see
the inside front cover of your text.
Think of MyEducationLab as an extension of the text. You will find practice test
questions, lists of children’s literature organized by topic, links to useful websites, classroom videos, and videos of John Van de Walle talking with students and teachers. Each
of the Blackline Masters mentioned in the book can be downloaded as a PDF file. You will
also find seven Expanded Lesson plans based on activities in the book. MyEducationLab is
easy to use! In the textbook, look for the MyEducationLab logo in the margins and follow links
to access the multimedia assignments in MyEducationLab that correspond with the chapter
content.
Go to the Activities and Application section of Chapter
3 of MyEducationLab. Click
on Videos and watch the
video entitled “John Van
de Walle on Teaching
Through Problem Solving”
to see him working on a
problem with teachers during a training workshop.
xix
xx
Preface
Big Ideas
Much of the research and literature espousing a student-centered approach suggests that
teachers plan their instruction around “big ideas” rather than isolated skills or concepts. At
the beginning of each chapter in Section II, you will find a list of the key mathematical ideas
associated with the chapter. Teachers find these lists helpful for quickly getting a picture of the
mathematics they are teaching.
Mathematics Content Connections
Following the Big Ideas lists are brief descriptions of other content areas in mathematics that
are related to the content of the current chapter. These lists are offered to help you be more
aware of the potential interaction of content as you plan lessons, diagnose students’ difficulties,
and learn more yourself about the mathematics you are teaching.
388
Z Activities
Chapter 19 Developing Measurement Concepts
of required precision. (Would you measure your lawn to
purchase grass seed with the same precision as you would
use in measuring a window to buy a pane of glass?) Students
need practice in using common sense in the selection of
appropriate standard units.
3. Knowledge of relationships between units. Students
should know those relationships that are commonly used,
such as inches, feet, and yards or milliliters and liters. Tedious conversion exercises do little to enhance measurement sense.
Developing Unit Familiarity. Two types of activities can
help develop familiarity with standard units: (1) comparisons that focus on a single unit and (2) activities that develop personal referents or benchmarks for single units or
easy multiples of units.
Activity 19.21
Developing Decimal Number Sense
333
About
One Unit
for students to learn from the beginning
that decimals
are
simply fractions.
Give students a model of a standard unit, and have
The calculator can also play a significant role in decithem search for objects that measure about the same
mal concept development.
as that one unit. For example, to develop familiarity
with the meter, give students a piece of rope 1 meter
long. Have them make lists of things that are about 1
meter. Keep separate lists for things that are a little
Recall how to make the calculator “count” by
less (or more) or twice as long (or half as long). Enpressing
1
. . . Now have students
students
to find familiar items in their daily
press
0.1
. . . When the courage
display shows
0.9,
stop and discuss what this means lives.
and what
Inthe
thedisplay
case of lengths, be sure to include curved
will look like with the next press. Many students will
or circular lengths. Later, students can try to predict
predict 0.10 (thinking that 10 comes after 9). This
a given
prediction is even more interestingwhether
if, with each
press, object is more than, less than, or
the students have been accumulating
base-ten
close
to 1strips
meter.
Activity 17.3
Calculator Decimal Counting
as models for tenths. One more press would mean
one more strip, or 10 strips. Why should the calculator
same
activity can be done with other unit lengths.
not show 0.10? When the tenth pressThe
produces
a display of 1 (calculators are not usually
set to display
Families
can be enlisted to help students find familiar distrailing zeros to the right of the decimal), the discustances that are about 1 mile or about 1 kilometer. Sugsion should revolve around trading 10 strips for a
a letter
square. Continue to count to 4 gest
or 5 byin
tenths.
How that they check the distances around the
many presses to get from one whole
number to the to the school or shopping center, or along
neighborhood,
next? Try counting by 0.01 or by 0.001. These counts
other frequently traveled paths. If possible, send home (or
illustrate dramatically how small one-hundredth and
use in10class)
one-thousandth really are. It requires
countsaby1-meter or 1-yard trundle wheel to measure
0.001 to get to 0.01 and 1000distances.
counts to reach 1.
For capacity units such as cup, quart, and liter, students
The fact that the calculator need
counts a0.8,
0.9, 1, 1.1 incontainer
that holds or has a marking for a single
stead of 0.8, 0.9, 0.10, 0.11 should give rise to the question
unit. They should then find other containers at home and at
“Does this make sense? If so, why?”
school
that hold
about as much, more, and less. Remember
Calculators that permit entry
of fractions
also have
a fraction-decimal conversion key.
On
some
calculators
that the shapes of acontainers can be very deceptive when
decimal such as 0.25 will convert to the base-ten fraction
estimating their capacity.
25
and allow for either manual or automatic simplification.
100
For
standard
Graphing calculators can be set so that
thethe
conversion
is weights of gram, kilogram, ounce, and
either with or without simplification.
Thestudents
ability of fracpound,
can compare objects on a two-pan balance
tion calculators to go back and forth between fractions and
with single copies of these units. It may be more effective to
decimals makes them a valuable tool as students begin to
work with 10 grams or 5 ounces. Students can be encourconnect fraction and decimal symbolism.
aged to bring in familiar objects from home to compare on
the classroom scale.
Standard area units are in terms of lengths such as
square inches or square feet, so familiarity with lengths is
important. Familiarity with a single degree is not as important as some idea of 30, 45, 60, and 90 degrees.
The second approach to unit familiarity is to begin with
very familiar items and use their measures as references or
benchmarks. A doorway is a bit more than 2 meters high
and a doorknob is about 1 meter from the floor. A bag of
flour is a good reference for 5 pounds. A paper clip weighs
about a gram and is about 1 centimeter wide. A gallon of
milk weighs a little less than 4 kilograms.
Activity 19.22
Familiar References
Use the book Measuring Penny (Leedy, 2000) to get
students interested in the variety of ways familiar
items can be measured. In this book, the author
bridges between nonstandard (e.g., dog biscuits) and
standard units to measure Penny the pet dog. Have
your students use the idea of measuring Penny to find
something at home (or in class) to measure in as many
ways as they can think using standard units. The measures should be rounded to whole numbers (unless
children suggest adding a fractional unit to be more
precise). Discuss in class the familiar items chosen and
their measures so that different ideas and benchmarks
are shared.
The numerous activities found in every chapter of Section
II have always been rated by readers as one of the most
valuable parts of the book. Some activity ideas are described
directly in the text and in the illustrations. Others are
presented in the numbered Activity boxes. Every activity
is a problem-based task (as described in Chapter 3) and is
designed to engage students in doing mathematics. Some
activities incorporate calculator use; these particular activities are marked with a calculator icon.
Apago PDF Enhancer
Of special interest for length are benchmarks found
on our bodies. These become quite familiar over time and
can be used as approximate rulers in many situations. Even
though young children grow quite rapidly, it is useful for
them to know the approximate lengths that they carry
around with them.
Teaching Con
siderations
Activity 19.23
Personal Benchmarks
Measure your body. About how long is your foot, your
stride, your hand span (stretched and with fingers together), the width of your finger, your arm span (finger to finger and finger to nose), the distance around
your wrist and around your waist, and your height to
waist, to shoulder, and to head? Some may prove to
be useful benchmarks, and some may be excellent
models for single units. (The average child’s fingernail
width is about 1 cm, and most people can find a
10-cm length somewhere on their hands.)
To help remember these references, they must be used
in activities in which lengths, volumes, and so on are compared to the benchmarks to estimate measurements.
Investigations in Number,
Data, and Space and
Connected Mathematics 3
In Section II, four chapters include features that
describe an activity from the standards-based curriculum Investigations in Number, Data, and Space (an
elementary curriculum) or Connected Mathematics
Project (CMP II) (a middle school curriculum). These
features include a description of an activity in the
program as well as the context of the unit in which it is
found. The main purpose of this feature is to acquaint
you with these materials and to demonstrate how the
spirit of the NCTM Standards and the constructivist
theory espoused in this book have been translated into
existing commercial curricula.
Grade 7, Varia
Context
Much of this
unit is built on
the context of
a group of stud
ents who take
a multiday bike
trip from Phil
adelphia to Wil
liamsburg, Virginia, and who
then decide to
set up a bike
tour business
of their own.
Students expl
a variety of func
ore
tional relation
ships between
time, distance
, speed, expe
nses, profits,
so on. When
and
data are plot
ted as discrete
points, students
consider wha
t the graph
might look like
between poin
ts. For example,
what interpre
tations could
be given to each
of these five
graphs showing
speed change
from 0 to 15
mph in the first
10 minutes of
a trip?
Investigation
bles and Patte
rns
3: Analyzing Gra
phs and Tables
Area
377
Task Descrip
tion
In this investig
ation, the ficti
onal students
in the unit beg
an gathering
data in preparation for setti
ng up their tour
business. As
their first task
, they sought
data from two
ferent bike rent
difal companies
as shown here
given by one
s, company in the
,
le
g
n
A
form
eter,and by the other in the form of of a table
a graph. The
task is interest
, Perim
3
e
ing
d
beca
use of the first
Gra
way in which
han
a
d
stud
ents eas rien
ce the value
of one represen ent of idexpe
and Are
tation overisan