Elementary and Middle School Mathematics Teaching Developmentally

(1)

(2)

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 New York San Francisco

Mexico City Montreal Toronto London Madrid Munich Paris Hong Kong Singapore Tokyo Cape Town Sydney

(3)

Apago PDF Enhancer

Editorial Assistant: Annalea Manalili

Senior Marketing Manager: Darcy Betts

Editorial Production Service: Omegatype Typography, Inc. Composition Buyer: Linda Cox

Manufacturing Buyer: Megan Cochran

Electronic Composition: Omegatype Typography, Inc. Interior Design: Carol Somberg

Cover Administrator: Linda Knowles

For related titles and support materials, visit our online catalog at www.pearsonhighered.com.

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner.

To obtain permission(s) to use material from this work, please submit a written request to Allyn and Bacon, Permissions Department, 501 Boylston Street, Suite 900, Boston, MA 02116, or fax your request to 617-671-2290.

Between the time website information is gathered and then published, it is not unusual for some sites to have closed. Also, the transcription of URLs can result in typographical errors. The publisher would appreciate notification where these errors occur so that they may be corrected in subsequent editions.

Library of Congress Cataloging-in-Publication Data

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. III. Title. QA135.6.V36 2008

510.71'2—dc22

2008040112

Allyn & Bacon is an imprint of

www.pearsonhighered.com

ISBN-10: 0-205-57352-5 ISBN-13: 978-0-205-57352-3 Printed in the United States of America

(4)

Apago PDF Enhancer

“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 proof that Dad was in fact a rock star, it came in the stories that poured in when he died—from countless 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)

“Believe in kids!”

—John A. Van de Walle

(5)

Apago PDF Enhancer

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.

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.

(6)

Apago PDF Enhancer

v

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 jour-nals 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.

Karen S. Karp is a professor of mathematics education at the University 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).

(7)

(8)

Apago PDF Enhancer

Brief Contents

CHAPTER 1

Teaching Mathematics

in the Era of the NCTM Standards 1

CHAPTER 2

Exploring What It Means

to Know and Do Mathematics 13

CHAPTER 3

Teaching Through

Problem Solving 32

CHAPTER 4

Planning in the

Problem-Based Classroom 58

CHAPTER 5

Building Assessment

into Instruction 76

CHAPTER 6

Teaching Mathematics

Equitably to All Children 93

CHAPTER 7

Using Technology

to Teach Mathematics 111

vii

CHAPTER 8

Developing Early Number

Concepts and Number Sense 125

CHAPTER 9

Developing Meanings for

the Operations 145

CHAPTER 10

Helping Children

Master the Basic Facts 167

CHAPTER 11

Developing

Whole-Number Place-Value Concepts 187

CHAPTER 12

Developing Strategies

for Whole-Number Computation 213

CHAPTER 13

Using Computational

Estimation with Whole Numbers 240

CHAPTER 14

Algebraic Thinking:

Generalizations, Patterns, and Functions 254

CHAPTER 15

Developing Fraction

Concepts 286

CHAPTER 16

Developing Strategies

for Fraction Computation 309

CHAPTER 17

Developing Concepts

of Decimals and Percents 328

CHAPTER 18

Proportional Reasoning 348

CHAPTER 19

Developing Measurement

Concepts 369

CHAPTER 20

Geometric Thinking

and Geometric Concepts 399

CHAPTER 21

Developing Concepts

of Data Analysis 436

CHAPTER 22

Exploring Concepts

of Probability 456

CHAPTER 23

Developing Concepts of

Exponents, Integers, and Real Numbers 473

APPENDIX A

Principles and Standards

for School Mathematics: Content Standards

and Grade Level Expectations A-1

APPENDIX B

Standards for Teaching

Mathematics B-1

APPENDIX C

Guide to Blackline

Masters C-1

SECTION I

Teaching Mathematics: Foundations and Perspectives

SECTION II

(9)

(10)

Apago PDF Enhancer

CHAPTER 1 Teaching Mathematics in the

Era of the NCTM Standards

1

The National Standards-Based Movement 1

Principles and Standards for School Mathematics 2 The Six Principles 2

The Five Content Standards 3

The Five Process Standards 3

Curriculum Focal Points: A Quest for Coherence 5

The Professional Standards for Teaching

Mathematics and Mathematics Teaching Today 5 Shifts in the Classroom Environment 5

The Teaching Standards 5

Influences and Pressures on Mathematics Teaching 6

National and International Studies 6

State Standards 7

Curriculum 7

A Changing World Economy 8 An Invitation to Learn and Grow 9

Becoming a Teacher of Mathematics 9

REFLECTIONS ON CHAPTER 1

Writing to Learn 11

For Discussion and Exploration 11

RESOURCES FOR CHAPTER 1

CHAPTER 2 Exploring What It Means to

Know and Do Mathematics

13

What Does It Mean to Do Mathematics? 13

Mathematics Is the Science of Pattern and Order 13

A Classroom Environment for Doing Mathematics 14 An Invitation to Do Mathematics 15

Let’s Do Some Mathematics! 15

Where Are the Answers? 19

What Does It Mean to Learn Mathematics? 20

Constructivist Theory 20

Sociocultural Theory 21

Implications for Teaching Mathematics 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

Writing to Learn 30

Online Resources 31

ix

SECTION I

Teaching Mathematics: Foundations and Perspectives

(11)

Apago PDF Enhancer

CHAPTER 3 Teaching Through Problem

Solving

32

Teaching Through Problem Solving 32

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

Selecting or Designing Problem-Based Tasks and Lessons 36

Multiple Entry Points 36

Creating Meaningful and Engaging Contexts 37

How to Find Quality Tasks and Problem-Based Lessons 38

Teaching about Problem Solving 42

Four-Step Problem-Solving Process 42

Problem-Solving Strategies 43

Teaching in a Problem-Based Classroom 43

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

A Three-Phase Lesson Format 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 Frequently Asked Questions 55

Writing to Learn 56

Online Resources 57

CHAPTER 4 Planning in the

Problem-Based Classroom

58

Planning a Problem-Based Lesson 58

Planning Process for Developing a Lesson 58

Applying the Planning Process 62

Variations of the Three-Phase Lesson 63

Textbooks as Resources 64

Planning for All Learners 64

Make Accommodations and Modifications 65

Differentiating Instruction 65

Flexible Groupings 67

Example of Accommodating a Lesson: ELLs 67 Drill or Practice? 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 Homework 71

Practice as Homework 71

Drill as Homework 71

Provide Homework Support 72

Writing to Learn 72

Online Resources 73

EXPANDED LESSON

Fixed Areas 74

CHAPTER 5 Building Assessment into

Instruction

76

Integrating Assessment into Instruction 76

What Is Assessment? 76

The Assessment Standards 76

Why Do We Assess? 77

What Should Be Assessed? 78 Performance-Based Assessments 78

Examples of Performance-Based Tasks 79

Thoughts about Assessment Tasks 80 Rubrics and Performance Indicators 80

Simple Rubrics 80

Performance Indicators 81

Student Involvement with Rubrics 82 Observation Tools 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

(12)

Apago PDF Enhancer

Writing Prompts and Ideas 85

Journals for Early Learners 86

Student Self-Assessment 87 Diagnostic Interviews 87 Tests 88

Improving Performance on High-Stakes Tests 89

Teach Fundamental Concepts and Processes 89

Test-Taking Strategies 89 Grading 90

Grading Issues 90

Writing to Learn 91

Online Resources 92

CHAPTER 6 Teaching Mathematics

Equitably to All Children

93

Creating Equitable Instruction 93 Mathematics for All Children 94

Diversity in Today’s Classroom 94

Tracking and Flexible Grouping 94

Instructional Principles for Diverse Learners 95 Providing for Students with Special Needs 95

Response to Intervention 95

Students with Mild Disabilities 96

Students with Significant Disabilities 100

Culturally and Linguistically Diverse Students 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 106

Possible Causes of Gender Inequity 106

What Can Be Done? 106

Reducing Resistance and Building Resilience 107 Providing for Students Who Are Mathematically

Gifted 107

Strategies to Avoid 108

Strategies to Incorporate 108 Final Thoughts 109

Writing to Learn 109

Online Resources 110

CHAPTER 7 Using Technology to Teach

Mathematics

111

Calculators in Mathematics Instruction 112

When to Use a Calculator 112

Benefits of Calculator Use 112

Graphing Calculators 113

Data-Collection Devices 114

Computers in Mathematics Instruction 115

Tools for Developing Numeration 115

Tools for Developing Geometry 116

Tools for Developing Probability and Data Analysis 117

Tools for Developing Algebraic Thinking 118 Instructional Software 118

Concept Instruction 118

Problem Solving 118

Drill and Reinforcement 118

Guidelines for Selecting and Using Software 119

Guidelines for Using Software 119

How to Select Software 119 Resources on the Internet 120

How to Select Internet Resources 120

Emerging Technologies 120

(13)

Apago PDF Enhancer

CHAPTER 8 Developing Early Number

Concepts and Number Sense

125

Promoting Good Beginnings 125

Number Development in Pre-K and Kindergarten 126

The Relationships of More, Less, and Same 126

Early Counting 127

Numeral Writing and Recognition 128

Counting On and Counting Back 128 Early Number Sense 129

Relationships among Numbers 1 Through 10 130

Patterned Set Recognition 130

One and Two More, One and Two Less 131

Anchoring Numbers to 5 and 10 132

Part-Part-Whole Relationships 134

Dot Cards as a Model for Teaching Number Relationships 137

Relationships for Numbers 10 Through 20 138

Pre-Place-Value Concepts 138

Extending More Than and Less Than Relationships 139

Doubles and Near-Doubles 139 Number Sense in Their World 140

Estimation and Measurement 140

Data Collection and Analysis 141

Extensions to Early Mental Mathematics 142

Literature Connections 143

CHAPTER 9 Developing Meanings

for the Operations

145

Addition and Subtraction Problem Structures 145

Examples of the Four Problem Structures 146 Teaching Addition and Subtraction 148

Contextual Problems 148

INVESTIGATIONS IN NUMBER, DATA, AND SPACE

Grade 2, Counting, Coins, and Combinations 150

Model-Based Problems 151

Properties of Addition and Subtraction 153

Multiplication and Division Problem Structures 154

Examples of the Four Problem Structures 154 Teaching Multiplication and Division 157

Contextual Problems 157

Remainders 157

Model-Based Problems 158

Properties of Multiplication and Division 160 Strategies for Solving Contextual Problems 161

Analyzing Context Problems 161

Two-Step Problems 163

SECTION II

Development of Mathematical Concepts and Procedures

(14)

Apago PDF Enhancer

CHAPTER 10 Helping Children Master

the Basic Facts

167

Developmental Nature of Basic Fact Mastery 167

Approaches to Fact Mastery 168

Guiding Strategy Development 169

Reasoning Strategies for Addition Facts 170

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 175

Subtraction as Think-Addition 175

Down Over 10 176

Take from the 10 176

Reasoning Strategies for Multiplication Facts 177

Doubles 178

Fives 178

Zeros and Ones 178

Nifty Nines 179

Using Known Facts to Derive Other Facts 180 Division Facts and “Near Facts” 181 Mastering the Basic Facts 182

Effective Drill 182 Fact Remediation 184

CHAPTER 11 Developing Whole-Number

Place-Value Concepts

187

Pre-Base-Ten Concepts 188

Children’s Pre-Base-Ten View of Numbers 188

Count by Ones 188

Basic Ideas of Place Value 188

Integration of Base-Ten Groupings with Count by Ones 188

Role of Counting 189

Integration of Groupings with Words 189

Integration of Groupings with Place-Value Notation 190 Models for Place Value 191

Base-Ten Models and the Ten-Makes-One Relationship 191

Groupable Models 191

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 197

Two-Digit Number Names 197

Three-Digit Number Names 198

Written Symbols 198

Patterns and Relationships with Multidigit Numbers 200

The Hundreds Chart 200

Relationships with Landmark Numbers 202

Number Relationships for Addition and Subtraction 204

Connections to Real-World Ideas 207 Numbers Beyond 1000 207

Extending the Place-Value System 208

Conceptualizing Large Numbers 209

CHAPTER 12 Developing Strategies for

Whole-Number Computation

213

Toward Computational Fluency 214

Direct Modeling 214

Student-Invented Strategies 215

Traditional Algorithms 217

(15)

Apago PDF Enhancer

Development of Student-Invented Strategies 218

Creating an Environment for Inventing Strategies 218

Models to Support Invented Strategies 218 Student-Invented Strategies for Addition

and Subtraction 219

Adding and Subtracting Single-Digit Numbers 219

Adding Two-Digit Numbers 220

Subtracting by Counting Up 220

Take-Away Subtraction 221

Extensions and Challenges 222

Traditional Algorithms for Addition and Subtraction 223

Addition Algorithm 223

Subtraction Algorithm 225

Student-Invented Strategies for Multiplication 226

Useful Representations 226

Multiplication by a Single-Digit Multiplier 227

Multiplication of Larger Numbers 228

Traditional Algorithm for Multiplication 230

One-Digit Multipliers 230

Two-Digit Multipliers 231

Student-Invented Strategies for Division 232

Missing-Factor Strategies 232

Cluster Problems 233

Traditional Algorithm for Division 234

One-Digit Divisors 234

Two-Digit Divisors 235

CHAPTER 13 Using Computational Estimation

with Whole Numbers

240

Introducing Computational Estimation 240

Understanding Computational Estimation 241

Suggestions for Teaching Computational Estimation 242

Computational Estimation from Invented Strategies 244

Stop Before the Details 244

Use Related Problem Sets 244

Computational Estimation Strategies 245

Front-End Methods 245

Rounding Methods 246

Compatible Numbers 247

Clustering 248

Use Tens and Hundreds 248 Estimation Experiences 249

Calculator Activities 249

Using Whole Numbers to Estimate Rational Numbers 251

CHAPTER 14 Algebraic Thinking:

Generalizations, Patterns,

and Functions

254

Algebraic Thinking 255

Generalization from Arithmetic and from Patterns 255

Generalization with Addition 255

Generalization in the Hundreds Chart 256

Generalization Through Exploring a Pattern 257 Meaningful Use of Symbols 257

The Meaning of the Equal Sign 258

The Meaning of Variables 262

Making Structure in the Number System Explicit 265

Making Conjectures about Properties 265

Justifying Conjectures 266

Odd and Even Relationships 266 Study of Patterns and Functions 267

Repeating Patterns 267

Growing Patterns 269

Linear Functions 274 Mathematical Modeling 276 Teaching Considerations 277

Emphasize Appropriate Algebra Vocabulary 277

Multiple Representations 278

Connect Representations 280

(16)

Apago PDF Enhancer

CONNECTED MATHEMATICS

Grade 7, Variables and Patterns 281

CHAPTER 15 Developing Fraction

Concepts

286

Meanings of Fractions 287

Fraction Constructs 287

Building on Whole-Number Concepts 287 Models for Fractions 288

Region or Area Models 288

Length Models 289

Set Models 290

Concept of Fractional Parts 291

Sharing Tasks 291

Fraction Language 293

Equivalent Size of Fraction Pieces 293

Partitioning 294

Using Fraction Language and Symbols 294

Counting Fraction Parts: Iteration 294

Fraction Notation 296

Fractions Greater Than 1 296

Assessing Understanding 297 Estimating with Fractions 298

Benchmarks of Zero, One-Half, and One 299

Using Number Sense to Compare 299 Equivalent-Fraction Concepts 301

Conceptual Focus on Equivalence 301

Equivalent-Fraction Models 302

Developing an Equivalent-Fraction Algorithm 304 Teaching Considerations for Fraction Concepts 306

CHAPTER 16 Developing Strategies

for Fraction Computation

309

Number Sense and Fraction Algorithms 310

Conceptual Development Takes Time 310

A Problem-Based Number Sense Approach 310

Computational Estimation 310 Addition and Subtraction 312

Invented Strategies 312

Developing an Algorithm 315

Mixed Numbers and Improper Fractions 317 Multiplication 317

Developing the Concept 317

Developing the Algorithm 320

Factors Greater Than One 320 Division 321

Partitive Interpretation of Division 321

Measurement Interpretation of Division 323

Answers That Are Not Whole Numbers 324

Developing the Algorithms 324

CHAPTER 17 Developing Concepts

of Decimals and Percents

328

Connecting Fractions and Decimals 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

Approximation with a Nice Fraction 335

Ordering Decimal Numbers 336

Other Fraction-Decimal Equivalents 337

(17)

Apago PDF Enhancer

Introducing Percents 337

Models and Terminology 338

Realistic Percent Problems 339

Estimation 341

Computation with Decimals 342

The Role of Estimation 342

Addition and Subtraction 342

Multiplication 343

Division 344

CHAPTER 18 Proportional

Reasoning

348

Ratios 348

Types of Ratios 349

Proportional Reasoning 350

Additive Versus Multiplicative Situations 351

Indentifying Multiplicative Relationships 352

Equivalent Ratios 353

Different Ratios 354

CONNECTED MATHEMATICS

Grade 7, Comparing and Scaling 356

Ratio Tables 356

Proportional Reasoning Across the Curriculum 358

Algebra 359

Measurement and Geometry 359

Scale Drawings 360

Statistics 361

Number: Fractions and Percent 362 Proportions 363

Within and Between Ratios 363

Reasoning Approaches 364

Cross-Product Approach 365

CHAPTER 19 Developing Measurement

Concepts

369

The Meaning and Process of Measuring 370

Concepts and Skills 371

Nonstandard Units and Standard Units: Reasons for Using Each 372

The Role of Estimation and Approximation 372 Length 373

Comparison Activities 373

Units of Length 374

Making and Using Rulers 375 Area 376

INVESTIGATIONS IN NUMBER, DATA, AND SPACE

Grade 3, Perimeter, Angles, and Area 377

Units of Area 378

The Relationship Between Area and Perimeter 380 Volume and Capacity 380

Units of Volume and Capacity 381 Weight and Mass 382

Units of Weight or Mass 383 Time 383

Duration 383

Clock Reading 383

Elapsed Time 384 Money 385

Coin Recognition and Values 385

Counting Sets of Coins 385

Making Change 386 Angles 386

Units of Angular Measure 386

Using Protractors and Angle Rulers 386 Introducing Standard Units 387

Instructional Goals 387

(18)

Apago PDF Enhancer

Estimating Measures 389

Strategies for Estimating Measurements 390

Tips for Teaching Estimation 390

Measurement Estimation Activities 391

Developing Formulas for Area and Volume 391

Students’ Misconceptions 391

Areas of Rectangles, Parallelograms, Triangles, and Trapezoids 392

Circumference and Area of Circles 394

Volumes of Common Solid Shapes 395

Connections among Formulas 396

CHAPTER 20 Geometric Thinking and Geometric

Concepts

399

Geometry Goals for Students 399

Spatial Sense and Geometric Reasoning 400

Geometric Content 400

The Development of Geometric Thinking 400

The van Hiele Levels of Geometric Thought 400

Implications for Instruction 404

Learning about Shapes and Properties 405

Shapes and Properties for Level-0 Thinkers 405

Shapes and Properties for Level-1 Thinkers 410

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 424

Location for Level-0 Thinkers 424

Location for Level-1 Thinkers 426

Location for Level-2 Thinkers 428 Learning about Visualization 429

Visualization for Level-0 Thinkers 429

CHAPTER 21 Developing Concepts

of Data Analysis

436

What Does It Mean to Do Statistics? 437

Is It Statistics or Is It Mathematics? 437

Variability 437

The Shape of Data 438

Process of Doing Statistics 439 Formulating Questions 439

Ideas for Questions 439 Data Collection 440

Using Existing Data Sources 440 Data Analysis: Classification 441

Attribute Materials 441

Data Analysis: Graphical Representations 443

Bar Graphs and Tally Charts 443

Circle Graphs 444

Continuous Data Graphs 445

Scatter Plots 447

Data Analysis: Measures of Center 449

Averages 449

Understanding the Mean: Two Interpretations 449

Box-and-Whisker Plots 452 Interpreting Results 453

(19)

Apago PDF Enhancer

CHAPTER 22 Exploring Concepts

of Probability

456

Introducing Probability 457

Likely or Not Likely 457

The Probability Continuum 459

Theoretical Probability and Experiments 460

Theoretical Probability 461

Experiments 462

Implications for Instruction 464

Use of Technology in Experiments 464

Sample Spaces and Probability of Two Events 465

Independent Events 465

Two-Event Probabilities with an Area Model 466

Dependent Events 467 Simulations 468

CHAPTER 23 Developing Concepts

of Exponents, Integers,

and Real Numbers

473

Exponents 473

Exponents in Expressions and Equations 473

Negative Exponents 476

Scientific Notation 477 Integers 479

Contexts for Exploring Integers 479

Meaning of Negative Numbers 481

Two Models for Teaching Integers 481 Operations with Integers 482

Addition and Subtraction 482

Multiplication and Division 484 Real Numbers 486

Rational Numbers 486

Irrational Numbers 487

Density of the Real Numbers 489

APPENDIX A

Principles and Standards for School Mathematics:

Content Standards and Grade Level Expectations A-1

APPENDIX B

Standards for Teaching Mathematics B-1

APPENDIX C

Guide to Blackline Masters C-1

(20)

Apago PDF Enhancer

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 com-ponents 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 learn-ing, 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 in-cluding 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.

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.

Preface

MyEducationLab

3 MyEd

NEW!

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, class-room 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 Ap-plication 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 dur-ing a traindur-ing workshop.

(21)

Apago PDF Enhancer

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.

Teaching Considerations 281

and begins the exploration of connecting equations or rules to the representations of graphs and tables. In the final investigation, students use graphing calculators to explore how graphs change in appearance when the rules that produce the graphs change.

Context

Much of this unit is built on the context of a group of students who take a multiday bike trip from Philadelphia to W

illiamsburg, V ir-ginia, and who then decide to set up a bike tour business of their own. Students explore a variety of functional relationships between time, distance, speed, expenses, profits, and so on. When data are plotted as discrete points, students consider what the graph might look like between points. For example, what interpretations 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?

Task Description

In this investigation, the fictional students in the unit began gathering data in prepa-ration for setting up their tour business. As their first task, they sought data from two dif-ferent bike rental companies as shown here, given by one company in the form of a table and by the other in the form of a graph. The task is interesting because of the firsthand way in which students experience the value of one representation over another

, depend-ing on the need of the situation. In this unit students are frequently asked whether a graph or a table is the better source of information.

In the tasks that follow

, students are given a table of data showing results of a phone poll that asked at which price former tour riders would take a bike tour . Students must find the best way to graph this data. After a price for a bike tour is established, graphs for estimated profits are created with corresponding ques-tions about profits depending on different numbers of customers.

The investigations use no formulas at this point. The subsequent investigation is called “Patterns and Rules”

Grade 7, _{Variables and Patterns}

Investigation 3: Analyzing Graphs and Tables

Source:Connected Mathematics: Variables and Patterns: Teacher Edition by Glenda Lappan, James T. Fey, William M. Fitzgerald, Susan N. Friel, &

Z

_Activities

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 activi-ties are marked with a calculator icon.

388 Chapter 19 Developing Measurement Concepts

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 impor-tant 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 mea-sures 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.

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.

Activity 19.23

Personal Benchmarks

Measure your body. About how long is your foot, your stride, your hand span (stretched and with fingers to-gether), the width of your finger, your arm span (fin-ger to fin(fin-ger and fin(fin-ger 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 com-pared to the benchmarks to estimate measurements. 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. Te-dious conversion exercises do little to enhance measure-ment sense.

Developing Unit Familiarity. Two types of activities can help develop familiarity with standard units: (1) compari-sons that focus on a single unit and (2) activities that de-velop personal referents or benchmarks for single units or easy multiples of units.

Activity 19.21

About One Unit

Give students a model of a standard unit, and have them search for objects that measure about the same 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 less (or more) or twice as long (or half as long). En-courage students to find familiar items in their daily lives. In the case of lengths, be sure to include curved or circular lengths. Later, students can try to predict whether a given object is more than, less than, or close to 1 meter.

The same activity can be done with other unit lengths. Families can be enlisted to help students find familiar dis-tances that are about 1 mile or about 1 kilometer. Sug-gest in a letter that they check the distances around the neighborhood, to the school or shopping center, or along other frequently traveled paths. If possible, send home (or use in class) a 1-meter or 1-yard trundle wheel to measure distances.

For capacity units such as cup, quart, and liter, students need a container that holds or has a marking for a single unit. They should then find other containers at home and at school that hold about as much, more, and less. Remember that the shapes of containers can be very deceptive when estimating their capacity.

For the standard weights of gram, kilogram, ounce, and pound, students can compare objects on a two-pan balance with single copies of these units. It may be more effective to work with 10 grams or 5 ounces. Students can be encour-aged to bring in familiar objects from home to compare on the classroom scale.

In Section II, four chapters include features that describe an activity from the standards-based cur-riculum 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.

Investigations in Number,

Data, and Space

and

Connected Mathematics

3

Area 377

two parts and reassembling it in a different shape can show that the before and after shapes have the same area, even though they are different shapes. This idea is not at all obvi-ous to children in the K–2 grade range.

Activity 19.8

Two-Piece Shapes

Cut a large number of rectangles of the same area, about 3 inches by 5 inches. Each pair of students will

This is an especially difficult concept for young children to understand. In addition, many 8- or 9-year

-olds do not

understand that rearranging areas into different shapes does not affect the amount of area.

Direct comparison of two areas is nearly always impos-sible except when the shapes involved have some common

dimension or property

. For example, two rectangles with the same width can be compared directly

, as can any two circles. Comparison of these special shapes, however

, fails h he attribute of area. Instead, activities in which gested. Cutting a shape into

knew when to use a “flexible” measuring tool such as the string or adding machine tape.

Grade 3, Perimeter, Angles, and Area

Context

The perimeter activity continues the development of ideas about linear measurement. At the point of this lesson, it is assumed that students understand the need for standard units and can use tools that measure length in both the metric and customary systems. Students are able to recog-nize that perimeter is the measure around the outside

edges of a two-dimensional shape.

Task

In this investigation, students select real-world objects and measure the perimeter or rim (Hint for students: the word rim is in perimeter). Key in this process will be the choices the students make, such as whether the object they choose has a perimeter that is regular

, like the top of a desk, or more challenging, like the top of a waste paper basket. They also need to choose the tool that will be best for measuring, given yardsticks or metersticks, or adding ma-chine tape or string. At first, students are asked as a group to suggest an object. Then one student traces with a finger the perimeter that will be measured. All students are asked to include this object in the exploration that follows. In this way, students will be able to compare the results of at least one common item, providing a basis for discussing any measurement errors. Students can be asked to include an estimate of the perimeter on their chart prior to actu-ally measuring. During the “after” period of the lesson, students should discuss how they measured objects that were larger than the tool they were using and how they

Source:Investigations in Number, Data, and Space. Grade 3— Perimeter, Angles, and Area,

Developing Decimal Number Sense 333 for students to learn from the beginning that decimals are simply fractions.

The calculator can also play a significant role in deci-mal concept development.

Activity 17.3 Calculator Decimal Counting

Recall how to make the calculator “count” by pressing 1 . . . Now have students press 0.1 . . . When the display shows 0.9, stop and discuss what this means and what the display will look like with the next press. Many students will predict 0.10 (thinking that 10 comes after 9). This prediction is even more interesting if, with each press, the students have been accumulating base-ten strips as models for tenths. One more press would mean one more strip, or 10 strips. Why should the calculator not show 0.10? When the tenth press produces a dis-play of 1 (calculators are not usually set to disdis-play trailing zeros to the right of the decimal), the discus-sion should revolve around trading 10 strips for a square. Continue to count to 4 or 5 by tenths. How many presses to get from one whole number to the next? Try counting by 0.01 or by 0.001. These counts illustrate dramatically how small one-hundredth and one-thousandth really are. It requires 10 counts by 0.001 to get to 0.01 and 1000 counts to reach 1.

The fact that the calculator counts 0.8, 0.9, 1, 1.1 in-stead of 0.8, 0.9, 0.10, 0.11 should give rise to the question “Does this make sense? If so, why?”

Calculators that permit entry of fractions also have a fraction-decimal conversion key. On some calculators a decimal such as 0.25 will convert to the base-ten fraction

25

(22)

Apago PDF Enhancer

Preface xxi

Chapter End Matter

3 NCTM Standards

3 Tech Notes

3 Assessment Notes

3

Assessment should be an integral part of in-struction. Similarly, it makes sense to think about what to be listening for (assessing) as you read about different areas of content devel-opment. Throughout the content chapters, you will see assessment icons indicating a short description of ways to assess the topic in that section. Reading these assessment notes as you read the text can also help you understand how best to help your students.

Throughout the book, you will see an icon indicating a reference to NCTM’s

Principles and Standards for School Mathe-matics. The NCTM Standards notes typi-cally consist of a quotation from the Standards

and/or a summary of what the Standards say about a particular topic. These notes correlate the content of this book with the Standards.

We hope you will find these quotations and statements helpful in understand-ing the vision for good mathematics instruction.

An icon marks each Tech Notes section, which discuss how technology can be used to help with the content just discussed. Descriptions include open-source soft-ware, interactive applets, and other Web-based resources.

Note that there are suggestions of NCTM e-Examples that connect to full lessons on the NCTM Illuminations website. (Inclusion of any title or website in these notes should not be seen as an endorsement.)

The end of each chapter is reorganized to include two major sub sections: Reflections, which includes Writing to Learn and For Discussion and Exploration; and Resources, which includes Literature Connections (found in all Section II chap-ters), Recommended Readings, Online Resources, and Field Experience Guide Connections.

Writing to Learn

To help you focus on the important pedagogical ideas, a list of focusing questions is found at the end of every chapter under the heading “Writing to Learn.” These study questions are designed to help you reflect on the main points of the chap-ter. Actually writing out the answers to these questions in your own words is one of the best ways for you to develop your understanding of each chapter’s main ideas.

For Discussion and Exploration

These questions ask you to explore an issue, reflect on observations in a classroom, compare ideas from this book with those found in curriculum materials, or perhaps take a position on a controversial issue. There are no “right” answers to these questions, but we hope that they will stimulate thought and cause spirited conversations.

Resources for Chapter 19 397

Writing to Learn

1. Explain what it means to measure something. Does your

ex-planation work equally well for length, area, weight, volume, and time?

2. A general instructional plan for measurement has three

steps. Explain how the type of activity to use at each step accomplishes the instructional goal.

3. Four reasons were offered for using nonstandard units

in-stead of standard units in instructional activities. Which of these seem most important to you, and why?

4. Develop in a connected way the area formulas for rectangles,

parallelograms, triangles, and trapezoids. Draw pictures and provide explanations.

5. Explain how the area of a circle can be determined using the

basic formula for the area of a parallelogram. (If you have a set of fraction circular pieces, these can be used as sectors of a circle.)

For Discussion and Exploration

1. Frequently

, a textbook chapter on measurement will cover length, area, volume, and capacity with both metric and cus-tomary units. Get a teacher’

s edition of a textbook for any grade level, and look at the chapters on measurement. How well does the book cover metric measurement ideas? How would you modify or expand on the lessons found there?

Reflections on Chapter

19

Literature Connections

How Big Is a Foot?

Myller, 1990

The story in this concept book is very attractive to young children. The king measures his queen using his feet and orders a bed made that is 6 feet long and 3 feet wide. The chief carpenter’

s apprentice, who is very small, makes the bed according to his own feet, demonstrating the need for standard units.

Every Minute on Earth: Fun Facts That Happen Every 60 Seconds

Murrie & Murrie, 2007

This is an amazing book that is not just about the concept of time. The authors provide fun facts about what can happen in 60 seconds: a snow avalanche travels 4.2 miles (6.8 kilome-ters); the adult heart pumps 3.3 liters (3.5 quarts) of blood; movie film travels 90 feet (27.4 meters) through a projector; a garden snail moves 0.31 inches (7.8 millimeters); people in the United States throw away 18,315 pounds (8325 kilo-grams) of food; and consumers spend $954.00 on chewing gum. Students can use the facts provided or identify others as they think and discuss these relationships.

Inchworm and a Half

Pinczes, 2001

In this wonderfully illustrated book, an inchworm happily goes about measuring various garden vegetables. One day a measurement does not result in a whole number

, and the worm gets very upset. Fortunately

, a smaller worm drops onto the vegetable and measures a half unit. Eventually

, other

3 is not a common measurement unit.) The story

pro-vides a great connection between fractions and measurement concepts, especially for the introduction of fractional units in measurement. Moyer and Mailley (2004) describe a nice series of activities inspired by the book.

Recommended Readings

Articles

Austin, R., Thompson, D., & Beckmann, C. (2005). Explor

-ing measurement concepts through literature: Natural links across disciplines.

Mathematics T

eaching in the Middle School,

10(5), 218–224.

This article includes a rich collection of almost 30 childr en’s books that emphasize overall systems of measur

ement, length, weight, capacity, speed, ar

ea, perimeter , and volume. Thr

ee books ar_e de-scribed in detail as the authors shar

e how to link measur ement to science, histor