GUIDING CHILDREN'S LEARNING OF MATHEMATICS

ELEVENTH EDITION

Guiding Children’s
Learning of
Mathematics

Leonard M. Kennedy

Professor Emeritus
California State University, Sacramento

Steve Tipps

University of South Carolina Upstate

Art Johnson

Boston University

AUSTRALIA BRAZIL CANADA MEXICO SINGAPORE SPAIN UNITED KINGDOM UNITED STATES

Guiding Children’s Learning of Mathematics, Eleventh Edition
Leonard M. Kennedy, Steve Tipps, Art Johnson

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To Rebecca Poplin for her inspiration
and constant support.
—S. T.

For my uncle Anthony Sideris.
His integrity and adherence
to the highest moral principles
continue to influence me.
—A. J.

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Brief
Contents

PART ONE • Guiding Elementary Mathematics with Standards 1

1 Elementary Mathematics for the 21st Century 3
2 Defining a Comprehensive Mathematics Curriculum 11
3 Mathematics for Every Child 27
4 Learning Mathematics 47
5 Organizing Effective Instruction 57
6 The Role of Technology in the Mathematics Classroom 79
7 Integrating Assessment 93

PART TWO • Mathematical Concepts, Skills, and Problem Solving 111

8 Developing Problem-Solving Strategies 113
9 Developing Concepts of Number 137
10 Extending Number Concepts and Number Systems 161
11 Developing Number Operations with Whole Numbers 187
12 Extending Computational Fluency with Larger Numbers 229
13 Developing Understanding of Common and Decimal Fractions 253
14 Extending Understanding of Common and Decimal Fractions 287
15 Developing Aspects of Proportional Reasoning: Ratio, Proportion, and Percent 333
16 Thinking Algebraically 357
17 Developing Geometric Concepts and Systems 389
18 Developing and Extending Measurement Concepts 437
19 Understanding and Representing Concepts of Data 489
20 Investigating Probability 523

Appendix A NCTM Table of Standards and Expectations 545
Appendix B Black-Line Masters 557

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Contents

List of Activities xv Reasoning and Proof 20 23
Communication 21
Preface xix Connection 21
Mathematical Representation 22
PART ONE • Guiding Elementary Integrating Process and Content Standards
Finding Focus 25
Mathematics with Summary 26
Standards 1 Study Questions and Activities 26
Teacher’s Resources 26
CHAPTER 1 • Elementary Mathematics for the
21st Century 3 CHAPTER 3 • Mathematics for Every Child 27
Solving Problems Is Basic 4
A Comprehensive Vision of Mathematics 5 Equity in Mathematics Learning 28
Gender 29
Principles of School Mathematics 6 Ethnic and Cultural Differences 30
The Mathematics Curriculum Principle 7
The Equity Principle 7 Multiculturalism 31
The Learning Principle 8 Ethnomathematics 31
The Teaching Principle 8 Students Who Have Limited English
The Assessment Principle 8 Proficiency 34
The Technology Principle 8 Technological Equity 37
Implementing the Principles and Standards 9 Students Who Have Difficulty Learning
Study Questions and Activities 9 Mathematics 37
Teacher’s Resources 9 Gifted and Talented Students 40
Individual Learning Styles 42
CHAPTER 2 • Defining a Comprehensive Multiple Intelligences 43
Mathematics Curriculum 11 Learning Styles 43
Connecting Concepts, Skills, and Application 12 Summary 44
What Elementary Children Need to Know in Study Questions and Activities 44
Mathematics 12 Teacher’s Resources 44
For Further Reading 45
Numbers and Number Operations 13
Geometry 14 CHAPTER 4 • Learning Mathematics 47
Measurement 15 Theories of Learning 48
Data Analysis and Probability 16
Algebra 17 Behaviorism 48
Learning Mathematical Processes 19 Cognitive Theories 48
Problem Solving 19

vii

viii Contents

Key Concepts in Learning Mathematics 48 Textbooks and Other Printed Materials 75
Focus on Meaning 48 Children’s Literature 77
Developmental Stages 49 Summary 77
Social Interactions 50 Study Questions and Activities 77
Concrete Experiences 50 Teacher’s Resources 78
Levels of Representation 50
Procedural Learning 51 CHAPTER 6 • The Role of Technology in the
Short-Term to Long-Term Memory 51 Mathematics Classroom 79
Pattern Making 51
Thinking to Learn 52 Calculators 80
Calculator Use 80
Research in Learning and Teaching Benefits of Using a Calculator 81
Mathematics 52 Research on Calculators in the Classroom 82

Reviews of Research 54 Computers 83
Teachers and Action Research 56 Research About Computers in the Classroom 83
Summary 56 Computer Software 83
Study Questions and Activities 56
Teacher’s Resources 56 The Internet 85
Virtual Manipulatives 86
CHAPTER 5 • Organizing Effective Instruction 57
Computer Games 88
What Is Effective Mathematics Teaching? 58 Video Technology 90
Using Objectives to Guide Mathematics Summary 90
Instruction 58 Study Questions and Activities 90
Becoming an Effective Teacher 60 Teacher’s Resources 90
For Further Reading 91
Using Objectives in Planning Mathematics
Instruction 60 CHAPTER 7 • Integrating Assessment 93

Long-Range Planning 61 Connecting Curriculum, Instruction, and
Unit Planning 61 Assessment 94
Daily Lesson Planning 62 Planning for Assessment 94
Varying Teaching Approaches 63
Informal or Exploratory Activities 63 Performance Objectives and Tasks 94
Directed Teaching/Thinking Lessons 63 Creating Assessment Tasks 96
Problem-Based Projects and Investigations 66 Collecting and Recording Assessment
Integrating Multiple Approaches 68 Information 96
Delivering Mathematics Instruction 68 Analyzing Student Performance and Making
Practice 69 Instructional Decisions 99
Homework 69 Implementing Assessment with Instruction 101
Grouping 70 Preassessment 101
Cooperative Group Learning 70 During Instruction 102
Encouraging Mathematical Conversations 72 Self-Assessment During Instruction 104
Managing the Instructional Environment 72 Writing as Assessment 104
Time 72 Assessment at the End of Instruction 105
Space 73 Interpreting and Using Standardized Tests in
Learning Centers 73 Classroom Assessment 107
Manipulatives 74 Summary 109
Study Questions and Activities 109

Contents ix

Teacher’s Resources 110 Teacher’s Resources 158
For Further Reading 110 Children’s Bookshelf 159
For Further Reading 159
PART TWO • Mathematical Concepts,
CHAPTER 10 • Extending Number Concepts and
Skills, and Problem Number Systems 161
Solving 111
Number Sense Every Day 162
CHAPTER 8 • Developing Problem-Solving Understanding the Base-10 Numeration
Strategies 113 System 163

Approaches to Teaching Problem Solving 115 Exchanging, Trading, or Regrouping 165
Teaching About, Teaching for, and Teaching via Assessing Place-Value Understanding 165
Problem Solving 115 Working with Larger Numbers 167
The Four-Step Problem-Solving Process 115 Thinking with Numbers 172
Eleven Problem-Solving Strategies: Tools for Rounding 174
Elementary School Students 116 Estimation 175
Other Number Concepts 178
Implementing a Problem-Solving Curriculum 132 Patterns 178
Take-Home Activities 133 Prime and Composite Numbers 178
Summary 135 Integers 179
Study Questions and Activities 135 Take-Home Activities 182
Teacher’s Resources 135 Summary 184
Children’s Bookshelf 135 Study Questions and Activities 184
For Further Reading 136 Using the Technology Center 184
Teacher’s Resources 185
CHAPTER 9 • Developing Concepts of Number 137 Children’s Bookshelf 185
Technology Resources 185
Primary Thinking-Learning Skills 138 For Further Reading 185
Matching and Discriminating, Comparing and
Contrasting 138 CHAPTER 11 • Developing Number Operations with
Classifying 141 Whole Numbers 187
Ordering, Sequence, and Seriation 142
Building Number Operations 188
Beginning Number Concepts 143 What Students Need to Learn About Number
Number Types and Their Uses 145 Operations 188
Counting and Early Number Concepts 146 What Teachers Need to Know About Addition and
Subtraction 190
Number Constancy 148 Developing Concepts of Addition and
Linking Number and Numeral 151 Subtraction 192

Writing Numerals 153 Introducing Addition 192
Misconceptions and Problems with Counting and Introducing Subtraction 196
Numbers 154 Vertical Notation 199
Introducing Ordinal Numbers 154 What Teachers Need to Know About Properties of
Other Number Skills 155 Addition and Subtraction 200
Skip Counting 155 Learning Strategies for Addition and Subtraction
One-to-One and Other Correspondences 155 Facts 201
Odd and Even Numbers 156
Summary 158
Study Questions and Activities 158

x Contents CHAPTER 13 • Developing Understanding
of Common and Decimal Fractions 253
Developing Accuracy and Speed with Basic
Facts 205 What Teachers Need to Know About Teaching
What Students Need to Learn About Multiplication Common Fractions and Decimal Fractions 255
and Division 208 Five Situations Represented by Common
What Teachers Need to Know About Multiplication Fractions 256
and Division 208
Multiplication Situations, Meanings, and Unit Partitioned into Equal-Size Parts 256
Actions 208 Set Partitioned into Equal-Size Groups 256
Division Situations, Meanings, and Actions 210 Comparison Model 257
Developing Multiplication and Division Expressions of Ratios 257
Concepts 212 Indicated Division 258
Introducing Multiplication 212 Introducing Common Fractions to Children 259
Introducing Division 216 Partitioning Single Things 259
Working with Remainders 218 Assessing Knowledge of Common Fractions 260
What Teachers Need to Know About Properties of Partitioning Sets of Objects 265
Multiplication and Division 219 Fractional Numbers Greater than 1 265
Learning Multiplication and Division Facts with Introducing Decimal Fractions 267
Strategies 220 Introducing Tenths 268
Division Facts Strategies 221 Introducing Hundredths 268
Building Accuracy and Speed with Multiplication Introducing Smaller Decimal Fractions 271
and Division Facts 224 Introducing Mixed Numerals with Decimal
Take-Home Activities 225 Fractions 272
Summary 226 Comparing Fractional Numbers 272
Study Questions and Activities 226 Comparing Common Fractions 272
Teacher’s Resources 227 Comparing Common and Decimal Fractions with
Children’s Bookshelf 227 Number Lines 272
Technology Resources 227 Equivalent Fractions 276
For Further Reading 228 Ordering Fractions 276
Rounding Decimal Fractions 277
CHAPTER 12 • Extending Computational Fluency Take-Home Activities 281
with Larger Numbers 229 Summary 282
Study Questions and Activities 282
Number Operations with Larger Numbers 230 Using Children’s Literature 282
Addition and Subtraction Strategies for Larger Teacher’s Resources 282
Numbers 231 Children’s Bookshelf 283
Multiplying and Dividing Larger Numbers 239 Technology Resources 283
For Further Reading 284
Number Sense, Estimation, and
Reasonableness 247 CHAPTER 14 • Extending Understanding of
Take-Home Activities 250 Common and Decimal Fractions 287
Summary 251
Study Activities and Questions 251 Standards and Fractional Numbers 288
Technology Resources 252 Teaching Children About Fractional Numbers in
For Further Reading 252 Elementary and Middle School 289

Contents xi

Extending Understanding of Decimal Take-Home Activity 328
Fractions 290 Summary 329
Study Questions and Activities 329
Decimal Fractions and Number Density 290 Using Children’s Literature 330
Teacher’s Resources 330
Expanded Notation for Decimal Fractions 292 Children’s Bookshelf 330
Technology Resources 331
Extending Concepts of Common and Decimal For Further Reading 332
Fraction Operations 292
CHAPTER 15 • Developing Aspects of Proportional
Addition and Subtraction with Common and Reasoning: Ratio, Proportion, and Percent 333
Decimal Fractions 292
Proportional Reasoning 334
Addition and Subtraction with Common What Teachers Should Know About Teaching
Fractions 293 Proportional Reasoning 334
Ratios as a Foundation of Proportional
Adding and Subtracting with Like Reasoning 335
Denominators 294
Meanings of Ratio 335
Using Algorithms to Compute with Fractional Teaching Proportional Reasoning 336
Numbers 294 Developing Proportional Reasoning Using Rate
Pairs and Tables 338
Adding and Subtracting When Denominators Are Working with Proportions 339
Different 297
Using Equivalent Fractions 339
Adding and Subtracting with Mixed Solving Proportions Using a Multiples Table 340
Numerals 299 Solving for Proportion Using the Cross-Product
Algorithm 343
Least Common Multiples and Greatest Common Developing and Extending Concepts of
Factors 302 Percent 344
Meaning of Percent 345
Using a Calculator to Add Common Fractions 304 Teaching and Learning About Percent 346
Working with Percent 346
Addition and Subtraction with Decimal Proportional Reasoning and Percent 351
Fractions 304 Working with Percents Greater Than 100 351
Take-Home Activities 353
Multiplication with Common Fractions 305 Summary 354
Study Questions and Activities 354
Multiplying a Fractional Number by a Whole Using Children’s Literature 354
Number 307 Teacher’s Resources 355
Children’s Bookshelf 355
Multiplying a Whole Number by a Fractional Technology Resources 355
Number 308 For Further Reading 356

Multiplying a Fractional Number by a Fractional
Number 308

Multiplying Mixed Numerals 312

Division with Common Fractions 315

Developing Understanding of Division
Algorithms 315

Developing Number Sense About Operations with
Common Fractions 318

Renaming Fractions in Simpler Terms 319

Multiplying and Dividing with Decimal
Fractions 320

Multiplying with Decimal Fractions 320

Dividing with Decimal Fractions 322

Relating Common Fractions to Decimal
Fractions 323

Using a Calculator to Develop Understanding of
Decimal Fractions 325

Using Calculators with Common Fractions 326

xii Contents Geometric Properties: Congruence and
Similarity 407
CHAPTER 16 • Thinking Algebraically 357 Geometric Properties: Symmetry 409
Teaching and Learning About Transformational
What Teachers Should Know About Teaching Geometry 411
Algebra 358 Teaching and Learning About Coordinate
Geometry 412
Patterning as Algebraic Thinking 358 Extending Geometry Concepts 414
Patterns in Mathematics Learning and Classifying Polygons 417
Teaching 359 Extending Geometry in Three Dimensions 420
Variables and Equations in Mathematics and Extending Congruence and Similarity 421
Algebraic Thinking 363 Extending Concepts of Symmetry 422
Equations in Algebraic Thinking 368 Extending Concepts of Transformational
Functions in Algebraic Thinking 369 Geometry 424
Extending Algebraic Thinking 373 Take-Home Activities 431
Extending Understanding of Patterning 373 Summary 432
Extending Understanding of Equations 378 Study Questions and Activities 432
Extending the Meaning of Functions 379 Using Children’s Literature 433
Take-Home Activities 384 Teacher’s Resources 433
Summary 386 Children’s Bookshelf 433
Study Questions and Activities 386 Technology Resources 434
Using Children’s Literature 386 For Further Reading 435
Teacher’s Resources 387
Children’ Bookshelf 387 CHAPTER 18 • Developing and Extending
Technology Resources 387 Measurement Concepts 437
For Further Reading 388
Direct and Indirect Measurement 439
CHAPTER 17 • Developing Geometric Concepts Measuring Processes 439
and Systems 389 What Teachers Should Know About Teaching
Measurement 440
Children’s Development of Spatial Sense 390
What Teachers Need to Know About Teaching Approximation, Precision, and Accuracy 440
Geometry 392 Selecting Appropriate Units and Tools 441
Development of Geometric Concepts: Stages of A Teaching/Learning Model for Measurement 444
Geometry Understanding 396 Activities to Develop Measurement Concepts and
Teaching and Learning About Topological Skills 446
Geometry 397 Nonstandard Units of Measure 446
Standard Units of Measure 449
Proximity and Relative Position 397 Teaching Children How to Measure Length 450
Separation 398 Perimeters 450
Order 398 Teaching Children About Measuring Area 451
Enclosure 399 Teaching Children About Measuring Capacity and
Topological Mazes and Puzzles 400 Volume 452
Teaching and Learning About Euclidean Teaching Children About Measuring Weight
Geometry 400 (Mass) 453
Geometry in Two Dimensions 400 Teaching Children About Measuring Angles 453
Points, Lines, Rays, Line Segments, and
Angles 403
Geometry in Three Dimensions: Space
Figures 406

Contents xiii

Teaching Children About Measuring Circle Graphs 504
Temperature 454 Stem-and-Leaf Plots 506
Teaching Children to Measure Time 454 Box-and-Whisker Plots 508
Extending Children’s Understanding of
Clocks and Watches 456 Statistics 510
Calendars 458 Take-Home Activity 517
Teaching Children to Use Money 458 Summary 519
Extending Measurement Concepts 460 Study Questions and Activities 519
Extending Concepts About Length 461 Using Children’s Literature 520
Estimation and Mental Models of Length 464 Teacher’s Resources 520
Extending Concepts About Area 465 Children’s Bookshelf 520
Measuring and Estimating Area 465 Technology Resources 521
Inventing Area Formulas 466 For Further Reading 521
Extending Capacity Concepts 470
CHAPTER 20 • Investigating Probability 523
Extending Volume Concepts 470
Extending Mass and Weight Concepts 473 Two Types of Probability 524
What Teachers Should Know About Teaching
Estimating Weights and Weighing with Probability 525
Scales 473 Extending Probability Understanding 528
Exploring Density 474 Probability Investigations 529
Expanding Angle Concepts 475
Expanding Temperature Concepts 478 Combinations 532
Extending Concepts of Time 479 Sampling 533
Measurement Problem Solving and Projects 480 Geometric Probability 533
Take-Home Activities 481 Expected Value 536
Summary 484 Simulations 539
Study Questions and Activities 484 Take-Home Activities 540
Using Children’s Literature 485 Summary 541
Teacher’s Resources 485 Study Questions and Activities 541
Children’s Bookshelf 486 Using Children’s Literature 541
Technology Resources 486 Teacher’s Resources 542
For Further Reading 487 Children’s Bookshelf 542
Technology Resources 542
CHAPTER 19 • Understanding and Representing For Further Reading 542
Concepts of Data 489
Appendix A NCTM Table of Standards and
Data Collection 490
What Teachers Should Know About Understanding Expectations 545
and Representing Concepts of Data 491
Appendix B Black-Line Masters 557
Organizing Data in Tables 491
Object and Picture Graphs 494 Glossary 565
Bar Graphs 495
Statistics 498 References 571
Extending Data Concepts 499
Using Graphs to Work with Data 499 Index 577
Histograms 502
Line Graphs 502

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List of Activities

8.1 Making People Patterns 118 10.7 Think of a Million 170
8.2 How Many Rectangles? 119 10.8 Spin to Win 170
8.3 Payday 120 10.9 Big City 171
8.4 Triangles Up and Down 121 10.10 How Many Beans? 176
8.5 Renting Cycles 121 10.11 Snack Stand Supply Problem 177
8.6 How Far? 124 10.12 Prime and Composite Numbers 179
8.7 Targets 128 10.13 Factor Trees 180
8.8 Pascal’s Triangle 130 10.14 Sieve of Eratosthenes 181

9.1 Matching Objects to Pictures 140 11.1 Solving Problems with Addition 193
9.2 Nuts and Bolts 140 11.2 More Cats 193
9.3 Sorting Boxes 142 11.3 Introducing the Equal Sign 195
9.4 Drawing Straws 143 11.4 Addition Sentence 195
9.5 Follow the Rules 144 11.5 How Many? 197
9.6 Counting with Anno 147 11.6 Counting On 199
9.7 Counting Cars 148 11.7 Vertical Form 200
9.8 Unifix Cube Combinations 149 11.8 Commutative and Associative Properties 202
9.9 Eight 149 11.9 Near Doubles 203
9.10 No More Flowers 149 11.10 Making Ten with the Tens Frame 204
9.11 Number Conservation 150 11.11 Subtracting with Hide-and-Seek Cards 205
9.12 Plate Puzzles and Cup Puzzles 152 11.12 Assessing with Flash Cards 207
9.13 Matching Numeral and Set Cards 152 11.13 Practice Addition and Subtraction Number Facts
9.14 Card Games for Numbers and Numerals 153
9.15 Patterns on the Hundreds Chart 155 with Calculators 207
9.16 Even and Odd 157 11.14 Repeated Addition 213
11.15 Color Combinations for Bicycles 216
10.1 Trains and Cars 164 11.16 Sharing Cookies 218
10.2 Train-Car Mats 166 11.17 Putting on the Nines 222
10.3 Beans and Sticks 167 11.18 Find the Facts 223
10.4 E-vowel-uation 168
10.5 Seven Chances for 100 168 12.1 Thinking Strategies for Two-Digit Addition 232
10.6 Place-Value Assessment 169 12.2 Decomposition Algorithm 237
12.3 Distribution 241

xv

xvi List of Activities 14.19 Measurement and Partitive Division 324
14.20 Operations with Common and Decimal
12.4 Multiplication with Regrouping 243
12.5 History and Multiplication Algorithms 244 Fractions 324
12.6 Division with Regrouping 246 14.21 Exploring Terminating and Repeating
12.7 Factorials 249
Decimals 325
13.1 Introducing Halves 261 14.22 Operations with Common and Decimal
13.2 Fractions on a Triangle 262
13.3 Cuisenaire Fractions 263 Fractions 327
13.4 The Fraction Wheel 264
13.5 The Fruit Dealer and His Apples 266 15.1 Qualitative Proportions 337
13.6 Exploring Fractions 267 15.2 Rate Pairs 338
13.7 Introducing Tenths 269 15.3 Ratios at Work 339
13.8 Fraction-Strip Tenths 269 15.4 Finding the Right Rate 340
13.9 Number-Line Decimals 270 15.5 Using a Multiples Table with
13.10 Introducing Hundredths 270
13.11 Decimal Fractions on a Calculator 271 Proportions 341
13.12 Fraction Strips 273 15.6 Racing for Fame 343
13.13 Common Fractions on a Number Line 274 15.7 Cuisenaire Rod Percents 346
13.14 Using Benchmarks to Order Fractions 275 15.8 Assessment Activity 347
13.15 Rounding Decimals to Whole Numbers 278 15.9 Hundred Day Chart 347
13.16 Rounding Decimal Circles 279 15.10 Elastic Percent Ruler 348
13.17 An Assessment Activity 280 15.11 Another Look at Percent 349

14.1 Place-Value Pocket Chart 291 16.1 Making Rhythm Patterns 361
14.2 Decimals on an Abacus 291 16.2 One Pattern with Five Representations 361
14.3 Place-Value Chart 292 16.3 Exploring Patterns with Pattern Blocks 362
14.4 Adding Common Fractions 295 16.4 Exploring Letter Patterns in a Name 363
14.5 Using a Calculator to Explore Addition with 16.5 What Number Is Hiding? 365
16.6 “How Many Are Hiding?” 365
Common Fractions 296 16.7 What Does the Card Say? 366
14.6 Adding with Unlike Denominators 297 16.8 Replacing the Number 367
14.7 Using Fraction Circles to Explore Adding 16.9 Multiple Value Variables 367
16.10 Balancing Bears 370
Fractions with Unlike Denominators 298 16.11 Weighing Blocks 371
14.8 Cakes at the Deli 301 16.12 What Does It Weigh? 371
14.9 Adding with Decimal Fractions 306 16.13 Magic Math Box 372
14.10 Adding Common Decimals with a Calculator 307 16.14 How Many Squares? 374
14.11 Multiplying a Fraction by a Whole Number 310 16.15 Building Houses 375
14.12 Multiplying a Whole Number by a Fractional 16.16 How Many Diagonals? 375
16.17 Calendar Patterns 376
Number 311 16.18 What’s Next? 377
14.13 Using Paper Folding to Multiply Common 16.19 Graphing Numerical Patterns 377
16.20 How Long Will It Take? 378
Fractions 312 16.21 Does It Balance? 379
14.14 Multiplying with Mixed Numerals 313 16.22 What Does the Scale Tell You? 380
14.15 Dividing by a Common Fraction 317 16.23 Find the Mystery Function I 382
14.16 Dividing a Common Fraction by a Whole 16.24 Mod(ular) Math Functions 382
16.25 Find the Mystery Function II 383
Number 318
14.17 Multiplying a Decimal by a Whole Number 321
14.18 Multiplying a Decimal Fraction by a Decimal

Fraction 321

17.1 Move Around the Solid City 393 List of Activities xvii
17.2 How Many Blocks? 393
17.3 Embedded Triangles 394 18.11 Name That Unit 462
17.4 Ladybug Maze 394 18.12 To the Nearest Centimeter 462
17.5 Simple Letters 400 18.13 The Shrinking Stirrer 462
17.6 Sorting Shapes 402 18.14 Step Lively! 464
17.7 Feeling and Finding Shapes 403 18.15 How Much Is 50 Meters? 464
17.8 What Is a Polygon? 404 18.16 Area of a Parallelogram 466
17.9 Lines, Segments, and Rays 405 18.17 Triangles Are Half a Parallelogram 467
17.10 Name My Angle 406 18.18 Exploring p 468
17.11 Blob Art 409 18.19 Finding the Area of a Circle 469
17.12 Geoboard Symmetry 410 18.20 Tile Rectangles 469
17.13 Coordinate Classroom 413 18.21 Volume of Cones and Pyramids 472
17.14 Find My Washer 413 18.22 Measuring Angles 476
17.15 The Mobius Strip 415 18.23 What Is a Degree? 476
17.16 Searching for Squares 416 18.24 Shrinking Angles 477
17.17 How Many Pentominoes? 417 18.25 Angle Wheel 477
17.18 Exploring Quadrilaterals 418 18.26 New World Calendar 479
17.19 Yarn Quadrilaterals 418
17.20 Categorizing Quadrilaterals 419 19.1 Picture Graph 494
17.21 Diagonally Speaking 420 19.2 Bar Graphs 496
17.22 Euler’s Formula 421 19.3 Reverse Bar Graphs 497
17.23 Similarity and Congruence 422 19.4 Stacking Blocks 501
17.24 Building Similar Triangles 423 19.5 Finding the Writer 502
17.25 Drawing Similar Figures 423 19.6 Histogram Survey 503
17.26 Symmetry Designs 424 19.7 Qualitative Graphs 504
17.27 Lines of Symmetry 424 19.8 Circle Graph Survey 507
17.28 Reflecting Points on a Coordinate Grid 425 19.9 Finding Mean Averages 512
17.29 Geometric Transformations 426 19.10 Over and Under the Mean 513
17.30 Exploring Dilations 427 19.11 Comparing Averages 514
17.31 Exploring the Pythagorean Theorem 429 19.12 Using and Interpreting Data 515
17.32 Using the Pythagorean Theorem to Find Right 19.13 Assessment Activity 516

Triangles 429 20.1 Guaranteed to Happen (or Not to Happen) 526
20.2 Maybe or Maybe Not 527
18.1 Station Time 447 20.3 Tossing Paper Cups 527
18.2 How Tall Are We? 448 20.4 Tossing Dice 531
18.3 Pencil Measurement 448 20.5 Designer Number Cubes 532
18.4 When Is a Foot a Foot? 449 20.6 Probability Tree 534
18.5 Big Foot 451 20.7 Sampling 534
18.6 Tables and Chairs 452 20.8 Multiple Drawing Sampling 535
18.7 Temperature 454 20.9 Spinner Probability 536
18.8 How Many Can You Do? 455 20.10 Design Your Own Spinner 537
18.9 How Long Was That? 455 20.11 Geometric Probability 538
18.10 Daytimer Clock 457 20.12 A Probability Simulation 539

xvii

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Preface

hen contemplating the 11th edi- of the first edition, which was published in 1970—to
tion of Guiding Children’s Learn- provide a readable and user-friendly textbook that
ing of Mathematics, the au- enables preservice and experienced teachers to de-
thors started by thinking about velop their own understanding of mathematics and
what has been happening in to offer them a wide array of experiential activities
as examples of active learning and creative teach-
mathematics education in the last ing. Through the years the book has changed as
five years. The No Child Left Behind chapters have been revised to reflect current issues
Act of 2002 (NCLB) has focused public attention on and emphases; but the philosophy of the book has
student performance and highlighted the demand remained constant:
for more and better mathematics instruction. The
act requires that teachers be “highly qualified” and • Mathematics enriches lives and expands worlds.
places an increased emphasis on their demonstrat-
ing their knowledge and skill in teaching. Certifica- • Mathematics is challenging, fun, and rewarding.
tion requirements have been influenced by NCLB,
with more states dividing early-childhood from ele- • Mathematics is a mission and a treasure shared
mentary licensure and creating middle-grade cer- by teachers and learners.
tificates. Many teacher certification programs have
also increased their requirements for mathematics We believe that teachers are the critical element in
background. The range of technology resources for creating exciting and successful mathematical ad-
teaching mathematics has likewise increased—es- ventures for learners, through
pecially those related to the Internet. Finally, pro-
viding appropriate instruction for all students has • Active engagement of students in worthwhile
become more challenging, as teachers work with mathematical tasks
students from diverse cultural, economic, and
language backgrounds and with varying degrees • Problem solving and thinking as central goals in
of ability. Each of these factors influences the life mathematics
space of teachers and, for that reason, the way in
which we have organized this edition and what we • Relating mathematical concepts and skills to life
have revised and changed. experiences
While acknowledging and incorporating
changes in mathematics teaching are important in • Communicating mathematical ideas in many
the 11th edition, however, the goal of Guiding Chil- forms
dren’s Learning of Mathematics is the same as that
Over the years, all of the chapters have been ex-
tensively rewritten, incorporating the feedback of
reviewers, editors, coauthors, and hundreds of in-
service and preservice teachers who have used the

xix

xx Preface

text. In addition, the authors have drawn from their verse student population: students with gifts and tal-
own experiences in order to continually improve ents, students who are culturally and linguistically
the book. Teaching with a textbook that you your- diverse, and students with special needs. In Chapter
self have written is an ongoing and humbling learn- 4, the topic of teaching and learning is explored by
ing experience. As a student once asked, “Do you way of learning theory, research, and professional
really agree with the textbook about . . . ?” guidelines for establishing an effective classroom.
Chapter 5 outlines the many decisions that teachers
Organization make every day with regard to planning and organiz-
ing the elements of instruction, including materials,
In the reorganization and revision for the 11th edi- grouping, time, and space. Chapter 6 presents new
tion, we endeavor to clarify and illustrate mathemat- and emerging technologies that impact mathemat-
ical and pedagogical issues without oversimplifying ics teaching and learning today. Chapter 7 describes
them. An obvious change is the restructuring of the the rationale for classroom assessment, using a va-
text content into smaller and more focused chapters. riety of techniques that teachers employ to enhance
In Part 1 of the book, the NCTM principles and stan- and adjust instruction.
dards provide the foundation for discussion of what
is important and how it can be taught effectively. Having a separate chapter for each principle pro-
Each of the six principles is treated in a separate vides modules that may be used in a variety of ways
chapter. by instructors and students. For example, the prin-
ciples could be used to introduce the course, or they
• Part 1: Guiding Elementary Mathematics with could be used at different times in connection with
Standards the chapters in Part 2.

• Chapter 1: Elementary Mathematics for the 21st In Part 2, important mathematical concepts are
Century defined and illustrated with problems and teaching
examples that now extend to grade 6. Part 2 bal-
• Chapter 2: Defining a Comprehensive Mathemat- ances the development of mathematical knowledge
ics Curriculum with methods of teaching the content and skills. In
each chapter, examples and activities illustrate ways
• Chapter 3: Mathematics for Every Child (NEW) in which teachers might engage children in active
mathematical thinking and problem solving. Activi-
• Chapter 4: Learning Mathematics ties in each chapter show how students can model
concepts with physical objects, and then record
• Chapter 5: Organizing Effective Instruction and communicate their actions informally as well
as with conventional symbolism. As they construct
• Chapter 6: The Role of Technology in the Math- meaning, students are encouraged to move toward
ematics Classroom (NEW) mental operations that require estimation, reason-
ableness, and application of mathematical concepts
• Chapter 7: Integrating Assessment to numerical and geometric situations.

The complexities of teaching and assessment Content standards have been presented in paired
are presented as choices and decisions that teach- chapters of “developing” and “extending” concepts
ers make. Rather than suggesting that one way is and skills over the K–6 continuum. The “develop-
the right way for teaching everything, we suggest ing” chapters emphasize content typically found in
that many approaches may be successful if they early childhood and primary grades. The “extend-
adhere to basic principles of effective teaching, ing” chapters focus on concepts and skills typical of
learning, and assessment. The NCTM principles are intermediate and upper elementary grades, through
introduced and the content and process standards grade 6, which many elementary schools include in
are described in Chapters 1 and 2 with classroom their building and curriculum. This organization al-
vignettes. The new curriculum focal points (2006) lows students and teachers to address the content
from NCTM are introduced in Chapter 2. These help appropriate to their needs and certification levels.
teachers identify and focus on critical elements of
the content at each grade level. Chapter 3 considers
the equity issues of teaching mathematics to a di-

Preface xxi

• Part 2: Mathematical Concepts, Skills, and Prob- attention. Multicultural Connections are suggestions
lem Solving for expanding subject matter to include topics and
content that will appeal to the diversity in classrooms.
• Chapter 8: Developing Problem-Solving Each chapter also contains representative end-of-
Strategies chapter problems from three highly-esteemed tests:
The National Assessment of Educational Progress
• Chapter 9: Developing Concepts of Number (NAEP), Trends in International Mathematics and
Science Study (TIMSS), and Professional Assess-
• Chapter 10: Extending Number Concepts and ment for Beginning Teachers (Praxis).
Number Systems (NEW)
Understanding mathematical concepts and
• Chapter 11: Developing Number Operations with building skills is within the capabilities of all future
Whole Numbers teachers, even if they have previously not enjoyed
or felt confident with mathematics. Using this text-
• Chapter 12: Extending Computational Fluency book invites preservice teachers to learn its content
with Larger Numbers (NEW) and methods through active engagement with the
text, the exercises, the activities, and their peers.
• Chapter 13: Developing Understanding of Com- The experience of learning via this textbook mod-
mon and Decimal Fractions els appropriate techniques that preservice teachers
can use with their students. Many new activities are
• Chapter 14: Extending Understanding of Com- presented in Chapters 8–20, and many others have
mon and Decimal Fractions (NEW) been revised. All of the activities and assessments
can be implemented in field settings.
• Chapter 15: Developing Aspects of Proportional
Reasoning: Ratio, Proportion and Percent In light of new and emerging Internet resources,
(NEW) each chapter features an Internet lesson plan, a
description of an Internet game that focuses on
• Chapter 16: Thinking Algebraically (NEW) improving mathematics skills, and references to In-
ternet sites with interactive activities through which
• Chapter 17: Developing Geometric Concepts students can explore chapter-related mathematics
and Systems concepts. Each chapter also includes activities that
are explicitly linked to each of the process standards
• Chapter 18: Developing and Extending Measure- highlighted by the National Council of Teachers of
ment Concepts Mathematics: communication, connections, reason-
ing and proof, and representation.
• Chapter 19: Understanding and Representing
Concepts of Data Technology itself is also used to provide many new
resources. The Guiding Children’s Learning of Math-
• Chapter 20: Investigating Probability ematics companion website (www.thomsonedu.com/
education/kennedy) for students and instructors offers
Chapters 10 and 12 extend the discussions of num- several features, including:
ber concepts and number operations, respectively.
Similarly, Chapter 14 extends the topics of common • Downloadable black-line masters for classroom
and decimal fractions. We now present two topical use
chapters—Chapter 15, dealing with ratio and pro-
portion, and Chapter 16, thinking algebraically. • Essential web links for math education

New Chapter Features • Activities Bank with a number of useful activities
not found in the textbook
In Part 2, the reader will also find increased empha-
sis on diversity, technology, and assessment. Intro- • PowerPoint® presentation with a talking-point out-
duced in Part 1, these topics are integrated through- line for each chapter
out the chapters in Part 2 via classroom vignettes
and activities. In addition, each chapter in Part 2 • NCTM Standards Spotlight, a correlation of activi-
contains new and exciting features related to as- ties in the textbook with NCTM standards
sessment, technology in mathematics, and diversity
in the classroom. Misconceptions highlight students’
typical misunderstandings, thus alerting teachers to
those concepts and skills that may need particular

xxii Preface

Acknowledgments Eileen Simons, Hofstra University
Fenqjen Luo, University of West Georgia
We thank the Wadsworth editorial and production Karla Karstens, University of Vermont
staff: Education Editor, Dan Alpert; Development Kyungsoon Jeon, Eastern Illinois University
Editor, Tangelique Williams; Editorial Production Lisa B. Owen, Rhode Island College
Project Manager, Trudy Brown; Assistant Editor, Ann Marina Krause, California State University,
Lee Richards; Editorial Assistant, Stephanie Rue; Long Beach
Marketing Manager, Karin Sandberg; and Advertis- Marshall Lassak, Eastern Illinois University
ing Project Manager, Shemika Britt. We also thank Mary Goral, Bellarmine University
the reviewers who provided invaluable feedback Nancy Schoolcraft, Ball State University
and guidance: Priscilla S. Nelson, Gordon College
Zhijun Wu, University of Maine at Presque Isle
Helen Gerretson, University of South Florida Rick Austin, University of South Florida

Barbara B. Leapard, Eastern Michigan University

Blidi S. Stemn, Hofstra University

Ed Dickey, University of South Carolina

Edna F. Bazik, National-Louis University

1P A R T

Guiding
Elementary
Mathematics
with Standards

1 Elementary Mathematics for the 21st Century 3
2 Defining a Comprehensive Mathematics Curriculum 11
3 Mathematics for Every Child 27
4 Learning Mathematics 47
5 Organizing Effective Instruction 57
6 The Role of Technology in the Mathematics Classroom 79
7 Integrating Assessment 93



CHAPTER 1

Elementary
Mathematics for
the 21st Century

lementary mathematics has been the subject of much discus-
sion, debate, and controversy in recent years. At the center
of this debate is whether children should focus on basic
computation skills or develop a wider range of knowl-
edge and skill in mathematics. The curriculum recom-

mended by the National Council of Teachers of Mathemat-
ics (2000) and adopted by many states emphasizes thinking and problem
solving related to all mathematical topics: numbers and operations,
statistics, measurement, probability, geometry, and algebra. The content
is connected to living, working, and solving problems in a technological
and information-based society. Computational skills are still important, but
students must know when and how to use numbers to solve problems.

The need for a well-balanced and coherent mathematics curriculum
prekindergarten (PK) through grade 12 was also emphasized with the pas-
sage of the No Child Left Behind Act of 2001. This law mandated each state
to adopt standards for academic performance and develop a testing program
to demonstrate student achievement.

Just as the focus of mathematics has changed, the strategies that teach-
ers use reflect new understanding of how students learn, based on research
on cognition—the process of learning. Teachers and parents are challenged
to consider mathematics differently from their school mathematics expe-
rience, which was dominated by calculations and procedures, drill and
repetition, and solitary work. An ideal classroom today finds students work-
ing together on challenging problems related to their lives, explaining their

3

4 Part 1 Guiding Elementary Mathematics with Standards

thinking to each other and their teacher, and using a variety of materials to
show what they understand and can do.

In this chapter you will read about:
1 Problem solving as the central idea in school mathematics
2 How the Principles and Standards for School Mathematics (National

Council of Teachers of Mathematics, 2000) serves as a model for
what mathematics should be taught and how it should be taught
3 Six principles that provide a foundation for school mathematics from
preschool through grade 12: mathematics curriculum, equity, teach-
ing, learning, assessment, and technology

Solving Problems Is Basic E XERCISE

Too many adults believe that they are not compe- Using the information in Figure 1.1, work in groups
tent in mathematics. They may even have math- of two or three to solve the cereal problem. Which
ematics anxiety just thinking about mathematics. box of cereal would you buy based on cost? Does a
At the same time, these people use mathematics coupon change your decision? If the store doubles
in their daily lives when they shop, cook, manage the coupon, does that change the decision? What
their money, work on home improvement projects, else would you consider when deciding which cereal
or plan for travel. Every citizen needs mathematical
concepts and skills for budgeting and saving, financ- to buy? •••
ing a house or car, calculating a tip at a restaurant,
or estimating distances and gas mileage. Often the Calculations are only part of solving problems
numerical answer is only one factor considered in a in mathematics. Reasoning is used to decide how
decision. Other issues may be more important than much cost, taste, and nutrition are considered in
the computed answer. Is a regular box of cereal for the final decision. Even after complex calculations,
$3.75 a better buy than the smaller box for $2.75 or such as the cost of remodeling or the various incen-
the giant box for $4.75 (Figure 1.1)? Does having a tives offered for buying a car, the numerical answer
“50 cents off” coupon change the answer? What is only one of many other factors involved.
other factors influence the choice of cereal?
Adults, even those who believe they were not
$2.75 $3.75 $4.75 good in school mathematics, often develop math-
16 oz 24 oz 30 oz ematical skills in their jobs. Carpenters and con-
tractors measure accurately and estimate job costs
Figure 1.1 Which cereal would you buy? and materials. Accountants, graphic designers, and
hospital workers use calculators and computers
routinely to record and analyze information and
designs. When mathematics is applied to realistic
life and work situations, many adults find that they
are capable in mathematics, despite their negative
attitudes toward mathematics in school. The need
to connect mathematics to realistic situations has
been one motivation for reform of the mathematics
curriculum and teaching.

The vision of mathematics has changed. Math-
ematics used to be a solitary activity done primarily

Chapter 1 Elementary Mathematics for the 21st Century 5

on worksheets. Now teachers ask students to work to- ers demand more accountability of schools and
gether to solve interesting problems, puzzles, games, teachers for student learning. The No Child Left
and investigations. When solving these problems, Behind Act of 2001 requires that states adopt stan-
students develop the concepts, skills, and attitudes dards-based curricula. The standards provide com-
needed for life and work. Numbers and calculation mon expectations for student performance, and
are still essential, but mathematical thinking and rea- statewide testing has been developed to measure
soning equip all children to solve a wide variety of student progress. Results are used to rate and rank
problems. schools based on student achievement. If schools
do not meet performance standards, sanctions may
Elementary mathematics teachers are on the be imposed under the act. Teachers also must dem-
front line of the effort to develop mathematical con- onstrate knowledge of content and teaching prac-
cepts and problem-solving skills. Classroom events tices to meet “highly qualified” requirements under
provide mathematical learning experiences. No Child Left Behind. The provisions of the act are
controversial, and debate continues on the local,
• If juice boxes are packaged in groups of three, state, and national levels.
how many packs are needed for 20 children?
A Comprehensive Vision
• How much money is needed to buy lunch, a of Mathematics
snack, and a pencil at the school store?
The effort to improve mathematics education was
• Is January a good month to take a field trip to the based on several factors. School mathematics has
zoo? Why or why not? been seen by many learners as irrelevant and bor-
ing when mathematics is actually useful and ex-
• Will this 12 ϫ 15 inch piece of paper be large citing when learned conceptually and practically.
enough to cover a cube for my art project? Research has accelerated the information available
about how mathematics is learned and effectively
• How many children prefer hamburgers to pizza? taught. Political demands for a standard curriculum
Hamburgers to hot dogs? Pizza to hot dogs? and increased assessment required a response from
the mathematics education community.
Students figure out what information they need and
how to use it to solve problems. The opportunity and challenge to formulate
this new vision of mathematics was met by the Na-
Problem-focused teachers ask, “Is there only tional Council of Teachers of Mathematics (NCTM).
one answer? Can anyone see another way to solve For nearly a century, NCTM has sought to answer
this problem?” As students discover problems with two central questions for teachers, parents, and
multiple answers and multiple solution paths, they policymakers:
become more flexible in their thinking. When prob-
lems serve as the context of teaching, children ask, • What mathematical concepts and skills are funda-
“Does this answer make sense?” Information from mental for students to know?
print and electronic sources is represented in many
forms: text, pictures, tables, and graphs. Students • What are the best ways to teach and learn these
learn as they write, draw, act, read about, and model essential concepts and skills?
their thinking. They engage in dialogue, demonstra-
tion, and debate about mathematical ideas. When NCTM, an international professional organization
children are actively engaged in doing mathemati- of more than 100,000 professionals in mathemat-
cal tasks, they are thinking about how mathematics ics education, provides resources and guidance to
works. This new vision of mathematics includes a teachers, schools, school districts, and state and
balanced mathematics program for students of all national policymakers through policy recommen-
ages that focuses on concepts, processes, and ap- dations and publications. In 2000, the NCTM Board
plications, with problem solving at its core. of Directors and members adopted Principles and

Public and political concern about student
achievement is another factor that influences the
development of new mathematics curriculum and
teaching. As a result, national and state policymak-

6 Part 1 Guiding Elementary Mathematics with Standards

Standards for School Mathematics. This comprehen- tent and learning needs of children throughout their
sive and balanced statement describes principles school years. Five content standards and five pro-
to guide mathematics programs and outlines the cess standards are common across all grade levels,
content and processes central to teaching and learn- showing the unity of mathematics knowledge and
ing mathematics (the report is available at http://www process.
.nctm.org).
Principles of School Mathematics
The new standards consolidated curriculum,
teaching, and assessment issues into one document. The principles surround all aspects of planning and
Core beliefs, called principles, are addressed directly teaching. Table 1.1 lists each NCTM principle and
in the new standards. Standards are organized into issues related to it. Teachers can ask themselves
four grade bands (PK–2, 3–5, 6–8, and 9–12) that whether they are following the principles as they
address unique characteristics of mathematics con- reflect on their teaching.

TABLE 1.1 • Six Principles for School Mathematics

Equity Principle

Excellence in mathematics education requires equity—high expectations and strong support for all students.
• Equity requires high expectations and worthwhile opportunities for all.
• Equity requires accommodating differences to help everyone learn mathematics.
• Equity requires resources and support for all classrooms and students.

Mathematics Curriculum Principle

A curriculum is more than a collection of activities:
• A mathematics curriculum should be coherent.
• A mathematics curriculum should focus on important mathematics.
• A mathematics curriculum should be well articulated across the grades.

Teaching Principle

Effective mathematics teaching requires understanding what students know and need to learn and then challenging and
supporting students to learn it well.
• Effective teaching requires knowing and understanding mathematics, students as learners, and pedagogical strategies.
• Effective teaching requires a challenging and supportive classroom-learning environment.
• Effective teaching requires continually seeking improvement.

Learning Principle

Students must learn mathematics with understanding, actively building new knowledge from experience and prior
knowledge.
• Learning mathematics with understanding is essential.
• Students can learn mathematics with understanding.

Assessment Principle

Assessment should support the learning of important mathematics and furnish useful information to both teachers and
students.
• Assessment should enhance students’ learning.
• Assessment is a valuable tool for making instructional decisions.

Technology Principle

Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances
students’ learning.
• Technology enhances mathematics learning.
• Technology supports effective mathematics teaching.
• Technology influences what mathematics is taught.

SOURCE: Reprinted with permission from Principles and Standards for School Mathematics (2000) by the National Council of Teachers of Mathematics. All rights
reserved.

Chapter 1 Elementary Mathematics for the 21st Century 7

• Curriculum: Do all children receive a well- TABLE 1.2 • Content and Process Standards
rounded and balanced program in mathematics? for School Mathematics

• Equity: Do all children have access and opportu- Content Standards
nities to be successful in mathematics?
• Numbers and operations
• Learning: Are the methods I use based on what is • Algebra
known about how children learn? • Geometry
• Measurement
• Teaching: Do the methods I use enhance learning • Data analysis and probability
by engaging children in mathematical thinking,
developing concepts and skills, and applying their Process Standards
knowledge to engaging problems?
• Problem solving
• Assessment: Do I use assessment to determine • Reasoning and proof
children’s strengths and needs on a continuous • Communication
basis and adjust my instruction accordingly? • Connection
• Representation
• Technology: Do I use technology to help children
explore and learn mathematical concepts? well-balanced mathematics curriculum includes
five content standards and five process standards
These six principles should be integrated into every taught across all grades, PK–12.
mathematics lesson (Figure 1.2).
Common standards emphasize the unity and
Learning Equity interrelatedness of mathematics knowledge and
process (Table 1.2). The content standards are or-
Teaching Standards Assessment ganized in four grade bands (PK–2, 3–5, 6–8, and
9–12) that address unique characteristics of content
Mathematics Technology and cognitive development of learners. The com-
curriculum plete curriculum with grade band expectations is
found in Appendix A. In this text, we emphasize the
Figure 1.2 Six principles for school mathematics first two grade bands, PK–2 and grades 3–5, with
additional attention to grade 6 because it is often
E XERCISE included in elementary certification and school or-
ganization. In Chapter 2 we present each of the con-
In a classroom you are observing, give examples tent and process standards with descriptions of how
of how you see the six principles at work or not in they are integrated into daily teaching. Standards
and expectations are also discussed in the appro-
evidence. ••• priate chapters in Part 2 of the text as content and
activities are introduced for each topic.
The Mathematics Curriculum Principle
The Equity Principle
The mathematics curriculum principle describes es-
sential mathematics concepts and skills that NCTM Equity means that the full mathematics curriculum,
believes are important for children to learn from pre- balanced with content and process, is available
school through grade 12. This comprehensive and to all students. Students in schools possess many
characteristics. They may be male or female; rep-
resent many different cultures, ethnic groups, and
languages; and possess a variety of background ex-
periences, mathematical interests, and abilities. De-
spite the differences among students, they all need
mathematics to carry out daily tasks, to work, and
to be informed citizens in a technological world.
Rather than being a gate or filter that allows only

8 Part 1 Guiding Elementary Mathematics with Standards

a few students to move forward, equity emphasizes of the classroom and the nature of the mathematical
that the mathematics door is open to all students conversation they encourage. Some choices limit
from the first day of their education and through- the conversation, interaction, and collaboration;
out their lives. All students need opportunities to other choices encourage dialogue, demonstration,
develop technical, vocational, and practical math- and engagement. In Chapter 5 we make sugges-
ematics and to develop reasoning skills. tions for developing a mathematics classroom that
encourages student involvement. In the chapters in
Equity does not mean that every student receives Part 2, we present lessons, activities, and ideas for
exactly the same program. It means that no child teaching content and process.
is deprived of the opportunity to learn; no arbitrary
limits are set on individuals. Because students have The Assessment Principle
different goals, abilities, and interests, mathemat-
ics must address individual needs as well. Gifted Instead of giving tests at the end of each reporting
students need opportunities to explore beyond the period, the assessment principle envisions continu-
basic curriculum. Students who experience difficul- ous interaction between learner and teacher before,
ties in mathematics need extra assistance to attain during, and after instruction. Assessment occurs
the knowledge and skills outlined in the standards. daily through asking questions and observing stu-
When instruction is flexible enough to allow chil- dent work, by reviewing written work done in class
dren the time and resources to explore topics of spe- or at home, by monitoring activities and exercises,
cial interest to them, it serves all students. In Chapter by reading and responding to journals, and by re-
3 we discuss how to provide a challenging and com- viewing portfolios and projects. Some assessments
prehensive mathematical program for all students. are snapshots of knowledge so that teachers can
make daily adaptations to instruction. Projects al-
The Learning Principle low students to integrate skills and knowledge in a
more integrated and practical situation. Portfolios
Understanding how children learn is the foundation and journals show growth over time. In Chapter 7 we
of teaching; teachers must understand learning the- present many different techniques that can be used
ories to design appropriate instructional activities. to collect information about student knowledge and
Current research supports effective teaching prac- skill, and we exemplify these techniques throughout
tices such as discussion and writing in mathemat- Part 2 of the text.
ics, active engagement with significant mathemati-
cal tasks, higher order thinking about mathematical The Technology Principle
problems, and use of models, materials, and mul-
tiple representations of mathematical concepts. Many parents and teachers fear that technology in-
In Chapter 4 we present knowledge from learning hibits children’s skill with computation. Technology
theories and research about effective teaching and is not a substitute for knowing numbers and opera-
learning in classrooms as a coherent guide for orga- tions or for building fundamental concepts; instead,
nizing instruction. technology is a tool for learning and applying math-
ematical concepts. Computers and calculators can
The Teaching Principle help children learn mathematics in powerful and
meaningful ways.
Teaching is directly related to learning. Knowing
how students learn leads to teaching techniques In a first-grade classroom, students dis-
that help students to develop an understanding of covered that when they keyed in 5 ϩ and
mathematical concepts, think strategically about repeatedly touched ϭ, the calculator dis-
problems, and develop strong process skills. Teach- played 10, 15, 20, 25, 30. . . . The students
ers consider many things as they make many deci- started skip counting by 5’s as the display
sions about instruction: grade-level standards, goals guided their chant. Irma noticed that all
and progress for each child, strategies and activities, the numbers ended in 0 or 5 and were in
materials, time and space, and grouping of children. two horizontal lines on the hundreds chart.
Their decisions and choices determine the climate

Chapter 1 Elementary Mathematics for the 21st Century 9

Some children experimented with 25 ϩ or what content is taught and the time needed for prac-
100 ϩ as the starting number. Racing for tice and reallocate time to teaching concepts and
the biggest number, they found that larger problem solving. In Chapter 6 we extend the discus-
starting numbers grew in length very fast. sion of technology use for learning mathematics.
One child got an error message; then they
all wanted to get “ERROR” and to learn what Implementing the Principles
that meant. and Standards

The children were deeply engaged with numbers: The principles and standards provide the vision of
how to count in different ways, when is a number what mathematics can and should be for elemen-
too big for the calculator, and multiplication. The tary students. Since adoption of the standards in
calculator became a tool for exploration of math- 2000, teachers have worked to implement the vi-
ematics concepts instead of being limited to check- sion through their instruction. The purpose of this
ing answers. The challenge is finding ways to inte- textbook is to provide content background and ex-
grate technology into a learning event. amples of teaching for beginning and experienced
teachers. In Chapters 2 through 7 we develop the
Technology also changes what is taught and six principles through discussion and examples
the time spent on topics. How many problems of of their classroom implementation. In Chapters 8
multiplication and division are required after stu- through 20 we focus on mathematics curriculum by
dents demonstrate understanding of the meaning describing activities and processes that give life to
and process? Do children really need to learn the the principles and standards each day for elemen-
algorithm for square root? When a calculator has a tary students.
built-in graphing program, how many hand-drawn
graphs are required? Teachers must reconsider both

Study Questions and Activities those that express reservations. What are the major
requirements of No Child Left Behind? What are the
1. Consider your own mathematics experiences in specific areas of disagreement and concern? Are
schools. Were your experiences consistent with the there any points of agreement?
NCTM principles?
Teacher’s Resources
• Did you have a well-balanced curriculum?
Ben-Hur, Meir. (2006). Concept-rich mathematics instruc-
• Were all students provided opportunity and sup- tion: Building a strong foundation for reasoning and
port to be successful? problem solving. Alexandria, VA: Association for Super-
vision and Curriculum Development.
• Did teachers use a variety of teaching techniques
that motivated and challenged each student’s Checkley, Kathy. (2006). Priorities in practice: The essen-
learning? tials of mathematics, K–6—effective curriculum, instruc-
tion, and assessment. Alexandria, VA: Association for
• Was technology used to enhance your learning Supervision and Curriculum Development.
mathematics?
National Council of Teachers of Mathematics. (2000).
• Did teachers assess your learning frequently with a Principles and standards for school mathematics. Res-
variety of techniques? ton, VA: Author.

2. Recall your experiences in mathematics. Does your
background illustrate positive or negative examples
of the principles?

3. Do a web search about the No Child Left Behind
Act, and locate sources that support the act and



CHAPTER 2

Defining a
Comprehensive
Mathematics
Curriculum

he vision of mathematics presented in the Principles and
Standards for School Mathematics (National Council of
Teachers of Mathematics, 2000) emphasizes thinking
and problem solving within a well-balanced curricu-
lum of content knowledge and process skills for PK–12

students. Students are actively engaged in interesting and
meaningful activities to develop mathematics concepts and build criti-
cal skills while solving problems. In this chapter we describe the National
Council of Teachers of Mathematics (NCTM) content and process standards
through classroom vignettes and examples that demonstrate how content
and process standards are developed in student-centered classrooms.

In this chapter you will read about:
1 Five content standards and classroom or real-life situations for each

standard
2 Five process standards and classroom or real-life examples for each

standard
3 How process and content standards are integrated into teaching

11

12 Part 1 Guiding Elementary Mathematics with Standards

Connecting Concepts, Skills, nections and relationships among concepts, skills,
and Application and applications in mathematics and other subject
areas. In one lesson or activity, several related con-
The NCTM curriculum standards present a compre- cepts or skills can be introduced, developed, or re-
hensive overview of what students should know and viewed. Classroom vignettes and examples in this
be able to do in mathematics as they progress in chapter illustrate how teachers connect mathemat-
school from PK through grade 12. Instead of mem- ics concepts with each other, with other subjects,
orizing procedures and definitions out of context, and with children’s interests and experiences.
students learn concepts and mathematical proce-
dures through problems that require mathematical What Elementary Children Need
reasoning. to Know in Mathematics

The NCTM standards provide a model for many The five content standards and five process stan-
state-adopted mathematics standards-based cur- dards in the NCTM standards identify the content
ricula. When first looking at state standards, teach- knowledge and the mathematical processes de-
ers may feel overwhelmed if they believe that each veloped over the PK–12 curriculum. In this chapter
standard is an isolated topic or skill. However, the we explain the importance and meaning of both
standards actually encourage teachers to build con-

TABLE 2.1 • Mathematics Curriculum Principle: Content Standards

Numbers and Operations

Instructional programs from PK through grade 12 should enable all students to
• Understand numbers, ways of representing numbers, relationships among numbers, and number systems
• Understand meaning of operations and how they relate to one another
• Compute fluently and make reasonable estimates

Algebra

Instructional programs from PK through grade 12 should enable all students to
• Understand patterns, relations, and functions
• Represent and analyze mathematical situations and structure using algebraic symbols
• Use mathematical models to represent and understand quantitative relationships
• Analyze change in various contexts

Geometry

Instructional programs from PK through grade 12 should enable all students to
• Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical

arguments about geometric relationships
• Specify locations and describe spatial relationships using coordinate geometry and other representational systems
• Apply transformations and use symmetry to analyze mathematical structures
• Use visualization, spatial reasoning, and geometric modeling to solve problems

Measurement

Instructional programs from PK through grade 12 should enable all students to
• Understand measurable attributes of objectives and the units, systems, and processes of measurement
• Apply appropriate techniques, tools, and formulas to determine measurement

Data Analysis and Probability

Instructional programs from PK through grade 12 should enable all students to
• Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
• Select and use appropriate statistical methods to analyze data
• Develop and evaluate inferences and predictions that are based on data
• Understand and apply basic concepts of probability

SOURCE: Reprinted with permission from Principles and Standards for School Mathematics (2000) by the National Council of Teachers of Mathematics.
All rights reserved.

Chapter 2 Defining a Comprehensive Mathematics Curriculum 13

TABLE 2.2 • Mathematics Curriculum Principle: Process Standards

Problem Solving

Instructional programs from PK through grade 12 should enable all students to
• Build new mathematical knowledge through problem solving
• Solve problems that arise in mathematics and other contexts
• Apply and adapt a variety of appropriate strategies to solve problems
• Monitor and reflect on the process of mathematical problem solving

Reasoning and Proof

Instructional programs from PK through grade 12 should enable all students to
• Recognize reasoning and proof as fundamental aspects of mathematics
• Make and investigate mathematical conjectures
• Develop and evaluate mathematical arguments
• Select and use various types of reasoning and methods of proof

Communication

Instructional programs from PK through grade 12 should enable all students to
• Organize and consolidate their mathematical thinking through communication
• Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
• Analyze and evaluate the mathematical thinking and strategies of others
• Use the language of mathematics to express mathematical ideas precisely

Connections

Instructional programs from PK through grade 12 should enable all students to
• Recognize and use connections among mathematical ideas
• Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
• Recognize and apply mathematics in context outside of mathematics

Representation

Instructional programs from PK through grade 12 should enable all students to
• Create and use representations to organize, record, and communicate mathematical ideas
• Select, apply, and translate among mathematical representations to solve problems
• Use representations to model and interpret physical, social, and mathematical phenomena

SOURCE: Reprinted with permission from Principles and Standards for School Mathematics (2000) by the National Council of Teachers of Mathematics.
All rights reserved.

content and process standards. Tables 2.1 and 2.2 ficiency is essential, understanding how numbers
provide an overview of the content standards and work and reasoning about numbers, called number
the process standards, respectively. In Appendix A sense and computational fluency, respectively, are
the content standards for students’ mathematical emphasized. Rather than memorizing isolated facts
growth and proficiency across four grade bands and procedures, students construct understandings
include expectations that provide benchmarks for about numbers and number systems as the founda-
teachers. In Part 2 of the textbook we provide activi- tion for learning operations, facts, and procedures.
ties, lessons, and examples that show how content
is taught within the framework provided by the pro- For children, understanding numbers progresses
cess standards. Rather than being separate topics, from counting objects and matching sets with nu-
the process standards are integrated into lessons merals, to using objects, pictures, and symbols that
and activities presented to children. represent place values for tens, hundreds, even mil-
lions and billions. Concepts of addition, subtraction,
Numbers and Number Operations multiplication, and division begin with stories about
everyday events using simple numbers. The stories
Learning about numbers and number operations provide the problems that are solved through ac-
has been and still is a central element of elementary tion, pictures, and number sentences. Stories with
school mathematics. Although computational pro- larger numbers or fractions draw on understand-

14 Part 1 Guiding Elementary Mathematics with Standards

ing of operations and allow students to solve prob- standing of number operations and algebra
lems. Computational fluency is the ultimate goal for but also integrates processes of problem
students, because they can solve problems using solving, representation, reasoning, and com-
paper-and-pencil techniques, technology, or esti- munication into the lesson.
mation or mental computation. They should know
which computational technique is most appropriate E XERCISE
and how precise they need to be to get a reason-
able answer. Throughout the development of num- Why would the teacher ask students to generate
ber sense, thinking about numbers is emphasized so rules for number operations? How is making a rule
that students can solve more complex problems, jus- different from learning the addition and subtrac-
tify their strategy, and explain whether the answer is
reasonable. tion facts? •••

While students learn about numbers, they can Geometry
also be learning about other standards. In this pri-
mary-grade vignette, a teacher bridges from number Geometry and spatial sense are central mathemati-
facts to an algebraic number pattern. cal ideas in our three-dimensional world. Many
activities—playing sports, driving, gardening, key-
Ms. Munoz shows two beads on a string boarding—require spatial sense. People use spatial
(Figure 2.1), then adds three more beads. awareness when they arrange furniture, pack bag-
She asks students what number sentence to gage, and wrap presents. Artists, architects, and en-
write, and students suggest 2 ϩ 3 ϭ 5. With gineers create sophisticated and aesthetic products
that use an understanding of geometry.
Figure 2.1 Beads showing 2 ϩ 3
Many mathematical concepts are represented
blocks at their desks, children create more with geometry. The number line is a one-dimen-
examples of adding 3, draw pictures of their sional model for addition and subtraction. A map is
blocks, and write other number sentences. a two-dimensional grid representation showing rela-
After recording several sentences on the tive locations of buildings, streets, and cities. The
board, she asks: “What is 5 plus 3? What is 8 same two-dimensional coordinate grid is used for
plus 3? What is 12 plus 3?” Using a hundreds graphing algebraic equations and showing area.
chart, students mark a starting number with
a green chip and put a red chip on “plus In elementary mathematics, children learn
3.” After several examples, Ms. Munoz asks about geometry and develop spatial sense in their
if they see a pattern for “plus 3,” and they environment. They find relationships among shapes
respond that plus 3 is like counting 3 more. and angles as they explore their world. Geometry
Finally, she says, “I am thinking of a mystery activities activate creativity, problem solving, and
number. Now I add 3 and have 26. What reasoning in students’ pictures and designs, such as
was the mystery number?” Children put the quilt patterns.
red chip on 26 and count down 3 to find 23.
They continue finding mystery numbers with Kai watches his grandmother use the “dia-
partners. The teacher asks children to gen- mond in a square” pattern to make a quilt
eralize a rule for adding 3 that encourages (Figure 2.2). He finds graph paper and starts
them to think about how numbers work. drawing the quilt designs with squares and
Asking about a mystery number, Ms. Munoz triangles. He finds that the triangles are re-
engages children in algebraic thinking x ϩ 3 ally larger squares rotated 45 degrees around
ϭ 26. The teacher not only develops under- the center square. He has to figure out how
long the side of each new square is and no-
tices that the sides get longer, but the angles
stay the same.

When Kai takes his quilt to school, his third-
grade teacher, Ms. Scott, recognizes that quilts are
a good way to integrate geometry, art, and measure-

Chapter 2 Defining a Comprehensive Mathematics Curriculum 15

E XERCISE

What is the approximate height of the tree shown in
Figure 2.3? How does this activity connect geometry

with measurement? •••

Figure 2.2 Quilt pattern: diamond in square Measurement

ment into a social studies unit on pioneers and the People measure length, area, volume and capacity,
westward movement. mass, angle, time, temperature, and money nearly
every day. Likewise, elementary school children
E XERCISE learn measurement concepts and skills through their
daily activities. They weigh fruit and vegetables on
Find three or four quilt patterns that show geomet- a variety of scales. They measure the length of the
hall, the room, and their bodies. They use television
ric concepts. Share them with classmates. ••• or bus schedules to plan activities. They measure
angles and areas when creating art projects. Mea-
In the sixth grade, Ms. Ledford finds quilt pat- surement is a natural activity for students, and the
terns with 3-4-5 triangles to introduce similar concepts and skills of measurement, such as itera-
triangles and the Pythagorean theorem. To tion, benchmarks for common units, approximation,
extend her students’ understanding, she takes and precision, develop through these experiences.
the class outside, where she has placed a 2-
meter upright bar in the ground that is casting Measuring the area of rectangles (Figure 2.4)
a 1.5-meter shadow (Figure 2.3). She asks the may begin as a counting exercise, but students soon
students how they could use this information realize that multiplication is a faster way to deter-
to estimate the height of the oak tree in the mine the area. When area is represented as a multi-
school yard. This lesson involves students in plication fact—4 ϫ 6 ϭ 24—they have the first step
problem solving, calculations, algebraic rea- of an algebraic formula (length ϫ width ϭ area) to
soning, and connecting geometric principles represent any rectangular area.
to the real world.
4
5

3 5 ϫ 3 ϭ 15 4 4 ϫ 4 ϭ 16

6ᐉ

4 6 ϫ 4 ϭ 24 w ᐉ ϫ w ϭ Area

Figure 2.4 Area of rectangles

2.0 m ??? m E XERCISE

1.5 m Measure the area of a book cover, desktop, or table
top using sticky notes or square pieces of paper.
5.8 m Use two or three different sizes of paper. Does this
activity help you understand length ؋ width ‫؍‬
Figure 2.3 Measuring a tree with similar triangles
area? •••

Classroom activities highlight the real-life con-
nection to measurement. A classroom savings ac-

16 Part 1 Guiding Elementary Mathematics with Standards

count shows compounding interest. Telling time is Data Analysis and Probability
a basic skill for organizing life activities, and ther-
mometers are essential tools for weather and per- In an information age, data are often organized and
sonal health. Comparison shopping is done with presented in the form of tables and graphs. Newspa-
flyers from newspapers, catalogs, and advertise- pers display information in charts, and content read-
ments. Sy Ning uses her knowledge of measurement ing in social studies and science depends on inter-
to decide which pizza is the best deal for the Liu pretation of graphical information. Graphs showing
family. average temperature and average rainfall help stu-
dents understand the climatic differences between
Sy Ning compares the price of small, me- a desert and a rainforest.
dium, and large pizzas (Figure 2.5). She
estimates the area of each pizza using pr 2, By organizing, displaying, and analyzing infor-
approximating p ϭ 3. She concludes that mation that they have collected, students learn how
the area of the 12-inch pizza is more than to interpret data and draw conclusions. Children
twice that of the 8-inch pizza, but the price collect information about birthdays, favorite pizza,
is less than double. Because one large pizza letters in names, hair color, and height of plants
costs almost $5.00 less than two small ones, grown from seeds. They turn these data into tables,
the family buys a large pizza with one-half charts, and graphs. When students have collected
cheese and pepperoni and one-half vegetar- information from their classmates and the students
ian toppings. in the next class, they can compare and contrast the
information gathered.

8Љ 10Љ 12Љ E XERCISE
$7.75 $9.75 $11.00
Find a table, chart, or graph in a newspaper. Write
Figure 2.5 Comparing pizzas three to five questions about the information
displayed, including both factual and inferential
Solving the problem involved more than calcula- questions. Trade with classmates to answer the
tions; favorite toppings and personal choices were questions they have written. What did you learn
also considered in the solution. Classroom activi-
ties highlight real-life connections to measurement. from the activity? •••
Making a schedule with routine and special events
uses the clock and calendar as tools. Charting Probability is the study of chance—how likely an
heights, temperature, and rainfall makes measure- event will occur. Forecasting weather involves prob-
ment meaningful and provides important data for ability based on atmospheric conditions. Health
analysis. and auto insurance premiums are calculated for
different people on the likelihood of their having
E XERCISE illnesses or being involved in an accident. Under-
standing chance is an important mathematical con-
What are typical high and low temperatures in cept for understanding many aspects of life. Prob-
your area in the winter? in the summer? What is ability experiments also provide opportunities for
the average annual rainfall? Where do you find this data collection.

information? ••• After the experiment of rolling one die 60 times
(Figure 2.6), students see that the expected and the
observed results are not exactly the same, but when
they add all the results from all the groups, they find
that the expectation of equal probability for each
number is quite accurate. Before students roll two
dice 60 times and record their results in a chart,
Mr. King asks them to predict how many of each
sum they will roll. Based on their previous experi-
ment, they predict 5 of each number 1 through 12.

Chapter 2 Defining a Comprehensive Mathematics Curriculum 17

By displaying the results in a table, chart, or graph, picked a red chip. After other sampling
students saw their results and interpreted them with exercises with objects and computer simula-
more confidence. They discovered that their predic- tions, students concluded that as the number
tions were not very accurate. of chips increases, the chance of winning is
reduced. When they talked about the lottery,
Roll 1 die Number Number Number Number Number Number they saw that the chance of picking six win-
ning numbers out of all the possible combi-
60 times of 1s of 2s of 3s of 4s of 5s of 6s nations meant that any one person would
have a very low probability of winning. The
Expected 10 10 10 10 10 10 lottery ticket states that the chance of win-
ning is 1 in 16,000,000, or 0.00000000625—
Observed 8 11 12 7 9 13 close to zero probability.

Figure 2.6 Expected and observed values of 60 rolls Activities and lessons with data and probability
of a die are engaging for students. Students enjoy taking sur-
veys and comparing results. They enjoy rolling dice,
E XERCISE spinning spinners, and drawing chips from a bag.
These activities are also easy to integrate into other
Conduct the experiments with one die and two dice subject lessons.
with two or three partners. How do your results for
rolling one die compare to those of the students?
What happens when you combine all the results in
the class? Based on your experiment with two dice,
what problems did the students find with their

predictions? •••

The newspaper reported that the state lottery E XERCISE

was worth $157 million, and fifth-grade stu- What is the probability of drawing any specific num-
ber from a bag of tiles numbered 1 to 50? What is
dents wanted to know how people could win. the probability of drawing any specific number af-
ter the first number has been removed? How would
Mrs. Imari asked students what they knew you calculate the probability of drawing those two

about the lottery. When she found they had numbers in sequence? •••

little understanding, she decided to introduce Algebra

probability experiments and simulations. Algebra has been the traditional topic for high
school mathematics, but the 2000 NCTM standards
The next day, included algebra as a priority for all grades. Although
algebraic concepts and thinking have been embed-
Mrs. Imari introduced a ded in elementary mathematics, they were not rec-
ognized or well developed. Concepts of patterns,
bag with 99 yellow chips relationships, change, variability, and equality are
fundamental to the study of numbers, operations,
and 1 red chip (Figure geometry, measurement, and data. Algebra is just
one more step when a teacher asks students to use
2.7). Each child removed their observations to create a general rule, predict,
estimate, and reason about missing information. A
a chip, recorded the kindergarten class performs algebra when they ana-
lyze a pattern and represent it in many forms.
color, and put it back in
Ms. McDowell’s children in kindergarten
the bag. Only 2 students can find, read, label, and extend patterns of
pictures and shapes. She wants them to rep-
out of 117 students in the resent patterns in other ways. She asks them
how to act out the shape pattern—circle-
fifth grade had drawn

a red chip. The ratio of

2 to 117 meant that the

chance of getting a red

chip was about 1 in 60

compared to the actual

ratio of 1 red to 99 yel-

low. Next, in a learning

center, students experi-

mented by drawing one

chip from a bag holding

Figure 2.7 Sampling 999 yellow chips and
exercises 1 red chip; no student

18 Part 1 Guiding Elementary Mathematics with Standards

Figure 2.8 Representations
of A-B-C pattern

circle square triangle circle square triangle

snap clap stomp snap clap stomp

red blue yellow red blue yellow

AB C AB C

square-triangle, circle-square-triangle—us- many total blocks would be in a tower hav-
ing actions and sounds. They respond with ing 10 rows?” without building all 10 rows.
snap-clap-stomp and animal sounds: “woof- The students organize their results in a table
meow-oink.” She asks how many different (Figure 2.9b) to find a pattern. The explora-
sounds or actions are in each sequence, they tion was not over because a student was
reply “three.” Finally, she asks if they could puzzled.
write the sequence using letters. They de-
cide they would need three different letters The next day, Yolanda
(Figure 2.8). (c) Stacked blocks asked why the top row was

Patterns are the beginning of algebraic thinking. missing. Mrs. Simmons
Collecting observations about concrete models in a asked if that would change
table makes number patterns and sequences more their results. When they put
apparent. one cube on the top (Fig-
ure 2.9c), they found a different pattern (Fig-
E XERCISE ure 2.9d). Sebastian said that the totals of the
blocks 1, 4, 9, 16, 25, . . . were squared num-
Work with a partner to represent a pattern of your bers because they were the result of multi-
own in five different ways: shape, picture, sound, plying a number by itself: 1 ϫ 1, 2 ϫ 2, 3 ϫ
3, 4 ϫ 4, . . . . Kaylee explained that when
action, symbols. ••• you got to the 10th row, the total number of

(a) Stacked blocks Students in the fourth (b) Table of number patterns (d) Table of number patterns
grade at Viejo Elementary
School build a triangle- Row No. of blocks Total Row No. of blocks Total
shaped tower with cubes 133 111
(Figure 2.9a). They count 258 234
3 7 15 359
the number of blocks on each level: 3, 5, 7, 4 9 24 4 7 16
9, . . . , and the total number. They note that 5 11 35 5 9 25
each row increases by 2 and predict that
the next row will have 11 blocks in it. They ... ... ... ... ... ...
predict the total will be 35 because 24 ϩ ... ... ... ... ... ...
11 ϭ 35; then they build the next level and 10 . . . . . . 10 . . . . . .
confirm their reasoning. Mrs. Simmons chal- ... ... ... ... ... ...
lenges them to answer the question “How 50 . . . . . . 50 . . . . . .
... ... ... ... ... ...
100 . . . . . . 100 . . . . . .

Figure 2.9 Finding a number pattern with blocks

Chapter 2 Defining a Comprehensive Mathematics Curriculum 19

blocks would be 100 because 10 ϫ 10 ϭ 100. Students need a variety of engaging experiences
Mrs. Simmons asked how many blocks it to develop proficiency in problem solving. Newer
would take to build a triangle shape with 20, elementary textbooks and supplemental materials
50, and 100 rows. Then she asked her stu- implement the standards with many stories and situ-
dents to compare their tables to see if they ational problems for students and teachers to solve.
could find a formula for the total number of Puzzles and games also develop problem-solving
blocks in the first tower. strategies. Interesting problems are also found in
newspapers or on the Internet. An article about the
E XERCISE world’s largest dormitory (Hoppe, 1999) included
statistics about the largest dormitory in the United
What would be the total number of blocks needed States at the University of Texas. A listing of the gal-
to complete the tower with 12 rows—without the lons of water used, pounds of cereal and dozens of
eggs eaten, and tons of garbage hauled away gen-
top block and with the top block? ••• erated questions about consumption and waste at
school and at home. These topics became math-
Through exploration with a concrete model, stu- ematical investigations connected to science and
dents found several patterns and ways to represent social studies.
the towers. Algebraic teaching and algebraic think-
ing occurs every time students ask questions about Many good problems are encountered in class-
combinations and possibilities. rooms throughout the school year. Teachers and
students identify problems and investigate alterna-
• What comes next? tive solutions.

• What is missing? • A class needs sashes for a school program. Each
sash has three stripes of different-colored mate-
• What would happen if . . . ? rial, and each stripe is 4 inches wide and 48
inches long. How many sashes are needed? How
• If this changed, what would happen? many yards of each color of material are needed?
What is the cost of 1 yard of material? What will
Opportunities for finding and extending patterns, be the cost of all the sashes?
using variables, observing change, and using sym-
bolic notation are found in almost every lesson. • Two computers are in the classroom. How can
everybody get a turn? What is the best way to
Learning Mathematical Processes schedule the computers with the least interrup-
tion of other activities?
The five process standards (see Table 2.2) empha-
size how children learn and think about mathemat- • A class is investigating climatic changes in its re-
ics. The process standards are threaded through the gion of the country. What is the average monthly
content standards emphasizing the context for learn- rainfall for the past 5 years? How do these aver-
ing mathematics. The five process standards intro- ages compare with the same averages for the past
duced in this chapter are integrated into Part Two of 100 years? Has the average for any given month
this book with content activities and lessons. changed? How has temperature changed over the
past 100 years?
Problem Solving
• A class is studying nutrition as part of a health
Many people recall learning mathematics by memo- unit. What number of calories is recommended
rizing facts followed with a few word problems at for girls in the class? for boys? What ratio of calo-
the end of the unit or chapter. In a modern math- ries from fats to total calories is recommended?
ematics program, problems are not an afterthought. What is the average daily intake of calories of
Instead, children begin with stories and situations each child in the class? How are the calories for
that require them to develop concepts and skills to each child distributed among the food types?
solve problems. In “learning through problem solv- How much does each child’s calorie intake vary
ing” (Schroeder & Lester, 1989), real-life and simu- from what is recommended?
lated problem situations provide context and reason
for learning mathematics. Problem solving is the
central goal of mathematics instruction.

20 Part 1 Guiding Elementary Mathematics with Standards

Realistic and open-ended problems invite realistic (a) Equilateral triangle
and open-ended problem solving. Nonroutine prob-
lems involve important mathematics concepts and (b) Other triangles
skills. Classroom investigations are called problem-
based learning. Thorp and Sage (1999) describe Figure 2.10 Triangles
problem-based learning as “focused, experiential
learning (minds-on, hands-on) organized around an equilateral triangle (Figure 2.10a). When
the investigation and resolution of messy, real-world introduced to another triangle, she refuses
problems.” By creating a “learning adventure,” teach- to name it a triangle. The name is limited by
ers work with students to solve problems with mean- her understanding of “triangles.”
ing. All the sources of problems contribute to the
development of problem solving and provide a vari- As students develop cognitively, they reason
ety of problems for student engagement. In Chapter more flexibly. Classification based on the relation-
8 we introduce some problem-solving strategies that ship between shapes is a powerful reasoning pro-
are used by children and adults when they encoun- cess. After working with a set of triangles, third grad-
ter problems. ers recognize that some attributes are common to
all triangles and that some characteristics identify
Reasoning and Proof specific triangles. One writes a journal entry ex-
plaining what the cooperative group decided about
Elementary students demonstrate reasoning and triangles.
proof when they explain their thinking. In the con-
text of mathematics-rich activities and problems, All triangles have three sides and three
students tell, show, write, draw, and act out what angles. Equilateral triangles are special
they are doing and explain why. By providing expe- because they have three congruent sides
riences for students to construct their own solutions and three congruent angles of 60 degrees.
and understandings, teachers encourage children’s Right triangles are special because they all
thinking. have one right angle. If right triangles have
two sides the same length, they are isosceles
The teacher may ini- right triangles. Scalene triangles have no
tiate problems that are congruent sides. Scalene triangles might
rich with mathematical also be right triangles.
concepts or follow up on
a student interest. The Teachers encourage mathematical discourse by
lottery vignette describes asking open-ended questions based on mathemati-
a real-life problem of in- cal tasks that engage students’ interest and thinking.
terest to students. Stu- Teachers must resist the temptation to provide quick
dents experienced simple and easy answers. Instead, they should ask students
samples, then more com- to explain their reasoning whether the answer is
plex samples, and drew correct or not. Errors are opportunities for students
conclusions based on the to develop reasoning and sound mathematical con-
data and evidence they generated. Throughout the cepts. Disagreements and arguments are important
investigations, they predicted, hypothesized, con- in a classroom that encourages thinking.
jectured, and reasoned using data.
During learning, children often have partial un-
derstanding or misconceptions. Young children
are preoperational, meaning that their thinking is
strongly influenced by visual impressions.

Mariel, age 5, has learned to identify
squares, rectangles, circles, and triangles.
The triangle shape used in her classroom is

Chapter 2 Defining a Comprehensive Mathematics Curriculum 21

Communication communicative enterprise challenges teachers to re-
think their ground rules for teaching mathematics.
The NCTM principles and standards challenge teach-
ers to organize classrooms so that they promote Connection
communication and collaboration. Mathematics is
often remembered as a solitary exercise of writing Connecting mathematics asks students and teach-
answers in workbooks or worksheets. Today, talk- ers to find mathematics in the real world, especially
ing, reading, writing, acting, building, and drawing things related to students’ lives and interests, asso-
are valued ways of learning mathematics. ciations among mathematical concepts, and ways
that mathematics are related to other school sub-
Communication is as fundamental to learning jects and topics. “When am I ever going to use this?”
mathematics as it is to reading and language arts. is answered through activities and problems that
In every lesson, children share their thinking and connect mathematics to real problems that make
improve their reasoning through oral discussions, sense to children.
written descriptions, journals, tables, charts, and
graphs. In the vignette about block towers, students School mathematics can be taught in connected
constructed models and counted blocks. Data ways or in disconnected ways. Disconnected math-
placed in tables led to conclusions, predictions, and ematics is sometimes directed by textbooks and
new questions. Students who snap, clap, and stomp other materials. Connected mathematics is still fo-
various patterns are communicating understanding cused on the development of concepts and skills
of sequence and repetition that they then symbol- but is open to many opportunities to build these
ize with pictures and symbols. As children listen to skills through problem solving. The problems can
explanations and solutions, they hear alternative be real, invented, based on classroom situations,
drawn from other subjects, or generated by student
ways of thinking and may interest. The lottery problem connected the interest
clarify their ideas. of students to mathematical concepts of probability,
data analysis, and number operations and to pro-
In the real world, peo- cesses of problem solving, reasoning, and commu-
ple often work together nication. Many classroom situations illustrate con-
on common goals or nections to children’s experiences.
problems. The elementary
classroom models coop- • A third-grade class sells pizzas at a Parents’ Night
erative efforts of informa- program. How much should the pizzas cost? How
tion sharing and task fo- many pizzas will be needed? How many slices
cus. A new play area calls can be cut from one pizza? What is a fair price to
on children to collaborate charge for one slice? What is a reasonable profit?
and negotiate the design.
Each grade level could • A science unit on plant growth includes experi-
work on the task and ments comparing growth under different light
make recommendations conditions. What measurements need to be
using diagrams and writ- made and how often? What units and measuring
ten explanations. instruments are required? What is the best way to
record observations? How should the results of
• A new surface is being put on the school’s play- the project be reported?
ground, and markings for old activities will be
covered. What game and activity spaces should • Ms. Wolfgang asks the students to use the news-
be laid out on the new surface? What is the best paper advertisements from the home improve-
way to lay out the spaces for the least possible ment store to determine how much it would cost
interference among players of different games? to carpet a platform being built for a stage and
reading area (Figure 2.11). The platform is 8 feet
Talking, sharing, discussing, and even arguing 6 inches by 11 feet 9 inches, but the cost of carpet
contrasts with traditional classrooms where stu- is given in dollars per square yard. One group of
dents work in isolation on getting the one right an- students decides to calculate the dimensions in
swer. Promoting mathematics as a collaborative and

22 Part 1 Guiding Elementary Mathematics with Standards

11Ј 9Љ ഠ 12Ј 3Ј block tower problem was represented concretely
$11.79 ഠ $12 with blocks, graphically in a table, and symbolically
with formulas. The lottery problem was acted out
3Ј 3Ј 3Ј through sampling, recorded in tables, and could be
simulated with computer programs. “Half” is shown
3Ј $12 $12 $12 $12 in pictures of areas and sets and symbolically as a
common fraction, a decimal fraction, or a percent.
8Ј 6Љ ഠ 9Ј 3Ј $12 $12 $12 $12 Although “mean” and “median” are computed in spe-
cific ways, students are sometimes confused by the
3Ј $12 $12 $12 $12 differences. When means and medians for various
data sets are displayed graphically, the differences
3 yd ϫ 4 yd ϭ 12 sq yd may be more apparent. Using different representa-
12 sq yd ϫ $12/sq yd ϭ $144 tions contributes to the meaning and complexity of
a concept.
Figure 2.11 Estimated area and cost of carpet
Place value representation is a complex notion
feet and then in square feet, or approximately that children develop over several years. Children
9 feet ϫ 12 feet ϭ 108 square feet, and then start with a naïve notion of one numeral or name
converts to square yards (108 Ϭ 9 ϭ 12 square for each amount. Two is one more than 1, 6 is one
yards). Another group converts feet to yards and more than 5, 11 is one more than 10. The idea that
approximates with 3 yards ϫ 4 yards ϭ 12 square numbers are just a never-ending sequence of words
yards. If carpet is priced at $11.79 per square yard, related by “plus 1” must change before students com-
they determine the cost as about $144, and then pute with larger numbers. They need to see and know
remembered the sales tax of 7.5% that must be that the number system is based on groups of 10 and
added. They decided $150 was a good estimate that number values can be represented in many
of the total cost. ways. The number 16 can be shown by 16 blocks ar-
ranged in many ways, pictures of 16 objects, 1 rod
Solving the carpet problem requires measure- and 6 cubes with place value blocks, a 4 ϫ 4 grid,
ment, addition, division, multiplication, estimation,
and percent. The diagram allows students to see
how multiplication is used to solve area (3 ϫ 4 ϭ
12 square yards) and find the total cost (12 yards ϫ
$12 per yard ϭ $144). Students may also want to look
at different flooring materials related to economics
topics in social studies.

Interdisciplinary teaching with integrated themes
and activities (Table 2.3) builds children’s sense of
how concepts in one subject link to concepts in
mathematics.

E XERCISE

Find an example of an integrated, interdisciplinary
unit on the Internet. How well are mathematics con-

cepts and skills developed in the unit? •••

Mathematical Representation 16 24 XVI IIII IIII
10 ϩ 6 42
When adults recall their experiences with math-
ematics in school, they often only remember fill- 8ϩ8 IIII I
ing out worksheets with numbers or writing on the 15 ϩ 1
board. Mathematical ideas can be expressed in
many ways: physical models, pictures, diagrams, Figure 2.12 Representations of 16
tables, graphs, charts, and a variety of symbols. The

Chapter 2 Defining a Comprehensive Mathematics Curriculum 23

TABLE 2.3 • Interdisciplinary Connections for Mathematics

Mathematics and Science

• Taking and recording temperature, wind speed, and air pressure in a weather unit
• Investigating the conditions for putting an object in orbit around Earth
• Using a scale of hardness to classify various kinds of rocks and minerals

Mathematics and Social Studies

• Investigating various devices for telling time, such as sundials, sand timers, and water clocks
• Investigating the mathematics used by ancient Egyptians during construction of the pyramids
• Studying Southwest Native American rugs, bowls, and baskets to understand the iconography and how symmetry

and tessellation create meaningful designs
• Comparing (1) the highest and the lowest places on land and (2) the highest place on land with the deepest place

underwater

Mathematics and Art

• Making a scale drawing of a backdrop for a class play and measuring and preparing the paper for the backdrop
• Planning and making an Escher-like tessellation (see Chapter 9)
• Creating Japanese origami

Mathematics and Health

• Keeping a height chart for a year
• Determining calories in school meals and home meals
• Measuring heart rate before and after exertion

Mathematics and Reading/Language Arts

• Looking for patterns in words, classifying words as rhyming and nonrhyming, and looking for palindromic words and
phrases

• Researching and writing about famous mathematicians
• Analyzing text to determine the frequency of letters (students can connect this to the television game show

Wheel of Fortune)

Mathematics and Physical Education

• Counting the number of hops while jumping rope
• Using movement activities to investigate geometric transformations: slides, flips, and turns
• Organizing games on a play area
• Timing races

and numerals and expressions such as 16, 42, 24, 15 cesses have been highlighted. A unit on statistics has
ϩ 1, 10 ϩ 6, 20 Ϫ 4, and XVI. Instead of being limited students measure and record their heights in inches
to one idea about 16 (Figure 2.12), children must un- (Figure 2.13a) and centimeters (Figure 2.13b). Then
derstand that multiple representations are essential they organize the measurements for the class in a
to developing a true understanding of number. table (Figure 2.13c) and a graph (Figure 2.13d). In-
formation from the table and the graph describes
Integrating Process the heights of children in the room and allows for
and Content Standards conclusions about heights of the group.

Mathematics content and processes develop through • “All the students are shorter than 70 inches (180
many experiences with mathematics-rich activities centimeters) tall, and everybody is taller than
and situations coupled with opportunities to com- 45 inches (110 centimeters).”
municate and connect those experiences. Effective
teachers integrate content and process standards • “Most students are about 55 inches (140 centime-
into every lesson and unit. In each of the vignettes ters) tall.”
and examples in this chapter, concepts and pro-
• “Most children are between 53 inches (130 centi-
meters) and 58 inches (145 centimeters) tall.”

24 Part 1 Guiding Elementary Mathematics with Standards

Height in inches (a) Heights of 29 students (inches) Height in centimeters (b) Heights of 29 students (centimeters)
80 180

Series 1 Series 1
70 160

60 140

50 120

40 100

30 80

20 60

10 40

20

0 0
1 4 7 10 13 16 19 22 25 28 1 4 7 10 13 16 19 22 25 28
Students Students

(c) Table of heights (inches) (d) Graph of heights (inches) median, and mean to data on rain-

Height Number Number of students fall and temperature. Some teach-
in inches of students
14
44 0
45 0 ers fear that the greater emphasis
46 1 12 on testing will limit their efforts to
47 1
48 1 10 do exciting projects and stimulat-

49 1 8 ing teaching. In reality, instruction
50 1
51 0 6 based on the NCTM principles and
52 0 standards develops essential con-
53 3
4
54 3
55 3 cepts and skills in the curriculum
56 2 2 and the demands for accountabil-
57 2
58 4 0 44 – 48 49 –53 54 –58 59 – 63 64 – 68 69 –73 ity. When children learn important
Height in inches mathematics in meaningful ways,
59 2 they learn the mathematics they
60 0
61 2 Figure 2.13 Data analysis and display of children’s need for their lives.
62 2
63 0 heights Many resources are available

for teachers. NCTM provides publi-

cations and materials that support

teachers’ efforts in implementing standards-based

64 0 instruction. Membership in NCTM includes a sub-
65 0 scription to one of three journals. For elementary
66 0 and early childhood teachers, Teaching Children
67 0 Mathematics is recommended. Mathematics Teach-
68 1

ing in the Middle School is written for middle-grade

Students note extremes or limits, variation in heights, teachers. The NCTM website http://www.nctm.org pro-

and a tendency of heights to cluster. These observa- vides many helpful links for teachers, such as sam-

tions introduce statistical concepts of range, mean, ple e-lessons, technology integration, and activities

median, and mode. The focus for this lesson is statis- in the Illuminations website. NCTM also sponsors lo-

tical concept, skills, and language, but students also cal, state, regional, and national conferences each

use knowledge of numbers and number operations, year with presentations and exhibits for teachers.

measurement, and algebraic thinking. Later in the Many other publishers and suppliers of mathematics

year, an integrated unit in social studies and science teaching materials have coordinated their materials

on climate provides another chance to apply range, and publications with the standards.

Chapter 2 Defining a Comprehensive Mathematics Curriculum 25

Working with teachers in your school or on your Finding Focus
team helps you to focus on the needs of the chil-
dren in your room or grade level. Sharing ideas, Teachers sometimes feel overwhelmed by national
articles, and materials with colleagues helps you to and state standards. In response to the concern that
grow professionally. Many school districts employ a the curriculum was “a mile wide and an inch deep,”
mathematics consultant or coordinator who knows the National Council of Teachers of Mathematics in
about available resources. An experienced men- September 2006 published Curriculum Focal Points
tor also can help a new teacher find resources and for Prekindergarten through Grade 8 Mathematics
learn how to use them. that highlights three core ideas, called focal points,

TABLE 2.4 • Curriculum Focal Points

Kindergarten Focal Points

Number and Operations: Developing an understanding of whole numbers, including concepts of correspondence, counting,
cardinality, and comparison

Geometry: Identifying shapes and describing spatial relationships
Measurement: Identifying measurable attributes and comparing objects by using these attributes

Grade 1 Focal Points

Number and Operations and Algebra: Developing understandings of addition and subtraction and strategies for basic
addition facts and related subtraction facts

Number and Operations: Developing an understanding of whole number relationships, including grouping in tens and ones
Geometry: Composing and decomposing geometric shapes

Grade 2 Focal Points

Number and Operations: Developing an understanding of the base-10 numeration system and place-value concepts
Number and Operations and Algebra: Developing quick recall of addition facts and related subtraction facts and fluency

with multidigit addition and subtraction
Measurement: Developing an understanding of linear measurement and facility in measuring lengths

Grade 3 Focal Points

Number and Operations and Algebra: Developing understandings of multiplication and division and strategies for basic
multiplication facts and related division facts

Number and Operations: Developing an understanding of fractions and fraction equivalence
Geometry: Describing and analyzing properties of two-dimensional shapes

Grade 4 Focal Points

Number and Operations and Algebra: Developing quick recall of multiplication facts and related division facts and fluency
with whole number multiplication

Number and Operations: Developing an understanding of decimals, including the connections between fractions and
decimals

Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes

Grade 5 Focal Points

Number and Operations and Algebra: Developing an understanding of and fluency with division of whole numbers
Number and Operations: Developing an understanding of and fluency with addition and subtraction of fractions and

decimals
Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, including

volume and surface area

Grade 6 Focal Points

Number and Operations: Developing an understanding of and fluency with multiplication and division of fractions and
decimals

Number and Operations: Connecting ratio and rate to multiplication and division
Algebra: Writing, interpreting, and using mathematical expressions and equations

SOURCE: From Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics (2006) by the National Council of Teachers of Mathematics.

26 Part 1 Guiding Elementary Mathematics with Standards

at each grade level. The principles and standards The publication as well as the website www.nctm
are still the framework for mathematics curriculum .org describe ways teachers can utilize the focal
and instruction with problem solving as the central points in planning and teaching. For example in
theme. The focal points listed in Table 2.4 identify grade 4, knowledge and skill with multidigit mul-
three critical knowledge and skills at each grade tiplication, area of two-dimensional shapes, and
level. The purpose was not to narrow the curricu- decimal fractions are emphasized. Students might
lum to three concepts, but to help teachers organize cover boxes with centimeter grid paper and find
instruction with attention to some essential develop- connections among multiplication, area, and deci-
mental knowledge. mal fractions.

Summary and process standards to your state or local math-
ematics curriculum. If you do not know the URL for
The worlds of 1910 and 2010 are different, and the math- your state’s department of education website, search
ematics that students need to live and thrive in the 21st for “mathematics standards state.” What similarities
century is different. A balanced program of number, ge- or differences do you find between the NCTM stan-
ometry, algebra, measurement, and data analysis con- dards and your school or state’s standards?
cepts and skills replaces a mathematics program focused 2. Observe in a mathematics classroom, and de-
primarily on computation. Development of thinking skills cide whether you see principles and standards in
and application of mathematics to solving problems practice.
breaks away from a focus on memorization. Understand- 3. Describe an ideal classroom that would exemplify
ing what problems to solve and when and how to solve the principles and standards. What would you ex-
them is the center of the modern mathematics program. pect to see and hear in the classroom?
4. Look through a catalog of mathematics materials.
NCTM recognizes the changing need of content and Identify materials related to each of the content and
context for school mathematics in The Principles and process standards.
Standards for School Mathematics (National Council of 5. Go to http://www.nctm.org and read about the prin-
Teachers of Mathematics, 2000). Five content standards ciples and standards. Look at the resources avail-
and five process standards provide a coherent vision of able to teachers on this website.
mathematics to help students become productive and 6. Not all educators and parents agree with the NCTM
thinking citizens. Five major content topics are devel- standards. Conduct an Internet search for sites that
oped across all grade levels: numbers and operations, raise objections to the standards. Do you agree or
algebra, geometry, measurement, and data analysis and disagree with their arguments?
probability. Learning content occurs when students are
communicating about mathematics, reasoning and solv- Teacher’s Resources
ing problems, connecting mathematics to their world,
and demonstrating their understanding in a variety of National Council of Teachers of Mathematics. (2000).
ways. The standards serve as a framework for state cur- Principles and standards for school mathematics. Res-
ricula in mathematics. They challenge teachers to focus ton, VA: Author.
on important mathematical content and skills that enable
students to become proficient problem solvers.

Study Questions and Activities

1. Find the curriculum standards for your school or
state on the Internet. Compare the NCTM content

CHAPTER 3

Mathematics
for Every Child

athematics knowledge and skill provide a key for entry into a 27
rapidly changing technological world. In the middle of the
last century the idea that only a few students, primarily
white male students, could be successful in mathematics
was widely accepted in American society. This perception

denied female and minority students, students with spe-
cial needs, and students with linguistic differences access to advanced
mathematics programs. As a result, females and minorities had fewer
opportunities to learn and faced limited opportunities in college and
vocational choices.

In 1998–1999, the NCTM Board of Directors renewed the challenge to
schools and educators to provide equal opportunities (National Council of
Teachers of Mathematics, 2000, p. 13):

The Board of Directors sees the comprehensive mathematics
education of every child as its most compelling goal. By “every
child” we mean every child—no exceptions. We are particularly
concerned about students who have been denied access in any
way to educational opportunities for any reason, such as language,
ethnicity, physical impairment, gender, socioeconomic status, and
so on. We emphasize that “every child” includes

• learners of English as a second language and speakers
of English as a first language;

• members of underrepresented ethnic groups and members
of well-represented groups;

• students who are physically challenged and those who are not;
• females and males;

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