Helping Children Learn Mathematics

by Reys, Robert; Lindquist, Mary; Lambdin, Diana V.; Smith, Nancy L.

• ISBN13: 9781118654101
• ISBN10: 1118654102
• Edition: 11th
• Format: Paperback
• Copyright: 2014-10-20
• Publisher: Wiley

Summary

The 11th Edition of Helping Children Learn Mathematics is designed to help those who are or will be teachers of mathematics in elementary schools help children develop understanding and proficiency with mathematics so they can solve problems. This text is built around three main themes; helping children make sense of mathematics, incorporating practical experiences and using research to guide teaching. It also integrates connections and implications from the Common Core Standards: Mathematics (CCSS-M).

Author Biography

Robert E. Reys began his career as a mathematics teacher at Horton Watkins High School in Ladue, Missouri. He has taught mathematics education courses at the University of Missouri since 1966. He has also been a visiting professor in Georgia, Canada, and Japan and was a Fulbright Fellow in Mexico and Sweden. Reys's work has earned him a widely recognized international reputation.

Table of Contents

Preface iv

Acknowledgments vi

Chapter 1 school mathematics in a changing world 1

Snapshot of a Lesson 1

Chapter Learning Outcomes 1

Introduction 1

What is Mathematics? 2

What Determines the Mathematics Being Taught? 2

Where Can You Turn? 6

What is Your Role Now? 9

Things to Do: From What You’ve Read 9

Things to Do: Going Beyond This Book 9

Chapter 2 Helping All Children Learn Mathematics with Understanding 11

Snapshot of a Lesson 11

Chapter Learning Outcomes 11

Introduction 12

What do We Know About Learning Mathematics? 12

How Can We Support the Diverse Learners in Our Classroom? 12

Helping Children Acquire Both Procedural and Conceptual Knowledge 17

How do Children Learn Mathematics? 18

How Can We Help Children Make Sense of Mathematics? 22

Cultural Connections 28

A Glance at Where We’ve Been 28

Things to Do: From What You’ve Read 29

Things to Do: Going Beyond This Book 29

Book Nook for Children 30

Chapter 3 Planning and Teaching 31

Snapshot of a Lesson 31

Chapter Learning Outcomes 31

Introduction 31

Preparing to Teach: Questions to Ask 32

Planning for Effective Teaching 43

Meeting the Needs of all Students 52

Cultural Connections 55

A Glance at Where We’ve Been 56

Things to Do: From What You’ve Read 56

Things to Do: Going Beyond This Book 57

Book Nook for Children 57

Chapter 4 Using Assessment to Enhance Learning and Teaching 58

Snapshot of a Lesson 58

Chapter Learning Outcomes 58

Introduction 59

Summative and Formative Assessments 59

Four Phases of Formative Assessment 59

Purposes for Formative Assessment 61

Ways to Assess Students’ Abilities and Dispositions 62

Keeping Records and Communicating About Assessments 74

Cultural Connections 77

A Glance at Where We’ve Been 78

Things to Do: From What You’ve Read 79

Things to Do: Going Beyond This Book 79

Book Nook for Children 80

Chapter 5 Practices and Processes of Doing Mathematics 81

Snapshot of a Lesson 81

Chapter Learning Outcomes 81

Introduction 81

Mathematical Practices from the Common Core State Standards for Mathematics—CCSSM 83

Mathematical Processes from NCTM’s Principles and Standards for School Mathematics 87

Cultural Connections 96

A Glance at Where We’ve Been 97

Things to Do: From What You’ve Read 98

Things to Do: Going Beyond This Book 98

Book Nook for Children 99

Chapter 6 Helping Children with Problem Solving 100

Snapshot of a Lesson 100

Chapter Learning Outcomes 100

Introduction 101

What is a Problem and What is Problem Solving? 101

Teaching Mathematics Through Problem Solving 103

Strategies for Problem Solving 112

The Importance of Looking Back 119

Helping All Students with Problem Solving 120

Cultural Connections 121

A Glance at Where We’ve Been 123

Things to Do: From What You’ve Read 123

Things to Do: Going Beyond This Book 123

Book Nook for Children 124

Chapter 7 Developing Counting and Number Sense 125

Snapshot of a Lesson 125

Chapter Learning Outcomes 125

Introduction 125

Number Sense 125

Prenumber Concepts 127

Early Number Development 131

Counting Principles 134

Cardinal, Ordinal, and Nominal Numbers 141

Writing Numerals 141

Cultural Connections 143

A Glance at Where We’ve Been 144

Things to Do: From What You’ve Read 145

Things to Do: Going Beyond This Book 145

Book Nook for Children 146

Chapter 8 Place Value and Decimals 147

Snapshot of a Lesson 147

Chapter Learning Outcomes 147

Introduction 147

Our Numeration System 148

Nature of Place Value 149

Beginning Place Value 152

Extending Place Value 155

Decimals 162

Connecting Decimals to Place Value 163

Rounding 165

Cultural Connections 167

A Glance at Where We’ve Been 168

Things to Do: From What You’ve Read 168

Things to Do: Going Beyond This Book 168

Book Nook for Children 169

Chapter 9 Operations: Meanings and Basic Facts 170

Snapshot of a Lesson 170

Chapter Learning Outcomes 170

Introduction 171

Helping Children Develop Number Sense and Computational Fluency 172

Developing Meanings for the Operations 173

Mathematical Properties 177

Overview of Basic Fact Instruction 178

Thinking Strategies for Basic Facts 183

Cultural Connections 193

A Glance at Where We’ve Been 194

Things to Do: From What You’ve Read 195

Things to Do: Going Beyond This Book 195

Book Nook for Children 196

Chapter 10 Computation Methods: Calculators, Mental Computation, and Estimation 197

Snapshot of a Lesson 197

Chapter Learning Outcomes 197

Introduction 197

Balancing Your Instruction 198

Calculators 199

Mental Computation 202

Estimation 207

Cultural Connections 213

A Glance at Where We’ve Been 214

Things to Do: From What You’ve Read 215

Things to Do: Going Beyond This Book 215

Book Nook for Children 216

Chapter 11 Standard and Alternative Computational Algorithms 217

Snapshot of a Lesson 217

Chapter Learning Outcomes 217

Introduction 218

Teaching Algorithms with Understanding 220

Addition 221

Subtraction 225

Multiplication 228

Division 232

Beyond Learning Computational Algorithms 237

Cultural Connections 238

A Glance at Where We’ve Been 239

Things to Do: From What You’ve Read 240

Things to Do: Going Beyond This Book 240

Book Nook for Children 241

Chapter 12 Fractions 242

Snapshot of a Lesson 242

Chapter Learning Outcomes 242

Introduction 242

Background Knowledge 243

Making Sense of Fractions 245

Extending Understandings of Fractions 249

Comparing Fractions and Equivalent Fractions 252

Adding and Subtracting Fractions 256

Multiplying and Dividing Fractions 258

Cultural Connections 261

A Glance at Where We’ve Been 261

Things to Do: From What You’ve Read 262

Things to Do: Going Beyond This Book 262

Book Nook for Children 262

Chapter 13 Ratio, Proportion, and Percent: Meanings and Applications 263

Snapshot of a Lesson 263

Chapter Learning Outcomes 263

Introduction 263

Ratios 264

Proportions 267

Percents 271

Cultural Connections 277

A Glance at Where We’ve Been 277

Things to Do: From What You’ve Read 278

Things to Do: Going Beyond This Book 278

Book Nook for Children 279

Chapter 14 Algebraic Thinking 280

Snapshot of a Lesson 280

Chapter Learning Outcomes 280

Introduction 280

Problems, Patterns, and Relations 281

Language and Symbols of Algebra 287

Representing, Generalizing, and Justifying 289

Another Look at Representing, Generalizing, and Justifying 297

Cultural Connections 298

A Glance at Where We’ve Been 299

Things to Do: From What You’ve Read 299

Things to Do: Going Beyond This Book 299

Book Nook for Children 300

Chapter 15 Geometry 301

Snapshot of a Lesson 301

Chapter Learning Outcomes 301

Introduction 301

Three-Dimensional Shapes 303

Two-Dimensional Shapes 309

Space 317

Transformations 319

Visualization and Spatial Reasoning 320

Cultural Connections 323

A Glance at Where We’ve Been 323

Things to Do: From What You’ve Read 323

Things to Do: Going Beyond This Book 324

Book Nook for Children 324

Chapter 16 Measurement 325

Snapshot of a Lesson 325

Chapter Learning Outcomes 325

Introduction 325

The Measurement Process 326

Other Aspects of Measuring 340

Connecting Attributes 344

Cultural Connections 346

A Glance at Where We’ve Been 347

Things to Do: From What You’ve Read 347

Things to Do: Going Beyond This Book 347

Book Nook for Children 348

Chapter 17 Data Analysis, Statistics, and Probability 349

Snapshot of a Lesson 349

Chapter Learning Outcomes 349

Introduction 349

Formulating Questions 351

Collecting Data 353

Analyzing Data: Graphical Organization 354

Misleading Graphs 361

Analyzing Data: Descriptive Statistics 363

Interpreting Results 368

Probability 369

Probability of an Event 370

Randomness 373

Cultural Connections 374

A Glance at Where We’ve Been 375

Things to Do: From What You’ve Read 375

Things to Do: Going Beyond This Book 376

Book Nook for Children 377

Chapter 18 Number Theory 378

Snapshot of a Lesson 378

Chapter Learning Outcomes 378

Introduction 378

Number Theory in Elementary School Mathematics 381

Divisibility 387

Other Number Theory Topics 389

Cultural Connections 393

A Glance at Where We’ve Been 393

Things to Do: From What You’ve Read 394

Things to Do: Going Beyond This Book 394

Book Nook for Children 394

References 395

Appendix 407

Index 409