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).
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
About the Author :
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.
