Helping Children Learn Mathematics

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.

• Preface ivAcknowledgments viChapter 1 school mathematics in a changing world 1Snapshot of a Lesson 1Chapter Learning Outcomes 1Introduction 1What is Mathematics? 2What Determines the Mathematics Being Taught? 2Where Can You Turn? 6What is Your Role Now? 9Things to Do: From What You’ve Read 9Things to Do: Going Beyond This Book 9Chapter 2 Helping All Children Learn Mathematics with Understanding 11Snapshot of a Lesson 11Chapter Learning Outcomes 11Introduction 12What do We Know About Learning Mathematics? 12How Can We Support the Diverse Learners in Our Classroom? 12Helping Children Acquire Both Procedural and Conceptual Knowledge 17How do Children Learn Mathematics? 18How Can We Help Children Make Sense of Mathematics? 22Cultural Connections 28A Glance at Where We’ve Been 28Things to Do: From What You’ve Read 29Things to Do: Going Beyond This Book 29Book Nook for Children 30Chapter 3 Planning and Teaching 31Snapshot of a Lesson 31Chapter Learning Outcomes 31Introduction 31Preparing to Teach: Questions to Ask 32Planning for Effective Teaching 43Meeting the Needs of all Students 52Cultural Connections 55A Glance at Where We’ve Been 56Things to Do: From What You’ve Read 56Things to Do: Going Beyond This Book 57Book Nook for Children 57Chapter 4 Using Assessment to Enhance Learning and Teaching 58Snapshot of a Lesson 58Chapter Learning Outcomes 58Introduction 59Summative and Formative Assessments 59Four Phases of Formative Assessment 59Purposes for Formative Assessment 61Ways to Assess Students’ Abilities and Dispositions 62Keeping Records and Communicating About Assessments 74Cultural Connections 77A Glance at Where We’ve Been 78Things to Do: From What You’ve Read 79Things to Do: Going Beyond This Book 79Book Nook for Children 80Chapter 5 Practices and Processes of Doing Mathematics 81Snapshot of a Lesson 81Chapter Learning Outcomes 81Introduction 81Mathematical Practices from the Common Core State Standards for Mathematics—CCSSM 83Mathematical Processes from NCTM’s Principles and Standards for School Mathematics 87Cultural Connections 96A Glance at Where We’ve Been 97Things to Do: From What You’ve Read 98Things to Do: Going Beyond This Book 98Book Nook for Children 99Chapter 6 Helping Children with Problem Solving 100Snapshot of a Lesson 100Chapter Learning Outcomes 100Introduction 101What is a Problem and What is Problem Solving? 101Teaching Mathematics Through Problem Solving 103Strategies for Problem Solving 112The Importance of Looking Back 119Helping All Students with Problem Solving 120Cultural Connections 121A Glance at Where We’ve Been 123Things to Do: From What You’ve Read 123Things to Do: Going Beyond This Book 123Book Nook for Children 124Chapter 7 Developing Counting and Number Sense 125Snapshot of a Lesson 125Chapter Learning Outcomes 125Introduction 125Number Sense 125Prenumber Concepts 127Early Number Development 131Counting Principles 134Cardinal, Ordinal, and Nominal Numbers 141Writing Numerals 141Cultural Connections 143A Glance at Where We’ve Been 144Things to Do: From What You’ve Read 145Things to Do: Going Beyond This Book 145Book Nook for Children 146Chapter 8 Place Value and Decimals 147 Snapshot of a Lesson 147Chapter Learning Outcomes 147Introduction 147Our Numeration System 148Nature of Place Value 149Beginning Place Value 152Extending Place Value 155Decimals 162Connecting Decimals to Place Value 163Rounding 165Cultural Connections 167A Glance at Where We’ve Been 168Things to Do: From What You’ve Read 168Things to Do: Going Beyond This Book 168Book Nook for Children 169Chapter 9 Operations: Meanings and Basic Facts 170Snapshot of a Lesson 170Chapter Learning Outcomes 170Introduction 171Helping Children Develop Number Sense and Computational Fluency 172Developing Meanings for the Operations 173Mathematical Properties 177Overview of Basic Fact Instruction 178Thinking Strategies for Basic Facts 183Cultural Connections 193A Glance at Where We’ve Been 194Things to Do: From What You’ve Read 195Things to Do: Going Beyond This Book 195Book Nook for Children 196Chapter 10 Computation Methods: Calculators, Mental Computation, and Estimation 197Snapshot of a Lesson 197Chapter Learning Outcomes 197Introduction 197Balancing Your Instruction 198Calculators 199Mental Computation 202Estimation 207Cultural Connections 213A Glance at Where We’ve Been 214Things to Do: From What You’ve Read 215Things to Do: Going Beyond This Book 215Book Nook for Children 216Chapter 11 Standard and Alternative Computational Algorithms 217Snapshot of a Lesson 217Chapter Learning Outcomes 217Introduction 218Teaching Algorithms with Understanding 220Addition 221Subtraction 225Multiplication 228Division 232Beyond Learning Computational Algorithms 237Cultural Connections 238A Glance at Where We’ve Been 239Things to Do: From What You’ve Read 240Things to Do: Going Beyond This Book 240Book Nook for Children 241Chapter 12 Fractions 242Snapshot of a Lesson 242Chapter Learning Outcomes 242Introduction 242Background Knowledge 243Making Sense of Fractions 245Extending Understandings of Fractions 249Comparing Fractions and Equivalent Fractions 252Adding and Subtracting Fractions 256Multiplying and Dividing Fractions 258Cultural Connections 261A Glance at Where We’ve Been 261Things to Do: From What You’ve Read 262Things to Do: Going Beyond This Book 262Book Nook for Children 262Chapter 13 Ratio, Proportion, and Percent: Meanings and Applications 263Snapshot of a Lesson 263Chapter Learning Outcomes 263Introduction 263Ratios 264Proportions 267Percents 271Cultural Connections 277A Glance at Where We’ve Been 277Things to Do: From What You’ve Read 278Things to Do: Going Beyond This Book 278Book Nook for Children 279Chapter 14 Algebraic Thinking 280Snapshot of a Lesson 280Chapter Learning Outcomes 280Introduction 280Problems, Patterns, and Relations 281Language and Symbols of Algebra 287Representing, Generalizing, and Justifying 289Another Look at Representing, Generalizing, and Justifying 297Cultural Connections 298A Glance at Where We’ve Been 299Things to Do: From What You’ve Read 299Things to Do: Going Beyond This Book 299Book Nook for Children 300Chapter 15 Geometry 301Snapshot of a Lesson 301Chapter Learning Outcomes 301Introduction 301Three-Dimensional Shapes 303Two-Dimensional Shapes 309Space 317Transformations 319Visualization and Spatial Reasoning 320Cultural Connections 323A Glance at Where We’ve Been 323Things to Do: From What You’ve Read 323Things to Do: Going Beyond This Book 324Book Nook for Children 324Chapter 16 Measurement 325Snapshot of a Lesson 325Chapter Learning Outcomes 325Introduction 325The Measurement Process 326Other Aspects of Measuring 340Connecting Attributes 344Cultural Connections 346A Glance at Where We’ve Been 347Things to Do: From What You’ve Read 347Things to Do: Going Beyond This Book 347Book Nook for Children 348Chapter 17 Data Analysis, Statistics, and Probability 349Snapshot of a Lesson 349Chapter Learning Outcomes 349Introduction 349Formulating Questions 351Collecting Data 353Analyzing Data: Graphical Organization 354Misleading Graphs 361Analyzing Data: Descriptive Statistics 363Interpreting Results 368Probability 369Probability of an Event 370Randomness 373Cultural Connections 374A Glance at Where We’ve Been 375Things to Do: From What You’ve Read 375Things to Do: Going Beyond This Book 376Book Nook for Children 377Chapter 18 Number Theory 378Snapshot of a Lesson 378Chapter Learning Outcomes 378Introduction 378Number Theory in Elementary School Mathematics 381Divisibility 387Other Number Theory Topics 389Cultural Connections 393A Glance at Where We’ve Been 393Things to Do: From What You’ve Read 394Things to Do: Going Beyond This Book 394Book Nook for Children 394References 395Appendix 407Index 409