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Contents ABOUT THE AUTHOR ix INTRODUCTION 1 Everyone Can Do mathematics 1 Why is Mathematics So Hard? 2 Response from Mathematics Educators 2 About This Book 3 Questions This Book Will Answer 3 Chapter Contents 4 Other Helpful Topics 5 Assessing Your Current Knowledge of How We Learn Mathematics 7 What?s Coming? 7 CHAPTER 1 ? Developing Number Sense 9 Babies Can Count 9 What Is Number Sense? 10 Animals Also Have Number Sense 11 Why Do We Have Number Sense? 12 Piaget and Number Sense 12 Learning to Count 13 Subitizing 14 Counting 15 How Language Affects Counting 18 The Mental Number Line 21 Expanded Notions of Number Sense 24 Can We Teach Number Sense? 26 Quantities to Words to Symbols 30 Gardner?s Logical-Mathematical Intelligence 31 What?s Coming? 33 Reflections on Chapter 1 34 CHAPTER 2 ? Learning to Calculate 35 Development of Conceptual Structures 36 Structures in Four-Year-Olds 37 Structures in Six-Year-Olds 38 Structures in Eight-Year-Olds 38 Structures in Ten-Year-Olds 39 Dealing With Multiplication 40 Why Are Multiplication Tables Difficult to Learn? 40 Multiplication and Memory 40 Is the Way We Teach the Multiplication Tables Intuitive? 41 The Impact of Language on Learning Multiplication 44 Do the Multiplication Tables Help or Hinder? 46 What?s Coming? 47 Reflections on Chapter 2 48 CHAPTER 3 ? Reviewing the Elements of Learning 49 Learning and Remembering 49 Memory Systems 50 Rehearsal Enhances Memory 53 The Importance of Meaning 54 How Will the Learning Be Stored? 58 When Should New Learning Be Presented in a Lesson? 60 Does Practice Make Perfect? 61 Include Writing Activities 64 Gender Differences in Mathematics 65 Consider Learning Styles 67 Consider Teaching Styles 69 What?s Coming 72 Reflections on Chapter 3 73 CHAPTER 4 ? Teaching Mathematics to the Preschool and Kindergarten Brain 75 Should Preschoolers Learn Mathematics At All? 76 Assessing Students? Number Sense 77 Preschoolers? Social and Emotional Behavior 77 What Mathematics Should Preschoolers Learn? 78 Preschool and Kindergarten Instructional Suggestions 79 General Guidelines 79 Suggestions for Teaching Subitizing 80 Learning to Count 85 An Easier Counting System 87 Teacher Talk Improves Number Knowledge 87 Questioning 89 Developing Sorting and Classifying Skills 90 What?s Coming? 94 Reflections on Chapter 4 95 CHAPTER 5 ? Teaching Mathematics to the Preadolescent Brain 97 What Is the Preadolescent Brain? 97 How Nature Influences the Growing Brain 97 Environment Influences on the Young Brain 101 Teaching for Meaning 103 What Content Should We Be Teaching 105 Teaching Process Skills 106 Does the Lesson Enhance Number Sense? 107 Does the Lesson Deal With Estimation? 111 From Memorization to Understanding 116 Multiplication With Understanding 119 Does the Lesson Develop Mathematical Reasoning? 123 Using Practice Effectively With Young Students 126 Graphic Organizers 128 Don?t Forget the Technology 129 What?s Coming? 131 Reflections on Chapter 5 132 CHAPTER 6 ? Teaching Mathematics to the Adolescent Brain 133 What Is the Adolescent Brain? 133 Overworking the Frontal Lobes 133 The Search for Novelty 136 Learning Styles and Mathematics Curriculum 139 Qualitative versus Quantitative Learning Styles 139 Developing Mathematical Reasoning 141 Instructional Choices in Mathematics 143 Graphic Organizers 149 Interpreting Word Problems 155 Making Mathematics Meaningful for Teenagers 156 What?s Coming? 161 Reflections on Chapter 6 162 CHAPTER 7 ? Recognizing and Addressing Mathematics Difficulties 163 Detecting Mathematics Difficulties 164 Determining the Nature of the Problem 164 Diagnostic Tools 166 Environmental Factors 170 Student Attitudes About Mathematics 170 Fear of Mathematics (Math Anxiety) 171 Neurological and Other Factors 178 Dyscalculia 179 Addressing Mathematics Difficulties 185 Research Findings 185 The Concrete-Pictorial-Abstract Approach 186 Using Process Mnemonics 188 Numeracy Intervention Process 190 Students With Nonverbal Learning Disability 190 Students With Both Mathematics and Reading Difficulties 192 Other Considerations 195 What?s Coming? 197 Reflections on Chapter 7 198 CHAPTER 8 ? Putting It All Together: Planning Lessons in PreK-12 199 Mathematics What Is Mathematics? 200 Questions to Ask When Planning Lessons 201 Is the Lesson Memory-Compatible? 201 Does the Lesson Include Cognitive Closure? 202 Will the Primacy-Recency Effect Be Taken Into Account? 203 What About Practice? 205 What Writing Will Be Involved? 206 Are Multiple Intelligences Being Addressed? 208 Does the Lesson Provide for Differentiation? 211 Simplified Instructional Model 214 Conclusion 215 Reflections on Chapter 8 217 GLOSSARY 219 REFERENCES 223 RESOURCES 235 INDEX 241

Library of Congress Subject Headings for this publication:

Mathematics -- Study and teaching -- United States.

Learning, Psychology of.

Learning -- Physiological aspects.