Table Of ContentHOW THE
Brain Learns
Mathematics
HOW THE
Brain Learns
Mathematics
David A. Sousa
=CORWIN PRESS
*A SAGE Publications Company
Thousand Oaks, CA 91320
Copyright © 2008 by David A. Sousa
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Library of Congress Cataloging-in-Publication Data
Sousa, David A.
How the brain learns mathematics/David A. Sousa.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4129-5305-4 (cloth)
ISBN 978-1-4129-5306-1 (pbk.)
1. Mathematics—Study and teaching—United States. 2. Learning, Psychology of.
3. Learning—Physiological aspects. I. Title.
QA13.S66 2008
510.71—dc22 2007024839
This book is printed on acid-free paper.
07 08 09 10 11 10 9 8 7 6 5 4 3 2 1
Acquisitions Editor: Robb Clouse
Editorial Assistant: Jessica Wochna
Production Editor: Sarah K. Quesenberry
Proofreader: Caryne Brown
Typesetter: C&M Digitals (P) Ltd.
Cover/Graphic Designer: Lisa Miller
Contents
ABOUT THE AUTHOR ix
ACKNOWLEDGMENTS xi
INTRODUCTION 1
Everyone Can Do Mathematics 1
Why Is Learning Mathematics So Hard? 2
Response From Mathematics Educators 2
About This Book 3
Questions This Book Will Answer 3
Chapter Contents 4
Other Helpful Tools 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
How Do You Think About Mathematics? 70
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
Using Cognitive Closure to Remember Meaning 104
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 to 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
About the Author
David A. Sousa, EdD, is an international educational consultant. He has made
presentations at national conventions of educational organizations and has
conducted workshops on brain research and science education in hundreds of
school districts and at several colleges and universities across the United States,
Canada, Europe, Asia, and Australia.
Dr. Sousa has a bachelor of science degree in chemistry from Massachusetts
State College at Bridgewater, a master of arts in teaching in science from Harvard
University, and a doctorate from Rutgers University. His teaching experience covers all levels. He has
taught high school science, served as a K–12 director of science, and was Supervisor of Instruction
for the West Orange, New Jersey, schools. He then became superintendent of the New Providence,
New Jersey, public schools. He has been an adjunct professor of education at Seton Hall University
and a visiting lecturer at Rutgers University. He was president of the National Staff Development
Council in 1992.
Dr. Sousa has edited science books and published numerous books and articles in leading
educational journals on staff development, science education, and brain research. He has received
awards from professional associations and school districts for his commitment and contributions to
research, staff development, and science education. He received the Distinguished Alumni Award and
an honorary doctorate in education from Bridgewater (Massachusetts) State College.
Dr. Sousa is a member of the Cognitive Neuroscience Society. He has appeared on NBC’s
Todayshow and on National Public Radio to discuss his work with schools using brain research. He
makes his home in south Florida.
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