# How the Brain Learns Mathematics

- David A. Sousa - International Educational Consultant

*To reach all your math students, use your brain—and theirs, too!
*

The bestselling and award-winning first edition of

*How the Brain Learns Mathematics*quickly revolutionized math teaching and learning. The second edition takes readers to the next level with new brain-friendly strategies backed by the latest research from education and neuroscience and even more ways to seamlessly incorporate what you learn about your students’ developing minds into your math classroom.

In this essential resource, you’ll discover the cognitive mechanisms involved in processing mathematical operations, while exploring the environmental and developmental factors that create learning difficulties.

*How the Brain Learns Mathematics*also presents a unique and simplified four-step teaching model that relates students’ classroom experience to concrete, real-world applications. Features of the new edition include

- More strategies for motivating adolescents
- Integration of the arts into mathematics instruction
- New information on how technology affects attention and memory
- Expanded sections on number sense and ELL instruction
- More than 160 new references and a greatly expanded index for readers’ convenience

No matter what grade you teach, your students are growing and changing. Understanding how their brains work is the key to reaching every one of them—and making math a positive part of their lives for years to come.

*“David Sousa’s book is a wonderfully readable presentation of how neuroscience and cognitive psychology can inform the teaching of mathematics in elementary and secondary schools. Sousa engages his readers intellectually with recent research on the brain and mathematics learning, and avoids pat answers where the evidence is suggestive rather than conclusive. The book is a valuable text for teachers who want a deeper insight into thinking processes behind the learning and teaching of math.”*

—Robert E. Slavin, Director, Center for Research and Reform in Education

Johns Hopkins University

*"Teaching mathematics without having read this book is like trying to master tennis without a coach. Sousa's book is a tour de force: It builds a solid bridge from cognitive neuroscience to daily classroom practice. Every teacher of mathematics will benefit from this well-researched, well-organized, thoughtful, and practical approach to making math instruction align with how brains learn."*

—Spencer Kagan, Publisher/Professional Developer

Kagan Publishing and Professional Development

Everyone Can Do Mathematics |

Why is Learning Mathematics So Hard? |

Response From Mathematics Educators |

About This Book |

Questions This Book Will Answer |

Chapter Contents |

Other Helpful Tools |

Assessing Your Current Knowledge of How We Learn Mathematics |

What's Coming? |

Babies Can Count |

What Is Number Sense? |

Animals Also Have Number Sense |

Why Do We Have Number Sense? |

Piaget and Number Sense |

Learning to Count |

Subitizing |

Counting |

How Language Affects Counting |

The Mental Number Line |

Expanded Notions of Number Sense |

Can We Teach Number Sense? |

Quantities to Words to Symbols |

Gardner’s Logical/Mathematical Intelligence |

What’s Coming? |

Reflections on Chapter 1 |

Development of Conceptual Structures |

Structures in Four-Year-Olds |

Structures in Six-Year-Olds |

Structures in Eight-Year-Olds |

Structures in Ten-Year-Olds |

Dealing With Multiplication |

Why Are Multiplication Tables Difficult to Learn? |

Multiplication and Memory |

Is the Way We Teach the Multiplication Tables Intuitive? |

The Impact of Language on Learning Multiplication |

Do the Multiplication Tables Help or Hinder? |

What’s Coming? |

Reflections on Chapter 2 |

Learning and Remembering |

Memory Systems |

Rehearsal Enhances Memory |

The Importance of Meaning |

How Will the Learning Be Stored? |

When Should New Learning Be Presented in a Lesson? |

Does Practice Make Perfect? |

Include Writing Activities |

Gender Differences in Mathematics |

Consider Learning Styles |

Consider Teaching Styles |

How Do You Think About Mathematics? |

What’s Coming? |

Reflections on Chapter 3 |

Should Preschoolers Learn Mathematics at All? |

Assessing Students’ Number Sense |

Preschoolers’ Social and Emotional Behavior |

What Mathematics Should Preschoolers Learn? |

Preschool and Kindergarten Instructional Suggestions |

General Guidelines |

Suggestions for Teaching Subitizing |

Learning to Count |

An Easier Counting System |

Teacher Talk Improves Number Knowledge |

Questioning |

Developing Sorting and Classifying Skills |

What’s Coming? |

Reflections on Chapter 4 |

What Is the Preadolescent Brain? |

How Nature Influences the Growing Brain |

Environment Influences on the Young Brain |

Teaching for Meaning |

Using Cognitive Closure to Remember Meaning |

What Content Should We Be Teaching? |

Teaching Process Skills |

Does the Lesson Enhance Number Sense? |

Does the Lesson Deal With Estimation? |

From Memorization to Understanding |

Multiplication With Understanding |

Does the Lesson Develop Mathematical Reasoning? |

Using Practice Effectively With Young Students |

Graphic Organizers |

Don’t Forget the Technology |

What’s Coming? |

Reflections on Chapter 5 |

What Is the Adolescent Brain? |

Overworking the Frontal Lobes |

The Search for Novelty |

Learning Styles and Mathematics Curriculum |

Qualitative Versus Quantitative Learning Styles |

Developing Mathematical Reasoning |

Instructional Choices in Mathematics |

Graphic Organizers |

Interpreting Word Problems |

Making Mathematics Meaningful to Teenagers |

What’s Coming? |

Reflections on Chapter 6 |

Detecting Mathematics Difficulties |

Determining the Nature of the Problem |

Diagnostic Tools |

Environmental Factors |

Student Attitudes About Mathematics |

Fear of Mathematics (Math Anxiety) |

Neurological and Other Factors |

Dyscalculia |

Addressing Mathematics Difficulties |

Research Findings |

The Concrete-Pictorial-Abstract Approach |

Using Process Mnemonics |

Numeracy Intervention Process |

Students With Nonverbal Learning Disability |

Students With Both Mathematics and Reading Difficulties |

Other Considerations |

What’s Coming? |

Reflections on Chapter 7 |

What Is Mathematics? |

Questions to Ask When Planning Lessons |

Is the Lesson Memory-Compatible? |

Does the Lesson Include Cognitive Closure? |

Will the Primacy-Recency Effect Be Taken Into Account? |

What About Practice? |

What Writing Will Be Involved? |

Are Multiple Intelligences Being Addressed? |

Does the Lesson Provide for Differentiation? |

Simplified Instructional Model |

Conclusion |

Reflections on Chapter 8 |

*"Teaching mathematics without having **read How the Brain Learns Mathematics is like trying to master tennis without a coach. Sousa's book is a tour de force: It builds a solid bridge from cognitive neuroscience to daily classroom practice. Every teacher of mathematics will benefit from this well-researched, well-organized, thoughtful, and practical approach to making math instruction align with how brains learn."*

**Kagan Publishing and Professional Development, San Clemente, CA**

*“David Sousa’s *How the Brain Learns Mathematics, Second Edition* is a wonderfully readable presentation of how neuroscience and cognitive psychology can inform the teaching of mathematics in elementary and secondary schools. Sousa engages his readers intellectually with recent research on the brain and mathematics learning, and avoids pat answers where the evidence is suggestive rather than conclusive. The book should be a valuable text for teachers who want a deeper insight into thinking processes behind the learning and teaching of math.”*

**Center for Research and Reform in Education, Johns Hopkins University, Baltimore, MD**

*“David Sousa has done it again! He has produced a highly-relevant, exceptionally practically, research focused book that will build better mathematics brains in classrooms and schools.”*

**James Madison University, College of Education Harrisonburg, VA**

*“Sousa nailed it with these powerful insights on mathematics instruction. Teachers simply have to understand how students learn in order to provide top-notch instruction, and the specific teaching suggestions herein are invaluable! I love the three tier structure, emphasizing differences in teaching Pre-K and K, Pre-adolescent brains, and adolescent brains, and the emphasis on number sense at all levels is essential in the classroom today. Math teachers will apply these critical lessons immediately in their classes, and I’d urge every mathematics teacher and every elementary teacher to get this book!”*

From a review in NCTM’s* Mathematics Teaching in the Middle School*:*“Classroom teachers, administrators, and math coaches will appreciate the research-based explanations for why mathematics instruction that focuses on meaning making, connections, and processes is so important.” *

**Montana State University Bozeman**

*“Few other books discuss the scientific way the brain is mathematically wired while maintaining relevance to those interested in K–grade 12 education. Readers can expect to gain a deeper understanding of why students learn certain concepts easily and struggle with others and why the battle for successful student learning in mathematics is ever-changing. This book is not merely a collection of lesson plans and activities; it is also a deeper investigation into the science of mathematical learning and inspires readers to continue their own learning into the fascinating world of education.” *

**Southwestern Oregon Community College**