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Developing Numerical Fluency

Making Numbers, Facts, and Computation Meaningful

By Patsy Kanter, Steven Leinwand

Numerical fluency is about understanding, not memorization.

In Developing Numerical Fluency, Patsy Kanter and Steven Leinwand take a fresh look at a commonly-asked question: “How do I teach number facts so my students know them fluently?” They apply their decades of experience teaching mathematics to rethinking effective fluency instruction.

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"This is a must-read book for any teachers of math."

—Jo Boaler, Professor of Mathematics Education at Stanford University and author of Mathematical Mindsets

 

Numerical fluency is about understanding

Numerical fluency is about understanding, not memorization. It comes over time as students engage in active thinking and doing, not endless worksheets and timed tests.

Classroom instruction and materials, however, often don’t feel aligned with these realities.

In Developing Numerical Fluency, Patsy Kanter and Steven Leinwand take a fresh look at a commonly-asked question: “How do I teach number facts so my students know them fluently?” They apply their decades of experience teaching mathematics to rethinking effective fluency instruction.

Classroom-tested ideas you can use right away

Each chapter introduces ideas, techniques, and strategies that contribute to meaningful fluency for all students. You’ll find:

  • pivotal understandings that illuminate what contributes to real numerical fluency
  • six instructional processes that support lasting fluency development
  • classroom structures and activities for building fluency in addition, subtraction, multiplication, and division
  • suggestions for creating a school-wide culture of numerical fluency.
Patsy and Steve remind us that, “Students do not develop numerical fluency by memorizing and regurgitating rules.” But many of us learned mathematics in exactly this way, making shifting our instruction challenging. Developing Numerical Fluency provides just the right support, offering big ideas for rethinking instruction paired with classroom-tested activities you can use right away.

Podcast

Contents

Introduction

  • Defining Numerical Fluency
  • The Challenge: Teaching in Ways We Were Not Taught
  • Our Goals for This Book

PART ONE: Building Understanding of Numerical Fluency

Chapter 1: Nine Pivotal Understandings for Numerical Fluency

  • Pivotal Understanding 1: Everything Begins with Counting
  • Pivotal Understanding 2: All Quantities are Composed of Parts and Wholes
  • Pivotal Understanding 3: Acquiring the Language of Operations Before Learning Facts
  • Pivotal Understanding 4: Properties of Operations
  • Pivotal Understanding 5: Using 5 and 10
  • Pivotal Understanding 6: 9 and 10-1 are the Same Number
  • Pivotal Understanding 7: Multiplication by 2, 3, 5, and 10
  • Pivotal Understanding 8: Equivalence in Equations are the Meaning of the Equal Sign
  • Pivotal Understanding 9: Place Value Understanding

Chapter 2: Six Processes for Developing Numerical Fluency

  • The Six Processes
  • Putting the Six Processes Together
  • Conclusion

Chapter 3: Classroom Structures for Developing Numerical Fluency

  • Guiding Principles for Teaching Structures
  • Number Talks
  • Games
  • Word Problem Banks
  • Problem-Solving Activities
  • Student-Made Materials
  • Conclusion

PART TWO: Developing Addition and Subtraction Fluency

Chapter 4: It All Begins with Parts and Wholes

  • Why Parts and Wholes Matter
  • Key Ideas and Skills
  • Classroom Activities

Chapter 5: Developing Fluency with Addition and Subtraction Facts

  • Key Ideas and Skills
  • Classroom Activities

Chapter 6: Developing Fluency Finding Sums and Differences

  • Key Ideas and Skills
  • Classroom Activities

PART THREE: Developing Multiplication and Division Fluency

Chapter 7: Giving Meaning to Multiplication and Division

  • Key Ideas and Skills
  • Classroom Activities

Chapter 8: Developing Fluency with Multiplication and Division Facts

  • Key Ideas and Skills
  • Classroom Activities

Chapter 9: Developing Fluency Finding Products and Quotients

  • Key Ideas and Skills
  • Classroom Activities

PART FOUR: Strategies for Creating a Schoolwide Culture of Fluency

Chapter 10: Activities and Resources for Establishing a Culture of Fluency and Fun

Chapter 11: It Takes a School to Develop Numerical Fluency

  • Making Better Use of Meetings
  • Teacher Collaboration
  • Garnering Administrative Support

Appendix

References

Samples

Companion Resources

Reviews

"This is a must-read book for any teachers of math. Patsy and Steve take us on a journey of all that needs to change in the development of numerical fluency and give critical advice along the way."

—Jo Boaler, Professor of Mathematics Education at Stanford University, co-founder of youcubed.org, and author of Mathematical Mindsets