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Building Powerful Numeracy for Middle and High School Students

By Pamela Weber Harris
Foreword by David Webb

Teachers of older students are often frustrated by the lack of number sense their students have.  Informed by research and classroom experience, this book details an instructional approach focused on developing number sense and understanding in higher math for all students.

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"I continue to be amazed at the power we can harness in our secondary students by teaching ourselves and our students real numeracy." Pamela Harris

As secondary math teachers, we’re often frustrated by the lack of true number sense in our students.  Solid research at the elementary level shows how to help all students become mathematically proficient by redefining what it means to compute with number sense.  Pam Harris has spent the past ten years scrutinizing the research and using the resulting reform materials with teachers and students, seeing what works and what doesn’t work, always with an eye to success in higher math. This book brings these insights to the secondary world, with an emphasis on one powerful goal:  building numeracy.

Developing numeracy in today’s middle and high school students is reflective of the Common Core State Standards mission to build “the skills that our young people need for success in college and careers.”  (CCSS 2010)  Numeracy is more than the ability to do basic arithmetic. At its heart, numeracy is the ability to use mathematical relationships to reason with numbers and numerical concepts, to think through the math logically, to have a repertoire of strategies to solve problems, and to be able to apply the logic outside of classrooms. 

How can we build powerful numeracy in middle and secondary students?  Harris’s approach emphasizes two big ideas:

  • Teach the importance of representation.   The representation of student strategies on models such as the open number line, the open array, and the ratio table promote discussion on relationships rather than procedures
  • Teach with problem strings.  Introduced by Catherine Twomey Fosnot and her colleagues in the Young Mathematicians at Work series, problem strings are purposefully designed sequences of related problems that help students construct numerical relationships. They encourage students to look to the numbers first before choosing a strategy, nudging them toward efficient, sophisticated strategies for computation.

Understanding numerical relationships gives students the freedom to choose a strategy, rather than being stuck with only one way to solve a problem.  Using the strings and activities in this book can empower your students to reason through problems and seek to find clever solutions.  They’ll become more naturally inclined to use the strategies that make sense to them.  Students become engaged, willing to think, and more confident in their justifications.  When we give secondary students this numerical power, we also help them learn higher mathematics with more confidence and more success.

Additional Resource Information

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Foreword by David C. Webb


Chapter 1 Numeracy

Chapter 2 Addition and Subtraction: Models and Strategies

Chapter 3 Addition

Chapter 4 Subtraction

Chapter 5 Multiplication and Division: Models and Strategies

Chapter 6 Multiplication

Chapter 7 Division

Chapter 8 Decimals, Fractions, and Percents: Models and Strategies

Chapter 9 Decimals and Fractions: Addition and Subtraction

Chapter 10 Decimals and Fractions: Multiplication and Division




“This book is an outstanding and welcome contribution to the field of mathematics education. It simultaneously addresses the development of numeracy and the extensions to the mathematical ideas taught on the secondary level and does so in a wonderfully engaging, coherent, and thoughtful way. Any secondary teacher reading this book will come away with a far deeper understanding of how to develop numeracy and fluent computation while integrating the more advanced topics that they are required to teach.” Cathy Fosnot, author of the Young Mathematicians at Work series

“What Pam Harris has offered in this book is a compelling approach for achieving greater coherence in the development of secondary students’ numeracy and computational fluency.” David C. Webb, University of Colorado at Boulder, Executive Director, Freudenthal Institute USA

“Pam Harris...suggests an excellent framework not only for building numeracy but also for extending mathematical ideas, such as the four operations, first presented in elementary schools.   As students apply the representation strategies outlined, they undoubtedly become more accurate, flexible, and effective mathematicians.” Mathematics Teaching in the Middle School

“This book is the expansion that I’ve been looking for.  Every move in this book is deliberate and calculated.  Pam Harris discusses numeracy and the importance of it in our students’ math education, in order for them to be successful for more advanced mathematical concepts.” Sadie Estrella, teacher, Hana, HI