Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense, Grades 3–5, Second Edition


AAP Pre K–12 Learning 2016 Revere Awards Winner

The focus of Beyond Pizzas & Pies, Second Edition, is on foundational understandings. The authors help teachers and students understand fractions as numbers, including the use of equivalency and thinking about benchmark numbers, so that when students start to compute with fractions (see the companion volume, Beyond Invert & Multiply), they have a solid foundation. You’ll find:

Classroom Scenarios that address common challenges students (and teachers!) encounter as they acquire an understanding of fractions;

What’s the Math? sections intended to clarify the mathematics;

What’s the Research? insights that show why “that’s the way we’ve always done things” might not be the most successful approach;

Classroom Activities with friendly, step-by-step instructions and video clips demonstrating how to teach the concepts;

Reproducibles that are also available in a downloadable, printable format;

Study Questions at the end of each chapter that encourage readers to set concrete goals for implementing what they’ve read; and

Connections to the Common Core for those states implementing the Common Core State Standards for Mathematics.

 (The accompanying DVD features 30 video clips filmed in actual classrooms. Clips range from 1 minute to 6 minutes in length with a total viewing time of approximately 70 minutes.)

Discussion questions for viewing each video clip

icon pdf How Beyond Pizzas & Pies and Beyond Invert & Multiply work together

Julie McNamara and Meghan M. Shaughnessy
Foreword by Francis (Skip) Fennell
192 Pages

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Julie McNamara is an Assistant Professor of Mathematics Education at California State University, East Bay in Hayward, California. She has also served as Mathematics Specialist with TeachingWorks at the University of Michigan. Julie spent two years supporting elementary and middle school teachers as a fulltime designer and provider of mathematics professional development for Math Solutions, and continues to consult for Math Solutions part-time. She has also taught mathematics methods in the elementary credential programs at the University of California, Berkeley and at Saint Mary’s College in Moraga, California. She is a former classroom teacher in the San Francisco Bay Area. Her experience in the classroom led her to pursue graduate studies in the Development in Mathematics and Science program at the University of California, Berkeley, and to earn her master’s degree and Ph.D. in mathematics education. Her research focuses on the teaching and learning of mathematics concepts that are foundational but considered "hard to teach and hard to learn."

Julie is co-author of Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fraction Sense, 2010, as well as Beyond Pizzas & Pies: 10 Essential Strategies for Supporting Fractions Sense, Second Edition, 2015. In addition, Julie has written a follow up fraction book, Beyond Invert & Multiply: Making Sense of Fraction Computation, Grades 3-6, 2015, all published by Math Solutions.

Meghan M. Shaughnessy is a postdoctoral research fellow at the School of Education at the University of Michigan. Her research focuses on the teaching and learning of elementary mathematics. She designs and studies innovative professional development materials for practicing teachers and teaches mathematics methods courses for preservice elementary teachers. Meghan received a Ph.D. in mathematics education from the University of California, Berkeley.

"The authors are right on—developing 'fraction sense' is one of the most difficult tasks of new (or experienced) teachers. This book provides a wonderful resource for helping teachers help their students become truly fluent with fractions."

Barbara Reys
President Association of Mathematics Teacher Educators

"Beyond Pizzas and Pies is smart, accessible, and usable. McNamara and Shaughnessy assemble the key resources and insights teachers need to help students develop robust understanding of fractions. Clear explanations of the specific mathematical ideas and research on key challenges for learners are coordinated with focused instructional tasks and detailed guidance for their use in the classroom. The tasks develop complex mathematical proficiency by combining work on concepts with skill development and the ability to use representations and to reason about fractions."

Deborah Loewenberg Ball
William H. Payne Collegiate Professor University of Michigan

"McNamara and Shaughnessy’s splendid work guides student understanding of fractions by engaging them in a relevant and thoughtful method. The authors bridge fundamental gaps between concept and application, using challenging and fun exercises that demonstrate these principals through visual and hands-on activities. Beyond Pizzas and Pies is an outstanding guide to a difficult unit on which so many mathematical concepts rely."

Rebecca Gris, GATE teacher, grades 2–3
Boise Schools, Boise, Idaho

"This book addresses not just what students should learn about fractions, but how children actually think, where that thinking comes from, and what teachers can do to make their instruction more effective. Each chapter is carefully structured to present the research base as well as classroom activities that can immediately be put to use. I can’t wait to use the book with my preservice teachers, since it summarizes the most essential (and usable) information about the teaching and learning of fractions."

Laurie Edwards, Professor of Education
Saint Mary's College of California

"Julie and Meghan have provided a resource rich in both mathematics content and pedagogical content. They clearly identify common student misconceptions about fractions and, even more importantly, offer practical ideas and activities to help deepen students' essential understanding of fractions!"

Josh Rosen, Math Specialist
Dobbs Ferry School District, New York

"This book lays out explicit strategies and activities for developing fraction understanding. The research, mathematics, and pedagogy are vividly brought to life through classroom scenarios and examples that will resonate with teachers."

Carne Barnett-Clarke, Director
Math Pathways & Pitfalls, WestEd

"Julie and Meghan's book forces educators to acknowledge that the traditional way of teaching fractions is limiting, lacking opportunity for higher-level application. They offer concrete solutions that can be easily integrated into any classroom curriculum."

Lindsay Phillips, eighth-grade teacher
Portsmouth Middle School, Portsmouth, Rhode Island

"Beyond Pizzas and Pies expertly presents the essential concepts that third-, fourth-, and fifth-grade teachers must fully understand so they can teach students to develop a deep conceptual understanding of fractions. By presenting ten critical strategies for developing fraction sense, McNamara and Shaughnessy provide a tightly scaffolded plan for both teachers and students that is easy to follow and goes well beyond the usual teacher manuals and math texts commonly used in upper elementary school classrooms."

Maggie Riddle, Principal
Jefferson Elementary, Berkeley Unified School District, California

"McNamara and Shaughnessy understand the difficulty of fractions in mathematics education and masterfully weave together classroom vignettes, the latest in education research, mathematics, and classroom lessons in easy-to-follow chapters that will challenge your assumptions about how to teach this critically important topic. This book is a goldmine of information delivered in simple, straightforward language."

Jennifer M. Langer-Osuna, Ph.D., Assistant Professor
University of Miami, Coral Gables, Florida

"Using these activities has increased my students' deep understanding of fractions and their math confidence, while decreasing their confusion when we discuss decimals. I needed this book when I started teaching thirty years ago!"

Morri Spang, Intervention and GATE teacher, grades 3–6
Micheltorena Elementary School, Los Angeles

"McNamara and Shaughnessy have created a gem of a resource for teaching the very difficult content area of fractions. This book is guaranteed to support teachers in developing deeper understandings of the complexities involved in reasoning about and computing with fractions, so that they in turn can support the developing understandings of their students."

Mary Q. Foote, Assistant Professor of Mathematics Education
Queens College, CUNY, New York

"Bravo to the clever authors of Beyond Pizzas and Pies for serving both teachers and students a practical menu with fresh recipes of applicable strategies that will help us think outside the pie. The explanations backed by research stimulated and altered my own math sense, empowering my instruction with more purpose and comprehension. My fifth graders will be challenged and charmed as we all deepen our fraction knowledge. "

Karen L. Thompson, fifth-grade teacher
Farrand Elementary, Plymouth, Michigan

Read a review of Beyond Pizzas and Pies from the June 2011 issue of the CMC Communicator

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