Children's Mathematics, Second Edition by Thomas P Carpenter, Elizabeth Fennema, Megan Loef Franke, Linda Levi, Susan B. Empson. Cognitively Guided Instruction
Children's Mathematics, Second Edition
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Children's Mathematics, Second Edition

Cognitively Guided Instruction

By Thomas P Carpenter, Elizabeth Fennema, Megan Loef Franke, Linda Levi, Susan B. Empson
Foreword by Mary M. Lindquist

This must-have resource for shifting your instruction to emphasize mathematical thinking and sense making explores an instructional approach unparalleled in its ability to help teachers develop children’s mathematical understanding. Online videos complement this examination of the development of mathematical thinking and problem solving.

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Full Description

Explore the new edition—including video clips, a sample chapter, and related blogs—at

The bestselling first edition of Children’s Mathematics  helped hundreds of thousands of teachers understand children’s intuitive mathematical thinking and use that knowledge to help children learn mathematics with understanding.  The highly anticipated Second Edition provides new insights about Cognitively Guided Instruction based on the authors’ research and experience in CGI classrooms over the last 15 years. Highlights include:

  • how children solve problems using their intuitive understanding of addition, subtraction, multiplication, and division
  • the development of children’s mathematical thinking throughout the primary grades
  • instructional practices that promote children’s active engagement in mathematics
  • connections between children’s strategies and powerful mathematical concepts

A new expanded collection of over 90 online video episodes illustrating children’s mathematical thinking, interactions between students and teachers, and classroom instruction that builds on children’s mathematical thinking.

Together, the Second Edition and videos provide a detailed research-based account of the development of children’s mathematical thinking and problem solving, and how teachers can promote this development in ways that honor children’s thinking.

Save money when you buy all three books in the Cognitively Guided Instruction series together: Children's Mathematics, Extending Children's Mathematics, and Thinking Mathematically


Foreword by Mary M. Lindquist
Introduction to the Second Edition

Chapter 1 Children's Mathematical Thinking
Chapter 2 Addition and Subtraction: Problem Types
Chapter 3 Addition and Subtraction: Children's Solution Strategies
Chapter 4 Multiplication and Division: Problem Types and Children's Solution Strategies
Chapter 5 Beginning to Use Cognitively Guided Instruction
Chapter 6 Base-Ten Number Concepts
Chapter 7 Children's Strategies for Solving Multidigit Problems
Chapter 8 Problem Solving as Modeling
Chapter 9 Developing Classroom Practice: Posing Problems and Eliciting Thinking
Chapter 10 Developing Classroom Practice: Engaging Students with Each Other's Ideas
Chapter 11 Mathematical Principles Underlying Children's Mathematics
Chapter 12 The Conceptual Basis for Cognitively Guided Instruction
Chapter 13 Conclusion: Keep on Learning

Appendix A The Research Base of Cognitively Guided Instruction
Appendix B Answers to Selected Questions for Reflection

How to Access Online Video Clips



The continuing popularity of this book is testimony to the importance of its central message: Young children can think mathematically and instruction that builds on their thinking takes them further and deeper into the core concepts of arithmetic than teachers might imagine.
James Hiebert, Robert J. Barkley Professor of Education, University of Delaware

Cognitively Guided Instruction not only changed the way I teach and restored my passion for teaching; it has had a positive impact on thousands of children.
Debbie Gates, elementary mathematics specialist

I am far more engaged and excited about teaching math myself because Children's Mathematics helped me understand what questions to ask and what student comments to listen for that will lead to student discovery.
Lesley Wagner, elementary school teacher

Children’s Mathematics is a must-read for all elementary teachers, coaches, and professional educators. The CGI frameworks help us attend closely to what children do and understand and provide a vision for how children’s ideas advance through well planned learning experiences.
Elham Kazemi, Professor of Mathematics Education, University of Washington

Children’s Mathematics has shown us that understanding “how” children think mathematically is so much more valuable to their ongoing development than trying to “give” them the way to think.  Our students have flourished! 
Kathy Goecke, elementary principal

There are some excellent videos to watch and some great examples of problems that get kids thinking...the examples really got me thinking about how I can ask more base ten multiplication and measurement division problems of kids at all grade levels.
Tara Trudo, mathematics specialist,

Children’s Mathematics is by far the most powerful and practical book that I have used in my work with teachers. The potential it has to not only deepen students’ conceptual understanding of mathematics but their teachers as well is unparalleled.
Andrew Jenkins, principal

Children’s Mathematics gives readers a fascinating glimpse into children’s mathematical ideas and their foundational role in instruction…This exemplary resource is essential for teachers, professional developers, and researchers who are interested in understanding, supporting, and extending children’s ways of reasoning.
Vicki Jacobs, Yopp Distinguished Professor of Mathematics Education, UNC Greensboro

This new edition continues to cognitively guide teachers and researchers to look at children as earnest thinkers and doers of mathematics.
James Brickwedde, Assistant Professor, Hamline University, St. Paul, MN

As anyone involved in leading change understands, the most difficult work is that of modifying deep-seated beliefs. CGI in mathematics, especially as so thoughtfully detailed in this most recent edition of Children's Mathematics, has the power to do just that. Children's Mathematics should be at the top of the resource list of any teacher/leader who believes in the power of children's mathematical capability. The authors thoughtfully and thoroughly allow us to peek inside children's minds as they construct flexible mathematical understanding. Everyone wins- the reader, their colleagues, and the children in their classrooms.
Turtle Toms, mathematics specialist

CGI gives us a concrete example of what it means to develop math instruction based on the ideas students already have. This principle should inform all mathematics instruction. Let's teach algebra based on the ideas students have about variables, geometry based on their ideas about shapes, and so on. Children's Mathematics shows us by example what this looks like in the classroom.
Christopher Danielson, Mathematics Instructor at Normandale Community College who blogs at Talking Math with Your Kids

This book is about helping students make sense of math.  But don’t be surprised if you start to make more sense of math yourself, which in turn will help you in teaching.
Susan Gehn, mathematics specialist

Through the second edition of Children’s Mathematics, the authors empower teachers to effectively transform their math instruction.
Tanya Vik Blais, elementary mathematics PD consultant

Children’s Mathematics guides teachers so we know how to create favorable conditions for students to make sense, what to look and listen for, and how to recognize important mathematics in our students’ thinking. Rather than prescribe how we should teach, the authors describe how students learn, and equip teachers to make effective instructional decisions grounded in research.
Tracy Johnston Zager, author of Becoming the Math Teacher You Wish You'd Had