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Visible Thinking in the K–8 Mathematics Classroom

The authors provide instructional strategies for maximizing students’ mathematics comprehension through interactive visual thinking. Included are grade-specific sample problems and concrete examples of each strategy.

Full description

Product Details
  • Grade Level: PreK-12, Elementary, Secondary
  • ISBN: 9781412992053
  • Published By: Corwin
  • Year: 2011
  • Page Count: 184
  • Publication date: January 21, 2011

Price: $39.95



"This book is a crucial tool for meeting NCTM mathematical content and process standards. Through the useful problems and strategies presented within, teachers will definitely know how well their students will comprehend. If comprehension is an issue in your class, this book is a must have!"
—Therese Gessler Rodammer, Math Coach
Thomas W. Dixon Elementary School, Staunton, VA

Seeing is believing with this interactive approach to math instruction

Do you ever wish your students could read each other's thoughts? Now they can—and so can you! Veteran mathematics educators Ted Hull, Don Balka, and Ruth Harbin Miles explain why making students' thought processes visible is the key to effective mathematics instruction. Their newest book contains numerous grade-specific sample problems and instructional strategies for teaching essential concepts such as number sense, fractions, and estimation. Among the many benefits of visible thinking are:

  • Interactive student-to-student learning
  • Increased class participation
  • Development of metacognitive thinking and problem-solving skills

Helpful features include vignettes, relevant word problems, classroom scenarios, sample problems, lesson adaptations, and easy-to-follow examples of each strategy in action. The authors also explain how students can demonstrate their thinking using calculators and online tools. The final chapter outlines steps math leaders can take to implement visible thinking and maximize mathematics comprehension for all students.


Key features

  • Clear explanation and examples of the instructional strategies, actions, and conditions necessary to begin the process of instituting visible thinking in the mathematics classrooms.
  • Concrete examples of each strategy in action
  • Specific discussion of visual thinking in elementary and middle school classrooms, including scenarios and strategies for working with students in these different groups either individually or in groups


Ted H. Hull photo

Ted H. Hull

Consulting Description

Ted H. Hull completed 32 years of service in public education before retiring and opening Hull Educational Consulting. He served as a mathematics teacher, K-12 mathematics coordinator, middle school principal, director of curriculum and instruction, and a project director for the Charles A. Dana Center at the University of Texas in Austin. While at the University of Texas, 2001 to 2005, he directed the research project “Transforming Schools: Moving from Low-Achieving to High Performing Learning Communities.” As part of the project, Hull worked directly with district leaders, school administrators, and teachers in Arkansas, Oklahoma, Louisiana, and Texas to develop instructional leadership skills and implement effective mathematics instruction. Hull is a regular presenter at local, state, and national meetings. He has written numerous articles for the NCSM Newsletter, including "Understanding the Six Steps of Implementation: Engagement by an Internal or External Facilitator" (2005) and "Leadership Equity: Moving Professional Development into the Classroom" (2005), as well as "Manager to Instructional Leader" (2007) for the NCSM Journal of Mathematics Education Leadership. He has been published in the Texas Mathematics Teacher (2006), Teacher Input Into Classroom Visits: Customized Classroom Visit Form. Hull was also a contributing author for publications from the Charles A. Dana Center: Mathematics Standards in the Classroom: Resources for Grades 6–8 (2002) and Middle School Mathematics Assessments: Proportional Reasoning (2004). He is an active member of Texas Association of Supervisors of Mathematics (TASM) and served on the NCSM Board of Directors as regional director for Southern 2.

Don S. Balka photo

Don S. Balka

Consulting Description

Don S. Balka, Ph.D., is a noted mathematics educator who has presented more than 2,000 workshops on the use of math manipulatives with PK-12 students at national and regional conferences of the National Council of Teachers of Mathematics and at in-service trainings in school districts throughout the United States and the world.

He is Professor Emeritus in the Mathematics Department at Saint Mary’s College, Notre Dame, Indiana. He is the author or co-author of numerous books for K-12 teachers, including Developing Algebraic Thinking with Number Tiles, Hands-On Math and Literature with Math Start, Exploring Geometry with Geofix, Working with Algebra Tiles, and Mathematics with Unifix Cubes. Balka is also a co-author on the Macmillan K-5 series, Math Connects and co-author with Ted Hull and Ruth Harbin Miles on four books published by Corwin Press.

He has served as a director of the National Council of Teachers of Mathematics and the National Council of Supervisors of Mathematics. In addition, he is president of TODOS: Mathematics for All and president of the School Science and Mathematics Association.

Ruth Harbin Miles photo

Ruth Harbin Miles

Ruth Harbin Miles coaches rural, suburban, and inner-city school mathematics teachers. Her professional experiences include coordinating the K-12 Mathematics Teaching and Learning Program for the Olathe, Kansas, Public Schools for more than 25 years; teaching mathematics methods courses at Virginia’s Mary Baldwin College; and serving on the Board of Directors for the National Council of Teachers of Mathematics, the National Council of Supervisors of Mathematic, and both the Virginia Council of Teachers of Mathematics and the Kansas Association of Teachers of Mathematics. Ruth is a co-author of five Corwin books including A Guide to Mathematics Coaching, A Guide to Mathematics Leadership, Visible Thinking in the K-8 Mathematics Classroom, The Common Core Mathematics Standards, and Realizing Rigor in the Mathematics Classroom. As co-owner of Happy Mountain Learning, Ruth specializes in developing teachers’ content knowledge and strategies for engaging students to achieve high standards in mathematics.

Table of Contents

Table of Contents



About the Authors

Part I. Preparing the Foundation

1. What Is Visible Thinking?

Understanding Mathematical Concepts

Thinking as a Mathematical Premise

Visible Thinking in Classrooms

Visible Thinking Scenario 1: Area and Perimeter


2. How Do Students Learn Mathematics?

What Is Thinking?

What Does Brain Research Indicate About Thinking and Learning?

What Is Mathematical Learning?

What Are Thinking and Learning Themes From Research?

Example Problems Revisited

Visible Thinking Scenario 2: Addition of Fractions


3. What Is Happening to Thinking in Mathematics Classrooms?

Improvement Initiatives and Visible Thinking

Visible Thinking Scenario 3: Subtraction With Regrouping


Part II. Promoting Visible Thinking With an Alternative Instructional Model

4. How Do Effective Classrooms Depend on Visible Thinking?

What Are Strategies, Conditions, and Actions?

Practice Into Action

Technology as Visible Thinking

Visible Thinking Scenario 4: Division


5. How Are Long-Term Changes Made?

Enhancing Student Learning

Teaching Approaches

Visible Thinking Scenario 5: Mixed Numerals

Visible Thinking Scenario 6: Place Value


6. How Are Short-Term Changes Made?

Pitfalls and Traps

Strategy Sequence

The Relationships Among the Strategy Sequence, Conditions, and Goals

Visible Thinking Scenario 7: Basic Addition and Subtraction Facts

Visible Thinking Scenario 8: Exponents


7. How Are Lessons Designed to Achieve Short-Term and Long-Term Changes?

The Current Approach to Teaching Mathematics

Elements of an Alternative Instructional Model

Types of Problems


Part III. Implementing the Alternative Model at Different Grade Levels

8. How Is Thinking Made Visible in Grades K–2 Mathematics?

Brainteaser Problem Example

Group-Worthy Problem Example

Transforming Problem Example


9. How Is Thinking Made Visible in Grades 3–5 Mathematics?

Brainteaser Problem Example

Group-Worthy Problem Example

Transforming Problem Example


10. How Is Thinking Made Visible in Grades 6–8 Mathematics?

Brainteaser Problem Example

Group-Worthy Problem Example

Transforming Problem Example


Part IV. Continuing the Work

11. How Do Teachers, Leaders, and Administrators Coordinate Their Efforts to Improve Mathematics Teaching and Learning?

Working With Administrators

Embedding Lessons Into the Curriculum

Providing Professional Development

Co-planning and Co-teaching


Appendix A: Research Support for Visible Thinking Strategies, Conditions, and Actions

Appendix B: Lessons Using Technology: Additional Materials





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