Preface
Acknowledgments
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
Summary
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
Summary
3. What Is Happening to Thinking in Mathematics Classrooms?
Improvement Initiatives and Visible Thinking
Visible Thinking Scenario 3: Subtraction With Regrouping
Summary
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
Summary
5. How Are Long-Term Changes Made?
Enhancing Student Learning
Teaching Approaches
Visible Thinking Scenario 5: Mixed Numerals
Visible Thinking Scenario 6: Place Value
Summary
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
Summary
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
Summary
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
Summary
9. How Is Thinking Made Visible in Grades 3–5 Mathematics?
Brainteaser Problem Example
Group-Worthy Problem Example
Transforming Problem Example
Summary
10. How Is Thinking Made Visible in Grades 6–8 Mathematics?
Brainteaser Problem Example
Group-Worthy Problem Example
Transforming Problem Example
Summary
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
Summary
Appendix A: Research Support for Visible Thinking Strategies,
Conditions, and Actions
Appendix B: Lessons Using Technology: Additional Materials
References
Index
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.
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 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.
"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*
"This book will help you, your students and your school. The
author merges what we know works in mathematical problem
solving, metacognition, social learning theory, and formative
assessment. The examples display grade-specific ways to help
individual students tackle brainteasers, whole-class
concepts, and adaptations of traditional textbook
exercises."
*Alan Zollman, President of School Science and Mathematics
Association*
"The author gives an excellent overview of what visual
thinking is, why it is important, and how to implement it in the
classroom. The text offers great advice for addressing many of
the Common Core State Standards for Mathematics Habits of Mind,
including making sense of problems and communicating mathematical
reasoning."
*Frederick L. Dillon, Mathematics Teacher*
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