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Visible Thinking in the K-8 Mathematics Classroom
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Table of Contents

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

About the Author

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.

Reviews

"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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