# Visible Thinking in the K–8 Mathematics Classroom

- Ted H. Hull - Educational Consultant, Texas
- Don S. Balka - Saint Mary's College, Notre Dame, IN
- Ruth Harbin Miles - Mary Baldwin College, VA

Published in Association with National Council of Teachers of 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

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.

Understanding Mathematical Concepts |

Thinking as a Mathematical Premise |

Visible Thinking in Classrooms |

Visible Thinking Scenario 1: Area and Perimeter |

Summary |

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 |

Improvement Initiatives and Visible Thinking |

Visible Thinking Scenario 3: Subtraction With Regrouping |

Summary |

Part II. Promoting Visible Thinking With an Alternative Instructional Model |

What Are Strategies, Conditions, and Actions? |

Practice Into Action |

Technology as Visible Thinking |

Visible Thinking Scenario 4: Division |

Summary |

Enhancing Student Learning |

Teaching Approaches |

Visible Thinking Scenario 5: Mixed Numerals |

Visible Thinking Scenario 6: Place Value |

Summary |

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 |

The Current Approach to Teaching Mathematics |

Elements of an Alternative Instructional Model |

Types of Problems |

Summary |

Brainteaser Problem Example |

Group-Worthy Problem Example |

Transforming Problem Example |

Summary |

Brainteaser Problem Example |

Group-Worthy Problem Example |

Transforming Problem Example |

Summary |

Brainteaser Problem Example |

Group-Worthy Problem Example |

Transforming Problem Example |

Summary |

Working With Administrators |

Embedding Lessons Into the Curriculum |

Providing Professional Development |

Co-planning and Co-teaching |

Summary |

*"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!"*

**Thomas W. Dixon Elementary School, Staunton, VA**

*"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."*

**Northern Illinois University**

*"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."*

**Strongsville City Schools, OH**