Visible Learning for Mathematics, Grades K-12: What Works Best To Optimize Student Learning

Rich tasks, collaborative work, number talks, problem-based learning, direct instruction…with so many possible approaches, how do we know which ones work the best? In Visible Learning for Math, six acclaimed educators assert it’s not about which one—it’s about when—and show you how to design high-impact instruction so all students demonstrate more than a year’s worth of mathematics learning for a year spent in school.

That’s a high bar, but with the amazing K-12 framework here, you choose the right approach at the right time, depending upon where learners are within three phases of learning: surface, deep, and transfer. This results in“visible” learning because the effect is tangible. The framework is forged out of current research in mathematics combined with John Hattie's synthesis of more than 15 years of education research involving 300 million students.

Chapter by chapter, and equipped with video clips, planning tools, rubrics, and templates, you get the inside track on which instructional strategies to use at each phase of the learning cycle:

• Surface learning phase: When—through carefully constructed experiences—students explore new concepts and make connections to procedural skills and vocabulary that give shape to developing conceptual understandings.
• Deep learning phase: When—through the solving of rich high-cognitive tasks and rigourous discussion—students make connections among conceptual ideas, form mathematical generalisations, and apply and practice procedural skills with fluency.
• Transfer phase: When students can independently think through more complex mathematics, and can plan, investigate, and elaborate as they apply what they know to new mathematical situations.

To equip students for higher-level mathematics learning, we have to be clear about where students are, where they need to go, and what it looks like when they get there. Visible Learning for Math brings about powerful, precision teaching for K-12 through intentionally designed guided, collaborative, and independent learning.

Table of Contents

List of Figures

List of Videos

About the Teachers Featured in the Videos

Foreword

About the Authors

Acknowledgments

Preface

Chapter 1. Make Learning Visible in Mathematics

• Forgetting the Past
• What Makes for Good Instruction?
• The Evidence Base
• Meta-Analyses
• Effect Sizes
• Noticing What Does and Does Not Work
• Direct and Dialogic Approaches to Teaching and Learning
• The Balance of Surface, Deep, and Transfer Learning
• Surface Learning
• Deep Learning
• Transfer Learning
• Surface, Deep, and Transfer Learning Working in Concert
• Conclusion
• Reflection and Discussion Questions

Chapter 2. Making Learning Visible Starts With Teacher Clarity

• Learning Intentions for Mathematics
• Student Ownership of Learning Intentions
• Connect Learning Intentions to Prior Knowledge
• Make Learning Intentions Inviting and Engaging
• Language Learning Intentions and Mathematical Practices
• Social Learning Intentions and Mathematical Practices
• Reference the Learning Intentions Throughout a Lesson
• Success Criteria for Mathematics
• Success Criteria Are Crucial for Motivation
• Getting Buy-In for Success Criteria
• Preassessments
• Conclusion
• Reflection and Discussion Questions

Chapter 3. Mathematical Tasks and Talk That Guide Learning

• Making Learning Visible Through Appropriate Mathematical Tasks
• Exercises Versus Problems
• Difficulty Versus Complexity
• A Taxonomy of Tasks Based on Cognitive Demand
• Making Learning Visible Through Mathematical Talk
• Characteristics of Rich Classroom Discourse
• Conclusion
• Reflection and Discussion Questions

Chapter 4. Surface Mathematics Learning Made Visible

• The Nature of Surface Learning
• Selecting Mathematical Tasks That Promote Surface Learning
• Mathematical Talk That Guides Surface Learning
• What Are Number Talks, and When Are They Appropriate?
• What Is Guided Questioning, and When Is It Appropriate?
• What Are Worked Examples, and When Are They Appropriate?
• What Is Direct Instruction, and When Is It Appropriate?
• Mathematical Talk and Metacognition
• Strategic Use of Vocabulary Instruction
• Word Walls
• Graphic Organizers
• Strategic Use of Manipulatives for Surface Learning
• Strategic Use of Spaced Practice With Feedback
• Strategic Use of Mnemonics
• Conclusion
• Reflection and Discussion Questions

Chapter 5. Deep Mathematics Learning Made Visible

• The Nature of Deep Learning
• Selecting Mathematical Tasks That Promote Deep Learning
• Mathematical Talk That Guides Deep Learning
• Accountable Talk
• Supports for Accountable Talk
• Teach Your Students the Norms of Class Discussion
• Mathematical Thinking in Whole Class and Small Group Discourse
• Small Group Collaboration and Discussion Strategies
• When Is Collaboration Appropriate?
• Grouping Students Strategically
• What Does Accountable Talk Look and Sound Like in Small Groups?
• Supports for Collaborative Learning
• Supports for Individual Accountability
• Whole Class Collaboration and Discourse Strategies
• When Is Whole Class Discourse Appropriate?
• What Does Accountable Talk Look and Sound Like in Whole Class Discourse?
• Supports for Whole Class Discourse
• Using Multiple Representations to Promote Deep Learning
• Strategic Use of Manipulatives for Deep Learning
• Conclusion
• Reflection and Discussion Questions

Chapter 6. Making Mathematics Learning Visible Through Transfer Learning

• The Nature of Transfer Learning
• Types of Transfer: Near and Far
• The Paths for Transfer: Low-Road Hugging and High-Road Bridging
• Selecting Mathematical Tasks That Promote Transfer Learning
• Conditions Necessary for Transfer Learning
• Metacognition Promotes Transfer Learning
• Self-Questioning
• Self-Reflection
• Mathematical Talk That Promotes Transfer Learning
• Helping Students Connect Mathematical Understandings
• Peer Tutoring in Mathematics
• Connected Learning
• Helping Students Transform Mathematical Understandings
• Problem-Solving Teaching
• Reciprocal Teaching
• Conclusion
• Reflection and Discussion Questions

Chapter 7. Assessment, Feedback, and Meeting the Needs of All Learners

• Assessing Learning and Providing Feedback
• Formative Evaluation Embedded in Instruction
• Summative Evaluation
• Meeting Individual Needs Through Differentiation
• Classroom Structures for Differentiation
• Adjusting Instruction to Differentiate
• Intervention
• Learning From What Doesn’t Work
• Grade-Level Retention
• Ability Grouping
• Matching Learning Styles With Instruction
• Test Prep
• Homework
• Visible Mathematics Teaching and Visible Mathematics Learning
• Conclusion
• Reflection and Discussion Questions

Appendices

• A. Effect Sizes
• B. Standards for Mathematical Practice
• C. A Selection of International Mathematical Practice or Process Standards
• D. Eight Effective Mathematics Teaching Practices
• E. Websites to Help Make Mathematics Learning Visible

References

Index