Description
Book SynopsisBeing an effective math educator is one part based on the quality of the tasks we give, one part how we diagnose what we see, and one part what we do with what we find. Yet with so many students and big concepts to cover, it can be hard to slow down enough to look for those moments when students’ responses tell us what we need to know about next best steps. In this remarkable book, John SanGiovanni and Jennifer Rose Novak help us value our students’ misconceptions and incomplete understandings as much as their correct ones—because it’s the gap in their understanding today that holds the secrets to planning tomorrow’s best teaching.
The authors lay out 180 high-quality tasks aligned to the standards and big ideas of Grades 6–8 mathematics, including number systems, integers, ratio and proportion, equations and expressions, geometry, and statistics and probability. The tasks are all downloadable so you can use or modify them for instruction and assessment. Each big idea offers a starting task followed by:
- what makes it a high-quality task
- what you might anticipate before students work with the task
- four student examples of the completed task showcasing a distinct "gap"
- commentary on what precisely counts for mathematical understanding and the next instructional steps
- commentary on the misconception or incomplete understanding so you learn why the student veered off course
- three additional tasks aligned to the mathematics topic and ideas about what students might do with these additional tasks
It’s time to break our habit of rushing into re-teaching for correctness and instead get curious about the space between right and wrong answers. Mine the Gap for Mathematical Understanding is a book you will return to again and again to get better at selecting tasks that will uncover students’ reasoning, better at discerning the quality and clarity of students’ understanding, and better at planning teaching based on the gaps you see.
Trade Review"Wow! This book is a treasure trove for middle school teachers and those who support middle level mathematics education. The tasks presented are top notch, the student work is illustrative of classrooms, and the suggestions for how to respond are incredible! Each of these three would make a great resource—to have them all blended together is outstanding." -- Kevin Dykema
"Mine the Gap for Mathematical Understanding provides the framework and guidance teachers need to move beyond finding correct answers to dig deeply into the mine of student thinking, to analyze misconceptions and gaps in understanding, and to develop and implement specific strategies to support every student in deep mathematics learning. Every teacher of mathematics needs this amazing resource to learn to mine the gaps for each of their students!" -- Becky M. Walker
"This work does what other books only attempt to do. It combines instruction, assessment, and practice with open-ended and rich tasks that allow for teachers to not only immediately implement the ideas but also understand the content and pedagogy behind them. The tasks, which are immediately implementable and customizable, engage each and every learner. They are based on cutting-edge and research-based instructional frameworks and provide countless learning opportunities for students." -- Zachary Champagne
"Mine the Gap is a great tool for teachers to use to grow their own understanding of student misconceptions and incomplete understandings and how to address them. This is an indispensable resource for all involved in supporting students’ growth in mathematics." -- Nathan Rosin
"More than just a nice collection of problems, this book shares a road map for teachers looking to enhance the quality of the math tasks they use with students. Teachers will appreciate the examples of actual student work paired with tips for analysis and instruction." -- Delise Andrews
"This book helps navigate how to use student work to drive instruction with rich engaging tasks, which will help all students become better mathematicians. The authors have done an excellent job of helping teachers to carefully look at student work to identify how students solved math problems, using this evidence to identify those students who understand the targeted skill, along with the misconceptions or misunderstandings of other students, with suggestions of how to move all students forward in their thinking." -- Cynthia Baumann
Table of ContentsBig Ideas and Tasks at a Glance Acknowledgments About the Authors Introduction Chapter 1. Rich Mathematics Tasks, Student Misconceptions, Using Tasks Prompts With Purpose: Using High-Quality Tasks Using Quality Tasks Reflecting on Chapter 1 Chapter 2. Number Systems Big Ideas 1. Addition and Subtraction of Fractions 2. Multiplication and Division of Fractions 3. Reasoning About Addition and Subtraction of Fractions 4. Reasoning About Multiplication and Division of Fractions 5. Problem Solving With Fractions 6. Decimals as Numbers 7. Addition and Subtraction With Decimals 8. Multiplication and Division With Decimals Chapter 3. Integers Big Ideas 9. Representing Integers 10. Representing Integers on Number Lines 11. More Representing Integers on Number Lines 12. Comparing Integers 13. Addition With Integers 14. Subtraction With Integers 15. Multiplication With Integers 16. Division With Integers Chapter 4. Ratio, Proportion, and Percent Big Ideas 17. Representing Ratios 18. Equivalent Ratios 19. Unit Rates 20. Using Ratios to Solve Problems 21. Reasoning With Percents 22. Unit Rate as Slope Chapter 5. Expressions, Equations, and Functions Big Ideas 23. Writing Expressions 24. Evaluating Expressions 25. Equivalent Expressions 26. Writing Equations 27. Solving Equations 28. Inequalities 29. Function Tables 30. Reasoning About Graphing 31. Comparing Functions 32. Systems of Equations Chapter 6. Geometry Big Ideas 33. Area of Composite Figures 34. Nets and Three-Dimensional Figures 35. Surface Area and Volume 36. Volume of Cylinders and Cones 37. Angle Relationships 38. Transformations, Similarity, and Congruence 39. Distance and Pythagorean Theorem Chapter 7. Statistics Big Ideas 40. Univariate Categorical Data 41. Univariate Quantitative Data 42. Displays of Univariate Quantitative Data 43. Deviations From the Mean 44. Bivariate Categorical Data 45. Bivariate Quantitative Data Chapter 8. What Do We Do Next? References and Additional Resources
