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Mine the Gap for Mathematical Understanding, Grades 6-8

Mine the Gap for Mathematical Understanding, Grades 6-8
Common Holes and Misconceptions and What To Do About Them

September 2017 | 408 pages | Corwin

Being 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. 

Big Ideas and Tasks at a Glance
About the Authors
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

“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
Mattawan Middle School, Mattawan, MI

“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
Cooperative Educational Services Agency #7, Green Bay, WI

“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
Florida Center for Research in Science, Technology, Engineering, and Mathematics at Florida State University

“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
Sun Prairie Area School District

“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
Lincoln (NE) Public Schools

“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
Omaha Public Schools

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ISBN: 9781506379821