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Mine the Gap for Mathematical Understanding, Grades 6-8
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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
 
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
Key features

·         180 Rich Tasks teachers can choose from and implement immediately;

·         Organized around math’s “big ideas” for grades 6-8 the number systems, integers, ratio and proportion, equations and expressions, geometry, and statistics and probability;

·         Written by a teacher’s teacher in a friendly, accessible style;

·         180 examples of student work used to walk teachers through analysis of misconceptions and incomplete understandings;

·         Highly templated and designed for easy use;

·         Additional rich tasks provided as blackline masters through a companion website;

·         Guidance around adapting tasks for differentiation and grade levels and avoiding common teacher misunderstandings and misinterpretations,


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ISBN: 9781506379821
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