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Mine the Gap for Mathematical Understanding, Grades K-2
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Mine the Gap for Mathematical Understanding, Grades K-2
Common Holes and Misconceptions and What To Do About Them



November 2016 | 384 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 helps us value our young learners’ 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.

SanGiovanni lays out 160 high-quality tasks aligned to the standards and big ideas of grades K-2 mathematics, including  counting and representing numbers, number relationships and comparison, addition and subtraction within 100 and 1000, money and time,  and multiplication and division. 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 taskwhat you might anticipate before students work with the task
  • 4 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 & Tasks at a Glance
 
Acknowledgments
 
Publisher’s Acknowledgments
 
About the Author
 
Introduction
 
CHAPTER 1. RICH MATHEMATICS TASKS, STUDENT MISCONCEPTIONS, USING TASKS
 
CHAPTER 2. COUNTING AND REPRESENTING NUMBERS
BIG IDEAS

 
1. Counting on From Any Number (0–120)

 
2. Counting on a Number Line

 
3. Skip-Counting

 
4. Representing Numbers

 
5. Representing Numbers to 1,000 With Models

 
6. Understanding Relationships Between Numbers

 
 
CHAPTER 3. NUMBER RELATIONSHIPS AND COMPARISON
BIG IDEAS

 
7. Comparing Numbers Within 100

 
8. Comparing Numbers Within 1,000

 
9. Comparing Numbers Within 1,000

 
10. Number Relationships Through Number Lines

 
11. Decomposing Numbers Less Than 100

 
12. More Decomposing Numbers

 
 
CHAPTER 4. ADDITION AND SUBTRACTION
BIG IDEAS

 
13. Representing Addition

 
14. Properties of Addition

 
15. Reasoning About Addition

 
16. Representing Subtraction

 
17. Connecting Addition and Subtraction

 
18. Reasoning About Subtraction

 
19. The Meaning of the Equal Sign

 
20. Problem Solving With Addition and Subtraction

 
 
CHAPTER 5. ADDITION AND SUBTRACTION WITHIN 100
BIG IDEAS

 
21. Addition Within 100

 
22. More With Addition Within 100

 
23. Reasoning About Addition Within 100

 
24. Subtraction Within 100

 
25. Reasoning About Subtraction Within 100

 
26. Problem Solving Using Addition and Subtraction Within 100

 
 
CHAPTER 6. ADDITION AND SUBTRACTION WITHIN 1,000
BIG IDEAS

 
27. Adding Within 1,000

 
28. Reasoning About Addition Within 1,000

 
29. Subtraction Within 1,000

 
30. Reasoning About Subtraction Within 1,000

 
31. Problem Solving Within 1,000

 
 
CHAPTER 7. MEASUREMENT WITH MONEY AND TIME
BIG IDEAS

 
32. Working and Problem Solving With Money

 
33. Working and Problem Solving With Money

 
34. Telling Time

 
35. Elapsed Time

 
 
CHAPTER 8. MULTIPLICATION AND DIVISION
BIG IDEAS

 
36. Representing Multiplication and Division

 
37. Reasoning About Multiplication

 
38. Properties of Multiplication

 
39. Representing Division

 
40. Problem Solving With Multiplication and Division

 
 
CHAPTER 9. WHAT DO WE DO NEXT?
 
References and Additional Resources

“Wow! Mine the Gap for grades K–2 teachers really unpacks! Not only does the book provide
dozens of great mathematical tasks, but—perhaps more important—it gives teachers the chance
to consider each task; anticipate student responses—a critical first step in formative assessment;
analyze actual student responses; and then consider what they would do in the classroom.
Thoughts about modifying each task provide another option for each task. At a time when teachers
are encouraged to regularly provide students with tasks that promote reasoning and problem
solving, this book will help teachers dig deeper as they mine for understandings.”

Francis (Skip) Fennell
National Council of Teachers of Mathematics

“Too often students (and parents and teachers) have the notion that the goal in math should
be to get the right answer. Yet insight into misconceptions and students’ thinking can tell us
much more about what students know (and don’t know) beyond simply a correct or incorrect
answer. SanGiovanni offers teachers a treasure trove of rich tasks and student work on those
tasks. Examples of how to analyze student thinking and next instructional steps make this a volume
that should be on every K–2 math teacher’s desk!”

Linda Gojak
National Council of Teachers of Mathematics

“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, Mathematics at Florida State University

“Mine the Gap for Mathematical Understanding is a much-needed and anticipated resource for
teachers, mathematics coaches, mathematics specialists, administrators, and other stakeholders.
The easy to follow, teacher-friendly format, the accompanying commentary for each student work
sample, along with the thoughtful reflection questions will quickly make this resource a ‘go-to’
professional development tool.”

Latrenda Knighten
East Baton Rouge Parish School System, Baton Rouge, LA

“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

“John SanGiovanni continues to provide teacher-friendly, must-have books. They empower teachers
by deepening their understanding of content and teaching.”

Megan Dooley
Indianhead Elementary, Charles County, Maryland

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