Mine the Gap for Mathematical Understanding, Grades K-2

Common Holes and Misconceptions and What To Do About Them
John J. SanGiovanni - Howard Public School System
Corwin Mathematics Series
Mine the Gap for Mathematical Understanding, Grades K-2
October 2016 | 384 pages | Corwin
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eBook
ISBN: 9781506382296
Available from January 0001
Paperback
ISBN: 9781506337685
Available from January 0001

Description

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. 

Contents

Big Ideas & Tasks at a Glance

Big Ideas & Tasks at a Glance

Acknowledgments

Acknowledgments

Publisher’s Acknowledgments

Publisher’s Acknowledgments

About the Author

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

References and Additional Resources

Resources

Description

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. 

Contents

Big Ideas & Tasks at a Glance

Big Ideas & Tasks at a Glance

Acknowledgments

Acknowledgments

Publisher’s Acknowledgments

Publisher’s Acknowledgments

About the Author

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

References and Additional Resources

Resources

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Mine the Gap for Mathematical Understanding, Grades K-2

Common Holes and Misconceptions and What To Do About Them

John J. SanGiovanni
Cancel Print

October 2016 | 384 pages | Corwin

Format Published Date ISBN Price
Paperback 01/02/2026 9781506337685 $40.95
Lifetime 01/02/2026 9781506382296 $37.00

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. 


Table Of Contents:

  • 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

Recent Product Reviews:

“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

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