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 San Giovanni 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.
San Giovanni 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.
About the Author
John San Giovanni is a mathematics supervisor in Howard County, Maryland. There he leads mathematics curriculum development, digital learning, assessment, and professional development for 41 elementary schools and more than 1,500 teachers. John is an adjunct professor and coordinator of the Elementary Mathematics Instructional Leader graduate program at Mc Daniel College. He is an author and national mathematics curriculum and professional learning consultant. John is a frequent speaker at national conferences and institutes. He is active in state and national professional organizations and currently serves on the Board of Directors for the National Council of Teachers of Mathematics.
Table of Contents
Big Ideas & Tasks at a GlanceAcknowledgmentsPublisher’s AcknowledgmentsAbout the AuthorIntroductionCHAPTER 1. RICH MATHEMATICS TASKS, STUDENT MISCONCEPTIONS, USING TASKSCHAPTER 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 NumbersCHAPTER 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 NumbersCHAPTER 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 SubtractionCHAPTER 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 100CHAPTER 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 1,000CHAPTER 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 TimeCHAPTER 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 DivisionCHAPTER 9. WHAT DO WE DO NEXT?References and Additional Resources