Mine the Gap for Mathematical Understanding, Grades 3-5: Common Holes and Misconceptions and What to Do about Them / Edition 1 available in Paperback
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 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.
San Giovanni lays out 180 high-quality tasks aligned to the standards and big ideas of Grades 3-5 mathematics, including addition and subtraction of multi-digit whole numbers, multiplication and division of single and multi-digit whole numbers, foundational fraction concepts, foundational decimal concepts, and operations with fractions and decimals. 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
- 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: ADDITION AND SUBTRACTION WITHIN 1,000 BIG IDEAS 1: Adding Within 1,000 2: Reasoning About Addition Within 1,000 3: Subtraction Within 1,000 4: Reasoning About Subtraction Within 1,000 5: Problem Solving With Addition and SubtractionCHAPTER 3: MULTIPLICATION AND DIVISION BIG IDEAS 6: Representing Multiplication 7: Reasoning About Multiplication 8: Properties of Multiplication 9: Representing Division 10: Reasoning About Division 11: Problem Solving With Multiplication and Division 12: Connecting Multiplication and Division 13: Representing Multi-Digit Multiplication 14: Reasoning About Multi-Digit Multiplication 15: Representing Multi-Digit Division 16: Reasoning About Multi-Digit DivisionCHAPTER 4: FOUNDATIONAL FRACTION CONCEPTS BIG IDEAS 17: Representing Fractions 18: Connecting Representations of Fractions 19: Fractions on a Number Line 20: Fractions Greater Than 1 on a Number Line 21: Decomposing Fractions 22: Equivalent Fractions on a Number Line 23: Comparing Fractions 24: Reasoning About Fractions 25: More Reasoning About FractionsCHAPTER 5: DECIMAL CONCEPTS BIG IDEAS 26: Representing Decimals 27: Representing Decimals as Numbers 28: Estimating and Rounding Decimals 29: Decomposing Decimals 30: Comparing Decimals 31: Addition With Decimals 32: Subtracting With Decimals 33: Problem Solving With DecimalsCHAPTER 6: ADDITION AND SUBTRACTION WITH FRACTIONS BIG IDEAS 34: Addition With Fractions on Number Lines 35: Reasoning About Addition With Fractions 36: Subtraction With Fractions 37: Reasoning About Subtraction With Fractions 38: Problem Solving With Addition and Subtraction of Fractions 39: Addition and Subtraction With Mixed NumbersCHAPTER 7: MULTIPLICATION AND DIVISION WITH FRACTIONS AND DECIMALS BIG IDEAS 40: Multiplication of Fractions 41: More With Multiplication of Fractions 42: Division With Fractions 43: Problem Solving With Multiplication and Division of Fractions 44: Multiplication With Decimals 45: Division With DecimalsCHAPTER 8: WHAT DO WE DO NEXT?References and Additional Resources