#
Mine the Gap for Mathematical Understanding, Grades 3-5
Common Holes and Misconceptions and What To Do About Them

- John SanGiovanni - Howard Public School System

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 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.

SanGiovanni 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.

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 Subtraction |

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 Division |

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 Fractions |

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 Decimals |

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 Numbers |

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 Decimals |

HEOA Compliance

####
**Available formats**

ISBN: 9781506337678 | Paperback | Suggested Retail Price: $34.95 | Bookstore Price: $27.96 |

Should you need additional information or have questions regarding the HEOA information provided for this title, including what is new to this edition, please email sageheoa@sagepub.com. Please include your name, contact information, and the name of the title for which you would like more information. For information on the HEOA, please go to http://ed.gov/policy/highered/leg/hea08/index.html.