#
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 |

“Wow! Mine the Gap for grades 3–5 teachers really unpacks! Not only does the book provide dozens

of great mathematical tasks, but—perhaps more importantly—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.”

**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 3–5 math teacher’s desk!”

**National Council of Supervisors 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.”

**Florida Center for Research in Science, Technology and 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.”

**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.”

**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.”

**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.”

**Indianhead Elementary, Charles County, Maryland**

“This book helps navigate how to use student work to drive instruction with rich engaging tasks,

which will help all students become better mathematicians. SanGiovanni has done an excellent

job of helping teachers to carefully look at student work to identify how students solved math problems,

using this evidence to identify those students who understand the targeted skill, along with

the misconceptions or misunderstandings of other students, with suggestions of how to move all

students forward in their thinking.”

**Omaha Public Schools**