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



© 2017 | 384 pages | Corwin

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 & Tasks at a Glance
 
Acknowledgments
 
Publisher’s Acknowledgments
 
About the Author
 
Introduction
 
CHAPTER 1: RICH MATHEMATICS TASKS, STUDENT MISCONCEPTIONS, USING TASKS
 
CHAPTER 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 Subtraction  
 
CHAPTER 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 Division  
 
CHAPTER 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 Fractions  
 
CHAPTER 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 Decimals  
 
CHAPTER 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 Numbers  
 
CHAPTER 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 Decimals  
 
CHAPTER 8: WHAT DO WE DO NEXT?
 
References and Additional Resources

Free resources

John SanGiovanni's The Opportunity of a Wrong Answer in K-8 Mathematics Webinar

John SanGiovanni's The Opportunity of a Wrong Answer in K-8 Mathematics Webinar

Learn from John SanGiovanni specific actions that teachers can take to close the gaps in student understanding of mathematics.

Watch now

“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.”

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 3–5 math teacher’s desk!”

Linda Gojak
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.”

Zachary Champagne
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.”

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

“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.”

Delise Andrews
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.”

Megan Dooley
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.”

Cynthia Baumann
Omaha Public Schools

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ISBN: 9781506337678
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