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Mine the Gap for Mathematical Understanding : Common Holes and Misconceptions and What to Do about Them (3-5) - John J. SanGiovanni

Mine the Gap for Mathematical Understanding

Common Holes and Misconceptions and What to Do about Them (3-5)

Paperback Published: 15th September 2016
ISBN: 9781506337678
Number Of Pages: 384

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Teaching elementary mathematics is complicated: teachers must not only understand the concepts, procedures, and application of seemingly "basic" mathematics, they must also understand mathematical misconceptions and apparently correct answers as well as the "logic" behind these answers.

Unfortunately, teachers are often trained to think about mathematics as basic procedures with correct or incorrect answers. Misconceptions and incomplete mathematical thinking often go undetected, as teachers are rarely trained to consider why or how an answer was generated. They often respond to incorrect answers by re-teaching the correct processes, and the underlying problem persists. Sometimes even students' correct answers do not indicate accurate reasoning. Unrecognized misconceptions then become permanent ways of thinking; each new layer of mathematics knowledge builds upon on flawed foundations.

This book addresses that problem head on. By covering the essential content or "big ideas" of 3-5 mathematics, it turns elementary teachers into experts in diagnosis, intervention, and evaluation (Hattie 2010), and delivers the tasks and tools to use in the classroom day-to-day. As a result of using these books, teachers will be able to: Select and employ up to 180 rich tasks for instruction or assessment; Consider what counts as mathematical understanding; Anticipate and plan for student misconceptions; Make instructional decisions based on specific misconceptions or incomplete understandings; Learn how to adapt tasks for differentiation, different grade levels; Avoid certain hazards such as teacher misconceptions when giving correct answers, or misinterpreting student drawings, writing, or equations; Use the book in a professional learning community, communicate with parents, and support parent conferences.

Tackling math's "big ideas" for grades 3-5 such as addition and subtraction, multiplication and division, fractions, and decimals, the book is organized into 45 Big Ideas by mathematical domain and grad band, each offering a rich task that walks teachers through the analysis of several student-work examples to identify: What is happening with this student's thinking? What do we want to ask this student? What doesn't this student understand about the mathematics? What might we do next for this student? Each section then offers three additional tasks and guiding information to use in the classroom for a total of 180 rich tasks teachers can implement immediately. Tasks are aligned to Common Core and other state and international standards.

About the Author

John SanGiovanni is a recognized leader in mathematics education and the current Mathematics Coordinator for Howard County Public School System. He is the co-author of three top-selling Heinemann books with Sue O'Connell, including Mastering Basic Facts: Addition and Subtraction (2011, 22,000 units sold), Mastering Basic Facts: Multiplication and Division (2011, 18,000 units sold), and Putting the Practices Into Action (2013, 30,000 units sold). John is currently serving on the Board of Directors for the National Council of Teachers of Mathematics (2015-2018). John is frequently a featured speaker at regional and national mathematics conferences and is also a national consultant providing professional development for elementary and middle school mathematics.

Industry Reviews

"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
"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
"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
"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
"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
"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
"John SanGiovanni continues to provide teacher-friendly, must-have books. They empower teachers
by deepening their understanding of content and teaching." -- Megan Dooley
"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

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 one 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 and fractions on number lines
35. Reasoning about addition with fractions
36. Subtraction with fractions
37. Reasoning about subtraction with fractions
38. Problem solving with additon and subtraction of fractions
39. Addition and subtraction of mixed numbers
Chapter 7 Multiplication and division with fractions and decimals
Big Ideas
40. Multiplication with fractions
41. More with multiplication of fractions
42. Problem solving with multiplication and division of fractions
43. Division with decimals
44. Multiplication with decimals
45. Division with decimals
What do we do next?

ISBN: 9781506337678
ISBN-10: 1506337678
Series: Corwin Mathematics Series
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 384
Published: 15th September 2016
Publisher: SAGE Publications Inc
Country of Publication: US
Dimensions (cm): 27.9 x 21.5  x 1.91
Weight (kg): 0.79
Edition Number: 1

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John J. SanGiovanni

About the Author


John SanGiovanni 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 McDaniel 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

Visit John J. SanGiovanni's Booktopia Author Page