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Math Decoded
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Math Decoded
Explaining the Why Behind the Numbers and How They Operate


Other Titles in:
Mathematics

December 2025 | 312 pages | Corwin

Experience the joy of answering "why" in math class like never before

Mathematics often feels like a collection of disconnected rules, but what if you could uncover their underlying story? In Math Decoded: Explaining the Why Behind the Numbers and How They Operate, author James Tanton provides educators with a compelling and accessible guide to understanding the logic, purpose, and poetry behind numbers and arithmetic.

Move away from simply teaching formulas and procedures. Instead, discover the coherent and human story of mathematics that will enliven your classroom and deepen your own mathematical knowledge.

By addressing the "why" questions students frequently ask, this book offers clarity on tricky concepts such as dividing by zero, fractions, and negative numbers through these key features:

  • Engaging Structure: Each chapter begins with a thought-provoking question, explores real-world scenarios, and culminates in bold insights and classroom activities.
  • Grade-Banded Tasks: The book includes classroom activities for elementary to high school, ensuring relevance for multiple grade levels.
  • Comprehensive Resources: The text is accompanied by videos, supplemental essays, and practical tools to deepen the learning experience.
  • Accessible Style: The book is written in a friendly, relatable tone to explain mathematics’ most mystifying and rigorous topics.

With this book, educators will gain a newfound appreciation for the beauty, structure, and coherence of mathematics. Math Decoded will help you inspire your students to see math not just as a subject, but as an invitation to explore, discover, and think deeply about the joy of numbers.

 
On the Companion Website
 
List of Videos
 
Acknowledgments
 
About the Author
 
Introduction
The Why of This Book

 
Who This Book Is For

 
The Story Behind the Story

 
What’s Inside

 
What’s Outside(!)

 
An Invitation

 
 
Chapter 1: Counting
Something to Mull On

 
Counting in the Real World

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and Some Classroom Tasks

 
A Task for Elementary School Grades

 
A Task for Middle School Grades

 
A Task for High School Grades

 
 
Chapter 2: Addition
Something to Mull On

 
Addition in the Real World

 
The Number Zero

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and Some Classroom Tasks

 
The Solitaire Game

 
Some Sophisticated Jargon

 
A Task for Elementary School Grades

 
A Task for Middle School Grades

 
A Task for High School Grades

 
 
Chapter 3: Negative Numbers
Something to Mull On

 
Negative Numbers in the Real World

 
Solitaire Again

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and Some Classroom Tasks

 
A Task for Elementary School Grades

 
A Task for Middle School Grades

 
A Task for High School Grades

 
 
Chapter 4: Distributing the Negative Sign
Something to Mull On

 
Distributing the Negative Sign in the Real World

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and Some Classroom Tasks

 
A Question on Notation

 
A Card Pile Mystery

 
A Task for Elementary School Grades

 
A Task for Middle School Grades

 
A Task for High School Grades

 
 
Chapter 5: Subtraction
Something to Mull On

 
Subtraction in the Real World

 
A Key Takeaway

 
Being Bold With the Math

 
Generalizing Our Thinking a Wee Bit

 
More Mulling and Some Classroom Tasks

 
Wishful Thinking Across the Curriculum

 
A Task for Elementary School Grades

 
A Task for Middle School Grades

 
A Task for High School Grades

 
 
Chapter 6: Multiplication
Something to Mull On

 
Multiplication in the Real World

 
Solitaire Yet Again!

 
The Number 1

 
A Key Takeaway

 
Being Bold With the Math

 
On the Question of What Multiplication Is

 
More Mulling and Some Classroom Tasks

 
A Task for Elementary School Grades

 
A Task for Middle School Grades

 
A Task for High School Grades

 
 
Chapter 7: The Distributive Property
Something to Mull On

 
The Distributive Property in the Real World

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and Some Classroom Tasks

 
A Task for Elementary School Grades

 
A Task for Middle School Grades

 
A Task for High School Grades

 
 
Chapter 8: Multiplication and Zero
Something to Mull On

 
Multiplication and Zero in the Real World

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and Some Classroom Tasks

 
A Task for Elementary School Grades

 
A Task for Middle School Grades

 
A Task for High School Grades

 
 
Chapter 9: Multiplication and Negative Numbers
Something to Mull On

 
Multiplication and Negative Numbers in the Real World

 
One Way Out

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and Some Classroom Tasks

 
A Task for Elementary School Grades

 
A Task for Middle School Grades

 
A Task for High School Grades

 
 
Chapter 10: Division
Something to Mull On

 
Division in the Real World

 
A Key Takeaway

 
Being Bold With the Math

 
Dividing Zero and Dividing by Zero

 
The Curious Case of 0 ÷ 0

 
More Mulling and Some Classroom Tasks

 
A Task for All Grades

 
For Extra Fun

 
How Some Curricula Address the Issue

 
 
Chapter 11: The Reciprocal Numbers (The Building Blocks of Fractions)
Something to Mull On

 
Reciprocal Numbers in the Real World

 
A Realistic Note

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and Some Classroom Tasks

 
A Task for Elementary School Grades

 
A Task for Middle and High School Grades

 
 
Chapter 12: Fractions Are Answers to Division Problems
Something to Mull On

 
Fractions as Answers to Division Problems in the Real World

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and a Classroom Task

 
The Definition of a Fraction

 
A Task for All Grades

 
 
Chapter 13: Multiplying Fractions
Something to Mull On

 
Multiplying Fractions in the Real World

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and a Classroom Task

 
A Task for All Grades

 
 
Chapter 14: Equivalent Fractions
Something to Mull On

 
Equivalent Fractions in the Real World

 
Two Pieces of Language

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and Some Classroom Tasks

 
A Task for Elementary School Grades

 
A Task for Middle School Grades

 
A Task for High School Grades

 
 
Chapter 15: Dividing Fractions
Something to Mull On

 
Dividing Fractions in the Real World

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and a Classroom Task

 
A Task for Upper Grades

 
 
Chapter 16: Adding and Subtracting Fractions
Something to Mull On

 
Adding and Subtracting Fractions in the Real World

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and a Classroom Task

 
A Task for All Grades

 
 
Chapter 17: Comparing Fractions
Something to Mull On

 
Comparing Fractions in the Real World

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and Some Classroom Tasks

 
Mixed Numbers

 
A Task for Elementary School Grades

 
A Task for Middle School Grades

 
A Task for High School Grades

 
 
Chapter 18: Exponents
Something to Mull On

 
Exponents in the Real World

 
A Key Takeaway

 
Being Bold With the Math

 
Understanding 212

 
More Mulling and Some Classroom Tasks

 
The Value of 00

 
A Task for Elementary School Grades

 
A Task for Upper School Grades

 
 
Chapter 19: Fractions as Decimals
Something to Mull On

 
Fractions as Decimals in the Real World

 
Leaning Into Reverse Multiplication

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling and a Classroom Task

 
On Multiplying by Ten

 
Understanding 0.9999 . . .

 
Believing 0.9999 . . . Is Meaningful

 
Final Conclusion

 
The Geometric Series

 
A Task for All Grades

 
 
Chapter 20: Not Every Number is a Fraction
Something to Mull On

 
Numbers That Are Not Fractions in the Real World

 
A Key Takeaway

 
Being Bold With the Math

 
More Mulling

 
 
Final Thought: Are there more Numbers
 
What's Next?
But How?

 
Now That You Have Read This Book

 
Be Bold!

 
 
References
 
Index

This is an amazing resource for anyone who wonders, ‘why?’ Mathematician James Tanton takes us into an exploration of mathematical conundrums
and curiosities like dividing by zero, multiplying two negatives, and fraction division, showing us why they work and giving us creative new ideas for helping learners. It is a must-read for all math educators and the mathematically curious.

Jo Boaler
Stanford University
Stanford, CA

Math often feels like a play we’ve seen too many times—we know every twist and turn. Or so we thought. In Math Decoded, James Tanton takes us
backstage, revealing surprising depth in even the most familiar concepts, and shows us not only new ways to understand mathematics, but also new ways to teach it more effectively. This is a backstage tour you don’t want to miss.

Peter Liljedahl
Simon Fraser University
Vancouver, British Columbia, Canada

James Tanton is a gift to mathematics education. There are few people who I consistently learn so much from, and Tanton is one of them. This book
is no exception. His playful and rigorous style makes important questions in K–12 math highly accessible. These are questions that are relevant whether you’re highly experienced or teaching mathematics for the first time. I dare you to read it and not be wowed!

Michaela Epstein
Maths Teacher Circles
Melbourne, Victoria, Australia

Math Decoded is James Tanton at his brilliant best: playful, profound, and passionately human. I’m constantly inspired by how he makes the abstract
feel alive, and the complex feel inviting. This book isn’t just a masterclass in mathematical insight; it’s a love letter to the ‘why’ that fuels every curious learner.

Eddie Woo
University of Sydney
Sydney, New South Wales, Australia

James Tanton is a master at making complex math ideas easy to understand as readers are presented with concrete examples that
extend to deep, mathematical ideas. Following each big idea, activities are provided to engage your students in topics they (and sometimes teachers) often find confusing. This book is a worthwhile read for any educator.

Abigail Bates
Bear Creek High School
Stockton, CA

This is a brilliant resource from a brilliant math educator who excels at unpacking complex mathematical concepts into simple, accessible ideas
for everybody. Tanton brings meaning, clarity, and joy to mathematics. The book masterfully addresses ‘why’ questions, providing profound
insights into numbers and their operations. A must-read for every educator!

Jennifer Chang Wathall
JCW Consulting
Hong Kong

Tanton brings to mathematics a quirky ‘what?,’ ‘why?,’ ‘how?’ approach that never fails to arouse curiosity. He’s grabbed my attention on issues
I thought I knew inside-out. For a school-aged audience, his approach can be dynamite. This book is a wonderful, inspiring resource of intriguing questions that bring ‘everyday’ math to life.

Keith Devlin
Stanford University
Petaluma, CA

A delightful journey into the playful depths of arithmetic. Tanton makes even classic stumpers—like ‘why can’t you divide by zero?’
and ‘what is infinity?’—feel like wondrous invitations to explore. This book is perfect for anyone who finds joy in the quirks and logic of mathematics.

Scott Baldridge
Louisiana State University
Baton Rouge, LA

In classic James Tanton fashion, this book takes readers through some of mathematics’ most perplexing questions. James masterfully tackles the timeless student query—’Why do we need to do this?’—and transforms it into an invitation to wonder, discovery, and delight. With his signature joy and clarity, he helps readers make sense of the very problems that once felt out of reach. A much-needed resource that brings the beauty and purpose of mathematics to life!

Sue Looney
Looney Math Consulting
North Easton, MA

As math teachers, how many times do we hear ‘Why?’ from our students? I have heard it too many times to count and have been frustrated when I didn’t have the answer. Not anymore! In Math Decoded, James Tanton takes the reader through the story of mathematics that illustrates the power and beauty behind the numbers. In this book, math transforms from stand-alone rules to intertwining concepts. I highly recommend this book to all teachers looking to deepen their own knowledge of mathematics and that of their students.

Robin Kubasiak
Sturgis High School
Bronson, MI

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