Developing Mathematical Reasoning by Harris, Pamela Weber;

Short description:

Math is not rote-memorizable. Math is not random-guessable. Math is figure-out-able.

Author Pam Harris argues that teaching real math—math that is free of distortions–will reach more students more effectively and result in deeper understanding and longer retention. This book is about teaching undistorted math using the kinds of mental reasoning that mathematicians do.

Memorization tricks and algorithms meant to make math “easier” are full of traps that sacrifice long-term student growth for short-lived gains. Students and teachers alike have been led to believe that they’ve learned more and more math as they move through the content, but in reality students are not necessarily progressing in their ability to reason mathematically.

Using tricks may make facts easier to memorize in isolation, but that very disconnect distorts the reality of math. The mountain of trivia piles up until students hit a breaking point. Humanity's most powerful system of understanding, organizing, and making an impact on the world becomes a soul-draining exercise in confusion, chaos, and lost opportunities.

In her landmark book Developing Mathematical Reasoning: Avoiding the Trap of Algorithms, Pam emphasizes the importance of teaching students increasingly sophisticated mathematical reasoning and understanding underlying concepts rather than relying on set rules for solving problems. Now, in this next companion volume, Developing Mathematical Reasoning: The Strategies, Models, and Lessons to Teach the Big Ideas in Grades 3–5 equips educators with practical tools to move beyond rote memorization toward true mathematical thinking for students in upper elementary grades. Focusing on additive and multiplicative reasoning, the book introduces strategies designed to improve mathematical reasoning, Problem Strings, and strategic modeling to strengthen student understanding.

Highlights include:

• Reasoning-based strategies: Replace traditional algorithms with approaches that build critical thinking while ensuring understanding.


• Problem Strings: Step-by-step guidance on walking students through a sequence of problems that spark insight.


• Grade 3–5 focus: Comprehensive coverage of additive and multiplicative reasoning tailored for upper elementary learners.


• Practical tools: Ready-to-use routines, discussion prompts, and modeling techniques for immediate classroom application.

Help students learn to think mathematically rather than memorize. Build confidence, deep understanding, and an appreciation for the logic and beauty of math.

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Long description:

Math is not rote-memorizable. Math is not random-guessable. Math is figure-out-able.

Author Pam Harris argues that teaching real math—math that is free of distortions–will reach more students more effectively and result in deeper understanding and longer retention. This book is about teaching undistorted math using the kinds of mental reasoning that mathematicians do.

Memorization tricks and algorithms meant to make math “easier” are full of traps that sacrifice long-term student growth for short-lived gains. Students and teachers alike have been led to believe that they’ve learned more and more math as they move through the content, but in reality students are not necessarily progressing in their ability to reason mathematically.

Using tricks may make facts easier to memorize in isolation, but that very disconnect distorts the reality of math. The mountain of trivia piles up until students hit a breaking point. Humanity's most powerful system of understanding, organizing, and making an impact on the world becomes a soul-draining exercise in confusion, chaos, and lost opportunities.

In her landmark book Developing Mathematical Reasoning: Avoiding the Trap of Algorithms, Pam emphasizes the importance of teaching students increasingly sophisticated mathematical reasoning and understanding underlying concepts rather than relying on set rules for solving problems. Now, in this next companion volume, Developing Mathematical Reasoning: The Strategies, Models, and Lessons to Teach the Big Ideas in Grades 3–5 equips educators with practical tools to move beyond rote memorization toward true mathematical thinking for students in upper elementary grades. Focusing on additive and multiplicative reasoning, the book introduces strategies designed to improve mathematical reasoning, Problem Strings, and strategic modeling to strengthen student understanding.

Highlights include:

• Reasoning-based strategies: Replace traditional algorithms with approaches that build critical thinking while ensuring understanding.


• Problem Strings: Step-by-step guidance on walking students through a sequence of problems that spark insight.


• Grade 3–5 focus: Comprehensive coverage of additive and multiplicative reasoning tailored for upper elementary learners.


• Practical tools: Ready-to-use routines, discussion prompts, and modeling techniques for immediate classroom application.

Help students learn to think mathematically rather than memorize. Build confidence, deep understanding, and an appreciation for the logic and beauty of math.

What happens when you shift math from being about rote memorizing and mimicking to focusing on strategies, thinking, and reasoning? There

is no one better than Pam Harris to guide you in this important path—and this one is for you, grade 3–5 teachers!

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Table of Contents:

Preface
About This Book
Language Use in This Book
Acknowledgments
About the Author
Part 1: Setting the Stage
Chapter 1: Mathematics for Teaching
What---s the Purpose of Learning Math?
The Development of Mathematical Reasoning
Spatial, Algebraic, and Statistical Reasoning
Major Strategies
Conclusion
Discussion Questions
Part II: Developing Additive Reasoning
Chapter 2: The Major Strategies for Addition
Additive Reasoning
Developing the Major Strategies for Addition
The Split by Place Value Strategy
The Add a Friendly Number Strategy
The Get to a Friendly Number Strategy
The Add a Friendly Number Over Strategy
The Give and Take Strategy
Comparing the Major Addition Strategies
Conclusion
Discussion Questions
Chapter 3: The Major Strategies for Subtraction
Developing the Major Strategies for Subtraction
The Remove by Place Value Strategy
The Remove a Friendly Number Strategy
The Remove to a Friendly Number Strategy
The Remove a Friendly Number Over Strategy
The Find the Distance/Difference Strategy
The Constant Difference Strategy
Comparing the Major Subtraction Strategies
Conclusion
Discussion Questions
Part III: Developing Multiplicative Reasoning
Chapter 4: The Major Strategies for Multiplication
Multiplicative Reasoning
Important Foundations
Developing the Major Strategies for Multiplication
The Smart Partial Products Strategy
The Smart Partial Products: Over Strategy
The Smart Partial Products: 5 Is Half of 10 Strategy
The Doubling/Halving Strategy
The Using Quarters and Scaling Strategy
The Flexible Factoring Strategy
Comparing the Major Multiplication Strategies
Conclusion
Discussion Questions
Chapter 5: The Major Strategies for Division
Important Foundations
Developing the Major Strategies for Division
The Smart Partial Quotients Strategy
The Smart Partial Quotients: Over Strategy
The Smart Partial Quotients: 5 Is Half of 10 Strategy
The Equivalent Ratios Strategy
Comparing the Major Division Strategies
Conclusion
Discussion Questions
Part IV: Putting It All Together
Chapter 6: Tasks to Develop Mathematical Reasoning
Sequencing Tasks
Problem Strings
Other Instructional Routines
Games
Hint Cards
Conclusion
Discussion Questions
Chapter 7: Modeling and Models
Strategies Versus Models
The Many Meanings of Model
Exploring Models by Their Best Uses
Our Modeling Framework
Conclusion
Discussion Questions
Chapter 8: Moving Forward
Mentor Mathematicians
Where to Start
Conclusion
Discussion Questions
References
Index

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