Developing Mathematical Reasoning: The Strategies, Models, and Lessons to Teach the Big Ideas in Grades K-2(Corwin Mathematics Series)

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, but in reality their brains never get any stronger. Using these tricks may make facts easier to memorize in isolation, but that very disconnect distorts the reality of math. 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 a set rule for solving problems. Now, in this first companion volume, Developing Mathematical Reasoning: The Strategies, Models, and Lessons to Teach the Big Ideas in Grades K-2, she demonstrates how counting and additive strategies serve as the foundation for creating efficient, accurate, and flexible thinkers. Everyone is capable of understanding and doing real math. This book: Gives step-by-step guidance on how to teach the strategies, models, and big ideas that foster confidence and long-term success, preparing students for increasingly complex mathematical challenges Offers the "what to do" to teach counting, addition, and subtraction in ways that promote reasoning over rote memorization Provides practical tools such as problem strings, models, classroom routines, and discussion questions designed to implement reasoning-based practices Includes supporting resources for creating a classroom culture where students see math as figure-out-able and gain confidence as mathematical thinkers By addressing common misconceptions about math and providing practical strategies for teaching real math, this book shows that everyone can use the mathematical relationships they already know to reason about new relationships. In other words, everyone can math-even the very youngest students!

Table of Contents:
Preface About This Book Language Use in This Book Acknowledgments About the Author PART I: SETTING THE STAGE Chapter 1: MATHEMATICS FOR TEACHING What’s the Purpose of Learning Math? The Development of Mathematical Reasoning Major Strategies Conclusion Discussion Questions PART II: DEVELOPING COUNTING AND COUNTING STRATEGIES Chapter 2: ALL ABOUT COUNTING The Difference Between Counting and Counting Strategies Foundations of Number How to Develop Counting The Number Sequence in the Teens The Number Sequence After the Teens Meaning of Decades Student Interview Conclusion Discussion Questions Chapter 3: COUNTING STRATEGIES About Counting Strategies Early Counting Strategies The Counting On, Counting Back Strategy Problem Types Developing Counting Strategies Conclusion Discussion Questions PART III: DEVELOPING ADDITIVE REASONING Chapter 4: THE MAJOR STRATEGIES FOR ADDITION WITHIN 20 Additive Reasoning Additive Strategies Developing Addition Within 20 The Get to 10 Strategy The Next Two Major Strategies The Using Doubles to Add Strategy The Add 10 and Adjust Strategy Comparing the Single-Digit Addition Strategies Conclusion Discussion Questions Chapter 5: THE MAJOR STRATEGIES FOR SUBTRACTION WITHIN 20 Developing Subtraction Within 20 The Remove to 10 Strategy The Next Two Major Strategies The Using Doubles to Subtract Strategy The Remove 10 and Adjust Strategy Finding the Distance/Difference Strategy Comparing the Single-Digit Subtraction Strategies Conclusion Discussion Questions Chapter 6: THE MAJOR STRATEGIES FOR DOUBLE-DIGIT ADDITION Developing Multi-Digit Addition Strategies The Splitting by Place Value Strategy The Next Two Major Strategies 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 7: THE MAJOR STRATEGIES FOR MULTI-DIGIT SUBTRACTION Developing Multi-Digit Subtraction Strategies The Remove by Place Value Strategy The Next Two Major Strategies The Remove a Friendly Number Strategy The Remove to a Friendly Number Strategy The Remove a Friendly Number Over Strategy Finding the Distance/Difference Strategy The Constant Difference Strategy Comparing the Major Strategies for Multi-Digit Subtraction Conclusion Discussion Questions PART IV: PUTTING IT ALL TOGETHER Chapter 8: TASKS TO DEVELOP MATHEMATICAL REASONING Sequencing Tasks Problem Strings Other Instructional Routines Games Hint Cards Conclusion Discussion Questions Chapter 9: MODELING AND MODELS Strategies Versus Models The Many Meanings of Model Exploring Models by Their Best Uses Our Modeling Framework Conclusion Discussion Questions Chapter 10: MOVING FORWARD Mentor Mathematicians Where to Start Conclusion Discussion Questions References Index

About the Author :
Pamela Weber Harris is changing the way we view and teach mathematics. Pam is the author of several books, including the Numeracy Problems Strings K-5 series, Building Powerful Numeracy, and the series Foundations for Strategies. As a mom, a former high school math teacher, a university lecturer, and an author, she believes everyone can do more math when it is based in reasoning rather than rote-memorizing or mimicking. Pam has created online Building Powerful Mathematics workshops and presents frequently at national and international conferences.  Her particular interests include teaching real math, building powerful numeracy, sequencing rich tasks to construct mathematics, using technology appropriately, and facilitating smart assessment and vertical connectivity in curricula in schools PK-12. Pam helps leaders and teachers to make the shift that supports students to learn real math because math is figureoutable!

Review :
Who better than Pam Harris to help you introduce K–2 students to mathematical reasoning—the language, the music, and the poetry of mathematics. A must-read book filled with teaching strategies and creative ideas. The abilities to count and to add are foundational to mathematics. All that follows is built upon these cornerstones. Get it right and math becomes ‘figure-out-able.’ In this book, Harris gives us the tools to get it right. Through real classroom examples, Harris takes us through strategies that are easy to adopt and effective in getting and keeping students engaged in the work of understanding mathematics. It is with great enthusiasm that I endorse this transformative book. At the heart of this work is a compelling discussion of reasoning. Through rich narratives and classroom vignettes, we see that math fact fluency is not only figure-out-able but enjoyable—sparking curiosity and confidence in every student. In short, this book is a masterclass in making math fact fluency meaningful and accessible for all. This book is a gift to primary teachers. It offers clear ideas we can use right away to help students build real understanding and develop as mathematical thinkers. From counting to additive reasoning, and through the power of models and Problem Strings, this book supports teachers in making instruction more purposeful and responsive. Finally, the book that K–2 educators have been waiting for is here! Harris wrote a book that explores the complexity of foundational numeracy skills and shares research-based approaches to develop mathematical reasoning with our youngest learners. This book will not only help teachers cultivate curiosity and confidence and build a community of mathers, but it will also help teachers become the mathers they were always meant to be. This book empowers K–2 teachers to deeply explore and understand the mathematics of the early grades, equipping them to foster students’ mathematical reasoning and conceptual understanding. Grounded in research and filled with practical classroom examples, it offers valuable guidance for transforming early elementary math instruction. What a powerful resource for equipping mathematics educators with the knowledge and skills necessary to develop their students’ mathematical reasoning and promote their confidence as mathematicians! If you’ve ever heard Pam Harris talk about the Levels of Sophistication in Mathematical Reasoning and thought, ‘I get it but what does that look like with MY students??’ This book delivers! It reveals how young children develop counting skills and thinking strategies for mathematical operations. You’ll gain insights into student development, practical tasks to showcase their thinking, and modeling techniques that benefit every child in your classroom. Pam Harris offers teachers another invaluable resource for transitioning from repetitive procedural instruction to helping students develop deeper, more conceptual understanding of mathematics. Through numerous examples of student thinking about mathematical concepts in the K–2 classroom, she provides extensive treatment of the ways children approach problem solving. This comprehensive approach will certainly prove helpful for teachers who are developing their understanding of how students learn to make sense of mathematics independently. This is a must-read for all K–2 teachers. Harris helps teachers understand how they can set their young learners up for long-term success in the math classroom. This book provides teachers with a resource that combines content knowledge, high quality instructional practices and ready-made instructional materials all in one! Upper elementary, middle school and even high school math teachers would all benefit from reading this book and better understanding the development of mathematical reasoning, too. This book is a game-changer for educators. With clear examples and practical strategies, it empowers teachers to transform their instructional practices. Packed with ‘aha’ moments and insights into the teaching and learning of mathematics, it inspires confidence in all of us. This is a must-read for anyone looking to make math instruction more meaningful for students while supporting the teacher with the ‘why’ behind it all. Math reasoning kicks algorithms to the curb when students engage their brains (not just their pencils) to solve problem strings using a variety of strategies. Harris takes the guess work out of teaching computation strategies intentionally by providing problem strings, teaching tips, and sentence frames for beginning and experienced teachers. This latest book from Pam Harris takes research related to the major milestones of mathematical development in the primary grades and transforms it into a language that is easy to understand. Using carefully chosen examples, real world experiences, and student voices, Harris has written a book that is illuminating and practical. Primary educators, whether new to the profession or seasoned experts, will find ideas here that resonate and challenge them to listen closely in order to further thinking.