Developing Mathematical Reasoning
Avoiding the Trap of Algorithms
Cameron Harris, Contributing Writer
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. 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.
Developing Mathematical Reasoning: Avoiding the Trap of Algorithms emphasizes the importance of teaching students increasingly sophisticated mathematical reasoning and understanding underlying concepts rather than relying on a set rule for solving problems. This book illuminates a hierarchy of mathematical reasoning to help teachers guide students through various domains of math development, from basic counting and adding to more complex proportional and functional reasoning.
Everyone is capable of understanding and doing real math. This book:
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Highlights the important mathematical relationships, strategies, and models for students to develop
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Offers personal stories, reflection sections, and extensive practical exercises for easy implementation
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Includes real math—a lot of it—to provide teachers with examples they can put to use in their classrooms immediately
This book is a valuable resource for educators looking to reach more students by building a strong foundation of mathematical thinking in their students. 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.
Are you teaching operations, fractions or functions? If so, Harris has some gorgeous ideas for you—showing us the ways they are all ‘figure-out-able’ with mathematical reasoning.
Developing Mathematical Reasoning is every teacher’s guide to breaking away from algorithmic-centered teaching. From the three distortions of mathematics to the hierarchies of mathematical reasoning, Harris helps us understand how math and math teaching have become entangled in a tension between algorithms and reasoning, and then shows us how to untangle this tension through a series of real classroom examples. In so doing, Harris shows us that math is, actually, ‘figure-out-able.’
Harris explores the limitations of an algorithm-centered classroom and emphasizes the need for true mathematical reasoning. By presenting a hierarchy of reasoning domains and advocating for a strategy-centered approach, this book equips educators with vital tools to empower students and deepen their understanding of mathematics.
Chock full of real stories about real people engaging with real math, Developing Mathematical Reasoning lives up to its title. Harris beautifully empowers educators with practical insights and steps to help students become true mathematical thinkers, not just mimickers—essential for a world that needs confident reasoners.
This book is a gem that should be read by every teacher of mathematics. Harris offers a K–12 continuum of narratives from classrooms and builds a strong argument for why algorithms should not be the focus of instruction if we truly want to produce numerate, mathematically empowered thinkers.
This book is a gift for all teachers, especially those of us raised in the era of algorithms and rote memorization. Harris walks you through how to help students reason their way to understand math conceptually. With each step in the progression, you learn how to help students graduate to more sophisticated ways of thinking and math-ing.
From the very first page, this book grabbed me and refused to let go. Harris’s insights into the challenges of learning mathematics, as well as her joyful explanations of what can be possible when we have the right attitude and mindset, are essential for today’s educators to absorb and integrate into their classrooms.
Harris critiques traditional math instruction by highlighting three key distortions about what math truly is. She encourages educators to move beyond algorithm repetition and instead promote real mathematical reasoning and problem solving, raising expectations and fostering deeper understanding for all students. A transformative read for anyone looking to elevate math instruction.
Harris takes you on an adventure that fast-tracks you along her journey of discovering how students learn best. A must-read for anyone wanting to open students’ horizons and get them to use what they already know to tackle new problems.
This is a timely and, ultimately, brave book about mathematics. Harris shines a light on ineffective practices and reminds us that math is so much more than memorized procedures. Her insights may ruffle some feathers about long held beliefs on math instruction. But the invitation to reach more deeply into real mathematics will open many eyes.