Take Math Students beyond Tired Algorithms

Developing Mathematical Reasoning: Avoiding the Trap of Algorithms (K-12) 
By Pamela Weber Harris
(Corwin, 2025 – Learn more)

Reviewed by Amy Leach

If you’ve ever sat in a Pam Harris workshop or listened to her Math is Figureoutable podcast and thought, “Wait, wait – I can’t write all this down fast enough!” then this book is your safety net. Developing Mathematical Reasoning is like having Pam Harris in your ear, but with a pause button and a whole lot of sticky notes.

It’s a resource that distills the powerful ideas from her workshops and podcasts, laying them out clearly and step by step. For instance, it explores the concept of relational thinking and offers practical examples of how to cultivate this type of reasoning in the classroom.

I’ve been teaching math long enough to know how easy it is to slip into algorithm-teaching mode, even when I know better. Kids want the “steps.” Parents want the math the way they did it. But as Harris reminds us – over and over and with compelling evidence – that’s not mathematical reasoning. That’s pattern-matching and memorization.

The book does not dismiss the importance of algorithms, but it argues that they should not be the sole focus of math education. This book is a call to action (and a how-to manual) for avoiding that trap.

What I found especially valuable was how thoroughly Harris defines types of mathematical reasoning. It’s not just “reason, don’t memorize” in the abstract. She explains specific kinds of reasoning – relational thinking, proportional reasoning, spatial reasoning, and more – with crystal-clear written examples.

Even better, she provides excellent classroom video links so you can see these ideas in action. That makes a huge difference. Reading about how a teacher facilitates relational thinking is one thing; watching it unfold in real time with real students is another.

I’ve tried to shift my own classes away from the “Here’s the formula, let’s practice it ten times” approach. But without solid models, it can feel wobbly. This book is giving me the confidence to try new structures and language in my lessons. For example, Harris models how to ask purposeful questions that don’t lead students straight to an algorithm but instead get them to notice and reason. I plan to borrow questions from the book and use them in my high school classes as brain warm-ups.

One of the most valuable aspects of the book is its practicality. It’s not just a theoretical argument against algorithms (though Harris makes that argument convincingly). It’s a comprehensive guide for teachers on what to do instead. Each chapter is filled with sample dialogues, student work, teacher strategies, and common pitfalls to avoid. Harris doesn’t sugarcoat it – this work is complex, sometimes messy, and not always instantly successful. But the potential benefits are enormous: students who truly comprehend math.

This book does not require you to buy into dumping algorithms completely. If you’re deeply skeptical of teaching without algorithms, Harris will convince you to want to change. If you have even a hint of doubt about the efficacy of teaching algorithms first, this is an eye-opener.

I’d especially recommend this book to:

• Elementary and middle school teachers trying to build conceptual understanding early.
• High school teachers who want to help students unlearn rote habits and actually think.
• Coaches or instructional leaders looking for concrete ways to support reasoning-focused teaching.
• Anyone who’s attended a Pam Harris workshop and wished they could bottle it up to revisit later.

In the end, Developing Mathematical Reasoning is a book that practices what it preaches. It doesn’t just tell you what reasoning is – it shows you. It doesn’t just argue against algorithms – it gives you an alternative. It’s one of those rare professional books you don’t just read – you use.

If you’re ready to see your students actually think mathematically (and to think a bit more like a mathematician yourself), I can’t recommend it highly enough.

With over 30 years in math education and assessment, Amy Leach teaches Algebra, Geometry, AP Precalculus, AP Calculus, College Algebra, and a self-developed Data Science course at Spring Valley (WI) High School. Her background includes two decades designing performance tasks and inclusive assessments, ten years in middle and high school classrooms, and service as a math curriculum coordinator. A lifelong learner, she recently deepened her expertise through the Data Science Summer Institute.