Avoiding the Trap of Algorithms
Developing Mathematical Reasoning: Avoiding the Trap of Algorithms (Grades K-12)
By Pamela Weber Harris
(Corwin, March 2025 – Learn more)
Reviewed by Kathleen Palmieri, NBCT
During the last trimester of school, I made plans to focus part of my summer professional reading on math techniques and strategies. As an educational researcher and classroom teacher, I have spent a lot of time digging into what makes my math students “tick.”
Forging ahead with my plan, I came across the Corwin title Developing Mathematical Reasoning: Avoiding the Trap of Algorithms by Pamela Weber Harris. Two things struck me in this title: “Mathematical Reasoning” and “The Trap of Algorithms.”
As with many books, I first thumbed through to see what jumped out at me. What caught my attention was the “Domains of Reasoning” and the many illustrations. With my interest already piqued, I then came across “Frequently Asked Questions,” QR codes leading to videos demonstrating the various domains of reasoning, and “Try it in your Classroom.” Forget waiting for summer, I dove into the book, reading it in two days and finding ways to try out the examples offered with my class.
The full spectrum of Mathematical Reasoning with Longitudinal Domains helped me to reflect on what I had observed with many of my students. “The domains are arranged left to right because it is critical to understand that one domain is not inferior to another, merely less sophisticated…each domain contains its own hierarchy of less to more sophisticated forms of that reasoning.” (p. 46) From left to right, the domains are Counting Strategies, Additive Reasoning, Multiplicative Reasoning, Proportional Reasoning, and Fractional Reasoning.
The Longitudinal Domains stretched to include Spatial Reasoning throughout all five domains, Algebraic Reasoning beginning from Additive Reasoning, including Fractional Reasoning, and Statistical Reasoning encompassing Multiplicative Reasoning and Fractional Reasoning. I liked how the word “sophisticated” was used to emphasize the growing mathematical process of reasoning.
As a teacher of intermediate (5th grade) students, I was very curious about “Multiplicative Reasoning” and “The Trap of Multiplication and Division Algorithms” found in chapter four. (p.95) Place value is the first topic in my school’s math curriculum and seems to be the sticking point for some students as the concepts get more complex. Reasoning about multiplication and division depends upon an understanding of place value and helps to avoid the “Trap of Algorithms” through the use of “Problem Strings” and ratios.
The example provided in the book started with 2400 ÷ 24. A student responds with the correct answer of 100, “with the reasoning that when multiplying by 100, the answer will be the original number ‘bumped up by two place values.’” Next, the teacher draws a ratio table on the board, not the standard division “house,” to “emphasize the scaling of 1 to 100 and 24 to 2400 as ‘x 100.’” Throughout this “Problem String” the teacher works from the “helper problem of 2400÷24” to move to the next problem of 1200÷24, 1800÷24, 1848÷24, etc. (pp. 96-102)
What ensues is an engaging math reasoning activity with students actively dividing using their reasoning, what they notice about the numbers, and what they know as math students. Trying this Problem String with my class proved to be both engaging and worthwhile, leaving students eager for more.
My classroom experience aligns closely with Weber Harris’s explanation: “As students realize they can group the groups, they chunk the problem into manageable parts” (p. 106). This approach fosters mathematical confidence, as students’ reasoning begins to resemble rich mathematical discourse rather than reliance on a single algorithm.
There are so many fine examples of ways to impart reasoning without algorithms in this book. “The Trap of a Traditional Division Algorithm” demonstrates the steps of the division algorithm and compares it to finding an equivalent ratio that’s easier to solve. Weber Harris also provides the three main traps of the division algorithm (p. 131), the third being “division becomes steps to memorize.” The “Try it in Your Classroom” suggested activities, especially “Doubling and Halving” (p.139), are highly engaging. Teacher tips throughout the book are easy to implement, and “Frequently Asked Questions” are thought-provoking.
Finally, “Developing High Leverage Teacher Moves” offers ideas such as students using private response signals and teachers using a neutral response. Weber Harris asks us to celebrate reasoning writing, “Everything in your classroom, from how you handle homework to grading tests to seating arrangements, should reflect that you value reasoning first and not merely answer-getting.” (pp.239-242)
Developing Mathematical Reasoning: Avoiding the Trap of Algorithms (Grades K-12) has been an enlightening mathematical experience for me as a math teacher. As Weber Harris states, “I’m telling you the goal of mathematics education is to develop mathematical reasoning. This is not just a fluffy ‘think better’ idea; it includes content. Learning the content is important. In fact, reasoning mathematically means owning content, deep down, inside and out, with a web of interconnected relationships.” (p. 223).
Each year, students bring diverse mathematical experiences and levels of understanding to the classroom. As educators, we must continue to grow in our practice and create opportunities for students to share their thinking and reasoning. If you’re looking for an eye-opening read that will reinvigorate your approach to teaching math, this book is a valuable resource. It offers fresh insights that will energize your instruction and deepen student engagement.
Kathleen Palmieri is a National Board Certified Teacher, NBCT Professional Learning facilitator and education writer. She is a fifth-grade educator in upstate New York who reviews and writes regularly for MiddleWeb. With a passion for literacy and learning in the classroom, she participates in various writing workshops, curriculum writing endeavors, and math presentations. As a lifelong learner, she is an avid reader and researcher of educational practices and techniques.
Kathie’s ongoing practice includes collaborating with colleagues and globally on X-Twitter and Bluesky @kathleenpalmieri.bsky.social and expanding her education adventures at www.kathleenpalmieri.com.
