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March 19, 2021

Less Is More: Increasing Mathematical Thinking

“All great changes are preceded by chaos.”  — Deepak Chopra

As a math teacher, currently in my first year of Common Core implementation, the above quote resonates with me. As educators, we are sometimes challenged with out-of-date curricula and little professional support. The chaos we might experience in attempting something as new and big as Common Core seems to bring more questions than answers.

The quote, though, reminds me that change is uncomfortable, and great changes can make us feel as though our foundation has been shattered. It’s imperative that out of the potential chaos that precedes change, we identify small, achievable goals. Narrowing our focus can increase our productivity, reduce stress, and engage learners.

Less Is More

Two years ago, I felt immense stress – and a sense of chaos – as I became increasingly aware that my classroom was a “sit and get” experience for students. I wanted them to think about math long after leaving my classroom, yet my students were working toward a right answer and never thinking about the task at hand again. I was so used to grabbing their pencils and reteaching students that they lacked independent thinking. Their dependency on me was overwhelming and I felt pressure to save every student.

When I stopped to reflect on my teaching, I realized that I was over-scaffolding tasks to the point where students thought math meant there was only one way to approach a task, and therefore only one right answer. My students were crippled by this idea of a right answer, which lacked emphasis on sense making or the process used to get there. I would teach an algorithm in isolation, make sense of it for the students, and move on to another strategy. My students saw math as a series of disjointed lessons without connections.

I decided to analyze a recent lesson and task – setting up proportions and solving with cross multiplication – and realized that as I was making meaning for them, by teaching this one strategy directly, I was creating disengaged learners. Sure, my presentation was broadcast worthy. I used amazing graphic organizers, songs – my cross-multiplication tune was an original rendition of “She Drives Me Crazy” – and flipped lessons. Yet, students were still distant from the math concept itself. Furthermore, their ability to transfer their skills from one month to the next was almost non-existent.

I finally realized that asking them to memorize the setup of a proportion and solving with cross multiplication, was asking them to memorize a format that didn’t make sense to them. This was, in fact, engaging students in the lowest level of cognitive demand: memorization.

So, I started removing from my curriculum any task that didn’t have multiple ways a student could approach it. In addition, I stopped introducing any tools and algorithms that the students couldn’t make sense of, or that I couldn’t really understand myself. I took a risk and started analyzing the cognitive demand of my tasks, and made an agreement with myself that I would stop asking my students to memorize math. I would engage them in learning math.

How I’ve Changed

What I used to do:
I gave direct instruction on how proportions were set up. We spent a few days setting up proportions and two days practicing over and over again the cross-multiplication algorithm. Practice was given in a drill format with multiple questions incorporating increasingly difficult numbers.

What I do now:
I utilize the richness of Inside Mathematics Photographs Task, focusing on one or two powerful questions.

Why I do it:
The questions posed in Photographs don’t ask for a predictable strategy. Students must not only find a solution, but they have to make sense of their answer and use their strategy to answer the question. I’ve seen students use common denominators, common numerators, tables, graphs, double number lines, etc., to answer the question. Student engagement increases when they can self-select a strategy and make sense of the problem with their innate problem-solving skills.

Your Challenge

Take away the scaffolding for a task and present a single question to your students. Ask them to use their creative thinking strategies to show you how they interpret or understand the task. Stick with the question, delving deep into the mathematics, for one to three days. As much as possible, recognize when you attempt to have students memorize, and be aware of why you asked them to do so. Whether right or wrong, celebrate risk taking with your students.


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