When you think back on what you learned in school, certain moments stand out. Maybe it was a lively debate in English class, or a science lab where you got your hands messy. For most of us, we remember the things we thought hard about. That’s no accident. Cognitive psychologist Dan Willingham has put it succinctly: We remember what we think about.1 And when it comes to learning math, the principle extends further: We remember what we think about and master what we practice.
But too often, math classrooms ask students to remember rules rather than wrestle with ideas. They’re told what to do, shown how to do it, and then asked to mimic. This approach might produce short-term performance, but it rarely nurtures the deep reasoning that helps students see themselves as true mathematical thinkers.
If every student is going to leave your classroom believing, “I am a math thinker,” then they need daily opportunities to do just that - to think. One of the most effective ways to accomplish this is by embedding routines that encourage reasoning into the fabric of your instruction.
Routines are short, structured activities that are repeated often enough to become predictable for students. They typically last just a few minutes, but over time they accumulate into a powerful culture of thinking. The predictability lowers anxiety because students know what to expect, while the open-endedness raises opportunity because students can contribute in many different ways.
Most importantly, routines help operationalize Willingham’s principle. They compel students to think about mathematical ideas rather than passively mimicking the teacher. And because routines recur, they give students repeated practice in reasoning, not just in computation.
In this article, we’ll explore three powerful routines: Notice & Wonder, Which One Doesn’t Belong?, and Number Talks and discuss practical tips for implementation.
What it is:
Created by math educator Annie Fetter, Notice & Wonder is deceptively simple. You show students an image, graph, equation, or problem situation, and ask two questions:
1. What do you notice?
2. What do you wonder?
Why it works:
Classroom example:
Project a scatterplot with no labels. Students might notice that most of the dots form an upward trend. Others might wonder what the axes represent or whether there are outliers. You haven’t said the word “correlation” yet, but students are already reasoning about it.
Implementation tips:
What it is:
Students are shown four options of numbers, shapes, equations, or graphs. Their task: decide which one doesn’t belong.
Why it works:
Classroom example:
Display four fractions: 2/5, 1/2, 4/7, and 5/5. One student says 5/5 doesn’t belong because it equals 1. Another argues 1/2 doesn’t belong because its denominator is an even number. A third claims 4/7 is the only fraction that is equivalent to a repeating decimal. Suddenly, a “simple” task has sparked a rich mathematical conversation.
Implementation tips:
What it is:
A short, daily routine (often 5–10 minutes) where students mentally solve a problem and then share strategies aloud. The goal is not speed but strategy sharing.
Why it works:
Classroom example:
Pose: What is 18 × 25? One student decomposes 100 as 4 × 25 and reasons that 18 × 25 = 450. Another sees 18 × 25 as (20 × 25) – (2 × 25) = 500 – 50 = 450. Students compare, notice efficiency, and expand their mental toolbox.
Implementation tips:
1. Launch Slowly - Introduce one routine at a time and let it become second nature before adding another. Students thrive when the structure feels predictable.
2. Set Clear Norms - Make participation low-risk: “Any noticing is valid,” “Mistakes are part of learning,” “Explanations matter more than answers.”
3. Celebrate Diverse Responses - Every response reveals thinking. Acknowledge unusual insights. Over time, students begin to value variety as a strength.
4. Reflect and Adjust - Build in quick debriefs: “What did we learn from today’s conversation?” Reflection helps students see the value of the routine beyond the immediate problem.
These routines are not just good pedagogy - they are equity strategies. Students who have historically been marginalized in math classrooms often internalize the message that they aren’t “math people.” Predictable reasoning routines disrupt that narrative:
When students are consistently asked to think, share, and justify, they begin to see themselves differently: not as outsiders to math but as participants in the mathematical conversation.
Take a moment to ask yourself:
Dan Willingham’s principle reminds us that memory follows thought. Students remember what they think about, and in math, they master what they practice. If our goal is to raise a generation of confident math thinkers, we must build classrooms where reasoning is not an occasional activity but a daily habit.
Notice & Wonder, Which One Doesn’t Belong?, and Number Talks are not add-ons or gimmicks. They are routines that weave reasoning into the fabric of instruction. With thoughtful implementation, they can transform not only how students learn math, but how they see themselves in relation to it.
Because in the end, every student is a math thinker, and they need the chance to practice thinking, day after day.
Pete Grostic, Ph.D
Executive Director
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1Why Don't Students Like School? - Daniel T. Willingham
2Ever Wonder What They’d Notice? [TEDx Talk] - Annie Fetter
3Which One Doesn't Belong? - Christopher Danielson
4Making Sense of Math - Cathy Seeley
10/06/2025