Every Student Is a Math Thinker Part 1: 3 Routines That Encourage Reasoning

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.

Why Routines Matter

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.

Routine 1: Notice & Wonder²

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:

Everyone can notice something, which makes this routine accessible.
Wondering naturally leads to questions that drive mathematical inquiry.
The shift from “What do you know?” to “What do you notice?” lowers the stakes and values observation over immediate correctness.

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:

Start small: use an image or problem with obvious features at first to build confidence.
Collect noticings and wonderings publicly on the board, in a shared doc, or on sticky notes.
Celebrate the breadth of responses, not just the “mathy” ones. A student who notices color or spacing is developing the habit of careful observation.

Routine 2: Which One Doesn’t Belong?³

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:

The brilliance is that all four can be justified as not belonging, depending on the reasoning.
The open-ended structure makes every answer potentially valid if supported with reasoning.
It builds argumentation skills: students must explain why.

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:

Begin with visuals (shapes, graphs) before moving to more abstract representations.
Model multiple answers yourself so students see that divergence is encouraged.
Push for justification: “Can you explain your reasoning?” should be your go-to follow-up.

Routine 3: Number Talks⁴

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:

Builds number sense and flexibility.
Encourages students to listen to and learn from one another’s methods.
Normalizes multiple solution strategies, elevating process over product.

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:

Choose problems with multiple solution paths, not just one “best” strategy.
Establish norms: no calculators, respect for thinking time, all answers welcomed.
Record strategies on the board in the students’ words and don’t translate into formal notation too quickly.

Keys to Implementation Across Routines

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.

Equity Through Routines

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:

Low-floor, high-ceiling access ensures that all students can enter, while advanced reasoning is still challenged.
Multiple perspectives are valued, not just the quickest or loudest answer.
Belonging is built daily, because every contribution is a piece of the collective learning puzzle.

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:

How do I make thinking visible in my classroom?
Do my routines invite every student to reason, or only those who already feel confident?
What small adjustment could I make tomorrow to normalize reasoning for all?

Final Thoughts

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.