Every Student Is a Math Thinker Part 3: Creative & Critical Thinking in Math

When students think mathematically, they’re not just recalling facts or repeating steps, they’re making sense of ideas, connecting patterns, and applying reasoning in new ways.

But for too many students, math still feels like a subject of repetition where success comes from mimicking procedures rather than creating or critiquing ideas.

As we’ve explored in Part 1 and Part 2 of this series, students remember what they think about and master what they practice. If our classrooms only ask them to regurgitate and mimic, that’s what they’ll master. Admittedly, becoming a master at regurgitating and mimicry can lead to top marks in certain environments, but it rarely leads to a lasting retention of knowledge and it certainly doesn’t build the skills we ought to care about. But if we invite our students to explore, invent, discuss, and question, they’ll master the building blocks of creativity and critical thinking, critical skills that support long-term memory.

Every student can be a creative thinker in math. Creativity isn’t a talent reserved for a few; it’s a way of engaging with ideas - a mindset of curiosity, flexibility, and persistence that can be cultivated through intentional classroom practice.

What Do We Mean by “Creative and Critical Thinking”?

In mathematics, creative thinking is the ability to generate ideas, make unexpected connections, and approach problems in novel ways. Creative thinking is propped up by the knowledge we have from other areas of our lives. Critical thinking complements creativity through its focus on evaluating, refining, and justifying those creative ideas with reasoning and evidence.

Together, they form a feedback loop:

Creativity generates possibilities.
Critical thinking tests and strengthens them.

Encouraging both requires classrooms where students feel safe to take intellectual risks and where divergent thinking is as valued as correct answers.

Strategy 1: Pose Open and Inviting Problems

Closed questions lead to closed thinking. Creative and critical thinking thrive when tasks are open-ended and when there are multiple valid strategies, pathways, or solutions.

Example: Instead of asking, “What is the slope of the line that passes through (2, 3) and (6, 7)?” - a question with one right answer - we can try:

“Create two different lines that both pass through (2, 3). What do you notice about their slopes?”

Or, instead of “Find the mean of this data set,” ask:

“Create two data sets that have the same mean but look very different.”

Open tasks like these push students to practice skills such as generating, comparing, and justification. Reminder: When using open tasks, it’s important to publicly value variety in responses as much as accuracy.

Strategy 2: Invite Students to “Play” with Math

Play is serious work. In mathematics, play means exploration - testing patterns, trying new approaches, and following curiosity.

Too often, we remove play from secondary classrooms under the assumption that rigor and joy are opposites. But they are partners. When students play, they engage deeply, and deep engagement leads to retention and mastery.

Examples of mathematical play:

Number or Shape Variations: “What happens if we change one condition?” For instance, “What if the triangle weren’t right-angled? Would the relationship still hold?”
“Always, Sometimes, Never” Statements: Students decide whether statements are always true, sometimes true, or never true, and justify their thinking.
Mathematical Puzzles: Pattern-finding activities or visual tasks like Which One Doesn’t Belong? promote flexible thinking.

Through play, students encounter uncertainty - a vital ingredient for creativity and problem-solving.

Strategy 3: Connect to Real Contexts and Curiosity

Students think more deeply when mathematics connects to questions they genuinely find interesting. Creativity thrives when math feels like a tool for making sense of the world, not a set of isolated exercises.

Practical ideas:

Data Stories: Let students collect or analyze data about something they care about such as music, sports, the environment, social media.
Design Challenges: “How would you design a fair scoring system for a new game?” or “How could you scale this model up to full size?”
Math Debates: Pose claims like “The median is always a better measure of central tendency than the mean.” Students use reasoning and evidence to defend a stance.

These experiences position students as users of mathematics. They strengthen both conceptual understanding and mathematical identity.

Why This Matters for Equity

In equitable classrooms, creativity and critical thinking are essential. But too often, when our students struggle with the content or when we struggle as leaders of the classroom, creative experiences are the first to go. Let’s try to see those challenges as a cue to invite more creativity, not less.

When we center creative reasoning:

Students from all backgrounds can contribute unique insights drawn from their lived experiences.
Diverse strategies are not just tolerated but celebrated.
Learning shifts from performance (“Did I get it right?”) to participation (“How did I think about it?”).

Creativity democratizes mathematics. It gives every student an entry point and affirms that there isn’t just one right way to think mathematically.

Equity, then, is more than who gets to take which courses, it’s about who gets to feel creative and capable while doing math every day.

Closing Thoughts: The Joy of Mathematical Thinking

As we wrap up this three-part series, let’s revisit where we began.

In Part 1, we explored routines that normalize thinking - structures like Notice & Wonder and Which One Doesn’t Belong? that make reasoning safe and visible.

In Part 2, we shifted the focus from correctness to reasoning, showing how justification and discussion help students practice the kind of thinking they’ll remember and master.

And now, in Part 3, we’ve extended that reasoning into the realm of creativity and critical thought. Such a place where students don’t just reproduce mathematics, they create it.

This is where belonging and brilliance meet. When students see themselves as capable of generating mathematical ideas - not just following someone else’s steps - they experience agency, joy, and confidence.

Willingham’s reminder echoes through this series: students remember what they think about and master what they practice. So let’s make sure they spend their time thinking deeply, exploring freely, and creating boldly.

Every student deserves to leave our classrooms not just knowing math but knowing themselves as mathematicians.

Pete Grostic, Ph.D

Executive Director

Please join Math Medic Foundation in our mission to improve math outcomes for all. You can contact us to get involved or donate here.

References

Boaler, J. (2019). Limitless Mind: Learn, Lead, and Live Without Barriers. HarperOne.

Gutstein, E. (2006). Reading and Writing the World with Mathematics. Routledge.

Smith, M., & Stein, M. (2011). 5 Practices for Orchestrating Productive Mathematics Discussions. NCTM.

Sullivan, P. (2011). Teaching Mathematics: Using Research-Informed Strategies. ACER Press.

12/01/2025