Talk to any math teacher for more than a few minutes, and “critical thinking” will probably come up. It’s one of those goals most educators agree on. Of course, we want students to think deeply and independently. But actually teaching that, especially in math, isn’t always straightforward.

There’s a lot to cover in a short amount of time. It’s part of the classroom routine to focus on simple problems that have clear instructions and only one correct answer. While there’s some value in that, it doesn’t represent the full range of real-world work of people using mathematics, work that requires exploration and critical decision-making.

That’s where math modeling comes in.

Problems That Require Real Thinking

Unlike typical exercises, math modeling problems aren’t fully defined from the start. Students need to figure out what a good question is before they can even begin to answer it. That requires a different kind of thinking.

They have to sort through information and test ideas. Instead of following a procedure, they’re asking themselves: What are we trying to figure out? What are we assuming? Is our idea reasonable?

This kind of thinking doesn’t happen all at once. Students might start down a path, realize it’s not working, and go back to rethink things. That process is where a true understanding starts to take shape.

There’s actually a great explanation of how this kind of challenge, what some call productive struggle in math, can be a real learning opportunity.

Math modeling lets students figure out new ideas, make mistakes, and learn by doing. It’s an iterative process that mirrors the real world where not every problem is solved in 60 seconds.

Math, Communication, and Collaboration

Math modeling also encourages students to explain their reasoning. And when they put their thinking into words, the real breakthroughs happen.

It’s different from the traditional “do the problem, check the answer” routine. Students talk through ideas, listen to their peers' ideas, and think about why they made certain decisions. 

That kind of interaction strengthens critical thinking in a way that isolated practice can’t always achieve.

These skills tend to stick. Students build confidence, not just in math, but in how they approach any challenge.  Teaching critical thinking is essential in mathematics, but the skills have value far outside the math classroom.

How It Works 

You’ll also see this kind of learning in action during COMAP’s HiMCM/MidMCM and MCM/ICM contests. Over several days, students work in teams to investigate a complex, real-world issue. They build a model, test it, and write a detailed paper explaining their approach.

There’s no single right answer, and students quickly learn that the process matters just as much as the outcome. This kind of problem solving isn't just about math. It teaches students how to stay with a problem and work through it.

Easy Ways to Get Started

The good news is, a single, open-ended question can get students thinking in new ways. Even just asking them to explain their process out loud, or compare different strategies, can start building those habits.

If you’re not sure where to begin, COMAP has shared a few approachable ways to bring math modeling into your curriculum without overhauling everything. No need to change everything, just shift the focus toward exploration and reasoning.

Math modeling is not just a teaching tool, but it’s a way to bring meaning and relevance into math education.

Critical thinking doesn’t grow from shortcuts or quick answers. It comes from giving students a chance to think for themselves, and modeling is one powerful way to do just that.