I spent nearly two decades teaching middle school math, which means I’ve seen students solve problems in just about every way imaginable.

Some were elegant. Some were efficient. Some took the scenic route.

And some made me stare at the paper for a moment and say, “Okay…show me what you did here.”

Those were often my favorites.

One student in particular comes to mind.

Math was hard for him.

Really hard.

But he worked. He tried. He was sweet and always appreciative of help. He was the kind of student you found yourself rooting for because you knew how much effort was behind every success.

One day, we were working on greatest common factor.

I had taught my students multiple strategies. They could list factors using the rainbow method. They could use factor trees and prime factorization. We had tools.

Then this student gave me the correct GCF for two numbers.

Great.

“How did you get that?”

“I simplified my fraction.”

What?

There was no fraction.

We were looking at two whole numbers.

I could have stopped him right there. I could have reminded him that we weren’t simplifying fractions. I could have redirected him to one of the methods we’d practiced.

Instead, I was curious.

“Come show me.”

He walked to the board and wrote one of the whole numbers as a numerator and the other as a denominator.

Okay.

Now we apparently had a fraction.

Then he started simplifying it.

He divided the numerator and denominator by a common factor. Then another. And as he worked, he kept track of the numbers he had divided by.

When the fraction could no longer be simplified, he multiplied those divisors together.

There it was.

The greatest common factor.

I remember looking at it and thinking:

Wait.

Does that actually work?

There was only one reasonable thing to do.

I gave him another problem.

It worked.

So I tried another.

It worked.

And another.

It worked.

Every. Single. Time.

At this point, my math-teacher brain was fully invested.

Then he told me it worked with three numbers too.

Now he really had my attention.

“Show me.”

He wrote three numbers in a vertical line and did exactly what he’d done before. He found a common factor that could divide all three numbers, simplified them step by step, kept track of those divisors, and multiplied them together.

Greatest common factor of three numbers.

It worked.

Of course it worked.

At some point during this interaction, the roles in the classroom had quietly shifted.

I wasn’t teaching him anymore.

He was teaching me.

And his method was beautiful.

Not just because it worked mathematically.

It made sense.

Students already knew how to simplify fractions. His process connected a new concept to something familiar. They didn’t have to generate long lists of factors and worry about missing a factor pair. They didn’t need to build multiple factor trees and then compare prime factors.

They could use something they already understood to make sense of something new.

So the next year, when I taught greatest common factor, I taught his method.

And the year after that.

And the year after that.

Most students preferred it.

A strategy discovered by a student who struggled with mathematics became one of the most useful strategies in my classroom.

I think about that often.

Because there’s a subtle danger in teaching something we know well.

We know the methods. We know the efficient processes. We know what we’re looking for.

And sometimes that makes it incredibly easy to look at student work and ask:

Did they do it correctly?

But that’s not quite the same question as:

What are they thinking?

If I had only looked for the process I taught, I would have missed his mathematics completely.

His work might have looked wrong because there wasn’t supposed to be a fraction.

But he wasn’t confused.

He was connecting.

He had taken something he already understood and used it to make sense of something new.

Isn’t that exactly what we want mathematicians to do?

Make connections. Look for structure. Recognize relationships. Use what they know to figure out what they don’t.

He didn’t reproduce my mathematics.

He made mathematics his own.

And I think there’s something important in that.

Creativity isn’t a subject.

We sometimes act as though students “do creativity” at certain designated times.

They create when they’re making art.

They create when they’re writing stories.

They create when they’re building something in STEM.

They create when we hand them an open-ended project and tell them to be innovative.

But I hadn’t given this student a creative math task.

I hadn’t said, Find an innovative new method for calculating GCF.

I taught him mathematics.

And then his brain connected it differently.

He took what he’d learned about simplifying fractions, recognized a mathematical relationship, tested it, generalized it to three numbers, and created a process that made sense to him.

That’s mathematical creativity.

Maybe creativity isn’t something we need to add to the curriculum.

Maybe it’s something we need to learn to recognize when it appears.

Because it can show up anywhere.

In mathematics.

In science.

In history.

In the way a child organizes information.

In the question they ask.

In the connection they make.

In a process no one taught them.

Meraki is about putting something of yourself into what you create.

Apparently, meraki can show up in GCF too.

I love that.

Because his process carried something of him. It reflected what he understood, how he made connections, and the way his mind had found a path through the mathematics.

And if I had been too focused on whether he followed the process I taught, I might have corrected the very thinking I should have been celebrating.

That’s stayed with me.

When a student’s work doesn’t look the way I expect, I try not to begin with:

“That’s not how you do it.”

I want to begin with:

“Show me.”

Because an answer can tell me whether a student landed in the right place.

Their thinking tells me something much more interesting.

It tells me how they traveled.

An answer tells me where you landed. Your thinking tells me how you got there.

That student showed me a path I hadn’t considered.

Then I spent years showing his path to other students.

I can’t help but wonder how many other beautiful ideas are hiding in student work that doesn’t look quite the way we expected.

What might we discover if we stopped looking for our process and started looking for their thinking?