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While that idea isn’t new — after all, humans have figured out every piece of math we now teach — it doesn’t always feel true to students. Especially when they’re handed a worksheet full of symbols and told to “just follow the steps.” But it shouldn’t be a secret that kids can actually figure out math for themselves.

This week, students in grades 2 through 10 took the MAP tests, a computerized assessment that adapts to each student's level to help teachers identify strengths and areas for growth in math, reading, and language usage. I was proctoring a group of 7th graders taking the math section and gave them my usual pre-test pep talk.

I said, “Even if you don’t know how to solve a problem right away, you have the ability to figure it out.”  I was met with scoffs and eye-rolls.  One student told me flat-out: “People way smarter than us already figured out math. We just have to learn it.”  And by learn, they meant memorize.  

While that belief is common, it’s one I’m determined to challenge. Later that same day, I sat down with some middle schoolers and tested whether or not they could figure out the math they were learning.

We tackled the topic of dividing fractions. My students know how to get an answer using the "keep-change-flip" algorithm, but they often don’t understand why it works. So instead of asking, “What’s ½ divided by ¾?” I asked, “How many groups of ¾ fit into ½?”

The students looked at the numbers and said, “None, because ¾ is bigger than ½.”

An excellent observation. I then asked, “Okay, so what fraction of a group of ¾ fits into ½?”  They paused and discussed for a bit, then drew a model like this one on the board:

They then described that two out of the three pieces that make up ¾ is the amount that would “fit into” ½.  So their (correct) answer was ⅔.  

I then asked them to solve ½ ÷ ¾ using the standard algorithm. They did. The answer was also ⅔.  After a few seconds of confusion and some awkward attempts to explain why,  they realized that the two questions were the same.  It’s not magic, it’s reasoning. And it’s figure-out-able, even for kids who don’t believe it yet.

Moments like these remind me why it’s so important to let students wrestle with ideas and not rush to the algorithm. When we create space for sense-making, students begin to realize that math is not a set of rules to follow, but a subject they can reason through. That mindset shift is where real confidence begins.