Since my previous column on the three layers of mathematical learning, I’ve gotten a lot of questions from parents about what mathematical maturity looks like. This column is my best effort to explain where it comes from and what it looks like, at least in my classroom.
I described the three layers of mathematical learning in the classroom and across the current research landscape. Routine expertise is the ground of basic facts, procedures, and automaticity that students use to solve ordinary problems logically and fluently. Adaptive expertise takes root in that soil when students are can recognize structural similarities even across unfamiliar problem types and adapt their routine expertise to new demands. Mathematical maturity grows out of this enriched foundation, as students learn to recognize higher-order situations that call forth new forms of mathematical agency, endurance, and intuition.
Each layer grows out of the one below, which means a missing layer is hard to backfill later. And a student who can’t see what they’re missing becomes vulnerable in future study as they attempt to progress with serious gaps.
Here’s what this looks like in the early stages.
As we move into the middle of the school year, we are working with parallelograms. This involves experiencing deeply what it means to start from a given definition, in which a quadrilateral is made out of two pairs of parallel sides. What does it mean to discover that a more complex figure inherits everything we know about parallel lines? What can we annotate? What can we combine? How do we know what we know?
A table of students gets stuck, and they call me over.
I ask them, “Why are consecutive angles supplementary?”
There is silence.
I suggest that they color-code the different parallel line pairs and angles. Some day, when they’ve achieved greater maturity, they’ll reach for the colored pencils and tracing paper themselves, but today is not that day. They trace, shift, compare.
Finally, someone observes, “Because they are same-side interior angles, formed by parallel lines, cut by this transversal.”
This is the beginning of mathematical maturity. Students learn to slow down and do the best they can, even if it seems overly simple or even ridiculous. I tell them all the time: The math won’t do itself. You have to do something. If you want to see the same diagram in different ways, you have to find ways to notice the different layers of reality that reveal different conditions.
We mine the diagram. We cultivate our minds.
Students unlock a new level of maturity whenever they start choosing practices because those practices strengthen their thinking.
In my experience, there’s a huge payoff to be gained by training students to run a four-step loop on any new situation they encounter: (1) spot a situation within the situation that you know something about; (2) generate as many insights as you can about that situation; (3) select the results that matter; (4) use this information to spot a new situation in the situation … and keep digging.
In this kind of investigation, the key unit of competence is the chain of reasoning itself — not any one individual “right answer.” The evidence I am looking for is that students saw something meaningful, did something with it, and used that to build more knowledge. They didn’t blindly check off one task without reflecting.
We mine the diagram. We cultivate our minds.
Students cultivating maturity are willing to try ideas and techniques that seem superficially inconvenient in the moment, but strengthen their thinking over the long run. They pick up the colored pencils. They trace and redraw. Their thinking builds value as they keep drawing. The page is the evidence.
A common way to think about mathematical maturity is to equate it with speeding up, but this is a misunderstanding. The key to maturity is to become fully present with whatever math is in front of you.
The opposite of maturity is treating math class as a series of trivial transactions rather than as the arena in which students strengthen their own thinking.
This transactionalizing mindset reduces effort to payment for tokens that accumulate toward a piece of paper, rather than an investment in thinking. A token is basically just a carnival ticket. The transactional student treats the thinking as the payment and the ticket as the goal.
Once effort becomes payment, the corresponding habits follow from the simplest consumer logic. Begging for points or for a do-over becomes asking for a discount. Taking a six-week summer geometry course instead of the full 38-week course is a cheap way to get ahead and collect a ticket. Asking for extra credit when you’re missing 50 of 70 assignments is an attempt to settle a large balance with a small payment. Outsourcing your thinking to AI or to a tutor is an attempt to buy answers without strengthening your mind. “Do we have to copy the diagram?” treats a tool as a toll.
None of these students is being lazy by nature. They’re responding sensibly to a gamified system that rewards token accumulation rather than deeper thinking.
The problem with that is, if you want to think better, you have to work for it and build it within yourself. There are layers and layers of maturity, and deep thinkers never stop growing.
