# The centrality of stupidity in mathematics

September 17, 2024*The Importance of Stupidity in Scientific Research* – published in 2008 in the Journal of Cell Science – is one of those wonderful essays that turns your understanding of “being smart” on its head. There are enough implications for education here that it’s worth reading the entire thing.

[W]e don’t do a good enough job of teaching our students

how to be productively stupid – that is, if we don’t feel stupid it

means we’re not really trying. —Martin Schwartz

[Hat tip to @KevinDKohl for the share on twitter. The original article can be found here.]

The idea of confronting one’s own stupidity and living with it productively – even joyfully! – feels even more central to me in mathematics than in science. The article touches on learning science, but it’s really about research. Stupidity is part of confronting the new, and being a capable science student means you might not have to face it that much when you’re merely absorbing the knowledge that previous scientists have already uncovered.

Math seems different to me. Everyone who learns math is familiar with the experience of being stuck on some new idea or problem, banging their head against it, and then, when they finally understand the answer (or having someone tell them), feeling stupid. There’s something fundamental in the nature of mathematics that makes it easy once you get it, and impossible before.

To wit: there’s an old joke where a professor is lecturing and remarks, “It’s obvious why this equation holds, of course.” A brave student raises her hand and says what everyone is thinking: “Could you please explain it? It doesn’t seem obvious.” The professor looks at the equation. “Isn’t it obvious?” He starts mumbling to himself. He starts to write on the white board, then stops. Finally he just stands there, looking at the board. Ten minutes pass. The students shift awkwardly and the seconds tick by. Suddenly the professor exclaims, “Aha! It *is* obvious!”

I’ve had numerous experiences in mathematics that left me feeling stupid. The one that made it clearest, I think, was taking real analysis in college. There’s a technical business with *epsilons* and *deltas*: one arbitrarily small value has to be smaller than another, thus proving that a function is continuous in a more rigorous way than high school calculus ever bothers to consider. It was arcane and baffling when I took the course. Six months later, it felt so simple that I couldn’t understand what my problem had been. Why had I been so stupid before, that this obvious mathematical idea had eluded me?

My personal theory about this (and this is untested, as far as I know) is that mathematics is somewhat unique in the way new learning develops in the mind. Piaget studied mathematical leaps in understanding in young children, and noticed that there are almost switches that flip at certain ages: before they’ve flipped, an idea is essentially impossible to understand; after, they’re obvious.

A classic example from Piaget is conservation of volume. (It’s worth checking out the video below if you’re not familiar with this one.) A child confirms two quantities of liquid are the same, sees one poured into a taller, thinner glass, and then says that the taller one has more water in it, *despite seeing it proved that the two have the same amount of liquid*.

Is the boy stupid? No. None of us were, and we all went through this phase. At some age, none of us could see that the amount of liquid stays constant. But once you see it, it’s impossible not to see it. And it’s almost impossible to imagine how it wouldn’t be obvious to anyone.

I suspect that mathematical understanding requires similar mental leaps, no matter how long you study it. It’s almost as if we construct new highways in our mind. As long as the construction goes on, no cars drive on it, and our mind hits the same confusion again and again. Then one day, the new road opens, and we “get it.” Once the cars can drive from one side of our mind to the other, the new idea feels obvious. And more, you are so changed by the new perspective that *you can’t even understand your own prior lack of understanding*. I don’t know any other field of study that so regularly gives you this experience. And if you track topics in mathematics, you can see this happening again and again: with base 10, fractions, algebra, calculus… it’s all hard, until it suddenly feels easy.

These jumps in comprehension can be thrilling, and they’re one reason math is so fun. But they do create a challenge for the student. The evidence that you learned something hard is that you feel like you’re stupid. That stupidity is essential to the process, not of pushing at the boundaries of what is known, but simply of getting your mind to take in tools and ideas more abstract than it was ever meant to learn. Students need to know that this feeling is the norm when it comes to learning math.

The centrality of stupidity can be especially challenging for the teacher. Being in the position of understanding means that we are distanced from someone who doesn’t understand. It takes an act of remembering that borders on wholesale invention to get a sense of how someone couldn’t see what seems so obvious once you see it. What’s happening in their mind? If you couldn’t understand your own inability from a few weeks ago, how do you understand theirs?

I’ve come to believe that one of the best ways to address the centrality of stupidity is to take on two opposing efforts at once: you need to assure students that they are *not* stupid, while at the same time communicating that feeling like they are stupid is totally natural. The message isn’t that they *shouldn’t* be feeling stupid – that denies their honest feeling to learning the subject. The message is that *of course* they’re feeling stupid… that’s how everyone *has* to feel in order to learn math!

It’s all just another reason math is deviously difficult, and spectacularly wonderful to learn. Is there any other study that changes you so deeply, and so often?

It resembles only ever doing easy runs. If a run isn’t some times harder than usual you aren’t going to improve. A lot of people quit running because it’s hard.

How you can learn anything unless you first clear out space in your mind for new ideas? Being aware that you don’t know is the first step toward learning something new. There’s Zen story in which the master pours a cup of tea and keeps pouring as tea spills over the sides of the cup. He then explains that a full mind cannot accept new ideas. In product design, which I coach, the #1 barrier to making progress is the inventor being overly attached to a clever solution they have created. The point of design is to serve the needs… Read more »

I’ve just started the climb toward a undergraduate in mathematics; I’m grateful for this insight that reinforces removing one’s own preconceptions to be humbled by every step of the way.

If not for people telling me that it was natural to feel stupid, I’m not sure how far I would have gotten in mathematics.

I urge us all to think about the language used to describe the moments you describe as feeling stupid. I’d urge ‘stupid’ to be replaced with ‘ I’m not making sense of that …..yet.” No one understands stuff they don’t understand until they do understand it !! In teaching teachers about the learning theories of Piaget he brilliantly coined not only the idea of limitations of thinking as a result of a child’s stage of learning ( cited by the liquid capacity assessment in this article ) but also the forever learning process he called ‘Constructivism’. In learning anything new… Read more »

I agree that it’s important to help kids (and adults) not feel stupid while they’re grappling with new ideas. I’m writing about something a little different here, though. The “stupid” feeling I’m referring to is the one you have AFTER you understand the new idea, and you apply it retroactively to yourself: “why did it take me so long to understand this obvious thing?” I think it is something particular about math that it’s so common to feel stupid after you’ve succeeded in learning something new. But I agree with the main point: feeling stupid doesn’t mean that you are… Read more »

When somebody turns this article into a ~6 minute video aimed at teenagers, I will show it to my 9th grade Algebra and 12th grade AP Stats students the next day!

I was just trying to figure out how to print this for my students to read! A video would be amazing!

Been saying this to my science students for years. Students that are good at science aren’t the ones that “get it”, they’re the ones that enjoy not getting it, being confused, finding answers they didn’t know. Stephen Hawking wasn’t the best because he knew about Black Holes – he was the best because he DIDN’T know, and spent his life determined to find out!!!

I am a teacher who supports math teachers in their efforts to improve math instruction. The biggest obstacle for most teachers to overcome is the fear of not knowing something, but the most incredible growth and joy for the teacher (and students) come when the teacher feels comfortable enough to see the unknown as an opportunity to grow—becoming a student alongside their own students.

The truth revealed I am most often a traveler into the unknown with both…. stupidly joyful as we go.

(Thank you, Dan)