> if we don’t feel stupid it means we’re not really trying
He’s really talking about curiosity. Calling it stupid is a cheeky glass-half-empty framing. But one of the big problems with math education is that we force students to struggle to a far greater degree than other subjects, under a belief that personal struggle, the “trying”, is the only way to truly get it. This is a pervasive cultural belief that extends to work and money too. And we force it from the beginning when kids are very young, without taking the time to develop their curiosity, and without setting up the system to gracefully nudge people who for whatever reason don’t see why they should try so hard.
Personally I suspect there are lots of things that could help motivate many more students than we do using our current system of demanding kids “try” to grok abstract rules using Greek letters. A combination of more visual storytelling, math history, and physical less abstract problems, along with a grading and progression system that ensures kids get it before moving to topics that depend on having got it might help a lot of people; too often kids are pushed to progress without ensuring they’re ready, and once that happens, “trying” is a fairly unreasonable expectation.
Think about how you learned your first language. Your mom taught it to you by rote repetition. She didn’t expect you to come to any of it on your own, and you weren’t expected to struggle with grammar or understand the rules or judged for getting them wrong, you were just gently corrected when wrong and celebrated when right. I don’t know if first language learning is a good way to learn math, but it is obvious that we have alternatives to today’s system and that today’s system isn’t serving everyone who’s capable of doing math.
> if we don’t feel stupid it means we’re not really trying
Sounds a bit like Kernighan's lever: "Debugging is twice as hard as writing the code in the first place. Therefore, if you write the code as cleverly as possible, you are, by definition, not smart enough to debug it. Yet." [0]
[0] https://www.linusakesson.net/programming/kernighans-lever/in...
In its impact on teaching, I'll say that based on teaching since 1979, students take feeling stupid as convincing evidence that their instructor is doing a bad job. No amount of assuring them that it is the gateway to enlightenment or however you put it will save you.
My current view on this is that it's a symptom of exactly what "expertise" means in academia. It does not mean expert judgement, nor expertise forged in experience.. no it means being an expert at giving accounts of one's knowledge in connection with other explicit accounts of knowledge.
Very little of anything worth knowing, in practice, can be given this account or a reliable one at least (physics sure,.. teaching?). Say, after decades of teaching, an exceptional teacher is not going to be able to (in general) report their methods in terms of the explicit accounts of methods as established in books. These are highly varied anyway, and full of rival theories.
Indeed, a person who could give such a count is most likely to be a poor teacher by comparison: since all their labour has been in the creation of these accounts, not in teaching (or far less).
You cannot do both. You cannot both acquire a vast depth of expertise that grounds good judgement (risk/reward, problems that arise in practice, context-sensitive question, intuitions for failure/sucess, etc.) -- and develop baroque accounts of that knowledge (its origins, remembering which papers you read, remembering all your projects, all the theories developed by academics, their history, and so on).
If knowledge is only, as academics say, just their own sort of accounting -- then one would feel stupid all the time. Since almost nothing can be thus accounted for.. and yet the world is replete with highly practiced experts in a very large number of domains.
Was anyone else very unimpressed with the video of the kid, water, and playdoh, until the very end? The whole time I was thinking that it's clearly a miscommunication, they are just assessing the kid's understanding of the word "more". If he thought it meant the tallest one, then all his answers would be correct.
But at the end, there is a question about sharing a graham cracker, which I am 100% sure a child of that age understands. They want at least the same amount of graham cracker as the other person. The kid also gets that one wrong, at the cost of his own bottom line. That really sold it.
This seems analogous to productive laziness in engineering. A good engineer is a bit lazy in a certain kind of way: they think about how to simplify or if that fails isolate or manage complexity. An insufficiently lazy engineer will create mountains of hideous complexity full of opportunities to show off but horrible to maintain and brittle.
The best feeling of my PhD was whenever two intuitions (or what I thought were facts) were predicting different outcomes. While maddening and nothing seemed to make sense, it was also the feeling of some big revelation waiting nearby to be found.
Indeed, and I find that my humanities/law-inclined smart friends don't reading math texts is supposed to make you feel this way. They read through a 100 pages of law textbooks and at no part do they feel dumbfounded by a paragraph or get stuck on a page for an hour. It's hard to learn it for sure, but you can read and read and read it. Reading math, on the other hand, is a staccato, a constant stop and go (and flip the pages back). One evening I might only progress 5 pages in the math textbooks because I stop after half a page to draw some sketches, some diagrams. Then I stand up and walk up and down the corridor for 10 minutes thinking things through and whether my current understanding makes sense and adds up to explain what I just read. But they aren't familiar with this mode of reading and working though a text, they think they are stupid or "non-math" people for not getting the meaning instantly, like they would in a book about law or marketing.
I think it’s mostly just that math education is largely suboptimal. It’s really an area where students hugely benefit from individual teaching. It’s cool that AI is making that accessible.
To an extent the techniques are still woefully primitive too. The standout for me personally is the calculational proof. It’s arguably the biggest advance in how math is done since the equals sign, but despite that it’s still rather uncommon. I suspect it will be another generation or two before it really catches on. Thankfully mechanical checking will drive adoption.
I had a very similar observation about engineering early in my career. The first project I worked on professionally felt vast compared to anything I'd seen in school. At first I was embarrassed to be new, to have to ask questions, to have to deal with solving problems in areas I didn't fully understand. It took months and months to "come up to speed", and I felt that I was drowning in complexity and unqualified for the work I was doing. Ultimately I came to understand that this is the normal state of engineering, especially when innovation is happening. The bulk of the work in engineering (not all of it, but the vast majority, especially in software) is fully understanding the problem space, the tradeoffs between alternative paths, understanding how your solution holds up and fixing bugs. In short, once you've gotten all your questions answered and finally feel fully qualified and no longer ignorant, you've also solved the problem you were working on. Time to move on to the next thing.
When I realized that, I realized that feeling dumb was actually normal, and that I should embrace it and expect to spend the majority of my career in that state. Not only did this dissolve my embarrassment, but it made me seek out ways to thrive in uncertainty and chaos -- which skills have been to my advantage for many years.
It is uncomfortable to admit you don't know things, or you don't know the best way to proceed, or you don't understand something. The temptation is to downplay that, to pretend you understand, to retreat toward the things you understand well. But poking at the unknown is how you get smarter, and ultimately how you solve problems. It takes courage, especially in a crowd, but it is also what solving problems normally feels like.
I want to distinguish two sources of "feeling of stupidity". One come from the challenge of grasping a difficult concept. The other is the smack on the head when you fail to see a simple but brilliant insight. In my view, you should not feel stupid in either situations, and the teacher should try to ward you against this feeling.
For the first type, I argue it's simply the resistance to a new mental model. The article's example of epsilon-delta language is a perfect example. It's a new way of thinking that takes time (and it did historically) to sink in. Competing on how fast you grasp this new concept is stupid. When the new mode of thinking becomes natural, it won't care how long you took to adapt to it.
For the second type, it's simply an impossible standard to reliably have eureka moments. Clearly, smarter people will have more of these than the average people, but no one can do this reliably. On the other hand, while it takes more work for us mortals to have these insights than a genius, there are plenty of ways to get there that don't require a super high IQ. Teachers should try to foster these moments because they are huge confidence builders, but try to minimise the impact of someone showing off their brilliance.
There is a related video by Prof. Courtney Gibbons about the feeling of "Math is hard" and how you just have to get used to it: https://www.youtube.com/watch?v=kenf8E1RuoA
I send this to students who feel discouraged by university-level topics. If math professors find things difficult, then you're probably OK... just keep hacking at it!
> ... if we don’t feel stupid it means we’re not really trying.
> Science involves confronting our 'absolute stupidity'.
I understand where the author is coming from, but these are just useless statements. Stupidity and knowing that you don't know stuff are not the same thing. The former involves an inability to understand or learn, whereas the latter involves an acknowledgment of our current state of ignorance and that we can do better.
I don't believe one can be successful in science by constantly feeling stupid and getting used to it. You have to be comfortable with not knowing stuff, but with the drive and self-confidence that you can discover new things and expand your knowledge, which is of course not easy either.
And how do we formalize stupidity aka human beehiveour?
We write linq-pad queries against the planetary NSA Meta+C db collected by those neat seeing stones everyone carries since 2008. Its anonymous, its collected unaware, its omni-present, its not deformed by the questionnaire, its obvious the conclusion of the science of neurons and the model of the mind.
And now for a word from our project lead: Peter.
I prefer the notion of Productive Failure [0].
Nice bookend to the other link. “All I want is a 17 sided polygon on my tombstone… here’s a simple guide on how to draw it.”
“Best I can do is a star.”
This is called “productive struggle” in the literature.
On the contrary, the problem with elementary mathematics education is that teachers don't tell students that they're supposed to not feel stupid at the point that they understand something. Students think it's fine that they mechanically do long division without understanding how it works. Then the next year, they have to be taught how to mechanically do long division again, and they still don't know why it works. Eventually, their foundation is so shaky that they don't understand why anything they're doing works.
I think this process of uncovering your stupidity / confusion could be gradual as Scott makes the point below. It need not be a single "Aha" moment.
https://scottaaronson.blog/?p=4974
> OK, but why a doofus computer scientist like me? Why not, y’know, an actual expert? I won’t put forward my ignorance as a qualification, although I have often found that the better I learn a topic, the more completely I forget what initially confused me, and so the less able I become to explain things to beginners.
I imagine that an issue for many isn't so much that feeling stupid is uncomfortable so much as it's a good heuristic for when you're in over your head in a way that could be dangerous to your life or livelihood. So then, it's actually a matter of trust: "I trust that wrestling with this problem for a few hours/days/weeks isn't going to disrupt my ability to get food/pay rent/be physically-safe." It's super easy to plow through feelings of insecurity when you can convince yourself that you're actually going to be secure, in the long run. If there are, however, negative and material consequences for getting things wrong...
The lawyer friend who dropped out went into a field where her "bag was secured", to use a contemporary phrase. The author acknowledges that she was capable; perhaps the root issue wasn't "feeling stupid", so much as "feeling like I'm going to be broke even if I crack this nut."
I equate feeling stupid with negative re-inforcment. When I find the solution I pat myself in the back (we are programmers, nobody cares). When I can't find the solution and then is shown to me I feel bad because I my skills proved insufficient.
Yet
That sense of feeling of success and failure are so fleeting I learn to let go of both.
They are the fuel that drive my quests but I don't sleep with them. They are both volatile.
> Students need to know that this feeling is the norm when it comes to learning math.
Cedric Villani, Field medalist, was saying the same thing. The problem is that not everyone is equally stupid, and if you're too stupid, you won't get the job or be a low performer in your field/team.
One of my wife's superpowers is that she isn't afraid to look stupid. When we were about to have a kid, my wife plied my mom with a lot of stupid questions. "Do I have to play with the baby all the time? What happens if the baby annoys me?" The result was that she now has an extremely solid baseline of knowledge about how to deal with babies.
Interesting
We don't really have a good name for the emotion that this article describes as "feeling Stupid".
You know what it feels like to be stupid? It feels like you are really smart! I feels like you already know all you need to know, about, say, vaccinations, or about hot to parent somebody else's children.
I'm currently a student teacher, and I'm really struggling to get this point across to my students. I'm asking them questions which make them really think, and since no other teacher has done that to them before, they feel really stupid. But they are not being stupid. If they were being stupid, they'd feel like they had it all figured out.
So yeah, there is this emotion, commonly but unfortunately called "feeling stupid", which you feel when you are trying to figure something out. What would be a good name for that emotion???
"Stupidity in math"??? Hmm ...!
The article for this thread has:
"First, I don't think students are made to understand how hard it is to do research."
Well, from my career and Ph.D., I found something different:
(1) In pure math, read some of the best materials, look for unanswered questions, and pick and try to answer one of those.
Took me 2 weeks. Paper got accepted in the best journal in the field.
(2) Applied Math. In practice, outside of math, see a need, problem, weaknesses in what is known/done, pick, and try to answer.
Another few weeks, then submitted to a journal and got published right away, no revisions.
(3) Engineering. In some real situation with some big, i.e., expensive, activity, see where a lot of money is being spent, pick, attack, maybe solve, and write it up.
Most of the work was done an airplane flight, and the results became dissertation in an engineering school.
One device: Pick some advanced topic in math, e.g., 'tightness' in probability theory, and use that as a tool to get some new results on the problem picked. There are problems where such a tool could be used but is poorly known by people concerned with the problem.
More difficult problems:
For the work in (1)-(3), still waiting for a check.
Problems:
(A) Getting hired for such work.
(B) When do get hired, get paid enough to buy a house and support a family.
Lessons:
(i) For academics, never saw where the professor got paid enough to buy a house and support a family.
(ii) For business, if do get hired and paid and are doing good work, then maybe, still, are not paid enough to buy a house and support a family. But the owners of the business are making more money from the good work than they are paying.
So, maybe start a business, do some good work key to the success of the business, and keep all the earnings for yourself.
Looking around, it appears that the people who make enough to buy a house and support a family own a successful business, although maybe just a successful LLC (limited liability corporation -- a relatively simple business type) as a physician, lawyer, tax expert, etc.
So, in simple terms, start a business, e.g., initially as just an LLC, and continue with hard work, good ideas, good insight, until have a successful business, and then sell it. Uh, if the money was from successful 'pesonal services', e.g., law, medicine, then may not have anything to "sell".
These days the Internet may provide some good opportunities.
"Science makes me feel stupid too. It's just that I've gotten used to it."
…And me too, especially when I think outside the orthodoxy, which I do all the time.
As I must be wrong of course I don't talk about such stuff, as one would be lauged at and ignored.
> Science makes me feel stupid too. It's just that I've gotten used to it.
When I started my PhD program, a group of us were given a little talk by the department secretary.
She told the story of how she went to audition for Jeopardy!, a trivia game show. She saw a whole bunch of other people at the audition get really nervous and choke up; her take on it was that they were used to being the most knowledgable in the room -- they were used to sitting in front of the TV screen with their friends or family and knowing every fact, and when they were suddenly confronted with a situation where everyone was as knowledgable as they were, they were suddenly very intimidated.
She, on the other hand, was completely relaxed -- she spent her days working with Nobel prize winners and loads of other people for whom she had no doubt were smarter than her. Being confronted with loads of people smarter than her was a daily experience.
She told this story to us to say, a lot of you will experience the same thing: You were used to being the smartest person in your High School, you were even used to being the smartest person in your classes at the prestigious university you attended. Now you'll encounter a situation where everyone is like you: the best and most driven people in your classes.
You'll feel stupid and inferior for a bit, and that's normal. Don't let it bother you. Eventually you'll notice while that most of these other people have areas where they're better than you, they have areas where you're better. And there will still be the occasional person who seems better than you at everything: that's OK too. You're not the best at everything, and you don't have to be.