> 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.
For every job I've ever had, I've been the "math and functional programming nerd", where I know lots of tricks in Haskell and F# and even concurrency theory within Java. I felt very smart.
I went to ICFP in 2019, and I can say with a high degree of confidence that I was the dumbest person there [1]. Everyone was speaking on four-syllable mathematical notation that I had never heard of, and talking about intricacies in GHC that I wasn't really familiar with, and different aspects of type theory that were completely foreign to me.
It was very humbling; it didn't depress me or anything, but made me realize that there's a lot to learn and improve on, and the people there were actually extremely nice and gave me some pointers so I can get incrementally closer to being as smart as they are.
I think 2024 Tom would be the second dumbest guy in the room if I went again.
[1] Knowledge-wise, I have no idea about IQ or anything.
Just imagine how much of a waste of time it would have been to attend a conference where everyone was pointing out mathematical concepts you had heard of, talking about intracacies in GHC you were familiar with, and discussing different aspects of type theory you thought were completely trivial!
In a sense, this is how being in a foreign country where people talk in other languages and act within their own culture. They are more productive than you because the environment fits them more than it does you at that time.
And this acclimation is also similar to https://en.m.wikipedia.org/wiki/Mathematical_maturity, how you talk with math people the from math village and feel you are stupid because you're not familiar with the symbols, the vocabularies. But when you have learned the language, you will see that you become better and better at learning, you have the means to gain more means--forming some sort of a positive feedback loop.
That secretary is probably hard funded, makes more money than 75% of the scientists she supported, enjoys vacations, and will have a comfortable retirement.
Yeah, I had a similar sentiment as I read the last sentence: "You're not the best at everything, and you don't have to be."
It really does get harder to internalize this when it starts to involve real, tangible outcomes like money and job security. No one would reasonably argue that what she said wasn't true on some spiritual or personal level, but it feels like a nothing-burger when people are clearly in a competitive environment, a competitive program, a competitive job, etc.
> 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.
I'm a college dropout who has managed to work professionally, in software, in RF / embedded development, medical robotics, all over the web, and most recently in AAA games. I've called PhDs colleagues, people with multiple Masters, people with decades of industry experience at the top of their fields...
I always feel like the odd-man out, and, while it used to bother me a bit, I'm pretty content with the fact that I probably always will. I frequently feel like I'm inferior to my colleagues, because of the sheer depth of their knowledge as it relates to whatever the particular domain is.
But I have the sense that I'm doing something right because I get great reviews, I frequently find moments where I can teach my colleagues something they didn't know, and they come to me for help, advice, and say good things about me (and vice versa).
But it is still a very odd feeling and I think it'll be with me for however long I work in this industry.
Athletes deal with it too. You go from being the best football player in your school, maybe even the best football player in your college. Then you go to the NFL and you're middle of the pack. Lots of guys get lost at the transition: a first round draft pick gets to the NFL and immediately loses his mojo.
I feel like this "get ready to be surrounded by peers for the first time" or the related "you aren't used to working hard, but now you will actually have to work hard" speech was given to me in some form at the start of high school, college, grad school, and in many other contexts and intermediate milestones. It wasn't ever completely true, but I think if I went for a PhD it would (obviously) have finally been true.
To be clear, I'm not saying I was always smarter than people around me, I just felt like I never had to work as hard as I suspected even through my Masters program.
> 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.
I don't know your background, but speaking as someone who has done a PhD in Pure Math, and working with a lot of people who have done PhDs in Pure Math, I disagree with you. It's very much a case of feeling stupid, and being able to embrace that and live with it.
The "being curious" thing is independent of the "feeling stupid" thing. It definitely exists, but it's absolutely not the same thing.
Looking at the rest of your comment, maybe we have a lot in common, but I definitely disagree with a lot of what you've written here, so no doubt our experiences are wildly different. Perhaps over a coffee[0] we could talk constructively about education, math, struggles to understand, and work ethics, but suffice to say here that even if we do have substantial common ground, I think we might have very different points of view.
Of course you disagree with me ;) You’re talking about the minority of people who’ve had incredible amounts of math success (yes despite sometimes legitimately feeling very stupid). I’m not talking about my own experience, FWIW, I’m talking about the majority of people who never get anywhere close to a PhD in math, because they were left behind by our math education system. Obviously I don’t know exactly what you disagree with since you didn’t elaborate, but we have bumped into each other in mathy threads fairly often, right? If we get the chance some time, it would be super fun to discuss math & education over coffee!
As a self-proclaimed cognitive engineer (it’s really easy to proclaim stuff these days!), I absolutely agree with you, for two basic reasons:
1. I don’t think the author was just saying that math is hard or “a struggle”, I think they were specifically pointing out the unusual nature of math(s) as a collection of meaningfully novel cognitive tools rather than facts or recombinations of existing tools. AKA “feel stupid” means “feel embarrassed you don’t understand earlier because now it’s obvious”, not “take a long time to understand” or other synonyms for struggle. We all agree on the vague shape of the proposed improvements to pre-graduate math education, I would guess!
2. That’s not how first language acquisition works, at all: the rules of grammar—not to mention etiquette-are far more complex than most laymen imagine, and intentional parental involvement via correction or the occasional picture book is absolutely the exception, not the rule. This is the core insight driving Noam Chomsky’s lifetime of scholarship, and I think he would agree that childhood linguistic development is more similar to mathematics education than practically any other activity, if we’re talking about “feeling stupid” like the author is.
Well, mathematics at a graduate level really is a subject that can only be self-taught, as are most subjects at the graduate level. Yes, some guidance can be available, but the pedagogical hand-holding that is undergrad is simply not possible. The analogy to language only really applies to mathematics that is well-understood and can be taught this way. In grad school, almost no mathematics you encounter is that well understood, so the teaching methods are absent.
There are lots of areas of math grad school that are well understood. Pretty much everything up to quals (and some beyond that) is well known and teaching methods are far from absent.
There are two authors here, since the post contains an inset about dealing with your own ignorance by another professor. They aren't saying quite the same thing. The inset is saying that every grad student will confront their "absolute ignorance" and it will be difficult, scary and possibly painful. The author of the post is saying it can be a source of joy. I suppose they can be reconciled. It could also be that so little of our behavior is based on knowledge that the only sane reaction is at least somewhat negative, whether characterized by being overwhelmed, or sad, or detached.
Reframing it as curiosity is a good point, although the essay as written resonated with me because it emphasizes the "productive ignorance" of research.
One of the central problems of our time in research and academics, I think, is an incentive to focus on areas that are well-established because we know they are likely to produce results that we have confidence in (according to whatever inferential criteria we use). The idea of it being ok to not know something a priori, to have lack of confidence in it, seems sort of discouraged in the current climate, because it's too risky.
Dancing around the elephant in the room, the problem is financial risk, e.g. this isn't really about research this is a business, and business must minimize risk to be profitable?
> 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.
I disagree. In fact, I found that often the better and more didactically streamlined the exposition is in a book, the less deeply I end up learning the material. It is precisely the personal struggle, having to make my own sketches and derivations, starting out with a misconception because of bad phrasing in the book and having to explore that misconception until I find what I misunderstood etc. makes the knowledge stick much better because it now feels my own, like an intimate friend.
Spoonfeeding may get people quicker to the point of solving the standardized quiz at the end of the chapter but that's not the same as learning and understanding. Another instance of metric-chasing in action.
It's a bit like how I learned MS Office or Photoshop by trial and error as a kid, or programming by mucking around trying to make a website do what I want. And you bet it was a struggle. Struggle but with reward at the end. Compare that with a handholding tutorial where you do learn how to do whatever the tutorial makers had in mind, but it won't generalize as much. Sounds totally dry. I loved computers, but hated school lessons that tried to teach us MS Office in the handholding spoonfeeding way. It's the death of the subject.
It's a safari in a safe car, looking at the animals through binoculars vs running around in the jungle in your own adventure amongst the beasts.
kids struggle with grammar, pronunciation, pragmatics, vocabulary, etc. they do so naturally, and maybe you've forgotten your own struggles, but they're very real
i think we can do better than we are doing at math education, much better, but there is no way to learn math, or anything else, without diligent effort. it won't happen by passive absorption. you can listen to people speaking spanish all day every day for years without learning more than a few words of spanish if you don't make any effort
curiosity is one possible motivation for making that effort, but the immediate result of the effort is, at first, failure. that's true of language, it's true of playing the guitar, it's true of programming, it's true of throwing clay on a potter's wheel. that failure feels like being clumsy, weak, or stupid, depending on the form it takes
and that's what the article is talking about. trying to do things that are beyond your mental ability makes you aware of, and frustrated with, that mental ability. that's not curiosity, it's feeling stupid. it's also how you increase your mental ability!
But he is not talking about education, about doing a course, where "getting the right answer makes you file smart". He is talking about research where nobody knows the right answer.
Martin absolutely was talking about education; research is education. Granted, not early education, but I’m not making any claims about what he said, I simply used his quote as a segue to make an observation about the connections between research thinking and today’s math curriculum. Research is a continuous spectrum. We are expecting kids in elementary, secondary, and early college to have a research mentality and research level motivation in order to succeed in math classes, unlike some other subjects. (Classes which, btw, were all research topics at some point in time and took tens, hundreds, even thousands of years to develop.) The mentality and motivation are important if you want to end up doing any of the actual graduate, post-graduate, or career research where nobody knows the right answer. The kids who are pruned out by our math system never make it there, and many don’t even make it to functional math literacy, even though many/most are perfectly capable, and that’s unfortunate and doesn’t reflect well on our education system. I’m suggesting we can do better.
> He’s really talking about curiosity. Calling it stupid is a cheeky glass-half-empty framing.
I disagree.
How can you be curious without something you don't understand?
The point, as I see it, is that if you find yourself in a position of complete understanding, then you must also have a complete lack of curiosity. If you think you are in that position, then the way to revive your curiosity is to deconstruct your position of expertise, i.e., recognize your position of stupidity.
---
Curiosity is step 2. Stupidity is step 1. Learning moves us from step 2 to step 3. The important thing to recognize is that step 3 is actually a new instance of step 1. Expertise is the base case of this recursive tree traversal: it's how you stop the learning process.
> How can you be curious without something you don't understand?
Good question. You might have discovered my point: curiosity comes with stupidity, implicitly by definition, right? I think that’s what you’re saying too. Maybe you don’t disagree after all?
You can’t have curiosity without stupidity, as you rightly point out. Ignorance is probably a better word than stupidity. Using “stupid” is imprecise and was used here for a bit of surprise and humor.
You can have stupidity (ignorance) without curiosity. When that happens, perhaps the expected result is no progress developing new understanding nor lessening of ignorance.
Given that curiosity implies ignorance, and that ignorance alone is not sufficient for learning, what justification is there for claiming curiosity and ignorance are separate steps or separate things when it comes to education or research? I’m suggesting they are two sides of the same coin, they must both exist before learning happens, and neither one can come before the other. Calling it curiosity instead of ignorance or stupidity is perhaps a kinder framing, especially for people who might not immediately get the self deprecating humor of “stupidity”.
As someone who is currently studing maths I strongly disagree with this
> He’s really talking about curiosity. Calling it stupid is a cheeky glass-half-empty framing.
One of the most important characteristics to succeed in maths is the ability to acknowledge things you don't understand, to fail, and to persist in spite of failure. Trying really is the only way to understand some hard things, because there are some things that are conceptually extremely difficult.
He's not talking about relative stupidity where there are other people you feel are smarter, he's talking about stupidity on an absolute basis. You don't know. You don't understand. But somehow you have to find a way to carry on, and then later on, looking back once you do understand, you're baffled by why you didn't know/understand or couldn't see some crucial things. You have climbed up a ladder and pulled it up behind you and it's hard now to imagine what it is like to be on the ground.
It's not about curiousity. Of course you have that - if you didn't you wouldn't be there in the first place.
I think the framing as "stupidity" is to highlight that you don't always chase creative questions. Quite often, you should chase the obvious or understood points.
The problem, I think, comes from the weaponization of "stupid" against people. The XKCD of the lucky 1000 plays a good role here. If you are constantly deriding others for stupid takes, then anyone that derides one of your stupid takes will hit hard. And that seems to be getting worse.
> 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]
“Simplicity is a great virtue but it requires hard work to achieve it and education to appreciate it. And to make matters worse: complexity sells better.” — Edsger Dijkstra
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.
I've become convinced that, in the end, no one really teaches you anything, you end up teaching yourself. That phrasing is a bit hyperbolic. It's more accurate to say a good teacher only gets you 50%, 60%, maybe 70% of the way there, and it's up to you to get you to 100%.
To be able to truly learn any given concept means being capable of answering a practically infinite number of different questions about that topic. The process of teaching is essentially trying to uncover which questions the student can't answer. The challenge, of course, is that the student doesn't know what questions they can't answer, because the questions haven't occurred to them. That is, until they start testing themselves, to see if they really do understand the concept.
Problem sets in textbooks are the canonical way of addressing this teaching challenge, but there are only so many pages in a textbook, and there are other concepts that need to be taught, so the scope of the problem sets are necessarily finite.
How many times have you nailed all the problems in a book, only to discover that there was some aspect of the topic you didn't understand, despite getting all the right answers?
The following two quotes from Martial Arts have been quite helpful to me in motivating my study efforts;
The Master shows the Gate, but it is the Student who has to walk through it.
To show one the Right Direction and Right Path, Oral Instructions from a Master are necessary but Mastery of the Subject only comes from one's own Incessant Self-Cultivation.
There is also a great inspirational story in the Mahabharata of "Ekalavya" (https://en.wikipedia.org/wiki/Ekalavya) who became an exceptional archer through self-training.
Eklavya inspires a life-long learning philosophy and his presence seems to be a celebration for the masses. In this EklavyaParv, the motto is 'You Create Yourself" and the legend of Eklavya is a testimony that is forwarded by many thinkers as well. The discipleship that Eklavya represents is the best for a student and enables one to be the creator of one's own destiny.
Adapting to current times, "The Master" can be a "Good Book" and you can have "Many Masters" but the effort and learning has to happen within the Student.
Source: I self-taught myself Martial Arts (JKD, Karate, Taijiquan) from books when i was young. Decades later when i did join a dojo to study under a Master, i was one of the top students with good skill and power.
This 100%. I taught myself *everything* in college. I relied on textbooks and online resources to teach myself the material outside of class. It worked pretty well. I wasn't the top of my class, nor have I retained all the things I've learned (but who does?).
Fortunately, what I have retained is the ability to pick up almost any subject and learn about it on my own. That's more important than anything they can teach you in a classroom.
I would claim there is no 100%. At least in engineering my professors were saying (paraphrasing) that it's all about trade-offs and most of the times there is not "one answer". I think many education systems (up to graduate at least) instill the idea that there is always "one answer" which has many bad repercussions later (people seeing things in black and white).
I don't think a student needs to always answer a question "on the spot". Being able to find an answer in a reasonable amount of time and explain an answer would be in my opinion more valuable. So then it's more about "how efficient can the student give the answer to the question" (answer on the spot, spend one hour, spend one week, etc.).
Tangentially, modern educational paradigms also characterize learning as a process of construction happening within the student's mind, rather than a transfer of knowledge.
Idk, one good prof made me a fifty percent better mathematician in a single course. I would have to slam my head in the wall for a year or two to do it without him.
I'm a recent graduate, but also worked in colleges for a couple of decades. Students I've interacted with almost universally blame themselves first when they don't get something. Only when they see many fellow students in the same boat do they tend to blame the teacher. From my vantage point it seems you're either assuming their frustration with the subject is frustration with the instructor, or overgeneralizing based on non-representative students, a non-representative subject, or... a non-representative instructor. I'm not trying to be a jerk, but career longevity wasn't a great indicator of pedagogical insight or its prerequisite EQ.
A good teacher can offer some tools and methods (as in methods of "learning to learn") to bridge the gap between stupidity and (for lack of a better meaning) "enlightenment" Something especially my math teachers in general lacked..
I disagree. I think good students judge their teachers based on how motivated and passionate the teachers themselves are about a topic.
Though I've had classes where I only realized their importance multiple semesters after having them. (Mostly the non technical, business classes)
I had a junior helpdesk employee that I was training/mentoring years back. He was 20 years old, fresh out of tech school. He was good at what he did, but he only did things he knew how to do. When he didn't know something, he'd ask me. Which is great. I'd say "Well, this sounds like DNS, it's like a phone book..." or "That's an APIPA address, it must not be getting DHCP. The computer shouts out to the network asking for an address..." and so forth. However, he kept asking the same questions.
After a few months in one of our monthly meetings he kind of broke "I don't understand what I'm doing out there, you need to train me! I need to be trained!" Completely perplexed I asked him what he was talking about. "You just answer my questions, but you're not training me!" I realized he was expecting me to learn him the answers to everything. I had to explain to him that the responsibility of learning was actually on him. "This isn't school, there's no study guide. We have documentation and Google. It's your responsibility to read it and make sense of it."
I told him that I can give him all the puzzle pieces but I can't put them together for him. To be fair, helpdesk is kind of about making things work and remembering the quick fixes and tricks for things to close out your tickets.
So I said, "Ok, I think you need a project. What do you do at home for fun?"
"Well I play a lot of video games."
"Perfect, we're setting up a Minecraft server". He laughed.
I said "No, I'm serious. We're using like 5% of this massively overblown server that was sold to us. Maybe this will help you put the pieces together."
I gave him a restricted vSphere account for his DMZ'd VM, sent him a guide and unleashed him.
"Well, I've never done this before..."
"Exactly. That's how you learn my dude."
"But..."
"RTFM"
"This VM doesn't do anything."
"Right, it needs an OS."
"We'll how do I install one?"
"Here's a guide."
"I installed the OS, how do I get into it?"
"SSH"
"No I mean the desktop."
"There isn't one."
And so he learned that a computer isn't the Windows desktop.
"I can't SSH in, it says connection refused."
"Right, that's the firewall."
"Well what do I do?"
"Google UFW"
"I can't SSH in anymore, it says connection timed out."
"Can you ping it?"
"No."
"Check the IP address in vSphere"
"It changed..."
"Why?" I asked.
"DHCP...! That's what a static IP is for!"
From then on he finally understood that learning actually takes a little effort and curiosity AND yes, it's OK to Google things. He had this idea that he had to know everything, memorize everything, and looking things up was "cheating". Not knowing something and feeling dumb is actually where learning happens rather than pure repetition.
About a year later he thanked me and said that he completely misunderstood my motivations initially and that he thought I was brushing him off and being lazy, when in reality I was giving him the opportunity to learn by not feeding him every detail. He felt like he was failing because he didn't know all the answers and said that he looked back at himself a year ago and couldn't believe what he was doing now and how far he'd come. "I had no idea what an IP address was but now I understand how the packets move through the switches, request an address..." etc.
We both ended up quitting and going our separate ways as the IT department there was an absolute shitshow. He's now a sysadmin and we chat now and then and he's mentioned that he's actually glad he learned in such a fucked up environment because you were absolutely forced to understand due to all the ridiculous hacks and workarounds that had been piled on over the years. Nothing could be taken for granted.
I learned in a similar way and I think trial by fire may be one of the best teachers. "Smooth seas never made a skilled sailor."
Learning only happens through a mish-mash of Trial-and-Error, Trial-by-Fire, Questioning, Curiosity, Reading/Copying/Mimicking, Thinking, Reflecting and finally Doing. All of the above are needed in some measure.
The trick is to do the above without losing our self-confidence in ourselves (we are guaranteed to feel "stupid" during the learning process) that "we can grok it" at some level and over a period of time. The problem today is that there is so many aspects and so much to learn about any one thing that students are trying to move very fast to learn everything which is an impossibility; they need to ruthlessly cut down on all inessentials and learn to focus on only one or two core things i.e. "sift the wheat from the chaff".
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.
My view is that the when academics call other academics “experts,” it’s just noting who works professionally on a topic. Usually those people will be able to give a reasonable account of their field. But a lot of the game is reviewing the specific subject matter before a presentation. Or steering a conversation toward familiar ground.
A teacher of topic X is not an expert in topic X. They are an expert in “teaching topic X.”
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.
I think this video, and most such experiments involving children, suffer from a central issue: children are extremely sensitive to what they think adults want to hear. The kid watched the woman manipulate the things. He probably figured it would be rude not to acknowledge her changes. They should figure out a way to have another kid as the experimenter and disguise the obvious test/interview situation somehow. Especially the cracker thing feels sooo odd. I can’t believe he would let a peer get away with it, but who’s he to argue with an adult, much less a stranger?
Also children are brought up with super obvious problems like “what object fits into which hole?”. I feel like some of these tests measure less the child’s understanding of the given problem per se and more whether they have previously been introduced to trick questions/illusions.
And even controlling for all that, you’re totally right. Even adults get confused by mass, weight, volume, apparent size etc. sometimes. The kid doesn’t even intellectually know those concepts. His only input here is by sight, but his answer may be different if he got to hold both objects and feel their weight.
The crackers question changes from amount to fairness. It's possible that the kid uses a different rule to evaluate it. Like "a small person needs small portions" or some variation.
I'd be convinced that the kid doesn't get it if they swapped crackers and then it stopped being fair.
> It's possible that the kid uses a different rule to evaluate it. Like "a small person needs small portions" or some variation.
Fair enough. It's definitely missing the opposing case where 1 graham cracker each is split on only one side and therefore the situation goes from fair to unfair, even though it's the same amount of graham cracker.
> I'd be convinced that the kid doesn't get it if they swapped crackers and then it stopped being fair.
I wouldn't count on that. I think the kid (and most adults) would claim it's fair if they thought they could get away with it. The deep intuition for "fair" that I expect from children would be derived from past experience negotiating with peers, not from any kind of moral theory.
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 analogy there isn't as direct as I'd like, because both engineers found a path between A and B[0], and I'd thought TFA was saying (because my experience has been) the initial feeling of stupidity comes from not seeing any path at all between them[1].
The way I currently think about it is that a learning space is a sort of skill tree (poset), and the easy concepts/skills are the ones where we can learn all the prereqs, and then just combine them (join reducible elements), whereas the tricky concepts/skills (the ones which make us feel stupid) are the ones that only have a single prerequisite, so we can't just combine things we already know, but have to do something novel[2] in order to acquire them (join irreducible elements).
[0] and both of them were probably confident all along that they'd make it, the former because they had already sketched out a few likely paths in their mind, the latter because they've always managed to muddle through before
[1] furthermore, having travelled from A to B multiple times, it's difficult for a teacher to empathise with those who are not following
[2] to https://news.ycombinator.com/item?id=41566945 I'd say maybe that's why traditionally we've waited until people are in their late teens/early twenties, significantly ego-invested in research as a career path, and have an experienced mentor, before we throw them overboard into the lake of obligate stupidity[3]?
(there are exceptions: Feynman habitually tested himself by attempting to self-derive [practising research mode] before allowing himself to read expository texts [entering spectator mode]. Somewhere he[?] claims something along the lines of him not being that smart, just that people were impressed after they asked him questions for which he could give answers he'd already spent hundreds of hours thinking about)
[3] everybody genius until it time to do genius shit, yo
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.
Readit News
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.
I went to ICFP in 2019, and I can say with a high degree of confidence that I was the dumbest person there [1]. Everyone was speaking on four-syllable mathematical notation that I had never heard of, and talking about intricacies in GHC that I wasn't really familiar with, and different aspects of type theory that were completely foreign to me.
It was very humbling; it didn't depress me or anything, but made me realize that there's a lot to learn and improve on, and the people there were actually extremely nice and gave me some pointers so I can get incrementally closer to being as smart as they are.
I think 2024 Tom would be the second dumbest guy in the room if I went again.
[1] Knowledge-wise, I have no idea about IQ or anything.
And this acclimation is also similar to https://en.m.wikipedia.org/wiki/Mathematical_maturity, how you talk with math people the from math village and feel you are stupid because you're not familiar with the symbols, the vocabularies. But when you have learned the language, you will see that you become better and better at learning, you have the means to gain more means--forming some sort of a positive feedback loop.
As a 50 years old person that some time ago was one of these brightest in class I can say that for most of us, people, it is:
You're not the best at anything, and you don't have to be.
"In a way, meeting those people was liberating. I will never be a world champion at anything, so I might as well play for the love of the sport."
It stuck with me and has become more meaningful as I get older and a bit slower.
[1] https://news.ycombinator.com/item?id=40557491
It really does get harder to internalize this when it starts to involve real, tangible outcomes like money and job security. No one would reasonably argue that what she said wasn't true on some spiritual or personal level, but it feels like a nothing-burger when people are clearly in a competitive environment, a competitive program, a competitive job, etc.
I'm a college dropout who has managed to work professionally, in software, in RF / embedded development, medical robotics, all over the web, and most recently in AAA games. I've called PhDs colleagues, people with multiple Masters, people with decades of industry experience at the top of their fields...
I always feel like the odd-man out, and, while it used to bother me a bit, I'm pretty content with the fact that I probably always will. I frequently feel like I'm inferior to my colleagues, because of the sheer depth of their knowledge as it relates to whatever the particular domain is.
But I have the sense that I'm doing something right because I get great reviews, I frequently find moments where I can teach my colleagues something they didn't know, and they come to me for help, advice, and say good things about me (and vice versa).
But it is still a very odd feeling and I think it'll be with me for however long I work in this industry.
To be clear, I'm not saying I was always smarter than people around me, I just felt like I never had to work as hard as I suspected even through my Masters program.
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.
The "being curious" thing is independent of the "feeling stupid" thing. It definitely exists, but it's absolutely not the same thing.
Looking at the rest of your comment, maybe we have a lot in common, but I definitely disagree with a lot of what you've written here, so no doubt our experiences are wildly different. Perhaps over a coffee[0] we could talk constructively about education, math, struggles to understand, and work ethics, but suffice to say here that even if we do have substantial common ground, I think we might have very different points of view.
[0] Other beverages are available.
1. I don’t think the author was just saying that math is hard or “a struggle”, I think they were specifically pointing out the unusual nature of math(s) as a collection of meaningfully novel cognitive tools rather than facts or recombinations of existing tools. AKA “feel stupid” means “feel embarrassed you don’t understand earlier because now it’s obvious”, not “take a long time to understand” or other synonyms for struggle. We all agree on the vague shape of the proposed improvements to pre-graduate math education, I would guess!
2. That’s not how first language acquisition works, at all: the rules of grammar—not to mention etiquette-are far more complex than most laymen imagine, and intentional parental involvement via correction or the occasional picture book is absolutely the exception, not the rule. This is the core insight driving Noam Chomsky’s lifetime of scholarship, and I think he would agree that childhood linguistic development is more similar to mathematics education than practically any other activity, if we’re talking about “feeling stupid” like the author is.
One of the central problems of our time in research and academics, I think, is an incentive to focus on areas that are well-established because we know they are likely to produce results that we have confidence in (according to whatever inferential criteria we use). The idea of it being ok to not know something a priori, to have lack of confidence in it, seems sort of discouraged in the current climate, because it's too risky.
I disagree. In fact, I found that often the better and more didactically streamlined the exposition is in a book, the less deeply I end up learning the material. It is precisely the personal struggle, having to make my own sketches and derivations, starting out with a misconception because of bad phrasing in the book and having to explore that misconception until I find what I misunderstood etc. makes the knowledge stick much better because it now feels my own, like an intimate friend.
Spoonfeeding may get people quicker to the point of solving the standardized quiz at the end of the chapter but that's not the same as learning and understanding. Another instance of metric-chasing in action.
It's a bit like how I learned MS Office or Photoshop by trial and error as a kid, or programming by mucking around trying to make a website do what I want. And you bet it was a struggle. Struggle but with reward at the end. Compare that with a handholding tutorial where you do learn how to do whatever the tutorial makers had in mind, but it won't generalize as much. Sounds totally dry. I loved computers, but hated school lessons that tried to teach us MS Office in the handholding spoonfeeding way. It's the death of the subject.
It's a safari in a safe car, looking at the animals through binoculars vs running around in the jungle in your own adventure amongst the beasts.
i think we can do better than we are doing at math education, much better, but there is no way to learn math, or anything else, without diligent effort. it won't happen by passive absorption. you can listen to people speaking spanish all day every day for years without learning more than a few words of spanish if you don't make any effort
curiosity is one possible motivation for making that effort, but the immediate result of the effort is, at first, failure. that's true of language, it's true of playing the guitar, it's true of programming, it's true of throwing clay on a potter's wheel. that failure feels like being clumsy, weak, or stupid, depending on the form it takes
and that's what the article is talking about. trying to do things that are beyond your mental ability makes you aware of, and frustrated with, that mental ability. that's not curiosity, it's feeling stupid. it's also how you increase your mental ability!
Feeling stupid is when you're confronted with an external you cannot understand why it works.
Curiosity can help push through feeling stupid, but you can be curious about something you won't struggle understanding it at all.
I disagree.
How can you be curious without something you don't understand?
The point, as I see it, is that if you find yourself in a position of complete understanding, then you must also have a complete lack of curiosity. If you think you are in that position, then the way to revive your curiosity is to deconstruct your position of expertise, i.e., recognize your position of stupidity.
---
Curiosity is step 2. Stupidity is step 1. Learning moves us from step 2 to step 3. The important thing to recognize is that step 3 is actually a new instance of step 1. Expertise is the base case of this recursive tree traversal: it's how you stop the learning process.
Good question. You might have discovered my point: curiosity comes with stupidity, implicitly by definition, right? I think that’s what you’re saying too. Maybe you don’t disagree after all?
You can’t have curiosity without stupidity, as you rightly point out. Ignorance is probably a better word than stupidity. Using “stupid” is imprecise and was used here for a bit of surprise and humor.
You can have stupidity (ignorance) without curiosity. When that happens, perhaps the expected result is no progress developing new understanding nor lessening of ignorance.
Given that curiosity implies ignorance, and that ignorance alone is not sufficient for learning, what justification is there for claiming curiosity and ignorance are separate steps or separate things when it comes to education or research? I’m suggesting they are two sides of the same coin, they must both exist before learning happens, and neither one can come before the other. Calling it curiosity instead of ignorance or stupidity is perhaps a kinder framing, especially for people who might not immediately get the self deprecating humor of “stupidity”.
But alas, I never thought as a kid that I didn't have time for other things. I was always into something.
> He’s really talking about curiosity. Calling it stupid is a cheeky glass-half-empty framing.
One of the most important characteristics to succeed in maths is the ability to acknowledge things you don't understand, to fail, and to persist in spite of failure. Trying really is the only way to understand some hard things, because there are some things that are conceptually extremely difficult.
He's not talking about relative stupidity where there are other people you feel are smarter, he's talking about stupidity on an absolute basis. You don't know. You don't understand. But somehow you have to find a way to carry on, and then later on, looking back once you do understand, you're baffled by why you didn't know/understand or couldn't see some crucial things. You have climbed up a ladder and pulled it up behind you and it's hard now to imagine what it is like to be on the ground.
It's not about curiousity. Of course you have that - if you didn't you wouldn't be there in the first place.
The problem, I think, comes from the weaponization of "stupid" against people. The XKCD of the lucky 1000 plays a good role here. If you are constantly deriding others for stupid takes, then anyone that derides one of your stupid takes will hit hard. And that seems to be getting worse.
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...
To be able to truly learn any given concept means being capable of answering a practically infinite number of different questions about that topic. The process of teaching is essentially trying to uncover which questions the student can't answer. The challenge, of course, is that the student doesn't know what questions they can't answer, because the questions haven't occurred to them. That is, until they start testing themselves, to see if they really do understand the concept.
Problem sets in textbooks are the canonical way of addressing this teaching challenge, but there are only so many pages in a textbook, and there are other concepts that need to be taught, so the scope of the problem sets are necessarily finite.
How many times have you nailed all the problems in a book, only to discover that there was some aspect of the topic you didn't understand, despite getting all the right answers?
The Master shows the Gate, but it is the Student who has to walk through it.
To show one the Right Direction and Right Path, Oral Instructions from a Master are necessary but Mastery of the Subject only comes from one's own Incessant Self-Cultivation.
There is also a great inspirational story in the Mahabharata of "Ekalavya" (https://en.wikipedia.org/wiki/Ekalavya) who became an exceptional archer through self-training.
Eklavya inspires a life-long learning philosophy and his presence seems to be a celebration for the masses. In this EklavyaParv, the motto is 'You Create Yourself" and the legend of Eklavya is a testimony that is forwarded by many thinkers as well. The discipleship that Eklavya represents is the best for a student and enables one to be the creator of one's own destiny.
Adapting to current times, "The Master" can be a "Good Book" and you can have "Many Masters" but the effort and learning has to happen within the Student.
Source: I self-taught myself Martial Arts (JKD, Karate, Taijiquan) from books when i was young. Decades later when i did join a dojo to study under a Master, i was one of the top students with good skill and power.
Fortunately, what I have retained is the ability to pick up almost any subject and learn about it on my own. That's more important than anything they can teach you in a classroom.
I don't think a student needs to always answer a question "on the spot". Being able to find an answer in a reasonable amount of time and explain an answer would be in my opinion more valuable. So then it's more about "how efficient can the student give the answer to the question" (answer on the spot, spend one hour, spend one week, etc.).
Or so I'm told.
I had a junior helpdesk employee that I was training/mentoring years back. He was 20 years old, fresh out of tech school. He was good at what he did, but he only did things he knew how to do. When he didn't know something, he'd ask me. Which is great. I'd say "Well, this sounds like DNS, it's like a phone book..." or "That's an APIPA address, it must not be getting DHCP. The computer shouts out to the network asking for an address..." and so forth. However, he kept asking the same questions.
After a few months in one of our monthly meetings he kind of broke "I don't understand what I'm doing out there, you need to train me! I need to be trained!" Completely perplexed I asked him what he was talking about. "You just answer my questions, but you're not training me!" I realized he was expecting me to learn him the answers to everything. I had to explain to him that the responsibility of learning was actually on him. "This isn't school, there's no study guide. We have documentation and Google. It's your responsibility to read it and make sense of it."
I told him that I can give him all the puzzle pieces but I can't put them together for him. To be fair, helpdesk is kind of about making things work and remembering the quick fixes and tricks for things to close out your tickets.
So I said, "Ok, I think you need a project. What do you do at home for fun?"
"Well I play a lot of video games."
"Perfect, we're setting up a Minecraft server". He laughed.
I said "No, I'm serious. We're using like 5% of this massively overblown server that was sold to us. Maybe this will help you put the pieces together."
I gave him a restricted vSphere account for his DMZ'd VM, sent him a guide and unleashed him.
"Well, I've never done this before..."
"Exactly. That's how you learn my dude."
"But..."
"RTFM"
"This VM doesn't do anything."
"Right, it needs an OS."
"We'll how do I install one?"
"Here's a guide."
"I installed the OS, how do I get into it?"
"SSH"
"No I mean the desktop."
"There isn't one."
And so he learned that a computer isn't the Windows desktop.
"I can't SSH in, it says connection refused."
"Right, that's the firewall."
"Well what do I do?"
"Google UFW"
"I can't SSH in anymore, it says connection timed out."
"Can you ping it?"
"No."
"Check the IP address in vSphere"
"It changed..."
"Why?" I asked.
"DHCP...! That's what a static IP is for!"
From then on he finally understood that learning actually takes a little effort and curiosity AND yes, it's OK to Google things. He had this idea that he had to know everything, memorize everything, and looking things up was "cheating". Not knowing something and feeling dumb is actually where learning happens rather than pure repetition.
About a year later he thanked me and said that he completely misunderstood my motivations initially and that he thought I was brushing him off and being lazy, when in reality I was giving him the opportunity to learn by not feeding him every detail. He felt like he was failing because he didn't know all the answers and said that he looked back at himself a year ago and couldn't believe what he was doing now and how far he'd come. "I had no idea what an IP address was but now I understand how the packets move through the switches, request an address..." etc.
We both ended up quitting and going our separate ways as the IT department there was an absolute shitshow. He's now a sysadmin and we chat now and then and he's mentioned that he's actually glad he learned in such a fucked up environment because you were absolutely forced to understand due to all the ridiculous hacks and workarounds that had been piled on over the years. Nothing could be taken for granted.
I learned in a similar way and I think trial by fire may be one of the best teachers. "Smooth seas never made a skilled sailor."
Learning only happens through a mish-mash of Trial-and-Error, Trial-by-Fire, Questioning, Curiosity, Reading/Copying/Mimicking, Thinking, Reflecting and finally Doing. All of the above are needed in some measure.
The trick is to do the above without losing our self-confidence in ourselves (we are guaranteed to feel "stupid" during the learning process) that "we can grok it" at some level and over a period of time. The problem today is that there is so many aspects and so much to learn about any one thing that students are trying to move very fast to learn everything which is an impossibility; they need to ruthlessly cut down on all inessentials and learn to focus on only one or two core things i.e. "sift the wheat from the chaff".
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.
A teacher of topic X is not an expert in topic X. They are an expert in “teaching topic X.”
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.
Also children are brought up with super obvious problems like “what object fits into which hole?”. I feel like some of these tests measure less the child’s understanding of the given problem per se and more whether they have previously been introduced to trick questions/illusions.
And even controlling for all that, you’re totally right. Even adults get confused by mass, weight, volume, apparent size etc. sometimes. The kid doesn’t even intellectually know those concepts. His only input here is by sight, but his answer may be different if he got to hold both objects and feel their weight.
I'd be convinced that the kid doesn't get it if they swapped crackers and then it stopped being fair.
Fair enough. It's definitely missing the opposing case where 1 graham cracker each is split on only one side and therefore the situation goes from fair to unfair, even though it's the same amount of graham cracker.
> I'd be convinced that the kid doesn't get it if they swapped crackers and then it stopped being fair.
I wouldn't count on that. I think the kid (and most adults) would claim it's fair if they thought they could get away with it. The deep intuition for "fair" that I expect from children would be derived from past experience negotiating with peers, not from any kind of moral theory.
The way I currently think about it is that a learning space is a sort of skill tree (poset), and the easy concepts/skills are the ones where we can learn all the prereqs, and then just combine them (join reducible elements), whereas the tricky concepts/skills (the ones which make us feel stupid) are the ones that only have a single prerequisite, so we can't just combine things we already know, but have to do something novel[2] in order to acquire them (join irreducible elements).
[0] and both of them were probably confident all along that they'd make it, the former because they had already sketched out a few likely paths in their mind, the latter because they've always managed to muddle through before
[1] furthermore, having travelled from A to B multiple times, it's difficult for a teacher to empathise with those who are not following
[2] to https://news.ycombinator.com/item?id=41566945 I'd say maybe that's why traditionally we've waited until people are in their late teens/early twenties, significantly ego-invested in research as a career path, and have an experienced mentor, before we throw them overboard into the lake of obligate stupidity[3]?
(there are exceptions: Feynman habitually tested himself by attempting to self-derive [practising research mode] before allowing himself to read expository texts [entering spectator mode]. Somewhere he[?] claims something along the lines of him not being that smart, just that people were impressed after they asked him questions for which he could give answers he'd already spent hundreds of hours thinking about)
[3] everybody genius until it time to do genius shit, yo