13 ms·

The centrality of stupidity in mathematics

284 points by ColinWright 2y ago

jimhefferon 2y ago
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.
- maroonblazer 2y ago
  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?
  - jplusequalt 2y ago
    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.
    - gtr 2y ago
      I think that is one of the things that you are supposed to learn at college, in my experience.
  - saintfire 2y ago
    That means you didn't write enough unit tests. Or so I'm told.
  - rramadass 2y ago
    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 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.
    - mbivert 2y ago
      > [...] my Rebbe was the geologist of the soul. You see, there are so many treasures in the earth. There is gold, there is silver, and there are diamonds. But if you don’t know where to dig, you’ll find only dirt and rocks and mud. The Rebbe can tell you where to dig, and what to dig for, but the digging you must do yourself.
      - rramadass 2y ago
        Also; "If" by Rudyard Kipling - https://en.wikipedia.org/wiki/If%E2%80%94 https://en.wikipedia.org/wiki/If%E2%80%94
    - marttt 2y ago
      "I self-taught myself Martial Arts (JKD, Karate, Taijiquan) from books when i was young." This is very interesting and impressive. How common would you think a complete self-study of martial arts actually is? I've always thought this cannot be done alone -- or, it would be extremely easy to get lost or head to some wrong direction, eventually harming yourself mentally or physically. Akin to how a common suggestion about yoga or meditation (I used to exercise vipassana daily for quite a while) is that the first, basic principles should be taught by a good master. Possibly due to personality type, I've always wanted to challenge this assumption, though. While learning martial arts on your own, what did you do to overcome more serious mental blocks or standstills (provided you had any)? Did you ever feel that "books are not enough"?
  - vladms 2y ago
    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.).
  - mrkandel 2y ago
    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.
  - Hunpeter 2y ago
    Tangentially, modern educational paradigms also characterize learning as a process of construction happening within the student's mind, rather than a transfer of knowledge.
- nicholasbraker 2y ago
  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..
- sureglymop 2y ago
  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)
  - zellyn 2y ago
    In college, there were people who rated professors negatively just because their classes were hard. Thirty years later, it still annoys me.
  - bee_rider 2y ago
    I don’t think you disagree. The original comment was about students in general, you are talking about good students.
- chefandy 2y ago
  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.
- bongodongobob 2y ago
  I have the perfect story to illustrate this. 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."
api 2y ago
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.
- 082349872349872 2y ago
  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 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
  - gradschoolfail 2y ago
    2][3 I have caught Feynman expositing in bad faith (second-][third-handedly)
User23 2y ago
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.
- minkles 2y ago
  > It’s cool that AI is making that accessible. If you want to be taught by a hallucinating crack head that gets only 60-70% at best of what it says right, then yes. As a qualified mathematician, I would suggest that the best path to teaching is small steps in a properly defined hierarchy of knowledge and practice practice practice. Most of the teaching is seen as rubbish because people didn't get enough practice further down the tree to be able to do it instinctively so higher level concepts can be retained.
  - criddell 2y ago
    > If you want to be taught by a hallucinating crack head that gets only 60-70% at best of what it says right, then yes. That description is only a slight exaggeration of some high school math teachers.
    - minkles 2y ago
      I don't disagree with that. I've argued with all my kids' ones at least once!
  - zellyn 2y ago
    I've had incredibly productive discussions with Claude about category theory. (I prefer Claude because it's the most pleasant to talk with; I think they optimized for that.) The ability to explain what I know already, hand-wave at what I think I understand about my question, and then get a description that meets me where I'm at is invaluable. Sure, occasionally Claude will tell me (incorrectly) that a CRDT's lattice operation needs an identity function: you absolutely have to go back and forth with wikipedia. LLMs are not a magic genie or oracle. But if you use them for what they're good at, they're amazing.
enriquto 2y ago
> 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/index.php https://www.linusakesson.net/programming/kernighans-lever/in...
- rossant 2y ago
  Unless "as cleverly as possible" is zero.
  - danbruc 2y ago
    Or negative.
- norir 2y ago
  For me the real lesson is that simple is hard.
  - davisp 2y ago
    “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
alphazard 2y ago
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.
- persnickety 2y ago
  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.
  - alphazard 2y ago
    > 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.
  - directevolve 2y ago
    The kid has the same portion of cracker before and after she splits it, though.
- wvbdmp 2y ago
  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.
hilbert42 2y ago
"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.
- deleted 2y ago
  [deleted]
- incognito124 2y ago
  If you think outside orthodoxy, why you care what others think?
  - hilbert42 2y ago
    Scientific orthodoxy - not societal orthodoxy. There's a vast difference.
harrigan 2y ago
I prefer the notion of Productive Failure [0]. [0] https://www.manukapur.com/productive-failure/ https://www.manukapur.com/productive-failure/
dahart 2y ago
> 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.
- Gupie 2y ago
  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.
  - dahart 2y ago
    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.
mjburgess 2y ago
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.
- directevolve 2y ago
  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.”
  - mjburgess 2y ago
    Sure. The way i think of it is that academic expertise is writing research papers in some domain.
javier_e06 2y ago
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.
gwd 2y ago
> 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.
- tapanjk 2y ago
  Well said. This should be part of orientation for every new college student.
  - bpshaver 2y ago
    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.
passion__desire 2y ago
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 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.
Dove 2y ago
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.
- setopt 2y ago
  I only feel dumb if I don’t know how to start looking at a problem, in some cases because I don’t understand the description of the problem either. But as long as I understand to some degree what we want to achieve, and have some vague idea of what corner I might start in, I usually don’t “feel” dumb even if I know very little about the final solution…
kzz102 2y ago
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.
- zeptian 2y ago
  the two types are nice ! the author likens your first type to building "mental" roads, which form new pathways of cognition, and takes time, and has emotional resistance, and requires conscious effort and practice to carve out. also to relate the roads to other roads correctly, so the mental map of roads is consistent, and can be traversed. the problem is that most students do not grasp ideas fully and develop facility with it. when this happens, the foundations are shaky, and facility is lost. then, they label themselves as incapable which leads to a vicious cycle where, the belief of being stupid leads to more stupidity. the second type is where the roads (ideas) are there, but a route from source to destination is not clear, and the aha moment is when you see the full path in the mental eye.
thomasahle 2y ago
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.
rhelz 2y ago
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???
- 082349872349872 2y ago
  anticipation of cluefulness? epistemic frustration? temporarily embarrassed expertise? perplexed?
  - gradschoolfail 2y ago
    Ideally it should cover TC-ST’s emotions as expressed in https://news.ycombinator.com/item?id=41548104 https://news.ycombinator.com/item?id=41548104 (Mulling/marvelling ATM how wikipedia talkpages, mastodon, o1, or HN are all inadequate for the kind of conversations we all hope for, something about the tension between sublimating koinonia & prowess-seeking) EDIT Getting a bit selfreferential here too https://news.ycombinator.com/item?id=41567018 https://news.ycombinator.com/item?id=41567018 EDIT 2: “barely bottled in-game insanity” could work
    - 082349872349872 2y ago
      on mulling/marveling: a) FPD read the Glass Bead Game, and instead of fixating on the game itself as I have done, noticed how the conversations tended toward an ideal (also, IIRC, recommended in the NE): https://franklin.dyer.me/post/123 https://franklin.dyer.me/post/123 b) etymologically, companions eat bread together, and symposia involve drinking. Maybe a problem with digital fora is that although we may have moved on from grooming and nitpicking, we still require at least a minial amount of analogue ingestion and imbibing for conviviality? (in the dating context, see https://news.ycombinator.com/item?id=910852 https://news.ycombinator.com/item?id=910852 ) EDIT: Re: "barely bottled in-game insanity", either someone needs to tell Stross that Series Landys don't have ignition buttons, or I need to accept the foreshadowing and realise that detail was an early in-story clue of power of eldritch horrors* to transmute and pervert even the most solid basis of innate rural goodness. * in the world of the Laundry; in our world it was Tata who added the buttons
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beryilma 2y ago
> ... 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.
- feoren 2y ago
  But when people quit STEM degrees, they don't say "it made me feel ignorant" or "I didn't have enough knowledge". They say "I was too stupid". The author is expressly trying to address those people, and the people who might be able to intervene in their lives. "Yes! We're all stupid! That's part of what it means to learn and research math and science!"
wileydragonfly 2y ago
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.”
bonoboTP 2y ago
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.
underlipton 2y ago
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."
yodsanklai 2y ago
> 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.
- 082349872349872 2y ago
  "Mathematics is the Poetry of Science" is a great title that also explains the phenomenon noted in https://news.ycombinator.com/item?id=41570976 https://news.ycombinator.com/item?id=41570976
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akerr 2y ago
This is called “productive struggle” in the literature.
sandspar 2y ago
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.
graycat 2y ago
"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.
InDubioProRubio 2y ago
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.
dirichletian 2y ago
Interesting
ivansavz 2y ago
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 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!
lern_too_spel 2y ago
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.

284 points by ColinWright 2y ago

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.

maroonblazer 2y ago
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?
- jplusequalt 2y ago
  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.
  - gtr 2y ago
    I think that is one of the things that you are supposed to learn at college, in my experience.
- saintfire 2y ago
  That means you didn't write enough unit tests. Or so I'm told.
- rramadass 2y ago
  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 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.
  - mbivert 2y ago
    > [...] my Rebbe was the geologist of the soul. You see, there are so many treasures in the earth. There is gold, there is silver, and there are diamonds. But if you don’t know where to dig, you’ll find only dirt and rocks and mud. The Rebbe can tell you where to dig, and what to dig for, but the digging you must do yourself.
    - rramadass 2y ago
      Also; "If" by Rudyard Kipling - https://en.wikipedia.org/wiki/If%E2%80%94 https://en.wikipedia.org/wiki/If%E2%80%94
  - marttt 2y ago
    "I self-taught myself Martial Arts (JKD, Karate, Taijiquan) from books when i was young." This is very interesting and impressive. How common would you think a complete self-study of martial arts actually is? I've always thought this cannot be done alone -- or, it would be extremely easy to get lost or head to some wrong direction, eventually harming yourself mentally or physically. Akin to how a common suggestion about yoga or meditation (I used to exercise vipassana daily for quite a while) is that the first, basic principles should be taught by a good master. Possibly due to personality type, I've always wanted to challenge this assumption, though. While learning martial arts on your own, what did you do to overcome more serious mental blocks or standstills (provided you had any)? Did you ever feel that "books are not enough"?
- vladms 2y ago
  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.).
- mrkandel 2y ago
  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.
- Hunpeter 2y ago
  Tangentially, modern educational paradigms also characterize learning as a process of construction happening within the student's mind, rather than a transfer of knowledge.
nicholasbraker 2y ago
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..
sureglymop 2y ago
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)
- zellyn 2y ago
  In college, there were people who rated professors negatively just because their classes were hard. Thirty years later, it still annoys me.
- bee_rider 2y ago
  I don’t think you disagree. The original comment was about students in general, you are talking about good students.
chefandy 2y ago
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.
bongodongobob 2y ago
I have the perfect story to illustrate this. 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."

api 2y ago

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.

082349872349872 2y ago
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 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
- gradschoolfail 2y ago
  2][3 I have caught Feynman expositing in bad faith (second-][third-handedly)

User23 2y ago

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.

minkles 2y ago
> It’s cool that AI is making that accessible. If you want to be taught by a hallucinating crack head that gets only 60-70% at best of what it says right, then yes. As a qualified mathematician, I would suggest that the best path to teaching is small steps in a properly defined hierarchy of knowledge and practice practice practice. Most of the teaching is seen as rubbish because people didn't get enough practice further down the tree to be able to do it instinctively so higher level concepts can be retained.
- criddell 2y ago
  > If you want to be taught by a hallucinating crack head that gets only 60-70% at best of what it says right, then yes. That description is only a slight exaggeration of some high school math teachers.
  - minkles 2y ago
    I don't disagree with that. I've argued with all my kids' ones at least once!
- zellyn 2y ago
  I've had incredibly productive discussions with Claude about category theory. (I prefer Claude because it's the most pleasant to talk with; I think they optimized for that.) The ability to explain what I know already, hand-wave at what I think I understand about my question, and then get a description that meets me where I'm at is invaluable. Sure, occasionally Claude will tell me (incorrectly) that a CRDT's lattice operation needs an identity function: you absolutely have to go back and forth with wikipedia. LLMs are not a magic genie or oracle. But if you use them for what they're good at, they're amazing.

enriquto 2y ago

> 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/index.php https://www.linusakesson.net/programming/kernighans-lever/in...

rossant 2y ago
Unless "as cleverly as possible" is zero.
- danbruc 2y ago
  Or negative.
norir 2y ago
For me the real lesson is that simple is hard.
- davisp 2y ago
  “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

alphazard 2y ago

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.

persnickety 2y ago
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.
- alphazard 2y ago
  > 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.
- directevolve 2y ago
  The kid has the same portion of cracker before and after she splits it, though.
wvbdmp 2y ago
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.

hilbert42 2y ago

"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.

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incognito124 2y ago
If you think outside orthodoxy, why you care what others think?
- hilbert42 2y ago
  Scientific orthodoxy - not societal orthodoxy. There's a vast difference.

harrigan 2y ago

I prefer the notion of Productive Failure [0]. [0] https://www.manukapur.com/productive-failure/ https://www.manukapur.com/productive-failure/

dahart 2y ago

> 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.

Gupie 2y ago
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.
- dahart 2y ago
  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.

mjburgess 2y ago

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.

directevolve 2y ago
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.”
- mjburgess 2y ago
  Sure. The way i think of it is that academic expertise is writing research papers in some domain.

javier_e06 2y ago

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.

gwd 2y ago

> 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.

tapanjk 2y ago
Well said. This should be part of orientation for every new college student.
- bpshaver 2y ago
  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.

passion__desire 2y ago

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 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.

Dove 2y ago

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.

setopt 2y ago
I only feel dumb if I don’t know how to start looking at a problem, in some cases because I don’t understand the description of the problem either. But as long as I understand to some degree what we want to achieve, and have some vague idea of what corner I might start in, I usually don’t “feel” dumb even if I know very little about the final solution…

kzz102 2y ago

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.

zeptian 2y ago
the two types are nice ! the author likens your first type to building "mental" roads, which form new pathways of cognition, and takes time, and has emotional resistance, and requires conscious effort and practice to carve out. also to relate the roads to other roads correctly, so the mental map of roads is consistent, and can be traversed. the problem is that most students do not grasp ideas fully and develop facility with it. when this happens, the foundations are shaky, and facility is lost. then, they label themselves as incapable which leads to a vicious cycle where, the belief of being stupid leads to more stupidity. the second type is where the roads (ideas) are there, but a route from source to destination is not clear, and the aha moment is when you see the full path in the mental eye.

thomasahle 2y ago

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.

rhelz 2y ago

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???

082349872349872 2y ago
anticipation of cluefulness? epistemic frustration? temporarily embarrassed expertise? perplexed?
- gradschoolfail 2y ago
  Ideally it should cover TC-ST’s emotions as expressed in https://news.ycombinator.com/item?id=41548104 https://news.ycombinator.com/item?id=41548104 (Mulling/marvelling ATM how wikipedia talkpages, mastodon, o1, or HN are all inadequate for the kind of conversations we all hope for, something about the tension between sublimating koinonia & prowess-seeking) EDIT Getting a bit selfreferential here too https://news.ycombinator.com/item?id=41567018 https://news.ycombinator.com/item?id=41567018 EDIT 2: “barely bottled in-game insanity” could work
  - 082349872349872 2y ago
    on mulling/marveling: a) FPD read the Glass Bead Game, and instead of fixating on the game itself as I have done, noticed how the conversations tended toward an ideal (also, IIRC, recommended in the NE): https://franklin.dyer.me/post/123 https://franklin.dyer.me/post/123 b) etymologically, companions eat bread together, and symposia involve drinking. Maybe a problem with digital fora is that although we may have moved on from grooming and nitpicking, we still require at least a minial amount of analogue ingestion and imbibing for conviviality? (in the dating context, see https://news.ycombinator.com/item?id=910852 https://news.ycombinator.com/item?id=910852 ) EDIT: Re: "barely bottled in-game insanity", either someone needs to tell Stross that Series Landys don't have ignition buttons, or I need to accept the foreshadowing and realise that detail was an early in-story clue of power of eldritch horrors* to transmute and pervert even the most solid basis of innate rural goodness. * in the world of the Laundry; in our world it was Tata who added the buttons

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beryilma 2y ago

> ... 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.

feoren 2y ago
But when people quit STEM degrees, they don't say "it made me feel ignorant" or "I didn't have enough knowledge". They say "I was too stupid". The author is expressly trying to address those people, and the people who might be able to intervene in their lives. "Yes! We're all stupid! That's part of what it means to learn and research math and science!"

wileydragonfly 2y ago

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.”

bonoboTP 2y ago

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.

underlipton 2y ago

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."

yodsanklai 2y ago

> 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.

082349872349872 2y ago
"Mathematics is the Poetry of Science" is a great title that also explains the phenomenon noted in https://news.ycombinator.com/item?id=41570976 https://news.ycombinator.com/item?id=41570976

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akerr 2y ago

This is called “productive struggle” in the literature.

sandspar 2y ago

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.

graycat 2y ago

"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.

InDubioProRubio 2y ago

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.

dirichletian 2y ago

Interesting

ivansavz 2y ago

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 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!

lern_too_spel 2y ago

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.