[HN Gopher] The centrality of stupidity in mathematics
___________________________________________________________________
The centrality of stupidity in mathematics
Author : ColinWright
Score : 196 points
Date : 2024-09-17 11:15 UTC (11 hours ago)
(HTM) web link (mathforlove.com)
(TXT) w3m dump (mathforlove.com)
| jimhefferon wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| I think that is one of the things that you are supposed to
| learn at college, in my experience.
| saintfire wrote:
| That means you didn't write enough unit tests.
|
| Or so I'm told.
| rramadass wrote:
| 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.
| mbivert wrote:
| > _[...] 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._
| marttt wrote:
| "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 wrote:
| 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.).
| nicholasbraker wrote:
| 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 wrote:
| 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 wrote:
| In college, there were people who rated professors negatively
| just because their classes were hard. Thirty years later, it
| still annoys me.
| bee_rider wrote:
| I don't think you disagree. The original comment was about
| students in general, you are talking about good students.
| chefandy wrote:
| 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.
| api wrote:
| 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 wrote:
| 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
| User23 wrote:
| 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 wrote:
| _> 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 wrote:
| > 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 wrote:
| I don't disagree with that. I've argued with all my kids'
| ones at least once!
| zellyn wrote:
| 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.
| minkles wrote:
| I have colleagues for that.
| zellyn wrote:
| I do too, but sometimes I read a blog post that makes me
| wonder about something and I don't want to schedule a 1:1
| with them and then wait to chat about it. We're mostly
| remote now...
| ocular-rockular wrote:
| Good luck finding that practice on certain topics. I hate to
| say it but GPT 4o has done a better job of breaking down
| problems and explaining them (granted at times incorrectly...
| thats where studying with other people comes in) for my qual
| practice than any of the profs or the useless texts ever did.
|
| We talk about scaffolding and the importance of pedagogy in
| math education yet none of that exists at higher levels. In
| my case it's literally been the blind leading the blind. It's
| a horrible environment to learn in. I say this as someone who
| has tackled some really tough material with no issue in the
| past thanks to having that hierarchy you mentioned. When that
| doesn't exist or there is nothing else, the process truly
| stalls. So sadly, I will take the crackhead over nothing.
|
| Then again, maybe I just hate what I'm studying which is it's
| own problem.
| enriquto wrote:
| > if we don't feel stupid it means we're not really trying
|
| Sounds a bit like Kernighan's lever: "Debugging is twice as hard
| as writing the code in the first place. Therefore, if you write
| the code as cleverly as possible, you are, by definition, not
| smart enough to debug it. Yet." [0]
|
| [0] https://www.linusakesson.net/programming/kernighans-
| lever/in...
| rossant wrote:
| Unless "as cleverly as possible" is zero.
| danbruc wrote:
| Or negative.
| norir wrote:
| For me the real lesson is that simple is hard.
| alphazard wrote:
| 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 wrote:
| 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 wrote:
| > 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 wrote:
| The kid has the same portion of cracker before and after she
| splits it, though.
| wvbdmp wrote:
| 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 wrote:
| _" 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.
| incognito124 wrote:
| If you think outside orthodoxy, why you care what others think?
| hilbert42 wrote:
| Scientific orthodoxy - not societal orthodoxy. There's a vast
| difference.
| harrigan wrote:
| I prefer the notion of Productive Failure [0].
|
| [0] https://www.manukapur.com/productive-failure/
| dahart wrote:
| > 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 wrote:
| 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 wrote:
| 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.
| Gupie wrote:
| I don't disagree with what you have just said except I
| understand education to refer to the transfer of knowledge
| while research is the discovery of new knowledge.
| JoeAltmaier wrote:
| Curiously the root of that word is 'to search again',
| meaning more like 'reviewing sources in the library' and
| less like 'doing experiments in the lab'.
| kragen wrote:
| re- in this case is probably an intensive prefix rather
| than indicating repetition. this is an uncommon re- in
| english, but does occur, for example in 'refried beans',
| a calque from spanish where intensive re- is still a
| productive prefix
|
| so it probably means 'search really hard' rather than
| 'search again'
| dahart wrote:
| Research is wholly about the transfer of knowledge, both
| before and after any associated experimentation.
| Sometimes there is discovery in between, but not always;
| survey papers and meta analyses are research, a very
| important part of research. Experiments that don't
| research previous work and don't communicate the results
| aren't research and usually don't result in the discovery
| of new knowledge. Can't know it's new unless you research
| what's already known.
| glitchc wrote:
| Research is discovery with a healthy dose of self-
| education.
| dahart wrote:
| I'm lucky enough to have had opportunity and
| encouragement to do research most of my career, and work
| closely with other researchers for decades.
|
| The first job of a researcher is to understand what
| others have done, before attempting discovery. Failure to
| do that critical step means it's not considered research.
| The second job is to build on the work of others. And the
| third job is to communicate those results to others.
| Discovery is the seeking of knowledge, which is
| education. Framing it as self-education is feeding a
| narrative of research as being an individual sport, but
| in reality research is entirely a collaborative team
| sport with incremental dependent results.
| derbOac wrote:
| 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.
| what-the-grump wrote:
| 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?
| wwarner wrote:
| 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.
| taeric wrote:
| 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.
| thomastjeffery wrote:
| > 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.
| dahart wrote:
| > 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".
| thomastjeffery wrote:
| My disagreement was semantic. Wasn't yours?
|
| My overall point is that the end of education is expertise,
| which itself is a form of ignorance. We generally consider
| stupidity and expertise to be antonyms, but they often
| exist as two opposing perspectives of the same experience.
| glitchc wrote:
| 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.
| j2kun wrote:
| It's possible, but counter to the point of a PhD:
| apprenticeship for research.
| l33t7332273 wrote:
| 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.
| grungegun wrote:
| In a sense, I spend most of my time on material after quals
| because it's so much harder to understand.
| ColinWright wrote:
| 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.
|
| [0] Other beverages are available.
| dahart wrote:
| 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!
| ColinWright wrote:
| I suspect part of the friction here is the difference
| between teaching math to people who will not go on to study
| it at an advanced level, versus teaching it to people who
| will.
|
| In that case, what is being taught is actually different,
| it's the similar in some senses to the difference between
| teaching someone to operate a machine, versus teaching them
| how to maintain, fix, and possibly improve said machine.
|
| I've said for a long time that if only we could identify
| early the students who will _not_ go on to study
| mathematics at an advanced level then we could better
| benefit them by having a substantially different curriculum
| taught in a completely different style.
|
| But even then, the willingness to be confused by something,
| and resilience in the face of being made to feel stupid not
| by the teacher[0] but by the current (temporary!) lack of
| understanding, is really, _really_ powerful.
|
| [0] Teachers who unnecessarily make students feel stupid
| should be prevented from ever, _ever_ going near students
| again. Ever.
|
| > _If we get the chance some time, it would be super fun to
| discuss math & education over coffee!_
|
| Where are you based? I travel a lot ... my email is in my
| profile, and you could always register with my "Meet With
| Me" system to get a heads-up if I'm going to be in the
| area.
| dahart wrote:
| > if only we could identify early the students who will
| not go on to study mathematics at an advanced level then
| we could better benefit them by having a substantially
| different curriculum taught in a completely different
| style.
|
| Yeah I'd very much agree with this. Actually strong agree
| with all your points there, it's well aligned with what I
| think I was trying to say.
|
| I'm US based, currently mostly Utah, sometimes
| California.
| ColinWright wrote:
| I was in CA for three weeks earlier this year, including
| getting to spend half a day with Cliff Stoll (Hi Cliff!).
|
| But I'm unlikely to be in the US now for a bit, but if
| you ping me an email then I can tell you how (if you
| choose) you can register with my "Meet With Me" system.
|
| Or not ...
|
| Regardless, yes, I think we do (or would) agree on most
| of these issues.
| btilly wrote:
| Your starting "of course" presumes that you are right. I'm
| disagreeing because I truly and honestly think that you're
| wrong.
|
| People who are struggling with math, even in elementary
| school, will often say things like, "That's so simple, why
| couldn't I get it? I must be stupid!" I've heard this over
| and over again in a wide variety of contexts. Including
| from my own children. We honestly believe that if it is
| simple, it should be easy to understand. And so the
| experience of having struggled with something simple,
| leaves us feeling stupid.
|
| _Even_ people with an incredible amount of math success,
| wind up feeling this. And pretty much universally have also
| developed coping mechanisms for it. But the experience
| itself is pretty much universal. My son felt it in middle
| school when he was struggling with long-division.
|
| Now does a poor education system make this experience more
| likely? Does it serve us poorly? Absolutely! I consider it
| a crime that Singapore has developed a better way of
| teaching elementary math, and we have not adopted it.
| Singapore is now moving to #1 across different subjects
| according to PISA, and Western education systems aren't
| even curious about how.
|
| However that is orthogonal to the key point here. Which is
| that math tends to be simple in a way that our brains
| aren't built for. And when we're confronted with how hard
| it is for us to learn something simple, it is easy for us
| to feel stupid. This can be very demoralizing. And it is
| very helpful for us to learn to accept and deal with that
| feeling.
| dahart wrote:
| My 'of course' to @ColinWright was intended to be tongue
| in cheek playful, and not presume anything. But I accept
| it might come off a different way than I intended.
|
| I think (like with others in this subthread) that I have
| mislead you or am being misunderstood or both. I'm
| familiar with the feeling of stupidity in math, and I'd
| agree that it's useful when/if harnessed. I'm calling the
| process of accepting it and dealing with it "curiosity",
| partly since if curiosity is missing then people tend to
| feel shame and anxiety with their stupidity and tend to
| avoid math and give up on learning it.
|
| I don't believe there is such a thing as right and wrong
| here. I'm making a point of view framing distinction, not
| disagreeing with the article or the quote in the article.
| I think that "stupidity" isn't the best word choice in
| general, even though it might work for math researchers
| in this case. That word comes with many overloaded and
| negative meanings that aren't accurate to what the
| article is really trying to say.
| bbor wrote:
| 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.
| kragen wrote:
| 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!
| bonoboTP wrote:
| > 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.
| spookie wrote:
| The struggle isn't necessarily bad for learning. It really is a
| good way to learn. I like it.
|
| But alas, I never thought as a kid that I didn't have time for
| other things. I was always into something.
| mjburgess wrote:
| 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 wrote:
| 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."
| javier_e06 wrote:
| 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 wrote:
| > 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 wrote:
| Well said. This should be part of orientation for every new
| college student.
| bpshaver wrote:
| 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.
| Noumenon72 wrote:
| Perhaps we should replace this messaging with "You may find
| that you won't have to work hard to get through X, but that
| doesn't mean you shouldn't." Educators don't have the time
| or ability to set up an incentive scheme that makes you
| "have to" work your hardest, but it gets more rewarding at
| each level.
| Loughla wrote:
| Bingo. You can breeze through all levels of education
| with a combination of personality and picking the right
| courses. Your faculty tend to be overworked and underpaid
| for the work they do. You are one of hundreds. They do
| not exist to make sure you're actually learning anything,
| just that you can spit back the course content
| appropriately.
|
| But once you get over the barrier to entry for most
| white-collar jobs (bachelor degree), what's the point? If
| you're not getting anything out of the education, you're
| only borrowing trouble from yourself in the future.
|
| My bachelor's was relatively easy. My masters was
| MASSIVELY difficult. The PhD was even harder.
|
| Because I sought out those difficulties.
| red-iron-pine wrote:
| along with basic hygiene
| andai wrote:
| My first day at computer science I saw a guy with a huge beard
| playing Dwarf Fortress, and I was like "oh crap, he's like ten
| times smarter than me."
| nickpeterson wrote:
| To be fair, he may have been a dwarf and had a natural edge.
| Ozarkian wrote:
| I would have assumed he was a UNIX system administrator.
| Everybody knows that only guys with enormous beards can
| properly tame a UNIX system.
| 0xEF wrote:
| And girls trapped in dinosaur theme parks, let's not forget
| knodi123 wrote:
| That was the exception that proves the rule.
| dwattttt wrote:
| Because the phrase has always bothered me: this means
| something other than what it's commonly understood to
| nowadays.
|
| An older use of the word "prove", as in to test, means it
| says "that's an exception that tests the rule, and finds
| it is incorrect"
| joseluis wrote:
| I agree in that the meaning of prove in that context is
| "put it to the test" but for me it doesn't go as far as
| finding the rule incorrect, because it's a general rule,
| not an absolute rule. A lot more exceptions would be
| necessary to make the rule incorrect for the general
| case.
| seszett wrote:
| I wouldn't be so sure, the same expression exists in
| French:
|
| _L 'exception qui confirme la regle_
|
| And there's no ambiguity about it, the exception is
| confirming that the rule is true.
|
| That's a stupid expression IMO, but I would be surprised
| if the English expression meant the exact contrary.
| ColinWright wrote:
| There is an alternate interpretation, that the existence
| of an explicit exception proves (confirms) the existence
| of a rule to which an exception can be made.
|
| So the (existence of an) exception proves (the existence
| of) the rule.
| knodi123 wrote:
| The phrase bothers me because it's often used to set up a
| cousin to the no true scotsman fallacy. If you can't find
| an exception, then it proves the nay-sayer right. If you
| CAN find an exception.... it still proves the naysayer
| right?!?
|
| I wouldn't use the phrase outside of silly internet jokes
| about 90s popcorn flicks.
| gopher_space wrote:
| I've always used it to mean that I don't care about your
| hypothetical edge case.
| underlipton wrote:
| You don't know enough transfem furries.
| madcaptenor wrote:
| So did she get to be on Jeopardy!?
| wileydragonfly wrote:
| That secretary is probably hard funded, makes more money than
| 75% of the scientists she supported, enjoys vacations, and will
| have a comfortable retirement.
| euvin wrote:
| 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.
| dgacmu wrote:
| It might feel that way, but I think it's really worth
| asking : is it?
|
| If I put humility aside for a moment, I'm awfully good in
| my field, and academia is hyper competitive, and yet I know
| people who are better at everything that I do -- and just
| like the secretary in that example, I know some people who
| I think could do everything that I can better than I do.
|
| But that really is okay, the world has room for all of us
| and more, and is much better off for having several of us
| applying our skills and abilities. We are not short on
| important problems to solve, we are short on solutions and
| solvers!
| askafriend wrote:
| It's advice you give other people but secretly don't take
| yourself. Because you want to be the very best - like no one
| ever was.
| mkleczek wrote:
| > You're not the best at everything, and you don't have to be.
|
| 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.
| quasse wrote:
| As another very insightful HN commenter said [1]:
|
| "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
| tombert wrote:
| 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.
| whatshisface wrote:
| 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!
| passion__desire wrote:
| I think this process of uncovering your stupidity / confusion
| could be gradual as Scott makes the point below. It need not be a
| single "Aha" moment.
|
| https://scottaaronson.blog/?p=4974
|
| > OK, but why a doofus computer scientist like me? Why not,
| y'know, an actual expert? I won't put forward my ignorance as a
| qualification, although I have often found that the better I
| learn a topic, the more completely I forget what initially
| confused me, and so the less able I become to explain things to
| beginners.
| Dove wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| anticipation of cluefulness? epistemic frustration? temporarily
| embarrassed expertise?
|
| perplexed?
| beryilma wrote:
| > ... 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| > 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 wrote:
| "Mathematics is the Poetry of Science" is a great title that
| also explains the phenomenon noted in
| https://news.ycombinator.com/item?id=41570976
| alexashka wrote:
| The article pre-supposes the lady who left science to do law did
| so primarily due to feeling 'stupid'. Is that what 'brilliant'
| people do? Switch fields of study due to feelings? At least _one_
| person in this story isn 't brilliant :)
|
| As for being comfortable with feeling 'stupid' - that's for
| children. Once you become an adult and start to piece together
| how the world works, you shift from a child seeking adult
| validation to 5 stages of grief - denial, anger, bargaining,
| depression and acceptance.
|
| Of course people in academia and the corporate world are
| notorious at being in denial, which is why they talk about
| 'centrality of stupidity', much like corporate NPCs talk about
| 'culture'.
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