[HN Gopher] You are not dumb, you just lack the prerequisites
___________________________________________________________________
You are not dumb, you just lack the prerequisites
Author : JustinSkycak
Score : 175 points
Date : 2024-08-24 13:57 UTC (9 hours ago)
(HTM) web link (lelouch.dev)
(TXT) w3m dump (lelouch.dev)
| bqc wrote:
| Would you like to recommend any resource?
| JustinSkycak wrote:
| Kind of depends what exactly you're trying to learn, but if
| you're working up to ML/AI, then there's _How to get from high
| school math to cutting-edge ML /AI_ from last week:
| https://news.ycombinator.com/item?id=41276675
| doctorpangloss wrote:
| This is really interesting but it seems too good to be true.
|
| You're going to reply that it's not for everyone, but you'll
| have nothing positive to say about the audience for whom this
| is a bad fit, which is also a suspicious form of
| generalization. It's kind of like how the chatbots and for
| that matter most online learning resources give the psychic
| feeling of learning.
| pests wrote:
| It seems impolite to make up your opponent's argument and
| then make commentary about it. Maybe let them respond?
| JustinSkycak wrote:
| What exactly is too good to be true? The idea that there's
| a path of knowledge leading from high school math to
| cutting-edge ML/AI?
| ydnaclementine wrote:
| I've been looking at this textbook for awhile. Very, very
| basics to calculus: https://www.amazon.com/Foundation-Maths-Dr-
| Anthony-Croft/dp/.... I wish there was a table of contents
| available
| billconan wrote:
| but when I read a paper, it's difficult to know what the
| Prerequisites are.
| JustinSkycak wrote:
| This is one reason why it's helpful to learn bottom-up when
| possible as opposed to diving straight into the deep end and
| trying to fill in missing knowledge as you go.
| chrisweekly wrote:
| I think there are certain circumstances where getting in over
| your head and digging your way out is a better approach --
| but I don't know how to distinguish those cases from the
| rest.
| JustinSkycak wrote:
| I don't think there's anything wrong with trying to jump
| headfirst into things that interest you. I would just
| recommend that you need to be honest with yourself about
| whether you're making progress -- and if you're starting to
| flail (or, more subtly, doubt yourself and lose interest),
| then it's an indication you need to re-allocate your time
| into shoring up your foundations.
| carlmr wrote:
| Hm, I've always felt bottom up more difficult to learn. I
| always found it helpful to first have an overview, a mental
| map of sorts, of the high-level details, so that when I
| looked at the details later I could make connections and know
| where to "put" this knowledge relative to other things.
|
| With bottom up I always feel lost because I don't know what
| it's useful for, the relationships to other pieces of
| knowledge, etc.
| JustinSkycak wrote:
| It can definitely be helpful to take a top-down approach in
| planning out your overarching learning goals.
|
| However, the learning itself has to occur bottom-up.
| Especially in math. Math is a skill hierarchy, and if you
| cannot execute a lower-level skill consistently and
| accurately, you will not be able to build more advanced
| skills on top of it.
|
| I wrote about this recently here if you're interested:
| https://www.justinmath.com/how-to-learn-machine-learning-
| top...
| SoftTalker wrote:
| It's good to have a high level view of what your ultimate
| goals are, but if you are lacking too much foundational
| knowledge you can't even conceive of it. Especially in a
| subject like math, everything builds from the bottom up.
|
| We don't give first graders an overview of differential
| equations and their applications when we start teaching
| them addition and subtraction.
| ninetyninenine wrote:
| The higher your iq the more easier it is to go top down.
| JustinSkycak wrote:
| Yes, which is why most people struggle so much with the
| top-down approach ;)
| tombert wrote:
| My trick is to find the paper you want to read, but immediately
| skip to the references; recurse until you get to a paper that
| you more or less understand.
|
| It's a bit time consuming but it makes paper reading a lot more
| fun.
| JustinSkycak wrote:
| Reminds me of my colleague's recent post on his experience
| getting up to speed on his dissertation topic while doing a
| PhD in mathematics:
| https://x.com/ninja_maths/status/1820583797491925386
|
| I'll quote a snippet below:
|
| _"My biggest mistake when starting my doctoral research was
| taking a top-down approach. I focused my efforts on a handful
| of research papers on the frontier of my chosen field, even
| writing code to solve problems in these papers from day one.
| However, I soon realized I lacked many foundational
| prerequisites, making the first year exceptionally tough.
| What I should have done was spend 3-6 months dissecting the
| hell out of all the key research papers and books written on
| the subject, starting from the very basics (from my knowledge
| frontier) and working my way up (the bottom-up approach)."_
| findthewords wrote:
| Finding a path starting backwards... also known as dynamic
| programming.
| cultofmetatron wrote:
| I would legit PAY for an app that managed dependencies for
| understanding a paper so that I could see the path I need to
| take to understand what I'm reading. Could apply to books
| too.
| dataf3l wrote:
| some time ago I was thinking about this issue, maybe the
| concept of "parametric books" will become popular in the
| future.
|
| https://www.youtube.com/watch?v=VXktVbeWAeM
|
| https://www.youtube.com/watch?v=Tpb2rXtBos4
|
| perhaps, with the advent of AI, one will be able to convert
| a current book into a more detailed book, and also a
| current book into a smaller book, so maybe this idea is
| even easier to implement than 4 years or so ago, before
| chatgpt (but after summarizers, which prompted the idea in
| my head).
|
| I would humbly appreciate any feedback on the concept of
| parametric books, for now, it's just an idea, but it's a
| free one, anyone is free to implement it.
|
| thanks in advance, for your comments on it.
| sn9 wrote:
| I mean if you're regularly trying to read papers in a
| particular field, just follow the curricula for
| undergraduate degrees in that field.
| IshKebab wrote:
| Papers are a terrible way to learn unless you are already an
| expert in the field, because the prerequisites tend to be "the
| entire rest of the field". It's a rare paper that actually
| assumes you might not know everything the authors knew.
| fuzzfactor wrote:
| Prerequisites are so important, and math is one of the things
| where it makes more of a difference.
|
| >You Are Not Dumb, You Just Lack the Prerequisites
|
| I know what you mean, after years of study I now feel confident
| that I don't lack the prerequisites to be as dumb as I could
| possibly want to be ;)
| asdf6969 wrote:
| me except I'm socially dumb because I didn't learn how to have
| friends in middle school
| willismichael wrote:
| Umm... most people are really bad at relationships in middle
| school.
| hypeatei wrote:
| But people still had them. GP is saying they didn't have
| _any_.
| toast0 wrote:
| Having friends in middle school can be pretty tough.
|
| Learning how to have friends in elementary school and high
| school is a lot easier and mostly unconnected with middle
| school friendship.
|
| Hopefully your social life doesn't revolve around friendship
| with middle schoolers, so no big deal if you miss out on that
| skill.
| elzbardico wrote:
| Not always, but a lot of times not having friends in Middle
| School is class based. When I had this age I moved schools, in
| the first one I was objectively poorer than almost everyone
| else and the subject of bullying and loneliness. I would hide
| in the library during reccess not only because I liked reading,
| but to avoid the pain.
|
| Moved to another school were I was absolutely average socio-
| economically speaking, had some of the best years in my life.
| Rich kids are incredibly cruel.
| anubhavs wrote:
| Some courses in my university were restricted to students in
| their third year or above because of "mathematical maturity",
| which I thought was complete BS because some of the courses had
| no other prerequisites. But after taking some of them, I get it.
| There's a general sense of problem solving flow that takes time
| to develop.
| HPsquared wrote:
| In mathematics in particular, one abstraction is built on top
| of another. It's like trying to read without knowing the
| alphabet.
| H8crilA wrote:
| I often refer to this quite explicitly, something like "my
| skills include reading and writing modern math", as if it was
| a foreign language, or musical notation.
| findthewords wrote:
| Most people are of average intelligence. Suppose that
|
| -a few IQ points here or there makes little difference in one's
| aptitude in real-world tasks
|
| -we then must accept that when most people think they are "dumb",
| there is some other effect going on such as:
|
| -lack of resources, hunger, mental distractions, illness, or
| motivating incentives.
| JustinSkycak wrote:
| Are you suggesting that when a student is struggling in
| calculus class due to not knowing their algebra, their issue
| can somehow be resolved without them having to learn algebra?
| danans wrote:
| I think they are suggesting that they might not have leaned
| algebra because of
|
| > lack of resources, hunger, mental distractions, illness, or
| motivating incentives
| Jensson wrote:
| Most people who are bad at math doesn't have those issues,
| lots of well off people are stupid.
|
| Edit: Also lots of people who lack many of those things
| still learn math very well. So it isn't a very good
| explanation. If you need optimal circumstances to learn
| then you have a problem.
| tjoff wrote:
| I'd argue motivation plays a much bigger role than
| intelligence.
| Jensson wrote:
| Everything I said is still right.
| tjoff wrote:
| Hardly.
|
| Yes, you can contort your definition of intelligence to
| be one where you have to be able to overcome whatever
| personal struggles one might have.
|
| But you wouldn't be actually measuring intelligence. And
| creating an environment that allows for more people to
| thrive isn't catering to the stupid either.
| Jensson wrote:
| Not sure what you are arguing, you didn't seem to read my
| post? I never said we shouldn't try to make more people
| succeed, or that everyone who fails are stupid.
| danans wrote:
| > lots of well off people are stupid
|
| Sure, but being well-off can also cause mental
| distractions and suboptimal motivations.
|
| Having resources, well-off parents can pay for help in
| addressing those problems, but they have to first
| recognize the problems, which is something they might be
| resistant to.
|
| In the extreme form of this (in terms of wealth, learning
| problems, denial thereof, and poor academic performance),
| you get very wealthy people paying/bribing for elite
| credentials and access for their children.
| dleink wrote:
| "Most people are bad at math" doesn't matter, "When
| someone has those issues they are more likely to be bad
| at math" is what we're talking about.
| brigadier132 wrote:
| I think a few iq points is actually much more impactful than
| you realize. 1 iq point boosts lifetime earnings by $50k.
|
| https://www.sebjenseb.net/p/how-profitable-is-embryo-selecti...
| fuzzfactor wrote:
| I would say the most valuable benefits would not be measured
| in currency.
|
| not my downvote btw
| max_ wrote:
| If you used the same techniques used by IQ "research"
| peddlers to invest money in the stock market, choose startup
| founders or business managers -- you would go bankrupt with a
| probability of 1
| ilrwbwrkhv wrote:
| That is why finding the right study materials for the
| fundamentals of a subject is so important. Take some time to find
| out the right method and material when learning something new
| till it speaks and inspires you. You will learn much better and
| faster.
| eesmith wrote:
| From Neal Stephenson's "Cryptonomicon", at
| https://archive.org/details/cryptonomicon0000step_b9v1/page/... :
|
| > "Shut up about Leibniz for a moment, Rudy, because look here:
| You--Rudy--and I are on a train, as it were, sitting in the
| dining car, having a nice conversation, and that train is being
| pulled along at a terrific clip by certain locomotives named The
| Bertrand Russell and Riemann and Euler and others. And our friend
| Lawrence is running alongside the train, trying to keep up with
| us--it's not that we're smarter than he is, necessarily, but that
| he's a farmer who didn't get a ticket. And I, Rudy, am simply
| reaching out through the open window here, trying to pull him
| onto the fucking train with us so that the three of us can have a
| nice little chat about mathematics without having to listen to
| him panting and gasping for breath the whole way."
| 65 wrote:
| The way I learned programming was to start with the absolute most
| basic thing I could think of - changing the color of a button on
| click in Javascript. With that I could start doing slightly more
| complicated things, until I could get a whole job as a
| professional software engineer.
|
| Learning is like climbing a staircase, and what you have to
| realize is you can't skip steps.
| ocean_moist wrote:
| I tend to slightly disagree. Some of my most valuable learning
| was taking on massive projects, and then just doing whatever I
| could to make it work. Of course, progress would be slow, and
| you'd need to chunk up the problem and take small steps, but it
| rarely felt like slowly building up or climbing a staircase.
|
| I think the trial by fire approach, just jumping into the deep
| end, is a much more effective way to learn. It's also more fun
| for me.
| pajeets wrote:
| I actually disagree with both of you. Programming is learning
| and the code is the side effect of what you have learned. If
| you jump at a large project but lack the fundamentals you are
| going to wasting energy on stuff you shouldn't be.
|
| Rather the best way to learn programming I find was to master
| the basics, memorize the most used routines and commands to
| avoid having to google it every time (ex. CSS)
|
| The problem now is we have LLM which kind of negate the need
| and we have lot of engineers who don't have good fundamental
| systems design.
|
| The other major issue is most of us engineers are scaling for
| a future that won't come. It's sufficient to squeeze ton of
| performance out of a single vertically scaled Postgres
| instance for example without the need to do exotic
| architectures.
|
| So prerequisites are important like in the article but less
| so now but the fundamentals and only learning what is needed
| is critical more than ever (ex. Kubernetes for a blog)
| akira2501 wrote:
| The best way to learn programming is to read and understand
| other programs.
|
| It's a really fun field, and you can isolate yourself and
| get very deep into thought, and then enter an exceptionally
| rewarding period of cycling between work and learning. I
| think a lot of programmers eschew more sensible things in
| favor of exclusively working in this mode.
| ocean_moist wrote:
| > If you jump at a large project but lack the fundamentals
| you are going to wasting energy on stuff you shouldn't be.
|
| "Wasting energy" in this case would be learning. This
| learning happens faster than following a tutorial and
| "mastering" the basics.
|
| > The problem now is we have LLM which kind of negate the
| need and we have lot of engineers who don't have good
| fundamental systems design.
|
| Yeah if you use LLMs for everything you are just going to
| struggle 10x harder when you encounter a problem. This does
| not discount my point.
|
| Also why is your name a slur.
| Jtsummers wrote:
| > I think the trial by fire approach, just jumping into the
| deep end, is a much more effective way to learn. It's also
| more fun for me.
|
| If people survive it this gets them past some substantial
| mental hurdles. People who take too long on the basics often
| get stuck in the beginner treadmill, not really progressing
| but getting better and better at those basics.
|
| Unfortunately, it's not perfect. Most of my colleagues who
| have taken this approach have deficient mental models of how
| computers work or are missing significant portions of
| background or fundamental knowledge.
|
| After the trial by fire, you have to go back and fill in the
| gaps, hopefully deliberately and not just by future trial by
| fire efforts.
| ocean_moist wrote:
| This isn't really my experience. I learnt ASM because I
| wanted to reverse engineer a C binary. I learnt kernel
| stuff because I needed to write a device driver. I learnt
| low level networking by writing an HTTP server in C. I
| learnt programming language design by writing my own
| programming language.
|
| This was, of course, before LLMs, but I don't see how I am
| missing "fundamentals." They generally come if you are
| building something non-trivial and are genuinely interested
| in technology.
| HPsquared wrote:
| Books and courses usually list their prerequisites. Or good ones
| do, anyway.
| JustinSkycak wrote:
| One problem is that it's easy to _think_ you know the
| prerequisites when in fact you don 't.
|
| For instance, a student struggling in calculus may _think_ they
| know algebra because they got a decent grade in algebra class,
| even though they struggle to solve a quadratic equation and
| they 've forgotten how trig works.
|
| -- Maybe they got saved by grade inflation, or
|
| -- maybe they did learn these things but they've gotten so
| rusty that they need to effectively re-learn them again, or
|
| -- maybe they learned and still remember everything from their
| algebra class, but the class was watered-down and cherry-picked
| the simplest possible cases of problems within each topic
| (e.g., quadratic equation always has leading coefficient of 1
| and is solvable via factoring) ...
|
| There's a million different ways that a student can look at a
| list of prerequisites and mistakenly think that they have
| learned them, especially if the prerequisites are listed as a
| handful of high-level categories as opposed to hundreds of
| granular atomic topics.
| nicf wrote:
| I'm a tutor, mainly working with adults who want to learn proof-
| based math, and the message behind this post definitely lines up
| well with my experience! If you're the sort of person who's
| animated by the idea of learning math but finding it challenging,
| it's worth considering that you might be missing some knowledge
| or skills that you'd be able to develop just fine if you knew to
| focus on them.
|
| There definitely is such a thing as "mathematical talent", but
| (a) if you're really excited by math then there's a decent chance
| your limiting factor is knowledge rather than talent, and (b)
| there's plenty to appreciate in the subject regardless of how
| much of it you have. My students come to me at all different
| levels but if they have enough time and motivation to work on it
| they all learn a lot of math!
|
| There are also plenty of people in the world who just aren't that
| into this stuff, but that's not really the population I'm talking
| about --- unless they _have_ to learn it for some reason, it
| probably doesn 't bother them that much that they don't know a
| lot of math! And I imagine a good chunk (though probably not all)
| of this group could probably find something to like in the
| subject if it was presented in an appealing way.
| JustinSkycak wrote:
| 100% agree. What I typically tell people is "your mathematical
| potential has a limit but it's likely higher than you think."
|
| Not everybody can learn every level of math, but most people
| can learn the basics. In practice, however, few people actually
| reach their full mathematical potential because they get
| knocked off course early on by factors such as missing
| foundations, ineffective practice habits, inability or
| unwillingness to engage in additional practice, or lack of
| motivation.
|
| (My comment here is basically the intro to a detailed article I
| wrote on the topic: https://www.justinmath.com/your-
| mathematical-potential-has-a...)
| randcraw wrote:
| That's an excellent essay. I especially liked this part:
|
| " Active learning and deliberate practice will be covered in
| more depth in later posts, but below are some key points:
|
| - Effective learning is active, not passive. It is not
| effective to attempt to learn by passively watching videos,
| attending lectures, reading books, or re-reading notes.
|
| - Deliberate practice requires repeatedly practicing skills
| that are beyond one's repertoire. However, this tends to be
| more effortful and less enjoyable, which can mislead non-
| experts to practice within their level of comfort.
|
| - Classroom activities that are enjoyable, collaborative, and
| non-repetitive (such as group discussions and
| freeform/unstructured project-based or discovery learning)
| can sometimes be useful for increasing student motivation and
| softening the discomfort associated with deliberate practice
| -- but they are only supplements, not substitutes, for
| deliberate practice.
|
| - Deliberate practice must be a part of a consistent routine.
| The power of deliberate practice comes from compounding of
| incremental improvements over a longer period of time. It is
| not a "quick fix" like cramming before an exam. "
| JustinSkycak wrote:
| Thanks! Yeah, I guess I should probably link to some of
| those later posts about active learning and deliberate
| practice in the article. If you want to read more about
| that part you liked, here's the main one I'd follow up
| with: https://www.justinmath.com/deliberate-practice-the-
| most-effe...
| wonger_ wrote:
| And when they're knocked off course, they often develop math
| anxiety. It's a very real sense of dread that's been
| conditioned over time from test taking pressure, missing
| foundations, and underperforming.
|
| Then they're not just lacking motivation, they're motivated
| to avoid math, which makes remediation more difficult. So
| sad.
| JustinSkycak wrote:
| Totally. It's a vicious cycle. Once you get knocked off
| course, you fall into this current that's pulling you
| further off course. And the further off course you go, the
| stronger that current is.
| magicalhippo wrote:
| I've had similar experiences helping my SO, my sister and a
| good friend with post-high school math in various forms.
|
| My SO had a teacher at school who'd determined she couldn't
| do math, and had the worst passing grade as a result. She
| wanted to go to engineering college and lacking the
| prerequisites she had to take their pre-course, ie all the
| math and physics required compressed in a year. She struggled
| hard from the get-go, and I had to go back to elementary
| algebra and build up. Yet after a few hard weeks the efforts
| started paying off, and in the end she nearly aced the pre-
| course.
|
| My sister had never been into math, and had taken a
| vocational route working in a kitchen. After some time she
| wanted to go to college doing something else, and that
| involved taking college level math. While not as strong as my
| SO, again similar story where persistence and working the
| foundations helped a lot, and she aced her somewhat easier
| college math course on time.
|
| My friend was a bit different, in that he'd never been
| interested in math but had to take it to get the required
| points to get into some uni program he wanted. He's fairly
| smart but struggled with motivation. So for him it required
| finding the right way of forming the questions so he got some
| motivation to solve it.
|
| These are my most direct experiences, though I've also helped
| others here and there. It led me to believe _most_ people
| could do reasonably well at entry-level college math (ie
| basic calculus, statistics etc). For some it might require
| quite a lot of effort to get there, but still doable for
| someone with motivation.
| SoftTalker wrote:
| > adults who want to learn proof-based math
|
| What is their usual motivation for this? Do they find they are
| running into regular work or life situations that require it?
|
| I think about all the math I took in high school and undergrad,
| and in my adult life I have not used anything more advanced
| than basic middle school algebra and occasionally some simple
| trigonometry. I don't even remember most of what I learned,
| other than very high-level concepts.
| nicf wrote:
| Motivations vary a bit, but most of them are just in it for
| personal enrichment, and the people who are in it for
| personal enrichment tend to be the most likely to stick with
| it. There are definitely jobs that require more math than the
| things you listed, but even if you have one of them the way I
| teach is usually more optimized for curiosity than
| professional goals.
| elawler24 wrote:
| I always thought I was bad at math. Then I decided to learn it
| again from the ground up when studying for the GMAT. I hired a
| tutor who completely re-taught me the basics and got me excited
| about it for the first time. I was amazed by how quickly I
| became comfortable with concepts as an adult, topics I assumed
| I was innately "bad" at. It made me realize how many things I
| could one day learn, given enough time and interest. Glad there
| are good tutors out there!
| scarmig wrote:
| Just wanted to chime in that Nic is an amazing tutor, and if
| you're someone who wishes they had studied math more
| rigorously, you should reach out! You'd be amazed how much you
| can learn in an hour every week or two that's focused entirely
| on your interests/strengths/weaknesses.
| KronisLV wrote:
| Lovely article, though honestly getting those prerequisites also
| takes a lot of time, effort and either motivation or discipline
| in ample amounts.
|
| As someone who was the "smart kid" growing up, going to the
| university without good work ethic was pretty eye opening, no
| longer being able to coast on intuitively getting subjects, but
| rather either having to put in a bunch of effort while feeling
| both humbled and dumb at times, or just having to sink
| academically.
|
| Even after getting through that more or less successfully and
| having an okay career so far, I still definitely struggle with
| both physical health and mental health, both of which make the
| process of learning new things harder and slower than just
| drinking a caffeinated beverage of choice and grokking a subject
| over a long weekend. Sometimes it feels like trying to push a
| rock up a muddy slope.
|
| And if I'm struggling, as someone who's not burdened by having
| children to take care of or even not having the most demanding
| job or hours to make ends meet, I have no idea how others manage
| to have a curious mind and succeed the way they do.
|
| Admittedly, some people just feel like they're built different.
| Even if I didn't have those things slowing me down as much
| (working on it), I'd still be nowhere near as cool as people who
| dive headfirst into low level programming, electrical
| engineering, write their own simulations, rendering or even whole
| game engines and such. Maybe I'm just exposed to what some
| brilliant people can do thanks to the Internet, but some just
| manage to do amazing things.
| vundercind wrote:
| > As someone who was the "smart kid" growing up, going to the
| university without good work ethic was pretty eye opening, no
| longer being able to coast on intuitively getting subjects, but
| rather either having to put in a bunch of effort while feeling
| both humbled and dumb at times, or just having to sink
| academically.
|
| Modern gifted education is very aware of this and is working on
| fixing that. How effectively, I dunno, probably not very, but
| at least it's a hot topic in the field this century.
|
| Damn near every person I've know who was "gifted" has a similar
| story, myself included. There's a lot of lost potential because
| we badly mis-handle those kids for years and years on end.
| dsign wrote:
| I thought that I was gifted, but in retrospect, maybe I
| wasn't. I couldn't play any games well; not the physical
| ones, not computer games, not card or board games. The other
| kids would get them fast, while I was always dumbfounded and
| it took me a long time to grasp the rules. Once, it became
| popular to play Warcraft in my circle of friends and, try
| hard as I may, I was the rag they mopped the floor with :-) .
| But math, physics and computer sciences were another story,
| because growing up I didn't have anything more interesting to
| do than reading and solving exercises from old soviet
| textbooks. So, I think I wasn't gifted, or just gifted by
| serendipity.
| Jensson wrote:
| Symbolic manipulation and intuitive understanding doesn't
| seem to be very correlated to me. I've met people who are
| very good at manipulating symbols, but they aren't very
| good at understanding less formal things like strategy
| games well, they tend to invent way too rigid rules for
| themselves and lose since they are bad at adapting.
|
| So maybe you are very smart with symbols, but less so with
| intuitive understanding? There is nothing wrong with that,
| makes it easier to decide what to work on, focus on your
| strengths and let others cover for your weaknesses.
| lamontcg wrote:
| My memory is kinda shit.
|
| So when I was in the gifted program, they got real
| concerned about how I didn't know my times tables and was
| super slow at a few of them. This was the 1970s, and
| before I knew binary, so 7s and 8s caused me issues. So
| I'd figure out 8x8 by going from 6x6=36 (real easy to
| memorize) and then adding +6+6 to mentally fill out the
| 8x6 block and then adding +8+8 to fill it out to a 8x8
| block (I'm visual/geometrical).
|
| I was the first kid in school to pass the AHSME to get
| invited to take the AIME though. Being before the
| internet, and being stuck on an island in Alaska, didn't
| get me enough exposure to higher math through my own
| self-direction to get anywhere on the AIME. If you don't
| live anywhere near a University library and/or don't know
| you can use it, that'll set you behind (at least back
| then, these days there's YouTube and sci-hub and
| friends).
|
| I still think I would have hit a wall anyway with Math,
| even with perfect exposure, because I'm visual and higher
| Math seems to require being very good with symbols and
| memory as well.
|
| I suspect lots of people still underestimate me because
| my memory is ass, and we associate memory with
| intelligence so much (e.g. Jeopardy).
| novok wrote:
| You might have certain targeted deficiencies that make
| processing those things well difficult. Like I have a hard
| time finding things in visual space, so I'm not good at
| jigsaw puzzles. I have slower reaction times & processing
| speeds also which makes me not as good at video games as I
| should be combined with the visual space deficiencies. When
| a shit ton of stuff is happening on screen I lose my place,
| while others might be able to manage it.
|
| But I also show an ability to think and learn deeply and
| score very well on symbolic and verbal intelligence, I also
| connect the dots very well and show a lot of skip thinking
| behavior. I call my brain a high torque, low RPM engine.
|
| Also you might not like anything that is 'competitive', so
| your brain shuts down in avoidance / disinterest. Or you
| might think deeply about everything, so it always takes you
| a while at first, but once you grasp it you grasp it at a
| deep level unlike others.
| dleink wrote:
| Hmm, this rings some bells for some people I know. Did
| you learn about those deficiencies and proficiencies in a
| systematic way or through experience? Any resources you'd
| suggest for thinking more deeply about these things?
| Log_out_ wrote:
| Somewhere sits a autistic gamedev, mulling over a beer:
| "everyone i know can easily dive headfirst into low level
| programming, electrical engineering, write their own
| simulations, rendering or even whole game engines, but talking
| to people or women ,some people are just built differently
| .Wish i was born on the side with the greener grass..
| veunes wrote:
| But it's important to remember that everyone has their own
| set of challenges
| veunes wrote:
| In university the environment changes dramatically
| vaylian wrote:
| Yes. This is mainly because of 2 reasons:
|
| 1. Things actually get more complicated
|
| 2. Most of your "teachers" know a lot less about teaching
| than your school teachers, because they never had formal
| training in teaching. However, there are some lecturers who
| are natural talents.
|
| The second point is why you have to do more work to actually
| learn a topic. Your lecturer won't meet you halfway, but you
| have to get a lot closer to them to grasp their explanations.
| whiterknight wrote:
| Formal teaching training never made a bad teacher good.
| sn9 wrote:
| https://en.wikipedia.org/wiki/Direct_instruction
| wenc wrote:
| Also gifted child here, but grew up with parents who drilled
| into me that "effort matters, you might not be the best, but
| with effort you can be better than who you were yesterday and
| that is a worthy endeavor" It's just a slight switch of
| mindset, but that small switch carried me through times when
| subjects got too hard and when I wanted to give up because I
| felt I wasn't talented enough.
|
| I knew so many gifted kids who got perfect scores in college
| freshman year (engineering) but started giving up when things
| didn't come as easily to them. They all ended up leading
| mediocre lives after.
|
| I wasn't as smart as them but I knew I could always do better
| if I tried harder (even though the ROI wasn't great at first).
| So I just kept grinding (16 hour days studying). My GPA rose
| very gradually until eventually I finished with a not-great but
| respectable 3.6 (the 3.4-3.7 range is typically attained by
| folks who maybe didn't have natural talent but worked hard).
| That GPA got me into grad school where I got to study what I
| loved (way fewer exams, more research).
|
| Carol Dweck wrote a book on "growth mindset" that talks about
| this. You don't have to read the book, but the idea of growth
| mindset, though simple, has been really transformative in my
| life.
| novok wrote:
| Sometimes the greatest gift you can give a gifted kid is
| failing early at something they want, so they learn the grit
| to achieve it.
| wenc wrote:
| Absolutely. No matter how gifted a person is, there is
| going to be some point in their lives or some domain where
| pure talent will be insufficient.
|
| Even super geniuses like Terence Tao struggled (he nearly
| did not pass his generals).
|
| https://terrytao.wordpress.com/2019/06/27/living-proof-
| stori...
|
| https://web.math.princeton.edu/generals/tao_terence
|
| The American Mathematical Society has a PDF of essays of
| brilliant mathematicians (including Terence Tao) who
| despite their abilities, faced obstacles and difficulties
| that their sheer talents were inadequate to overcome, and
| they had to persevere.
|
| https://blogs.ams.org/inclusionexclusion/2019/06/26/living-
| p...
| nuancebydefault wrote:
| This seems to be a story of a kind that shows up regularly at
| HN. People who are smart in several ways but because they
| cannot do what some other (1 out of 10000) people-with-or-
| without-children can, they don't feel smart and are a bit
| frustrated by it.
|
| Those others make a 'tutorial for creating a raytracer from
| scratch in a weekend', invent a format for a binary that runs
| as-is on several processor architectures, or are maintaining
| parts of the linux kernel.
|
| I easily recognize such stories, perhaps because that story
| might apply to myself as well. This unfulfillment treat applies
| to this post as well as the writer of the article (the writer
| indicated that before his 150 day submersion he felt a bit dumb
| at mathematics).
|
| The thing I want to bring across is now... we should not strive
| for such capability. There are many things to learn and ways to
| grow as a person, why beat ourselves up for not being very good
| at this or that particular thing?
| nsxwolf wrote:
| Well, we let kids believe they're just not good at math, and
| tell them that's ok I'm sure you're good at other things, and
| they believe it, and wish we'd stop doing that.
| nuancebydefault wrote:
| While they are not studying maths, they can grow and
| improve themselves in other ways, not?
| dwaltrip wrote:
| There's a world of difference between that and beating
| yourself up for not being in the top 0.01% of performance.
| KronisLV wrote:
| > There are many things to learn and ways to grow as a
| person, why beat ourselves up for not being very good at this
| or that particular thing?
|
| I'd say that this takes a lot of work to unlearn, be it
| social media or whatever else seems to teach us to compare
| ourselves against others. Even though there are people way
| more brilliant than me out there (maybe they're naturally
| gifted, maybe they have a better work ethic, or different
| circumstances), it is definitely possible to be happy for
| their success, rather than lean into being jealous or what
| have you.
|
| Of course, they will often achieve more than I will and will
| lead better lives as a result of that, but that's also
| something to accept and take in stride, rather than for
| example believing that I'm some temporarily embarrassed soon-
| to-be millionaire who's one good idea away from a lavish
| lifestyle. Not that it should discourage me from being
| curious about new ideas, even if writing my own particle
| simulation quite quickly ran into the n-body problem and also
| the issues with floating point numbers when the particles get
| close and the forces between them great.
| austin-cheney wrote:
| Math is hard but fortunately programming is easy. I am just a
| dumb soldier who taught themselves to program while traveling
| around Afghanistan.
|
| The down side of being a dumb soldier programmer is that it's
| really really hard to find sympathy when people complain about
| how hard life is when they are utterly reliant on a bunch of
| abstractions and clutter to do their jobs for them.
| brigadier132 wrote:
| Don't sell yourself short, programming is just as difficult as
| math. I'm confident saying if you are able to code you would
| also be able to learn pretty advanced math.
| YZF wrote:
| I also taught myself how to program at a younger age.
|
| You'll be surprised how hard this is for many people. I was
| surprised by how many really struggled through the first
| introductory programming course in university.
|
| Math isn't that hard. But for almost everyone it requires a lot
| of work to get to the next level. Some of the same skills
| transfer. Something too removed from where you are feels very
| alien, partly because of language, symbols, conventions, etc.
| and also not having the building blocks/theories etc.
| casebash wrote:
| Just thought I'd add a comment as someone who came top of the
| state in my grade in multiple olympiad competitions:
|
| I always felt that a large part of my advantage came from having
| a strong understanding of maths from the ground up.
|
| I felt that a lot more people could have gained the same level of
| understanding as I did if they had been willing to work hard
| enough, but I also felt that almost no-one would, because it'd be
| an incredibly hard sell to convince someone to engage in years-
| long project where they'd go all the way back to kindergarten and
| rebuild their knowledge from the ground up.
|
| In other words, excellence is often the accumulation of small
| advantages over time.
| ofrzeta wrote:
| I can only blame teachers. In primary school after four years
| they finally managed that there's only on kid per class left
| (that would be you I guess) has fun with math. At home I am
| fighting an uphill battle because I know it can be fun (my kid
| even likes logic puzzles). Living in Germany, for the record.
| TheRealPomax wrote:
| I'd rather blame the system that teaches the teachers. I'm
| certainly not going to blame someone for not knowing how to
| teach an onramp that they don't even know exists because they
| themselves were never taught properly.
| doubled112 wrote:
| Sometimes it's just the teacher.
|
| I loved reading until my grade 4 teacher decided we would
| all write a book report every week. Haven't read a story
| book since. It's been 20+ years.
|
| Forced fun is never fun. The other grade 4 class wrote
| three that year.
| pastage wrote:
| People react differently to task and teachers, there is
| no one way to do it. I got good grades because a teacher
| let us repeat a task like "write a report" seven-fourteen
| times. The feedback was given by him in class
| highlighting the important points of getting a good
| grades, and then 24 hours after handing in the report we
| got it back with notes mentioning which important parts
| we had missed.
|
| This thought me the rather simple lesson that getting it
| right on the first try is really hard.
|
| Writing a book report is completely different from
| reading a book. I have heard people doing literature in
| university being sick of books because of the same issue.
| dleink wrote:
| That sounds like a dream. I have a distinct memory of
| doing in class writing in 3rd grade where the teacher
| would force us to redo it if there were mistakes and give
| minimal feedback. As far as my 3rd grade memory can
| recall, I rewrote it several thousands times and never
| got it all the way right.
| veunes wrote:
| The idea of starting from scratch and rebuilding one's
| knowledge, especially when it means going back to the basics,
| is daunting
| randcraw wrote:
| I think much of that 'daunt' comes from the lack of
| instructional resources needed to support a solo journey
| through higher math. Yes, there are some great illuminating
| sources (like Kahn Academy and 3blue1brown), but if you're
| embarking on an epic quest (like recapping a BA in math), the
| essential guidance needed for coherent and graceful passage
| through all the requisite concepts simply does not exist --
| short of reading 20 HS and college textbooks, which will
| subject you to a maddening amount of redundancy while leaving
| many fundamental concepts underexplained.
|
| The day that large language models can capably tutor me
| through the many twisted turns of higher math -- that's when
| I'll believe that deep AI has achieved something truly
| useful.
| fragmede wrote:
| Can you link a chat and show specifically where one falls
| off explaining, eg complex numbers or integration by parts?
| it's been a while since my math minor, but ChatGPT seemed
| to be able to guide me through what I recall of those
| topics.
| novok wrote:
| I think mathematics education is pretty horrible this way. You
| only start actually learning the foundations of math in your
| 3rd or 4th year of undergrad.
|
| At least nowadays there is a shit ton of youtube resources and
| more, so a self interested kid can learn it far easier. I tried
| and the books that were out there were... sparse and textbooks
| are written for other professors, not students.
| ocean_moist wrote:
| Some credit needs to be given for just jumping in. Just
| analytically breaking down complex problems into pieces that are
| understandable.
|
| It's often faster to work top-down and turn unknown unknowns ->
| known unknowns -> known knowns.
| JustinSkycak wrote:
| It can definitely be helpful to take a top-down approach in
| planning out your overarching learning goals.
|
| However, the learning itself has to occur bottom-up. Especially
| in math. Math is a skill hierarchy, and if you cannot execute a
| lower-level skill consistently and accurately, you will not be
| able to build more advanced skills on top of it.
| MichaelNolan wrote:
| I think these conversations of top down vs bottom up, are
| frequently a case where people are talking past each other
| due to using different definitions, and having different
| goals. And I think this is especially common when one of the
| people is a software engineer, and the other is not.
|
| For software, the tooling and the abstractions are so
| powerful that you can make incredible software without
| knowing what's really happening under the surface. Imagine an
| "abstraction ladder", with each level building on the
| previous level. Frequently in software development, you only
| need to understand a single level n, and maybe a little of
| n-1, to make great software. So the advice of "just jump in"
| is often great advice because you're jumping into a "shallow
| pool".
|
| (P.S, I love the work you're doing with MathAcademy, though I
| wish there was discounted price for "casual" learners that
| only have an hour or two a month)
| JustinSkycak wrote:
| Agree, it depends highly on goals. _Using_ off-the-shelf ML
| /AI models (to make great software) requires far less
| background knowledge than _implementing_ new models being
| introduced in papers, which in turn requires far less
| background knowledge than _producing_ new models that
| improve upon the state-of-the-art.
|
| Thanks for the kind words about Math Academy! It's true
| that we focus on students who are trying to acquire math
| skills to the highest degree possible -- we teach math as
| if we were training a professional athlete or musician. We
| maximize learning efficiency in the sense that we minimize
| the amount of work required to learn math to the fullest
| extent.
|
| I realize that there are many learners who only want to
| devote an hour or two per month, but, at least right now,
| such learners would be better served elsewhere. It's a
| totally different optimization problem -- maximize surface-
| level coverage subject to some fixed, miniscule amount of
| work -- and as a result it would require different
| different curriculum and possibly different training
| techniques (or at least, differently calibrated
| techniques).
|
| But it's definitely an idea to think about in the future.
| :)
| ocean_moist wrote:
| > Especially in math. Math is a skill hierarchy, and if you
| cannot execute a lower-level skill consistently and
| accurately, you will not be able to build more advanced
| skills on top of it.
|
| To be frank, I am only a math hobbyist and studying CS. I
| haven't taken any proof based courses (only calc 3, linear
| algebra).
|
| However, it is my experience that once you get the initial
| proof based stuff down and are familiar with common proof
| writing techniques, you can pretty much learn anything.
|
| I often start on an nLab page for a mathematical structure
| and just click on links in the definition till I see
| something I know. I then try and make sense of it and go back
| up the chain. I try to solve some problems and see some
| invariants. If I want to go more in depth I watch lectures on
| youtube and then read a book about it and do some more
| problems.
|
| I'm not really sure if this is a top down or bottom up
| process, but it seems more top down to me. Of course, if I
| encounter a whole field I don't know then I need to start
| from scratch and go bottom up. This is also for essentially a
| late-undergrad math major understanding of topics at most.
| photochemsyn wrote:
| Intelligence is far too complex to be meaningfully described with
| a single number like IQ. Measures of physical capability don't
| suffer from this issue - a person might be strong, or they might
| be fast, and everyone knows that the power lifter and the
| marathon runner use wildly different training regimens to improve
| their abilities.
|
| If physical capabilities are highly trainable, up to some genetic
| limit that the vast majority of people never even get close to,
| then it seems that intelligence must work the same way - e.g.
| prodigious feats of memorization can be achieved via training
| regimes (memory palaces etc.), as can one's three-dimensional
| visualization skills (e.g. a chessboard layout, or rotating a
| platonic solid, etc.) or the ability to rapidly construct
| arguments using logic and reason - but we don't seem to be able
| to classify different areas of mental ability as easily as with
| physical abilities.
|
| Sadly, this is one of those politically difficult topics as the
| blank slatists and the genetic determinists (Lysenko vs. Galton)
| have tried to use all kinds of pseudoscience to support their
| ideological arguments, when the underlying point is just that
| training your mind is as beneficial as training your body, and
| everyone should do it at least to some extent.
| ninetyninenine wrote:
| However iq is the most correlated number in psychology. It's a
| heavy predictor of many many things. It's the one thing in
| psychology that has the most scientific validation.
|
| Yes iq can't measure everything that intelligence represents.
| But it does measure something extremely important and
| meaningful to us.
| eesmith wrote:
| I think wealth might be more correlated, as it can be
| estimated without testing based on tax data. It's also a
| heavy predictor of many, many things, with scientific
| validation. Including IQ test scores.
| captainclam wrote:
| A really fascinating corollary I've observed since I've gotten
| into more advanced maths, or even doing actual research as a PhD
| student, is that there's nothing special going on at the higher
| levels. You're just working with different materials. Materials
| that require more time and effort to 'get', but once you get them
| they are just another tool at your disposal.
|
| I was similar to the author in that, throughout high school and
| undergrad, I presumed that the mind that could comprehend
| advanced math or do novel research (in any field) was truly
| unknowable. Like there was this x-factor they had that wasn't
| there for me.
|
| I've long enjoyed puzzle games (like The Witness or Stephen's
| Sausage Roll). It turns out that problem solving in non-trivial
| domains is never terribly different than problem solving in those
| games, or any other domain really. Like my brain isn't doing
| anything _different_ than the usual tree-search algorithm that
| any chess player performs when they are projecting moves ahead
| into the future.
|
| Its just iterating on concepts that seem abstruse to most people.
| But at the end of the day, deep problem solving in math or AI
| research tends to be the same moving-shapes-around-in-my head
| that I would do if I was trying to move an awkwardly shaped couch
| through a narrow doorway.
| dwaltrip wrote:
| That's a really cool observation, thanks for sharing
| ninetyninenine wrote:
| No, this isn't universally true. Your intelligence does affect
| your ability to learn math. It is not always just a lack of
| prerequisites.
| JustinSkycak wrote:
| Sure, there is such a thing as mathematical talent. However,
| few people actually reach their full mathematical potential
| because they get knocked off course early on by factors such as
| missing foundations, ineffective practice habits, inability or
| unwillingness to engage in additional practice, or lack of
| motivation. Basically, your mathematical potential has a limit,
| but it's likely higher than you think.
| Barrin92 wrote:
| If you want to compete with Terence Tao, maybe. But everyone
| with a functioning brain can get to a high level of
| proficiency, I know because I've taught people who were
| convinced math is an alien language and they lack the math
| gene.
|
| It's the same with any skill, lots of people are convinced they
| can't make music, or learn Mandarin or what have you. 99% of it
| is preconceived notions that they don't belong into a certain
| club and because we keep telling people that if something's
| hard for you, you're untalented and you should focus on
| something else. But it's just effort, even if it's more effort
| for some in the beginning, there's no magic in any of it.
| ninetyninenine wrote:
| Sure but it absolutely matters heavily if not just as much
| prereqs.
|
| I met a girl who struggled with trig and another guy who
| basically trivially studied the subject and aced it. IQ
| matters a lot.
|
| We are talking in circles. Both matter but I'm saying with my
| example above that even at basic math, both still matter.
| It's not some thing where iq only matters for phd level math
| karaterobot wrote:
| True, but then there's nothing wrong with being dumb. I know a
| lot of smart people, and they're all dumb about most things. Like
| the universe, we're mostly empty, with some hot, bright spots.
| What I mean is, don't think of yourself as fundamentally smart or
| dumb, think of yourself as having a lot to learn, no matter who
| you are or how others perceive you.
|
| But sure, this is a good reminder of how you go about learning
| new things. It's the Julie Andrews method of pedagogy: "start at
| the very beginning (a very good place to start)"
| password54321 wrote:
| The real truth: if you aren't good, there is nothing wrong with
| that and there are more than enough developers in the world and
| people who are good with math. What we need is more people
| creating real and interesting jobs for these skills.
|
| Also most people aren't great with spatial reasoning. Chess
| requires zero prerequisites yet the average level of chess on
| chess.com is constant one turn blunders. It took only a year of
| playing on and off to get to 98th percentile and up to maybe 70th
| percentile most of it is capitalising on basic mistakes. We need
| to stop deluding people with feels good content, that's how you
| get memes like imposter syndrome.
| pajeets wrote:
| Streams of tears roll down my cheek as I write this because this
| article perfectly highlighted that it wasn't the laziness but
| rather what was causing it , mainly the lack of prerequisite
| fundamentals needed to thrive in math field. Had I known this my
| life trajectory would've been different instead of self loathing
| and inferiority complex I built up around something so innocently
| simple.
|
| The rush of epiphany and self-forgiveness that overwhelms me
| after all these years. I realize now that learning grade school
| math in French and then started to learning algebra and calculus
| in Japanese abruptly moving to an English speaking institution to
| continue math degree (which i abandoned for reasons in the
| article i realize now ) screwed me up big time because neither
| French nor Japanese nor English is my first language.
|
| For instance I would store numbers in French in my head and
| perform arithmetic in French but to do any sort of additional
| algebraic or calculus I would need to switch to Japanese
| internally and finally write out response in English. Learning
| the advanced topics in English was never going to work out, it
| was like building a castle on sand and the stones are made out of
| mud. I always thought I was too "dumb" to
| understand math. During my school years, it was evident to me
| that for some kids math was easy, and for others like myself:
| painfully difficult. This belief shadowed me for
| years, a constant reminder that while believe I am smart... I'm
| not THAT smart. Recently, after 150 days immersed
| in learning math, I had a stark realization.
|
| The struggle wasn't because I wasn't capable, but rather, I was
| simply missing a shit-ton of pre-requisite knowledge.
|
| I wish I could show this article and translate it into other
| languages. There are lot of young kids in schools who tell
| themselves they are dumb or lazy because they can't do well in
| math and sciences.
|
| God knows how many of us are walking around feeling inadequate or
| frustrated at ourselves because we convinced ourselves we are not
| worth it or capable when in reality its the prerequisites both
| conscious and subconscious, overt and covert we fail to realize
| as fundamental stepping stones to success.
|
| It might as well be that failure in startups or business ventures
| or relationships even also stem from this principle: that the
| fundamental prerequisites were not taught or caught early on
| (either due to environment, upringing, socioeconomic constraints)
| have solidified into bad habits, bad model of world, bad model of
| others that ultimately transpire into bad thoughts, bad words,
| bad actions and opposite outcomes of what we set out to
| accomplish.
|
| Going forward I must make it my mission to realize what
| fundamentals and prerequisites I do not have and instead of brute
| forcing and letting my ego ignore it, I have to put aside time to
| build those basic building blocks.
|
| A cathartic angst feels deep in me. Might be too late for me due
| to my age and I fear I will ignore my own writing here and others
| will too. It's truly sad that we are all realizing it this late
| and will forget whatever lessons were learned. I wish society and
| people would stop pointing fingers at people and rather realize
| build tolerance from the fact that not everybody gets to build
| the same prerequisites as humans cannot be the same, some are
| innately inclined to better at certain things while others are
| not.
|
| Equal outcomes is a failure in the making and schools need to
| stop and focus on helping students build prerequisites on their
| own schedule and pace.
| whackyMax wrote:
| As a side note, the headline seemed a fun one, it seemed to me to
| say "you are not dumb, but you just lack a few prerequisites to
| be dumb".
| veunes wrote:
| > It's like trying to defeat a Elden Ring boss... at level 1.
| Just in love with this comparison
| GeoAtreides wrote:
| You know how there's a window for learning to speak?
|
| https://en.wikipedia.org/wiki/Critical_period_hypothesis
|
| I posit there's a similar window for highly abstract thinking,
| like math or logical thinking or, controversially, for learning
| how to learn.
| hervature wrote:
| Why are you stating a hypothesis as if it was fact? I'm 30 and
| started learning Spanish (1,000 hours so far) and think I'll be
| near-native level in another 1,000 hours and maybe actually
| native in 5,000 total hours which I might get to by the time
| I'm 40. As we age, we simply have other commitments that we
| cannot devote this much time to language learning. Kids
| "easily" learn language because they can easily put 1,000 hours
| of exercise each year for 5 years.
| GeoAtreides wrote:
| This is not about learning a second language, this is about
| learning to speak in the first place
|
| for example: https://en.wikipedia.org/wiki/Critical_period_hy
| pothesis#Dea...
| hervature wrote:
| The leading sentence is:
|
| > The critical period hypothesis[1] is a theory within the
| field of linguistics and second language acquisition that
| claims a person can only achieve native-like fluency[2] in
| a language before a certain age.
|
| Granted, I do admit I missed the subtlety that it applies
| to learning a first language as well.
| dranudin wrote:
| I have lived in a foreign country for about 3 years. My wife
| is from there and I speak with her in that language every
| day. I also took courses until the B2 level. My communication
| in that language amounts to easily more than 5000hrs. I am
| fluent, but no native speaker would call me native and that
| will never happen. My wife learned my language in school
| since elementary school up to graduation. She lives in my
| country since about 5 years. In her job she has to talk to
| people for basically 8hrs a day in the local language. She is
| fluent but nobody would call her native. Instead people
| wonder where she is from because they cannot match her accent
| to a particular country. Most likely we will never have
| native competency in the foreign language. So if you make it
| to native in 5000hrs, you are way above average.
| dleink wrote:
| I wonder if there's some kind of plateau you reach in a
| learning a language that way. Like, what if you started
| working with a hollywood dialect coach?
| sn9 wrote:
| Yeah there's definitely a gap between functionally fluent
| and being mistaken for a native that requires some
| intentionality and effective study that is unlikely to be
| crossed accidentally/passively or by focusing on the
| wrong things/methods.
|
| Some combination of learning the phonetics of the target
| language, 1000s of hours of comprehensible input, singing
| to music in the target language, and doing impressions of
| native speakers are all things that can help.
| Dr_Birdbrain wrote:
| I would be curious to hear what "better learning methods" the
| author used. He calls them out but doesn't describe them.
| rod_o wrote:
| Pretty sure he means https://mathacademy.com/
| DantesKite wrote:
| According to this tweet, he used Math Academy.
|
| " Excited to see a @_MathAcademy_ student blowing up Hacker
| News with some insights they've realized about the process of
| learning math!"
|
| https://x.com/justinskycak/status/1827397012758917622?s=46&t...
| j45 wrote:
| Math as taught by one of my teachers made a lot of sense.
|
| Some topics will come easier and click. Others will need to be
| brute forced by practicing examples.
|
| I can see how that generates pre-requisite knowledge one way or
| the other.
| atoav wrote:
| As an educator, this is the thing I always say. If you try to
| teach someone programming for example, try to make a honest list
| of all required prior knowledge. This is usually stuff that is
| totally _obvious_ to anybody in the field, but if you don 't know
| it it might give you a hard time. E.g. that programs run within
| the context of an operating system and the OS provides interfaces
| for interaction with hardware.
|
| Not all of that is needed upfront, but certain explanations just
| won't make any sense if required knowledge is missing.
| closetnerd wrote:
| The source of about 30% of the hits I took in homework and tests
| was, to this day, not having memorized the basic values of
| Sin/Cos(pi, pi/2, pi/4).
| YZF wrote:
| Instead of memorizing the values you can learn the shape of the
| graphs and/or learn the meaning of sin/cos on a right angle
| triangle.
|
| I think I got ok through a math/comp.sci. degree without
| memorizing values but I will say the "mechanical" aspects of
| math including trig functions and things like integrals and
| derivatives involve drilling those enough to make these things
| automatic (which is sort of memorizing but not exactly). I was
| always lazy so I never got really good at that (and my calculus
| related grades are evidence of that ;) ).
| sn9 wrote:
| Here's an Anki deck for the unit circle:
| https://ankiweb.net/shared/info/1813562307
| digger495 wrote:
| The author's point about Elden Ring is especially on point..
|
| Because not only do you need to be level 50, but you need to try
| and fail five times before you see any kind of success.
|
| Failure is _inevitable_. Quitting is optional.
|
| You have to learn from each mistake.
| nsxwolf wrote:
| I wish someone had told me this in school.
| w10-1 wrote:
| For many practical applications, slow learners do just as well as
| fast ones, once they're up to speed.
|
| Math teaching is mostly playing hide-the-ball, which teachers
| justify by saying people learn more deeply when they figure it
| out for themselves. But really that just shifts the burden of
| backfilling prerequisites to the student.
| Jensson wrote:
| > But really that just shifts the burden of backfilling
| prerequisites to the student.
|
| Not if the students were taught that way from the start, learn
| for yourself at every step and you are never behind.
| whiterknight wrote:
| Are you suggesting that you don't learn as much when you figure
| our the puzzle as oppose to being told the answer?
| neom wrote:
| Anyone here with developmental dyscalculia managed to overcome it
| and get into math? I have no clue how to deal with numbers, I
| know this is weird to say but it feels like they don't exist for
| me.
| CM30 wrote:
| This tracks pretty well with my experiences learning programming,
| both when developing websites and making video game mods.
|
| The times I failed, I was looking at other people's work and
| trying to figure things out too quickly and in an unstructured
| way. I saw the complexities of a program that was in development
| for weeks/months/years, then basically panicked and thought I'd
| never be able to make something like that.
|
| When I learnt the basics, I then saw how these problems could be
| broken down into their simplest forms, and ended up learning a
| lot more efficiently as a result.
|
| Of course, having examples of what to do helps a lot, it's just
| your examples need to be merely a tad more complex than what you
| already know, not a masterpiece from some genius that spent the
| last decade working on it. Or if they are from that sort of
| person/company, you should try and break down sections of the
| work at a time to understand where they're coming from, not the
| whole thing at once.
|
| It's much more reasonable to try and figure out how someone like
| Facebook or Netflix implemented a profile page or edit button
| than say, how the whole system works on a greater level.
___________________________________________________________________
(page generated 2024-08-24 23:01 UTC)