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.
> 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.
As a gifted kid, when you get to university, you are supposed to learn work ethics practically overnight. Until then, between your 6 and 18 years, the school kept telling you that you should slow down and wait until the average kids also get it.
That's like failing at a sprinting competition, when your entire training consisted of walking really really slowly.
My wife has a nephew that is insanely smart. Like, aced the SATs at 15 smart.
His mother insisted that he have a fairly normal-paced education, and it seems to have paid off.
He’s a really decent chap, who teaches at a university, is married, and has a kid. His college career was at top schools, though. They pretty much threw scholarships at him.
My university experience was very much “submit! work!” - I was on “keeping of term” almost permanently and had the threat of rustication hanging over my head for the duration of my time there as I flat out refused to attend lectures, tutorials, or labs - I was having way too much fun running a bar and the debating society.
Anyway. What I learned was that people rarely make good on their threats, that charm and doing the absolute bare minimum to not “get fired” will get you through - and that I can cram a three year physics course into a month of intense study and still pass with a 2:1, which they demoted to a Desmond as they didn’t feel they could in good conscience reward me with a 2:1 - which also taught me that institutions can’t be trusted and are ultimately run by opinion.
This lead to a career path of opportunistic system-hacking and an early retirement to a cabin in the woods. I never had a work ethic, apart from in that which interests me. If something bores me, it’s for the birds.
I’m not quite sure what I’m trying to say here, other than that at no point in my education was I given any useful guidance or advice, just expectations of prodigy, and was left to figure things out for myself - which I did, but not as I think others would have hoped. I now, in my forties, know that I have raging ADHD - it wasn’t even a consideration as a kid - just that I was “brilliant but bone-idle”.
Do you think yours is a life well spent for anything or anyone other than your own enjoyment, even if you were clever enough to save yourself the stress suffered by many?
I would say that it has not provided as much utility value to mankind as it could have. Sure, I’ve created jobs, generated wealth, given to philanthropic causes - and in excess of the lives lived by most - but I would say with confidence that had I perhaps had some guidance other than the eternal threat of punishment, I would have developed something other than a frankly criminal instinct, and might have been more able to give more to society and fulfil my potential.
Instead, I learned to avoid the consequences, not to avoid the crime, and can’t deny that I have chosen a selfish path as a result.
There are gifted schools that don't force kids to slow down to the lowest common denominator. I went to one. In theory, they also enforce a work ethic. My problem was the reverse: I started working after school when I was 15, and when I got to university I found my peers had absolutely slovenly work ethics and no experience, and I was being taught things that were laughable in a real work environment, by professors who hadn't had a real job in decades (if ever). And I was supposed to be paying $30k a year for the privilege. I realized at 19 that I was better off at any pay rate in the private sector ;)
I had a similar experience perhaps in some ways. I graduated high school and initially started university at 14. (I was tall, which is probably good in this case, as it let me pass for a very young looking maaaaybe 18, in the setting).
My “work history”
Has been 8 ish, doing yard work around town with 2 other kids (nothing crazy, weeding, push mowing, leaves, clearing brush, etc), and then working at a stables for a bit when 12.
I ended up getting an hourly part time position at campus, at 16 ish I took a break for 2 years, worked in grocery, and then in pharmacy.
18, I ended up back at school. Embarrassingly it took me 3 more years to finish my undergrad. But I worked full time or nearly full time the entire time. (Floater pharmacy tech, part time backup event photographer, and then as a store manager at a hobby store).
I found it very hard to make friends with people my own age, but found it incredibly easy to make friends with returning students - either they had worked a career and came back to school, or had done some time in the military etc., I also made some great friends who had moved to the U.S., usually as late teens, from developing countries.
Knowledge, perspectives, etc I learned from these people … and from some people I ended up being very lucky to work with, honestly proved in hindsight to be much more impactful and valuable to my life and career then I could have ever imagined.
I will say, tongue in cheek, but also some truth:
You could probably apply the anna karenina principle to all of the people I can think of who were impactful in some way… either through the lens of trauma/ struggle / or dysfunctional family. (This would also apply to myself!)
I believe real world friends or colleagues are your alternative family. (Online friends, not so much). In any case, its so critical to have your eyes open to other traditions and creeds and ways people live their lives. Without that, we would just judge without knowing why people did things.
The person you need to show mercy to most is yourself. And to do that you have to understad how everyone else lives.
When I was at university, some of my classmates were admitted right after high school, others failed, waited for one year while having a job, and then applied again. The difference between those who had one year of work experience and those who had none was profound; the former were adults, and the latter were kids; it was as if they had five years of age difference, not only one.
I wouldn't underestimate the professors though. Doing research (that is their actual work; teaching is just a hobby) can be hard work.
Which school did you go to, and are you aware if it's kept the same high standards? I believe there are interesting challenges for any mind, the hard part is the match-making customized for each person. It'd be cool if there was an equivalent Facebook/Netflix algorithm for learning.
I always thought the alternative to "gifted programs" is not having a program which is even worse. At some point optimizing teaching becomes unaffordable.
The elementary school I went to operated somewhat as a petri dish for psychiatric experiments, as I realized later, with class sizes around 15 students and extremely personalized teaching. My 8-12 grades were at a private prep school. Both could be scalable models, but it would require a moonshot level of public funding to go into hiring potential teachers away from more lucrative jobs. (Personally, I'd love it if my taxes went to that).
But I don't credit those schools for my success. Nor do I credit native intelligence. My two elder brothers are lawyers whose names you likely know; think of the largest case in recent history. One is severely dyslexic and the other I'd wager is mildly autistic. I'm a college dropout. Oddly, I earn more per hour designing databases than the famous one does taking down large companies. What we had in common was a pattern of learning how to think, how to be curious and ask questions, and how to separate wheat from chaff. All of which came from our grandfather, who was forced to leave a yeshiva at 12 years old, and his father, and his grandfather.
I truly believe that almost all formal education is bunk. It's a useless plaster on a gaping social wound, namely that parents don't have the toolset to teach children a love of learning throughout their lives, along with the methods and skills to do so for themselves. All the information taught in K-12 schools is readily available, yet most adults can't remember a thing about the most basic aspects of history, math or science. The reason being, they weren't interested when they learned it the first time, and they weren't raised to be curious enough to answer their own questions or (re)fill holes in their own knowledge. This is why most people can't utter the words, "I don't know, let's look it up." Moreover, most people don't believe it's their obligation as a person to be as well-rounded as they can make themselves, because no one ever told them that was important, even crucial to their survival.
Learning itself should be taught. And it can be taught at home. The major obstacle would be how to overcome, obliterate and shame the intellectual laziness of most people that's built into most cultures - including those of most who go to college. Everything else, all concepts and facts, can be learned later, and are ephemeral.
Half of all the people in the world are below average. I know, it's crazy right? Confirmation bias makes us expect others to be like ourselves, but they are not, and you're going to have a bad time when you expect them to be like yourself. This is probably going to sound harsh, but it's true: Children and teenagers innately sneer at others that are not like themselves (so much bullying in schools). Part of maturing as a person is to learn that other people are different and have different needs. Many people don't learn this, and they seem immature as a result, and they are ineffective at dealing with other people.
It does if you consider the part going to your county:
“In 2022, the federal government spent … This is 13.6% of the total spent on elementary and secondary education in 2022. The remaining funding comes from state and local governments, which contribute 43.7% and 42.7%, respectively.”
You've misread that. It is not saying that 43.7% of the state budget goes to elementary and secondary education. It is saying that 43.7% of the funding for education comes from the state. Those are completely different.
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.
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.
Things improve with practice though, at least they did for me. I largely sucked at action video games as a kid, I also didn’t have interest probably because I wasn’t good. I took to learning musical instruments and now in my 40s I accidentally found Im pretty good at videogames, much much better than when I was young. Coincidentally I got better at sports as well.
This reminds me of how a lot of top video game speedrunners (e.g. Portal) are professional musicians because they are very good with accurate timings lol
There's also some anecdata floating around in the -- what would you call it, the "competitive speed typing" community? Or "people who play TypeRacer a lot". Some seemingly significant correlation between skill in typing and piano. Or at least the appearance of skilled typists having some interest or experience in piano. This could probably be generalized to being good at fingering or timing, or maybe even further, to having effective mindsets or attitudes regarding performance of skills (like the need for a presumptive confidence of sorts), or being aware of good practice methods.
I accept your hypothesis, but allow me to suggest an alternative hypothesis for you to consider (and maybe reject after consideration).
I am in my late 40s. Modern videogames are a great deal easier than games I played as a child. I tried playing a few games from my mispent youth recently, and was absolutely amazed at just how much harder they were.
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?
Gifted/high IQ/whatever is not a blanket pass to excel. We'll suck at a lot of things, we're subject to plenty of the same mental illnesses and struggles, etc.
Gifted education is intended to address the needs of the students beyond what can be provided in a standard classroom. That's not just more worksheets or harder textbooks; it should also cover students who are able to coast through the advanced classes and make sure they know that they, like every other human, won't have everything easy. A lot of school programs have missed the mark and a lot will, but education is a process of improvement. Many of the teachers today are doing better for the kids because of the lessons learned from our teachers.
Special education (including gifted education) isn't legally mandated in schools to make the students prove they're eligible for the label. If you got value out of the program, it was meant for you.
Why do you think that after “no child left behind” the situation is better for talented kids? IMO the situation is much much worse. Many gifted programs are being eliminated and all classes are being slowed down to accommodate the bottom 33%. Even slightly above average kids are going to feel like 2 standard deviation genius doing basic school work because compared to the curriculum they are.
> Even slightly above average kids are going to feel like 2 standard deviation genius doing basic school work because compared to the curriculum they are.
They'll be treated like geniuses too. I used to get treated like one because of my incredible ability to plug numbers into formulas and write down the answers that came out in a piece of paper. I simply did not understand how people could possibly have any problem with it. I found it so dull I ended up in a computers course where I learned to automate that kind of human computer nonsense away forever.
They treated me like a genius for working out some basic math, and the truth is I suck at math. I actually like math, but I suck at it. I used to get away with never needing to do homework as a kid. As a result I never developed the discipline necessary to hone math skills. Now I want to learn something interesting like queueing theory but I barely understand the papers and articles because I'm missing numerous prerequisites.
Your post seems contradictory in that you could plug numbers into formulas and get answers. Isn’t that a fundamental skill? Is there an implied value in understanding the formula and how to apply it via math to obtain a correct answer?
The question then is how did you develop the ability.
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.
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).
I used to believe that I was good at math because my memory was shit. So I had to actually understand everything, because I couldn't rely on memorizing it.
But some random facts were easy to memorize, for example that the chessboard has 64 fields. I had problems with 6×9 and 7×8 though, always confused about which one was 54 and which one was 56.
I can't tell you what 6x9 is. But I can tell you what 9x6 is. I always turn that one around. My mind immediately jumps to a visual 60 coz 10x6 is super easy of course and subtracting 6 from it is easily 54.
Similarly 8x7 I can't tell you but 7x8=56 in my brain feels like a little "rhyme" I just need to repeat and I have the answer.
That's also how I remember (somewhat) arbitrary passwords. If it "flows" well almost like a rhyme and can be typed fluently I'll remember. Actual arbitrary ones don't work as well.
Yeah, after a while I also learned that 9x6 = 10x6 - 6 = 54.
And then I just remembered that 7x8 is "that other difficult number", because by that time I already remembered that 54 and 56 are the two most difficult numbers in the multiplication table. :)
Btw, same here, 9x6 and 7x8 feels much more natural than the other way round.
So, fun fact from calculus (though you can easily prove this with basic algebra as I do below):
- You want to compare two products: in this case 6x9 and 7x8.
- And in each product, if you add the two numbers together, you get the same result. In this case, 6+9 = 7+8.
Then the product will be larger for the pair of numbers that are closer together. So 7x8 > 6x9. That might help you remember which is 56 and which is 54.
You typically see this in a word problem where you are given a fixed amount of fence and you have to enclose the largest rectangular area. The answer is to use a square area (two sides being equal). If the problem has constraints that prevent the sides from being equal, then you pick the length and width to be as close to each other as possible.
In case you want to transfer the geometric intuition to an algebraic proof: If the sum of the two side lengths is 2m, then the two side lengths can be written as (m+n) and (m-n) for some positive n. If you multiply the two, you get (m+n)(m-n) = m²-n². To maximize the product, you need n to be as close to 0 as possible - i.e. for both sides to be as close to each other as possible.
Oh! So that’s why my teachers made us memorize row by row: For me 54 and 56 are in an entirely different category (resp the 6 and 8 categories), didn’t even realize they landed in the same dozen when learning my tables!
Roguelikes like Nethack/Slashem/Dungeon Crawl Stone Stoup and strategy/RPG games such as Liberal Crime Squad or Battle for Wesnoth would be easier for you.
What high school makes money from their sports programs? Even in basketball or football-obsessed towns, I doubt the ticket sales and concessions revenue from home games would even cover the coaches' salaries, transportation to away games, and uniforms. Most schools near me have players sell candy or coupon books to pay for their teams' expenses.
It's not revenue. I wasn't in the US. People sink money into high performance athletes. They lose money. In Australia only the AFL and Rugby League make much money. That is also only at the high levels.
There are places in Australia, such as Sydney, that have a network of selective high schools for kids who do well. But outside of Sydney it's much weaker.
There is more effort put into kids who are not doing well at school to improve their performance than in getting the most from kids who can do well. Perhaps it makes sense.
Sports only make money at the point of the most elite college programs, usually mainly its the football or basketball program that subsidizes the rest of the athletic program.
I think its more you become aware of where you stand fast. You go to meets or games of a bunch of different schools and see who is literally the best in the area. You go to state competitions and see who is best in the entire state. And then there are nationals where you see who is best in the entire nation. Throughout this your stats are posted online where you and college scouts can see them.
Academics have no comparison. We struggle to even compare grades because of grade inflation.
Probably more that lots of kids will try super hard to be good at sports. Being good at school carries a certain level of stigma and lots of kids who could be “smart” choose to slack instead.
I'm not sure being "gifted" has anything to do with it. High school is easy, and they keep dumbing it down over time. Little to zero effort is required to pass. This can fool people with a minimal amount of intelligence into thinking they're Very Smart. Especially since everybody keeps telling you that you're the smartest kid around and treating you like a nerd. I was able to get near max grades with little study in the subjects I cared about, and passable grades with zero effort in the ones I didn't. It can seriously warp one's perception of reality.
When I got into medical school I straight up failed a class for the first time in my life right in my first semester. I got my ass kicked so hard it's not even funny. I had to put in actual effort into learning stuff for the very first time in my life. I had to spend all of my waking hours in the laboratories to learn this stuff. I met people who had zero issues studying 5-10 times as much as I did. In the middle of it all I got diagnosed with ADHD by the neurologist I was shadowing.
This is when I finally understood the point of school. One of the most bitter complaints from students is the fact most people don't use the knowledge they learn. That's not really the point of school. The point is to just show that you can learn. The point is teaching you how to study and apply yourself so that you don't get your ass kicked later in life when the really difficult stuff starts. Perseverance and mental resillience.
I'm not really a "gifted" person but people treated me like one and in retrospect it was quite detrimental to me. If anything they probably need to identify the "smart" kids and kick their asses harder because the existing classes aren't getting the job done. But if teachers do that people accuse them of tracking...
I wonder how many of these "work ethic" stories are just undiagnosed ADHD stories in disguise...
I mean, this is the second time I see ADHD mentioned in this thread, and the story of somebody who just can't keep focus study for long hours kind of fits textbook symptoms.
To be clear, I'm not saying everyone commenting here with similar stories have ADHD, but uh... if it rings a bell, maybe think about it.
Oh lots. Not all but lots. It's not like I have a study to point to but I still have no doubt about that. You'll see lots of ADHD types here and in similar forums. For some reason, ADHD people seem to be really attracted to technology. They have "attention deficit" and yet don't have any trouble at all concentrating for 10 hours straight on things they care about and I often find that technology in general is in that list. Remember that "bipolar lisp programmer" article? It's more like ADHD lisp programmer. Actual bipolar patients I've seen weren't like that.
When I talk to an ADHD patient, I end up getting this distinct feeling they're talking about me instead. I run them through the diagnostic criteria and they match. I still refer them to a trusted psychiatrist regardless so that a proper differential diagnosis can be made, which does include bipolar disorder. Psychiatry is hard and I'm not about to underestimate the difficulty of it, especially since I could kill the patient if I'm wrong. So far getting this feeling seems to be virtually pathognomonic though.
Maybe because tech is responsive. In most cases the time required to compile, run and test code is a couple minutes. A couple hours at worst. The immediate gratification is really attractive for ADHD people.
^ This was what I intended to comment, and then I thought I should look up the "bipolar lisp programmer" article since I haven't read it before.
And I'll just note that, while the article doesn't mention it, a major feature touted by Lisp enthusiasts is the REPL :) Talk about immediate gratification...
By the way, the story sounds like mine too. Except I went to law school (law is undergrad in where I live). Everyone assumed I did poorly because I wasn't interested in the subject (a reasonable assumption since I was clearly oriented towards programming), but I think the real problem was the "artifice" mentioned in the article. Today, I still read legal cases/materials with interest from time to time, but if I were put back in that artificial law school environment I don't think I'd survive.
I used to be one of the math/science wiz in grade school. I also got hammered on the work ethic part, multiple times. Unfortunately studying/working 12+ hours a day in the name of "work ethic" impacts my body beyond what I can handle, and mental health as well. That's not the way I operated growing up, and my body isn't going to handle it all of a sudden now.
Here I am, 3 cardiac arrests later, trying to figure out how to fit into a society where everyone seems to be hellbent on working every waking hour and eating UberEats while I'm trying to stay alive with immense amounts of self-care in my off-work hours (cooking healthy, hiking, actually disconnecting from the internet, etc.).
Not the OP, but there’s another poster in this whole thread who says that he studied “16 hours per day” and that his former college mates, much gifted than him but who hadn’t studied as much, now live “mediocre” lives.
Which is to say that this is the competition that lots of people who study in a professional environment (i.e. not for fun) have to grapple with, I feel sorry for them because I don’t see an easy solution for all this madness (because it is definitely madness to study 12-16 hours per day).
In high school I could sleep well AND ace everything. I didn't have to study that much to ace honors and AP courses.
Partially because I aced everything, I got into one of the universities considered "top". Although I was excited about the research part, I quickly found out that many courses were hard af. I had to study 12+ hours a day to get good grades there. I did get good grades after the initial shock, but it was hard, I slept very little, and I fucked up my health doing it, without realizing it at first.
Tech companies I have worked at, including the one I just left, routinely don't give you the option to work 8 hours. It's either you work 12+ hours to meet performance expectations or they ask you to leave. My body, unfortunately, cannot tolerate that and "needing to work normal hours" isn't generally one of the available disability accommodations.
Hey, screw modern education. Particularly if you went to a "gifted school". I was stupid for going to college for a year. At least I dropped out and saved myself a lifetime of debt.
Think of the time you waste with that garbage versus how much you can "grok over a weekend" and the math is definitely in favor of the latter.
The major flaw in the educational system in the US is that it's run for profit and it wrongly informs people they need to stay in it, rather than gaining real world experience. Coding, in particular, but also design and web are trades. Like bartending, plumbing, or fixing cars. They're really only learned on the job, and every school that promises to place you in them is a racket. (I also wasted $500 on two weeks of "bartending school" when I was 21, just to see a roster of horrifically shitty bars that were supposedly hiring. All lessons in who's scamming you should be that cheap. What does higher ed cost these days?)
Yes those people who went to school for dentistry or medicine or law or engineering are all idiots who just should have spend a weekend reading a Wikipedia page.
Dentists and doctors in general is complicated because it's impossible to train "on your own", but law school...come on, you just need a good brain, patience and reading a lot.
You have some valid points like college won’t necessarily teach you what your job needs you to do. There have been some partnerships with companies where they have asked colleges to step in a build out degreed training programs.
People with degrees regardless of degree do tend to make more money in the long run.
>> People with degrees regardless of degree do tend to make more money in the long run.
In aggregate. But that's wildly skewed by people who end up with higher degrees. If you look at bachelors, for every 2 people with a degree who make $20/hr bartending or selling used cars, there may be one person with no degree earning $150/hr coding or plumbing or fixing machines.
There's "gifted" and there's gifted. The truly gifted often think they _have_ learned how to work hard because their classmates in the "gifted" classes got praised for their hard work. There's a top echelon of kid who is both common enough that every school district has 1–10 in each cohort but notably more advanced than what district standard gifted education is designed to handle.
I was an inner city “gifted kid” - although I’m proud of what I have accomplished so far, I feel like my potential was stifled, as you said mis-handled. I’m very interested in helping inner city gifted kids today unlock their full potential. How could I start?
The guys who did well at my uni (a very good one) in my experience really were consistent in putting the work in. It's not that they had to really struggle with the material and homework problems, but they consistently did a good shift every day, and seemed to quite enjoy doing past exam questions when exam season approached, and all the rest. There was of course the odd mathmo savant who could see the matrix, but they were the exception. This was a good university that you've heard of, it had the best of the best aswell as well as the people like me. There were correspondingly people who I thought were brilliant and had serious flare, and have gone on to have excellent careers, but they didn't really Do The Work and their exams results reflected that. Thankfully, it doesn't really matter in the long run provided you survive it all psychologically.
This rings so true to me. In fifth grade after being given an IQ test, I took Algebra I early, only to retake it when I moved to junior high the next year. Then I had to retake trig because I did no homework, but aced the tests. Luckily the teacher recognized my ability and allowed me to take pre-calc while I retook trig.
By the time I was in undergrad I had no motivation to study. I'd skip classes and cram the night before for any class that was memorization based. I didn't even buy the books, just went to the library and checked the book out for a couple hours.
Fast forward a bit and I ended up dropping some classes because I was paying so little attention, I didn't even know when the tests were supposed to be. Came in one day and the prof said, "You missed the test last week, I presume you want to retake it?"
That said, I've done well enough having gone back later in life to get a PhD, but I do wonder sometimes if I would have accomplished more if I was forced to push the boundaries of my ability, thus leading me to develop a work ethic earlier on.
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?
The difference is that math is the only field of study of things that are 100% true about the universe. It's the most pure knowledge that humanity has, so it's normal people recognize it. You can read 100 philosophy books and maybe you will learn a few things that are correct among all the rambling, but everything you learn from the basics of math proof by proof all the way "up", you can be assured everything is true. There's something special about this field of knowledge that nothing else has.
In a sense even this is not true, as in any sufficiently complex (which turns out to be quite simple) formal system you can create proofs that are true and untrue at the same time creating a contradiction. In other words, mathematics works by setting up useful axioms and following up on the logical consequences, but they usually can be used to create contradictory proofs even if useful in many problems.
I recommend learning about Gödel’s incompleteness theorem behind it all.
For a pop science book that explains it nicely I recommend ”I am a strange loop”. The wiki intro is also quite good
Wait a minute, the universe as we know it, is a model of a universe, the way we humans understand it today.
That model is flawed, this is in fact the basis of science.
The latest scientific finding is considered true until... a more advanced model proves that there are cases where it is not true. A thrown object on earth follows a parabole? ... no it follows a straight line, the space around it is bent by a force that is known as the gravitational field.
What about math itself, no universe considered?
Math itself follows axioms which cannot be proven. Since we found no contradictions in the maths built upon those axioms, we consider those axioms to be true. You can think however of axioms on which you can build equations that are true and untrue at the same time.
My point being, whether we like it or not, the Truth with a capital T does not exist or at least cannot be proven.
Cannot be proven, physically. Mathematical truths prove themselves with their application to physical motion, e.g. Fourier Transformation, such extension of logical principles unto physical body sensations in contact with an external world is great evidence for myself that mathematical reasoning self-reliance in its ordering is great.
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.
To coast through a serious program (physics, engineering physics, pure math, etc…) at a major university while barely doing any work and also simultaneously getting high marks takes a lot more than 0.01% performance.
More like 0.001%, at the very least, with some extra luck needed too.
We have not acknowledged an approach that teaches the majority math properly. Find approaches that do and also have some problems that resonate to children in an age appropriate way set for their environment.
Worse still, we lead kids to believe that 'doing math' is performing computations. And so even many kids that can calculate passably grow up thinking that 'math' is boring and they hate it.
Pretty early after kids learn about numbers and computations, they learn about sets, units, lengths, surfaces, weights etc.
Where i live, mathematical formulas are already thought to 7-8 year olds. Also real world questions are asked for which they need to find a solution, and where they need to explain how they found it.
> grow up thinking that 'math' is boring
How can it be thought in a less boring way, i do not immediately see it.
> real world questions are asked for which they need to find a solution, and where they need to explain how they found it.
This could be fun if the questions are predictable enough, but typically in my own early education 'word problems' just ended up being computation problems with the single extra step of translating the question into one of a very small handful of known formulas.
> mathematical formulas are already taught to 7-8 year olds
That's great, but applying memorized formulas is still just computation.
> How can it be thought in a less boring way, i do not immediately see it.
Mathematics is the work mathematicians do. That work is fundamentally creative: it's about exploring, defining, and constructing abstract structures and conducted through writing. Mathematics as it's taught before college is typically presented almost exclusively as a mere instrument in service of engineering or the empirical sciences. This is like teaching physics purely as a parade of unexplained facts about past results and never giving students a chance to conduct experiments!
There should be way less emphasis on the notion of a single track of linear progress from arithmetic to calculus. Formulas should be derived by students rather than just presented to them for memorization. Formal logic should be introduced about as early as arithmetic; first-order logic certainly isn't any more complicated than addition and multiplication. Teachers should prove the principles they expect students to rely on. Mathematical topics which do not require great facility with engineering computations (e.g., calculus, linear algebra, trigonometry), like propositional logic, discrete math, and basic geometry, should be used as opportunities to get students reading and writing proofs for themselves in multiple mathematical contexts as early as possible.
The more students have a sense of the foundations of what they're learning, the more meaningful it can be to them. The more deeply they understand their formulae, the less memorization is required. And the sooner they engage with proofs, the sooner they have a chance to engage with mathematical as a creative and collaborative process.
> 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.
What made you believe you were in the 99.999th percentile when going into university? (As opposed to something more realistic like the 99th percentile)
Unless you were literally outsmarting your teachers every day at age 16, it seems difficult to successfully fool yourself in this way.
There are a lot of things that one can be in the absolute top of, and overall academic achievement need not be one of them.
Speaking personally, when these articles come out, there are always a lot of comments about "I didn't really try super hard in high school, but college was a huge wakeup call for me and I had to learn to learn."
That wasn't me at all. I somewhat lazily skated through high school, and got a mix of 4s and 5s on AP exams. I did the exact same thing in college, with no change to my work/learning ethic, and lazily skated my way to finishing my 4-year molecular bio degree in 3 years, with a GPA of like 3.5 or so. Then I went to grad school, did more of the same for two years, and won an award for having the 2nd best masters thesis produced by the university that year.
Then I got a great job in my field doing cancer research, did that for 5 years, then jumped careers entirely and now work in robotics.
But you know what? I feel like I'm constantly surrounded by people smarter than me. I'm not some brilliant person, I'm just some dude that when presented with some problem, things just seem to make sense for a path forwards, and maybe my special thing is that I just always go explore that path and learn that either I was right or why I was wrong and that just pays dividends. When I see people around me who work hard at things, who study and memorize and read papers, they impress the heck out of me, because I really struggle to do the same thing. And when I do, I really struggle to absorb any information; if something doesn't make sense to me, it's like it just passes out of my head. I have to do/build/try it to make it make sense a lot of the time, or at least have things framed in a way that just intuitively makes sense for me.
Anyway, my point is that maybe I was the top 99.99% of something, because, clearly I was/am pretty good at some things that apparently most "gifted" people struggle with. But I never got a 4.0, I never aced all my classes, and I never really cared to as long as I felt I was getting what I needed to out of the classes. I did the work I needed to do to gain the information and skills I felt I was there for, and as long as the number assigned to me by the professor for doing so was at least an 80, I was happy.
> What made you believe you were in the 99.999th percentile when going into university?
Oh, nothing at all. I'm just a case of suddenly discovering in university that you also need good work ethic and that showing up alone is no longer enough (as a sibling comment points out) and you can't always cram all of the topics for exams in your head in a single night before the exam. In my case, calculus introducing new concepts (for which I didn't have a practical use, so it was even more confusing) and probability theory get less intuitive was that wake-up call. Well, that alongside an ASM course with a toolchain that I couldn't easily get working on my computer, or working with Prolog in similar circumstances, or understanding that I've underestimated how long making a 2D simulation project in C++ for extra credit would take, if I need to have collision detection and some physics for a soccer example.
That said, in my Master's studies, once I got to specialize in the things that were of more interest to me, I ended up graduating with a 10/10 evaluation for the thesis and 9.87/10 weighted average grade across the subjects. That's not like a super big achievement from a small regional university, but definitely goes to show that learning some things was easier for me than others. I probably need to venture outside of my comfort zone occasionally though and not just do the things that are comfortable.
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.
> 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"
Can confirm that getting lauded for effort encouraged doing hard things and pushing the limits of my abilities, rather than focusing on things that came easily.
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.
> 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.
Unfortunately, for some this comes in university where we see peers with worse grades getting access to better graduate roles and work placements because of who they are related to, and you can't study your way to having the right surname.
>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"
This is also taught heavily in middle/high school sports; it's a great life lesson.
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..
> Lovely article, though honestly getting those prerequisites also takes a lot of time, effort and either motivation or discipline in ample amounts.
Exactly. It's like saying that winning a marathon is easy, you just need to run fast enough.
The entire problem is that many people won't be able to learn the prerequisites of advanced math. There are always exceptional cases, like someone really smart who just totally slacked off during school because they spent all their time building apps or websites or tinkering with games or electronics. Or someone who was never given the opportunity to learn certain subjects because they had to work as a teen because their parents died etc. But the vast majority of people are presented with the prereq material they just can't absorb it.
> 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.
Parent here. Raising children has a way of making you more efficient. In my pre-child years there were days where I could putter around and relax because I knew I could make up the work later in the evening or even a weekend. Or at least that’s the lie we tell ourselves in the moment.
Post-kid, things come into focus. You learn how to do the work now whether you feel like it or not, because the price for doing it later becomes much higher.
I also was misled by all of the internet comments about how parenting and raising kids is awful and everyone secretly hates it despite their fake happy social media posts. After having kids you realize you actually like your kids and want to spend more time playing with them. That alone is motivation to get work done now so you don’t have to sacrifice the valuable kid time later.
It’s hard to explain until you get there, but I’ve talked to many other parents who went through the same growth phase. I’ve also caught up with some (not all, some) of my old friends from high school who were academic superstars but who did not have kids, and it’s remarkable to observe how some (again, don’t flame me, not all) of them are stuck in the low motivation/low effort loop and cite that as one of their reasons for not having kids. To each their own, but I for one am glad I ignored the internet/Reddit rhetoric about how kids are an impossible burden that will only make your life worse.
Also, having a kid is great to get you to learn new things. I want my son to learn to play an instrument, so I've finally been taking piano classes because I know that very few children of completely non-musical parents actually succeed in learning to play instruments. Learning piano has been a lot of fun, I have a blast playing simple things with him (he's still young) and learning some of my favorite music.
Likewise, after reading "Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers", I've been having a lot of fun getting back into Mathematics because it's fun doing things like this together.
I had the complete opposite experience, never gifted. Bullied by my teachers in middle school for my lack of intelligence. It was always an effort battle. The only way to get the pre-reqs was to literally drill 100s of problems. I remember I did for one math test in college and I studied consistently for hours for weeks only to get an 89. I earned that and my college roommate coasted in and got a 96. Work ethic wasn’t a choice it was the only way. My perspective on gifted students not utilizing their full potential is smallest violin. I wish I had what you had. But I don’t and all I had was motivation and discipline. I struggled all the time in school but work has been easy by comparison.
I’m emotional about this and don’t mean to attack anyone, but all my friends in the same state are all depressed and on SSRIs. It is sad for me to see them. For me the conflict was never being good enough, for them it’s the same but they seemingly chose not to do anything about it. That might be the difference simplistically but life isn’t that simple. There is no conclusion for this antidote or argument. Maybe just find a problem where you get pushed to your absolute limit, might be at academia or startup or something. You find some truths at the end of that.
I was that kid, now I have 3 sons and I am homeschooling my oldest because he is similar. The hack is essentially to praise the work, not the outcome. On some level you have to be unfair to your kid to be fair with him.
Although I'm not on the super sweaty side of trying to grok some of those items in your list, I have been going back to square one from the perspective of having started with and followed the whole dependency tree of frontend but never understanding the really low level.
Over time, I've learned to appreciate getting "reps" in whatever subject, and just sitting with it for ages and grinding your brain into dust trying. Something I've also been noticing now in my 30s, is that your and my ability to do that is something to be proud of—even if it won't come as fast as for someone innately faster or with more prereqs—because many people are truly not curious and don't push themselves. The amount of otherwise smart people even that won't or can't sit down to learn something for curiosity sake, and I'm not talking something very long-term like game engines, is... like next to zero people.
And it not just true for learning hard or impractical subjects, but the overwhelming majority of the surprisingly high number of decent friends I have, do not have the patience for pushing themselves and will bail on hard physical activities as quick as possible, even if they're pretty fit. It's quite eye opening. They tell me they want to go do __, but not if they can't squeeze it in to an afternoon, even if they definitely have the time, and it really detracts from trying to share the experience or curiosity.
Just accept that this is the case and meet people where they are. And collect some friends that share your curiosities and passions. People can be good and reliable friends even if they are content living a simple life in simple ways as it's always been, without pushing for abstract novelty and learning.
People try to blame school for killing the natural curiosity of kids, but I've come to be very skeptical about this. It's an idealized romantic notion. Most mammals lose their playfulness in adulthood and most humans are like this too, even if not to the same extent (some say humans retain more neoteny, similar to domesticated animals like dogs that also remain more playful than wolves as adults).
These two types of people often misunderstand each other. For you a job might become boring if it stays the same year upon year. But for most people the thought of having to keep up with ever changing knowledge requirements after school and even once they are settled in their jobs is utterly terrifying.
Ya, I'd actually agree with your whole comment, and for that reason I was somewhat hesitant to frame my last sentence as I did, because what you suggested is what I've come to learn to do anyway. Part of the reason I was hesitant is because it's not about me, so much as it is about reconciling that ambiguous difference in expectation when they express that they have an interest in something.
Unfortunately, there's some amount of skepticism and doubt I have to embrace, being careful what I express an interest in, and letting them be as serious about whatever commitment as they're authentically prepared for. I typically only do things for myself now, and extend an invitation to people I think might want to join, but I don't bet on it, and only rarely plan more involved activities with people who've clearly put in some organic initiative. On the extreme shallow end of this paradigm, I'm sure as hell not going to agree to go on a hike "sometime" with anyone who's hopped up on coke or drunk at a party, obviously. But sometimes it's just less clear, and I have to ask myself "how likely is it that this person would plan this themselves and take the initiative, and do they even have appropriate footwear or baseline level of fitness?", and I might ask that explicitly and go from there. I also do the same for the them, such as if I'm asked if I want to "get into Tennis" or something. I just say "nah, not really, I might join if you have a spare racket but right now I couldn't give it the investment it might deserve" and we can spend time doing other things we already enjoy together.
Otherwise though, I try to keep a small list of easy stuff in my back pocket to accommodate people with more rigid schedules and less innate drive for that specific outing; if after a few they want to take things further, I'll extend myself, and that's how good friendships last.
Sometimes people are just bad at predicting their own level of commitment or disentangling desires and reality. Or they know they are "supposed to" do something so they promise it or even believe that this time it will work... This is often the case with diet and exercise and studying hard. Then sweets, sitting on the couch and procrastination happen. Then the cycle repeats.
Also in some cultures it's rude to reject invitations so people always say yes but it's understood implicitly by both parties that it's just politeness. The US tends to be like this, always saying stuff like "you should come over for dinner sometime" etc and you must reply enthusiastically but both the invitation and the acceptance count only in reality if a date and time is attached. This is often baffling to continental Europeans for example who tend to be more direct and would follow up with a message next day, asking when they should come over. Which is awkward because it wasn't a real invitation.
The coasting until real work was required part sounds very familiar, and also the getting by in objective terms but being unhappy and struggling mentally part too.
After embarrassingly many years I've learned that there's a little voice inside that says "you can't", "that won't work", "that was shit". Succeeding at anything, for anybody, is stacking a whole lot of little failures and frustration, but crucially, being able to ignore that little voice. I, and I assume many of us, have burned half of my energy and mental health fighting that little voice. To me, it seems that most of those manage-anything people are not just generally gifted intelligence-wise or physically, but have learned to silence that inner voice and just do things until they succeed, while still avoiding to do "stupid" things.
As a distance runner, I've learned a bit of dopamine trickery, but managing that inner voice, and even being aware of it, is turning out to be a life-long project. It becomes a mission of figuring out what you're burning your energy on, and why. You can't strive for happiness or success, but it should be possible to get enough stuff done to find contentment and acceptance.
I have just recently learned that the little voice can be amplified by people around you. Being removed from those re-enforcing that little voice is an amazingly freeing mentally.
> 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.
Paradoxically, I _feel_ like I have more time after I had kids than before. This is of course, after YOB (year of baby- first 12mo).
You see, I experienced such small slices of free time during YOB, that I became way more efficient at a ton of stuff, and dropped things that were time wasters.
Because I reproduced, I needed more dough. After working 3 jobs, 90 hrs a week for long enough, I decided to study programming.
Went to a code boot camp and walked out with a job. But the grinding didn’t stop there. What followed that was years and years of grinding, and studying as much as possible outside of programming at work.
Until finally, I got where I was going. I climbed up the ladder until I got tired of climbing, and avoided more stressful and time-demanding roles like management. I get senior dev compensation, don’t work more than 40hr/week, and I don’t commute. This is the life I built.
For everything I pursue, there are others who can run circles around me. But I can still look down from that ladder and see how far I came.
Wait, what am I talking about? Ok, the whole point is I became efficient and middle-class because I have mini-mes that deserve it.
Also, you never know what’s on the other side of that wizard level person you see. Everything I said sounds nice, but I wasn’t taking care of myself. Last year I had a mental breakdown, and am just now getting out of it.
So yeah, you’re not alone, etc etc, you know. There’s probably more people that can relate than you think.
It's wild how much kids teach you about yourself and your time on this earth, let alone your time available each day.
It's hard work but I can't think of a better motivation to improve one's self. And the best part is you don't even necessarily realize that it's happening.
I think one factor that people forget when they see someone else that dive headfirst into something like low level programming or electrical engineering and then wonder why they haven’t is because maybe you just don’t care.
I did personally get a degree in EE and have worked on low level programming (and funnily enough I do a lot of frontend these days), but there are things that still wow me like people writing game engines. I tried working on game engines briefly and I never got anywhere.
I eventually realized that I just don’t care about game engines or making games for that matter. I wasn’t willing to put in the effort. Watching other people build game engines is more of a spectator sport for me. And that’s fine.
But when I do find something that makes me really happy, it keeps me up thinking all day and night about it.
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.
I always thought the real reason was that the cadence of the lessons more closely matched what is possible for a person of average or above intelligence who was motivated to learn. The rate at which information is taught to children in school is kept at a low level because most of the students are not motivated to learn the subjects and instead need to be guided to learn those specific subjects.
The example I was always given for this was the rate at which those same "non-gifted" students would learn subjects they actually care about--like dinosaurs or sports facts. Kids will soak that information up and spend tons of time learning more, but it just isn't useful. Instead we force them to spend less times on those subjects they love and guide them into things society views as more important.
DI is rarely used to train teachers, unfortunately, and even if it were, the education-government complex designs school procedures and discipline rules that would make it very hard to implement.
For me things got much better when I got to uni. In high school I didn’t care about most of the subjects, but at uni I was studying what I had chosen for myself.
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.
To see the brilliance of advanced ideas, you need to be quite advanced yourself.
What you wrote is analogous to saying that playing football in the Premier League is nothing special compared to me and my weekend buddies playing in the yard, it's still human beings kicking a ball, just with better coordination and strategy. Yeah. Coming up with new math at a research level is also just manipulating conceptual objects in your mind. Except this insight doesn't make the average person capable of either.
For some reason people have a big hangup around admitting that some people are more talented at abstract thought and advanced math than others, even though this is absolutely obvious even to a schoolkid. The existence of the Premier league in no way diminishes the value of kicking the ball with friends over the weekend. Math can be fun in the same way, but not everyone will reach the same levels.
Hmmm...this wasn't at all my point, but I understand why I my comment could be read this way (the failure to communicate is mine).
I am not suggesting an equivalency between all efforts at all levels, or that innate talent doesn't exist. I was not at all speculating about the nature of preternatural talent. The expertise-gap I was invoking was more like early undergrad to late grad school levels, not pick-up football to Premier League.
The concept I'm getting at is that of a mental block I have observed in myself, and I suspect resides in others, which is really subtle but ultimately quite limiting. I'd have to think harder than I want to at the moment if I were to try to articulate it more clearly, but I do want to be clear that my comment wasn't about innate talent.
Yes, maybe I have experienced something similar in my research area in machine learning. Fundamentally, the best people of the best labs use the same tools in roughly the same ways as I do, there's no set of magical secret tools on that tier. They also aren't really using any math I don't know about.
They just comprehend papers quite fast, understand what's relevant and what is not with great intuition, have a lot of prior work in their memory, have gone through lots of projects so they know how to approach the new problem, they are aware of what actually are interesting open questions, they have refined "research taste", (and of course stuff like understanding the academic system and how to make the most of it etc). But the day to day activity is not fundamentally different than for lower tier researchers. They just have better ideas, better overview, and better execution. When they explain it all in hindsight it appears simple and as if you could have done it too. And of course they also run up against bugs and hardware issues and big messes but often just plow through and grind it out because they can see the light at the end of the tunnel well.
It's kind of a bummer in a way. Life is just less magical than we imagine. When I was much poorer as a kid than now as an adult, I used to imagine that being rich must be so different. But actually people are just people, there is no discontinuity anywhere. I've flown business class and been to fancy dinners, it's not fundamentally different from taking a bus trip or throwing a student party at the dorm. The people talk about somewhat different subjects, or are more polite, but overall it's nothing that a poor person wouldnt be able to comprehend.
It also reminds me how there's no "arrival" in life. There's alsays a next thing to do. But for example as a student you think graduation is "making it" but then you realize it continues in somewhat different form, but it's now about acing job interviews and you think landing a job will be the goal line, but then you see it goes on and now you chase promotions and good evaluations and so on. Similar things about private life.
> Coming up with new math at a research level is also just manipulating conceptual objects in your mind. Except this insight doesn't make the average person capable of either.
Yes, it doesn’t make them capable, but it can motivate them to learn how to manipulate conceptual objects. It doesn’t take 1 in million genius to use these things. The path has already been paved by so many people you can follow in their footsteps or blaze your own trail.
> Materials that require more time and effort to 'get'
They really don't. Each rung in the ladder takes about as long to step through as the previous. The challenging/annoying/discouraging aspect is that a lot of research (especially in pure math) has so many rungs to step through.
The truth is many of those aren't useful at all - distinctions invented purely for the sake of being able to say you invented something. Even when I was doing really "deep" research I could explain it to anyone by just dropping all the technical jargon and using goal oriented language instead. When I would do this most people would naturally try to reason about it in exactly the same way "researchers" would. The proof is they'd come up with the exact same solutions to the problems that were legit solutions, they had just already been discovered ie the low-hanging in fruit solutions.
The central thing to me is - "quality of efforts" more than "quality of results". For anything sufficiently complicated or "difficult" - one needs to persevere. There's nothing magically special in any place, like a university, except perhaps a few understanding people around to egg you on (that can make a big difference in the right conditions). But end of the day, it is the individual's effort that matters. Effort can make up for many, many deficiencies. More quality effort held steady over a long-enough time should produce at least tolerable results, if not exceptional results.
But at higher you also learn how to solve problems. Suppose you see a problem which is quite unfamiliar, you have tools and you know how to use tools- combine tools, separate those and use for separate parts, invent tools, etc.
This is a good argument for, "why do I need to learn math/science? I'll never use it."
Maybe you won't need to factor polynomials daily, but knowing how to solve that kind of problem can be applied to other scenarios outside of Algebra II.
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.
It's not just working hard enough, but also doing the right kind of work. Many people make the mistake of trying to memorize things without understanding. Which may be easy at the beginning when you memorize a fact or two, but it gradually accumulates, especially in math when the old topics never go away as the new ones are introduced. And then the memorizers are actually working much harder, and even that is not enough, so they fail.
So why the aversion to understanding? I suspect part of that is generational; if your parents sucked at math because they relied on memorization, they probably won't introduce you to math as an something worth understanding. It will either be "give up", or "work harder" but in the sense of memorizing harder. Not just your parents, but the entire culture around you will be like that. Another part is that most math teachers at elementary schools actually suck at math; because teachers are many, but people good at math are few and they have many better careers available. But another problem is the insistence of school system on everyone going forward at a predetermined speed -- sometimes understanding takes time, and when you don't have the time, you are forced to memorize; but once you start memorizing, you usually need to keep memorizing, because understanding can only be built on understanding the prerequisites.
Properly taught elementary-school math should be fun, like this: https://www.matika.in/en/ Fun makes people think.
A lot of people don't understand what understanding really even entails. They don't know that some people actually understand a topic/idea/whatever, can play around with the ideas in their head, think from first principles on the topic. They've never understood a topic in their life.
If passion, or own experience, is missing it may be a case of unknown unknowns for both parents and teachers.
The Matika site looks really nice but I have difficulties comprehending the instructions. Even the very first one for first grade. “Children step by record.” What does that mean? I tried the next one. “During addition we write addends below each other…” What? If all addends are below, no addend is on the top. It makes no sense. Then, “…and the sum below the line” with no line in sight. What, where, which line, how? That was frustrating.
Wow, the English translation sucks much more than I noticed. :(
The whole "stepping" thing is a reference to how they (in the web page author's country) teach basic addition and subtraction at first grade. There is a carpet with numbers on the floor, you start at number zero, and do addition like "2+3" means "two steps forward, pause, three steps forward, now look at the number you are standing on". The carpet is situated so that from the sitting kids' position the zero is on the left, and the numbers increase to the right.
The idea is to turn integers into something "tangible", in a way that can later be extended to negative numbers.
So the instructions should be like: "You start at given number. Right arrow means a step forward to a greater number. Left arrow means a step backward to a smaller number. What number you end at?"
Sorry, I already know all these things by heart, so I didn't notice how the English instructions don't make sense. Guess I should contact the author about it.
I feel the same way when I'm on hold and a recording tells me "your call will be answered in the order it was received". This isn't about grammatical pedantry -- I don't care that they didn't say in which -- it's about it not making sense. Which, as I said, isn't grammatical pedantry. But it probably is still a bit pedantic. Still, though, how can one thing have an order? What order was my call received in? Is it before or after itself? I get the sense that whoever recorded that didn't spend any time actually thinking about it, or they would have said "Calls are answered in the order [in which] they are received" or something.
I unfortunately spent the entire introduction to calculus in hospital, so missed it - when I came back to school, I was dropped straight into “differentiate this” and “integrate that”. There was no explanation of what either operation was, just the rules that you followed to obtain the result. I had no idea that we were looking at rates of change or at areas under curves. For the first time in my life, I found myself bewildered, and struggling - until a month later I happened to find myself reading a biography of Newton which actually explained what the purpose of calculus/fluxions was - and then it became easy, as it was obvious if a result was nonsensical.
I knew people who somehow missed the information that fractions are the same as division. So they could e.g. reduce the fraction 40/20 to 4/2, then after thinking about it longer also to 2/1, and... then had no idea what to do next.
For me it was completely mind-blowing, how someone could do fractions without understanding what they are? But I imagine, at school they probably could solve some problems, couldn't solve others, got C, moving on to the next lesson.
I just think for some people math is very fun since the very young age and so they of course practice it. For others it may not be, so it is hard work for them. E.g. I have always enjoyed ever since I can remember doing these exercises. When walking home I used to multiply different numbers as a past time in my head. Most people are not going to do those things, and it didn't feel hard work at all.
In first grade I used to run through workbooks being addicted to solving those problems like some addictive mobile or video game and at that point teacher had to stop me and I was frustrated.
I only had this addiction to math and physics - a bit to chemistry, and I couldn't really focus on other text / memorization based subjects.
And it makes sense to me that genetically in a population you will have brains out of the box that are naturally optimized for different specialities, since having a specialized brain allows you to have more power in that specific area. Problem is when you force those specialised brains into the same way of studying.
Exactly. We enjoy different activities. For math oriented kids it's not a grind, it's interesting and fun. For me, reading novels was much more of a grind, as I just wasn't that interested in people and their conflicts and condition.
It took until my twenties that I could realize the value in humanities and "social" topics.
Similarly most people will naturally learn about countless types of fashions and connotations of liking various music bands etc which is actually quite a lot of information to memorize. But it's fun and feels relevant while math feels disembodied and irrelevant to their social goals in life.
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.
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.
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.
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.
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.
It was a dream for ME. I always think of that teacher when I do code reviews. The important part is how you manage to communicate things effectively. I had one friend who never managed to get better and complained, not loudly but it was clear it felt like hell to him.
I do not know if the instructions would have worked for you, maybe it just worked on the ones that really saw a need to improve in this specific task. I know most people missunderstand me when I give out instructions.
Yet understanding is necessary but not sufficient when you read university math, especially advanced courses.
Proofs assume you have the elusive thing referred to as ”mathematical maturity”, which means many algebraic manipulation steps are skipped because it’s assumed you can just see the result straight away.
This ability to see the connection is not understanding but learning by rote, having done the same tricks with similar equations a thousand times.
This is what makes advanced math books/courses slow for me as a CS phd researcher. I can very slowly progress through, but it takes a massive amount of time to work through what just happened. If you take 60 instead of 20 courses on math the routine you have is just completely different. I guess you can call it fluency in the language.
(For example now I’m reading optimal control & variational calculus along with the functional analysis it needs, its heavy.)
Very well put. Many people are very blind for this, they forget that everything they can do they at some time had to learn as well. And not everyone learns everything at the same time.
Anecdotally, something I can actually confirm from personal experience with math. As long as I could remember, I had trouble with a lot of it.
Then during the last years of high school I had an excellent teacher and a lot of concepts actually did start to click on some level. Frustratingly, I still had a lot of trouble. While I understood the abstract concepts much better.
In order to solve issues, I still had to apply a lot of concepts I was supposed to have learned in all the years previously.
How would you approach rebuilding foundational knowledge from the kindergarten level? I have completed all the courses on Khan Academy from kindergarten through 6th grade and have also practiced with more challenging problems beyond those provided on Khan Academy. I'm trying to find the most effective strategies to solidify these fundamental skills.
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.
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.
I always sucked at math, even though I did it in undergrad. I basically did this over the course of the last five years to try get better. It went something like this:
Spivak - Calculus. This was a bad idea. Got maybe 30% of it. Gave up at Taylor series.
Hammack - Book of Proof. Finally understood how to prove things, and induction arguments.
Abbot - Understanding analysis. Got far, things fell apart around the Gamma function.
Apostol - Volume I. Got better at calculus. Also trigonometry. Exercises were easy. Skipped differential equations. It was too hard.
Hoffman/Kunze - Linear Algebra. Gave up after a few chapters, too hard.
Friedman/Insel - Linear Algebra. Much better, got to the Spectral Theorem and gave up.
Rudin - Principals of Mathematical Analysis. Absolutely brutal, probably got 30% of it.
Abbot, round 2. Much easier this time, got through the whole book.
Spivak, round 2. Much better, got through the whole book. Actually found it easy.
Hubbard - Vector Calculus. Gave up early, it was too hard.
Apostol - Volume 2. Much better. Stopped somewhere in the middle when it got too focused on differential equations and physics stuff.
Back to Friedberg / Insel - Made it through the spectral theorem.
In between I was doing a lot of mathematical statistics and probabilty stuff like Casell-Berger (I did this book twice, each time going back to the math where I floundered). I’ve worked through just about every exercise in the above books and watched YouTube video lectures where they exist (there is a good one for Rudin). Solution manuals sometimes exist, sometimes you have to find university courses based on the books and look for homework assignments where they have posted solutions, Quizlet has ok solutions, some are buggy. Apostol volume I some dude worked through and posted online.
Anyway point is I refused to accept how stupid I am and I brutally forced myself to become better at math. My attitude was I don’t give a fuck how long it takes, I will keep going until I get better.
I think I’m better now, although I’m still shit. It’s true what von Neumann said: In mathematics you don’t understand things, you just get used to them.
I'm not going to try to recap all of that, but, as an example, if you have a sufficiently strong understanding of arithmetic, learning basic modular arithmetic should be effortless, pigeonhole principle completely obvious.
I was quite surprised when I tried applying for a Microsoft internship in uni and they gave me a question on the pigeon-hole principle.
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.
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.
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.
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.
"
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...
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.
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.
Why do you think there is a limit at all? What is it about higher level math that is intrinsically incomprehensible to a subset of people?
I suspect that the limit is actually in research and discovery, not comprehension. Calculus took some brilliant minds to develop but now it can be taught to most high schoolers.
As detailed in the article, my conclusion of there being a limit does not rest on the assumption that higher math is intrinsically incomprehensible to a subset of people (though, unrelatedly, I would expect that to be true in some cases).
In the article, the key underlying assumption is that the further you go in math, the more energy it requires to learn the next level up -- and everyone's "energy vs level of abstraction" curve is shifted based on their cognitive ability and degree of motivation/interest.
Here is a quote from the article that gets at the main argument:
"As Hofstadter describes, the abstraction ceiling is not a “hard” threshold, a level at which one is suddenly incapable of learning math, but rather a “soft” threshold, a level at which the amount of time and effort required to learn math begins to skyrocket until learning more advanced math is effectively no longer a productive use of one’s time. That level is different for everyone. For Hofstadter, it was graduate-level math; for another person, it might be earlier or later (but almost certainly earlier)."
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!
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.
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.
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.
> What is their usual motivation for this? Do they find they are running into regular work or life situations that require it?
I have a chip on my shoulder. In university I was depressed and didn't even bother attending lectures let alone doing the work in the first couple of years, couple that with professors who when contested were off by a 20-40% bc they cba to care for a secondary course in another department...
When looking for thesis advisors, I found one interested in the things I was. They made a comment asking whether I had an issue with mathematics. Over the year I learned enough mathematics to get to what I was interested in and understand the bleeding-edge literature (calc, linalg, vec calc, prob theory, etc). I corrected some of his proofs in his classes by the end of the thesis.
Still, my early grades haunt me, and parts of me wants to get a math degree just to prove that it is not a skill (read, intellect) issue.
Good question! No, I don't usually work with college students. Most of my students are actual grown-ups with jobs. I've found quite a few people who just wish they'd been able to study this stuff in college but didn't get the chance for whatever reason and still have some unresolved curiosity about it. It's a very fun group to work with; they're very motivated!
> “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.”
certainly your math skill level neither makes you "smart" or "dumb" (which really aren't opposites, either).
prerequisites are (ahem) required. not having them does imply having a bad time.
what's missing is that different people's brains work differently and people have different talents.
if you learn differently, that can factor into that lack of prerequisite knowledge - perhaps the way it was taught didn't work for you.
but some people's brains just don't like math. other's are gifted at it. you can have all the prerequisite knowledge needed, be the best most diligent student, be wildly intelligent in general, and still not just "get" math.
so this article was about someone who actually did have a decent proclivity to math, but was robbed of it because of some missing foundation. and then said "ah ha!" there's the problem! but that doesn't mean that's the case for everyone else - far from it.
it smacks of the "affirming the consequent" fallacy:
Sounds like perverse socialization to me. What actually justifies this claim? Do some people's brains "just not like reading?" Do some people's brains "just not like music?" There's nothing special about math here.
yes, certainly. pretty common - outside of a disability, we can all learn to read, but it's hard and not enjoyable for many.
> Do some people's brains "just not like music?
yes. (when i say "like" btw. i meant roughly "natural ability", not "enjoy") of course most people love to listen to music, but many people do not have much of a musical ability - and you can't work, learn, or practice your way into it beyond a certain point, either.
> There's nothing special about math here
yes.
some people are terrible at dealing with socialization and others are brilliant. we are NOT all wired the same way. we all have different abilities. what justifies that claim is several billion examples.
A lot of people on HN just don't really know any stupid people. There are many adults in the US who struggle to work a grocery store self-checkout machine or remember a string of numbers longer than 5 or 6 digits. (I'm 70% sure the correct order of magnitude for this group is "millions" and 99% sure it's at least "hundreds of thousands").
Maybe with better nutrition, childhood conditions, and healthcare a good portion of this group could have been promoted into a different group, but the idea that everyone just needs better prereqs and they'd learn math better isn't right. The article itself was written by someone who manages their own website.
It is possible to teach math to this population, I've done it briefly when I was a teacher in an adult education school.
It's significantly slower than for average people and requires a ridiculous amount of effort on the part of the teacher and student, but it's possible.
Due to limited working memory many of these students will make mistake no matter how difficult the material is, and they will be slow at resolving problems, but I've never had a student without serious mental health issues be unable to get to precalculus level, and in the ones I've seen try they've been able to get to university math.
Again, many will always make more mistakes and/or be slow, but the concepts themselves are generally within grasp.
This is not the most popular idea these days but talent differences exist and they make a huge difference in the outcome. In my profession and in my hobbies I've seen plenty of people with 20+ years of experience who are quickly overtaken by a talented newcomer in less than a year. And I've also seen people who put a lot into a field they really have no proclivity towards and they get absolutely nowhere.
I see talent as the base of the pyramid. The limit to how high you can build your pyramid is the talent. There's certainly plenty of people who never reach their full potential for lack of prerequisites, opportunities, resources, etc. But there's a clear difference in how high and how fast different people can build even with equal circumstances.
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.
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.
Even for talented people, learning a skill to a high level takes years.
Mere mortals don't really have that many years of free time. Choosing a path that one seems talented at is usually the right path.
It's one thing for the person to decide they want to take the risk knowing that they might not have the talent to learn the skill in a "reasonable" time, but it's another to thing to pretend there's no difference and cheer on a person chasing an improbable dream and waste a persons' life.
To be or not to be. Basically.
So basically the disagreement is to the approach to default to when the future is unclear. Taking the right action requires a degree of foresight that people generally don't have. (I'm not sure everyone has the same skill to intuit the future, but I'll grant you that it's just effort to learn it, even if it requires significantly more effort for some...)
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
I have heard of a few philosophers of their day years back being considered "the most learned man in Europe". Not the smartest, but the most learned. Learned implies agency, smart implies something innate.
With the advent of IQ tests and the computer you get the brain as computer metaphor gone much too far.
Terence Tao has a 64 core threadripper, so if you are just using a 4 core i5 don't even bother.
Then of course if you believe an i5 is basically worthless it becomes a self fulfilling prophecy even though for most tasks both would be plenty fast enough.
many people will never be able to dunk a basketball no matter how hard they train. Some people will be able to dunk at age 14 without doing any training.
We accepts these types of basic inequalities when it comes to physical ability without blinking an eye, yet if you apply the exact same logic to the brain (a physical organ) people go into hysterics.
Most people could probably get a lot closer to dunking than they think they are capable of, but theres a very obvious hard limit.
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
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?
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.
I see it with people who "want to learn programming" but they fizzle out.
That is not a single or 2 persons it is basically dozens of people whom I gave materials or tried to guide. It never is that they are too dumb to learn programming it is that they rather do something else than sitting in front of computer hunting down why their program doesn't compile.
For math I got good enough to barely pass, for electronics I know basics, so I am "want to learn more math"/"want to learn more electronics" person but never get to really spend time on it.
DevOps stuff, programming, databases, web frameworks is something I can fiddle with all day and not get bored, I can spend all day hunting that misconfig somewhere.
On the face value all looks the same it is fiddling with stuff and solving puzzles, but somehow one type of puzzles is more interesting for me and since I do it a lot also much easier for me.
I remember listening to a podcast where a psychiatrist was talking about something similar to what you are describing.
Take something a person could desire like wanting to learn how to play the piano, for example. Often times, what people desire is not the learning part but the end result. People are fantasizing about drilling scales and chords for days on end. People are more likely to fantasize about creating music, playing music for the enjoyment of others, the praise, etc.. So, people tend to fizzle out when the reality does not meet the expectation.
The brain tends to fantasize about all the good that something can bring, but the brain also tends to vastly underestimate the work required to achieve said goal.
I think for programming it is mostly idea of "easy job" like sit behind the keyboard and get paid loads of money, as soon it turns out it is not sitting but quite exhausting mental gymnastics they are out.
I also heard that loads of times: "get a real job" (maybe not exactly in those words but hey, that was the gist) then I got people who would try out some basic things for an hour or two feeling mentally exhausted.
Not saying they were not capable or stupid - just that they underestimate how much taxing it would be for them to do something like programming for couple hours.
It is easy for me but I am doing it for 10+ years professionally and good couple years while I was a kid.
The more I observe about people and how they learn and do things, the more I suspect that motivation plays a much bigger role than we think in intelligence.
Looking at back to high school and then university. I fully believe I would have been capable of learning maths, electronics and so on... But I just did not care enough to put any actual work into the process. Still graduated. But well I really just did not care to put in effort. Thus lacked the motivation...
Same can be said about any learning including languages. It really comes down to motivation, outside sufficient immersion.
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.
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.
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.
> lack of resources, hunger, mental distractions, illness, or motivating incentives
Sure, that's a factor, but people can also simply be dumb. There's no quality equal for all humans, whether it's length, strength, weight, hair color, or intelligence, regardless how you measure it. The (rather superficial) article looks at it from the first person stance, but individuals are bad at estimating their own cognitive capacities.
People often, but not always, lack the prerequisites because they weren't able to learn them when it was taught in school. And that was because they lacked the pre-pre-requisites because they in turn didn't pick those up when that was the material being taught.
In math, things build upon previous things to much greater degree than in other subjects. If you get off track once, it's hard to catch up.
But if you lack prerequisites because it was never taught in high svhool etc, that's a failure of the curriculum.
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.
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.
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)
> If you jump at a large project but lack the fundamentals you are going to wasting energy on stuff you shouldn't be.
On the other hand, for many people (like me!) it can be hard to feel motivated to learn a new programming language/framework/etc just for the sake of it (especially when you've got plenty of stuff going on already). (Note that I use "motivation" here in the immediate, executive function sense, not in the broader desire to do a thing sense.)
In such cases, I have found that the best way to learn how to do what a large project requires is to first define the project that you want to accomplish that uses them, and then break that down into all the parts you need to learn to make it happen.
The outcome is still "mastering the basics" before you actually take on the large project as a whole, but it can still look very much like trying to tackle the large project from some angles.
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.
> 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.
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.
From what I've observed there's different ways that people cope when they find themselves in the deep end. Some try to learn and understand everything they can about their new environment (this seems to describe your experience), while others just try to find a working solution for their immediate task.
In my experience, the first approach is extremely effective, even if it can sometimes result in analysis paralysis. However workplaces almost always prefer the latter so it can be hard to fill in the gaps later.
Yes. I learned that about myself a long time ago. Sometimes it takes me a week to figure out "how to change the color of a button" but after that I can ramp up pretty fast. As I get older it just becomes a question of targeted time investment strategies. Where do I want to invest that week?
The issue is mostly when the steps are not conventional, too high, tilted, slippery, behind thick fog. Suddenly you have a new problem, learning to climb in new ways.
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.
Related to this, it is why when writing documentation for something, it can be extremely useful to also list the prerequisites someone needs to pick up the information in the document before them. Possibly with also linking to resources about them, depending on your audience.
Not only that, writing out the list of prerequisites also helps the author write a better document. Because thinking about what knowledge is required serves the same function as thinking about a good unit test does. It makes you stop to consider "the obvious" and sometimes realize you have overlooked something.
Because when you are thinking about these prerequisites, you are likely also thinking about why they are needed and what challenges come with them. This in turn might lead you to revise aspects of the documentation to make them clear as well.
Because at 7 you obviously also already have 7 years of experience behind you. Sure, it is not as much as an adult, but it still matters a lot. Different environments and stimuli make it so that also for 7-year-olds, they can have vastly different prerequisites for anything.
Often aptitude is not aptitude at all, but all of the above.
Then, during learning anything, the same thing also applies. How much support you get from teachers and parents. What sort of environment you have available to practice in. And a lot more.
Finally, when all other factors are equal, aptitude can play a certain role. But one that in an educational setting is largely irrelevant. Because if everything else is perfectly in place to teach something, including learning any prerequisites, aptitude is only something that matters in hindsight.
Which is extremely important as well. Labeling someone as lacking aptitude can be highly discouraging. Repeatedly hearing that they aren't “naturally” good at something can lead them to stop trying, even if it's not true.
I really don’t understand this cope. It’s scientifically established that intelligence is highly heritable, especially the analytical kind. It also agrees with experience, we all know people who have a very hard time understanding mathematics, while others sail through it.
Of course it’s not fair, life isn’t fair. But the good news is that you can quite easily compensate for lack of aptitude with more work, and that is most definitely the case for mathematics, up to and including undergraduate level.
I grew up in Sweden where everyone goes to the same kind of pre-school, that does very little math teaching. Still, the difference in aptitude when we started school was significant. But we all know this.
Oh boy, I don't even know where to start here. It isn't a cope, it is a more nuanced take.
Your take is so black and white that it only holds up if you almost willfully ignore all other context. Obvious things like different kids having a different home experience, exposure to different things outside of school, exposure to different things in the years leading up to pre-school, etc. These are just a few factors that actually heavily influence where someone starts and how easily they pick up some subjects. Then there is the fact that following the same curriculum or even being in the same class doesn't mean getting the same attention from teachers. In fact, ironically those with "more aptitude" sometimes get more attention further increasing their headstart.
I honestly want to invite you to go back, read my other comment again, actually take the time to internalize it then reply back again.
Because you are very close to actually agreeing with me. Specifically because you mention the practice and extra work bit. You just don't realize it yet.
No, I am very far from agreeing with you. I am saying that if you keep all other conditions the same, you will still see vast differences in the ease of understanding mathematics. This is borne out both by science - there is strong consensus that intelligence is highly heritable, and everybody's experience.
So even if we limit "aptitude" to a strictly genetic sense, it will still explain most of the difference in math ability at 7. All other factors related to growing up will add up to less than half of that.
Regarding practice compensating for genetics, I am not talking about having more supportive parents or more demanding pre-school, I am talking about Asian level hardcore drilling. That can certainly make up for most of the difference, at least when it comes to basic mathematics. But that means that the concepts that a child with math aptitude will pick up in 5 minutes will take 5 hours of drilling for another child.
> This is borne out both by science - there is strong consensus that intelligence is highly heritable
That is again a simplification of reality, leaving out a lot of context and nuance.
1. You are right that research seems to indicate that intelligence is heritable, meaning that genetic factors play a role in individual differences in intelligence. Estimates of the heritability of intelligence typically range from 50% to 80%, depending on the study, age of the participants, and the methods used. I am guessing that this is where your "All other factors related to growing up will add up to less than half of that" remark comes from. However, that 50%-80% is in relation to the inheriting intelligence from the parents. It does not mean that it influence more than half of your intelligence. It also highly depends on the specific aspects of intelligence that is being measured.
2. If we are throwing in statements as borne out by science then you can't ignore that studies also show that factors such as education, nutrition, and socioeconomic status significantly impact cognitive development. In fact, some of the most critical periods for brain development occur in early childhood. Things like:
a) Prenatal environment: Factors such as maternal nutrition, stress levels, and exposure to toxins can affect fetal brain development.
b) Early childhood nutrition: Proper nutrition in the first few years of life is crucial for optimal brain development.
c) Stimulating environment: Exposure to a variety of experiences, toys, and learning opportunities in early childhood can enhance cognitive development.
d) Physical activity: Regular physical activity from an early age can promote brain health and cognitive function.
e) Parental interaction: The quality and quantity of interactions with caregivers, including talking, reading, and responsive care, significantly impact cognitive development from infancy.
f) I could go on for a while, but the picture should be clear enough.
Again, aptitude can be a thing. But all things considered, it is really not all that relevant when we are talking about the development of people and them learning things. Anyway, at this point your use of phrases like "cope" and a somewhat fatalistic view ("life isn’t fair") already suggests to me that you actually have no interest in actually expanding your view and scope on these matters. In fact, it could easily be seen as you arguing in bad faith. Which is ironic given we are talking about intelligence and aptitude to picking up things. So I suppose this reply is more for other people to read. I am certainly done with this conversation now. Regardless, have a good day :)
I've read that in some families, especially from Asian cultures, you're started off with math tuition as soon as you can read.
And some people are just lucky to have the right environment. I happened to have had access to some more advanced books from older relatives as a kid and I noticed that it gave me an edge over my peers in some areas.
Asian families, for sure. But that’s doesn’t explain that vast chasm we all observed between non-Asians in school growing up. Why not just accept that it’s possible to be born with an aptitude for math?
If We closely guard this secret instead of telling people on online forums, We could gaslight people into thinking they are at fault for not being good enough at math. Then We mathematize everything that can be mathematized in higher education and research, and gatekeep it by requiring math even when it's not strictly needed.
(I mean, that's not my opinion, that's just how modern society has been running for... a century or two?)
It's not the "access" to those books that gave you an edge, 95% of children would be completely uninterested in those books. As you know, today 100% of Western children have free and instant access to the best math teaching material you could imagine, right from the phones, but math grades keep falling.
Smarter people will get the prerequisites faster. Nothing wrong with that, I'm more of a persistent person myself, which is a quality in itself. The really accomplished people are both smart and persistent.
But otherwise I agree with the article. I have zero basics in physics because my first teacher was generally senile and there was noone else (small town), and it was always something where I automatically tried my best to just get a passing grade.
The point about persistence is really important. I've worked with a lot of smart people who aren't persistent and, frankly, quite often lazy and/or uninterested. They don't work through problems by themselves and are quick to seek help or quick answers at every hurdle. Needless to say they aren't going and filling in the gaps in the requisite knowledge to better themselves either.
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.
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.
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 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)
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. :)
> 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.
My step-mom tutors. A high schooler was having a terrible time in algebra. She quickly realized there was foundational numeracy missing. The next lesson they walked out to the residential street and asked: how many tires are there. The kid froze and guessed. Time to start counting. A couple months later, the kid was able to raise their grade to passing in algebra
This article makes me think of the skill tree or roadmap posts we’ve seen here on HN or Reddit over the years. For example https://roadmap.sh/frontend …
I feel like we could do a better job of providing ourselves fundamental tools like this in helping ourselves and others learn. Not just in tech, but in life overall. The “dev” tree above is embedded in the life skill tree that should start in elementary school.
Haha, even the life skill tree has “fictional” branches that intercept the game world skill tree … you really do have to learn all of the dependencies necessary to case 5th level fireballs … there are real rules to be learned in the games their usefulness is just siloed into the fictional realm.
I’m curious about what that “preliminary knowledge” is? I’ve read stuff like a mathematician delight and the Joy of X and it’s such a beautiful, attractive but seemingly unattainable realm of knowledge.
As an example, this is the math that I’m aware of and have been exposed to:
Arithmetic
Algebra
Geometry
Trigonometry
Calculus
I’m vaguely aware of linear algebra but haven’t studied it (it also seemed unattainable)
I’m also aware of discrete mathematics and even bought the book concrete mathematics by Knuth, only to be totally stuck in the very first example of recursion and the tower of Hanoi…
So, what is that preliminary knowledge and how does one goes about acquiring it?
From where I sit sometime it feels like I don’t what I don’t know and I don’t even know how to ask how to learn what I don’t know I don’t know
Probably figuring out math paths. I've always been bad at remembering formulas or theorems, but that's because I remember how to get to that formula or prove the theorem instead. E.g. in grade school, I never memorized the sum of cubes formula, but I knew
a^3 + b^3 = a^3 - (-b)^3
and that the difference of cubes looks somewhat like the difference of squares, so I guessed
a^3 + b^3 = (a - (-b))(<and figured out the rest.>)
Having the whole path in your head makes it so you don't get stuck when you forget one step, and also makes it easier to make new connections.
Same, I never could remember formulas at school. What I did during exams was to "reconstruct" them visually: I'd sketch a few graphs at various random points and try to derive a formula from there. I also remember successfully solving problems by sketching geometrical shapes (triangles, lines etc.) and just trying to reason with basic logic + trial and error.
I realized back then that a lot of math (at least school-level math) can be grokked if you visualize it geometrically/spatially, as manipulatable objects in space. I don't know why our teachers rarely explained it like that. For most students, it was like strange symbol manipulation rules that you must remember by heart and can't derive from scratch.
Currently I'm fascinated with the way neural networks can be understood as a problem of trying to untangle tangled manifolds (to make them linearly separable) by "folding"/distorting space, using basic matrix manipulations... That way it's not magic anymore, it's something which appears so straightforward.
I like this idea. I'm guessing that it might be trying to learn math from proofs? or from first principles? I'm trying to figure out how could I find where to learn math in this way?
Yes, essentially proofs. I didn't use that word because people usually only write proofs for each other, so they end up being much more formal. When it's just yourself, chicken scratch in the margins is good enough.
How you learn it this way is by always asking yourself how something came to be. Why is the area of a triangle base * height / 2? Why are you allowed to subtract two equations (but not inequalities)? What is this curly thing in Green's theorem? How did someone come up with that alternating sign formula for the determinant? And so on.
This is an optimistic take on things. With years you understand there is a small cohort that is not capable of learning the basics even if they try, then there are folks who do not even know they are lacking.
This is a half truth. Part of the reason some people take a lot longer to grok the prerequisites and get left behind in class is because of cognitive ability.
Working memory (WAIS) digit span, and broader performance IQ (as opposed to verbal IQ), generally indicates how many conceptual 'items' you can have in your head at once. With more advanced math, this becomes _critical_ to forming the coherent plumbing between concepts in your head, leading to understanding.
Incidently, ADHD is largely an expression of specific personality traits and low working memory.
I felt that during college. Suddenly I couldn't hold enough abstract terms or variables at once to be able to reflect.
It slowed me down near zero, but weirdly, if you keep trying, your brain may evolve some new abilities and maybe as important.. keep feeling joy about learning even if slowly.
I wonder how one can raise this aspect of thinking.
This is where LLM's have been the most helpful for me. When I am engaging with an entirely new subject and have a bunch of questions I need to pepper someone with, I can ask as many clarifying follow ups as I need without getting self conscious or worry about annoying whoever I'm speaking with. The LLM is infinitely patient and able to easily handle beginner level prereq questions.
This is the best blessing the internet has given to worldwide curiosity and intelligence. The ability to ask "why" ad infinitum, without pissing off parents or taking up the entire class's time.
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.
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.
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.
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?
I’m bilingual since a child but didn’t really use one of the languages much after age 12 to about 30. When I do speak the second language I can fool locals into thinking it’s my first language, but it takes exactly 1 mistake or mispronunciation and they ask “oh are you actually English?”. The bar is that that high for passing native fluency in another language, if you somehow could fake the accent perfectly as well (there is absolutely no way).
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.
> 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.
I suspect this highlights are more serious issue which is that most of our training methods are not adaptive. They work well only if the students arrives at the right phase in their understanding and otherwise make poor use of everyone's time. Yet assuming the student has something to learn, and the teacher knows about it, this does not have to be the case.
One discussion of training I found eye opening was Pat McNamara's thoughts on what I believe he said was called "skills-based training" versus "performance-based training". With skills-based training, instructor start out the training session with the idea in mind to cover certain skills. A lesson is successful if it covers the skills the instructor wanted to cover. Performance-based training is geared towards improving the students' performance, so skills are introduced based on the students' actual level of ability and the relevance of training in a particular skill for improving their performance.
One motivation for adopting performance-based training is the lack of success of skills-based training in many contexts. Why is skills-based training sometimes unsuccessful? One reason is that the skills may be too hard -- the instructor chooses the skills with imperfect information on the students' level, and they choose the wrong skills. The students receive the training but their abilities do not actually improve; they don't know what's going on. Another reason can be that the skills are too easy -- the students receive the training and actually meet all the standards, but it doesn't actually help them get better.
Pat McNamara discusses these concepts in the context of being a shooting instructor for police departments and military units. It seems that one often doesn't know what these units know before one shows up, and the officers and soldiers in any one unit can be quite different individually, so the instructor has frequent occasion to think about the relationship between what they planned to teach and what they actually did when prompted by the students' questions and challenges.
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.
If you choose the right kind of problem that the children can figure out on their own -- it helps a lot. The will remember it better, and they will also feel better about their own skill at math.
But if you incorrectly estimate the difficulty of the problem... and instead of noticing your mistake and fixing it, you just wait for a miracle to happen -- the kids will only waste time and get frustrated.
The problem is teachers who believe they can do this trick, but fail to notice that they are doing it wrong, and then they blame the kids.
A lot of people hate rote learning - but elementary math classes, meaning all the way up to calculus (or prior to more rigorous proof based classes), do require a bunch of memorization. That's on top of getting an intuition...the "Aha!" moments.
I've observed that many math students in those types of elementary classes struggle because they're unable to recognize identities. They get some problem which involves substituting hard parts with easier parts using identities, but don't recognize them. So they try to solve the problem directly, and end up writing pages upon pages, before either getting stuck or doing some error that follows them until they get stuck.
Once they're showed what identities to use, they say "of course, I should have known that!" - but they never put in the time to solve all the problems.
And I was like that, too. I always thought that math would be a nifty because you'd "only" need to learn the various theorems, and if you understood those, then that should have been enough. It didn't really hit me that I'd need to put in hard work solving problem sets until I started recognizing patterns and knowing what to use, and where to use it.
Same goes for those that don't really understand the theory. Lots of math problems later will be of the type "Here's a difficult looking problem, is [statement x] true or false?" - and because they don't understand what math theorem to use, and all its properties, they'll try to brute force it by jumping into calculations.
You see it all the time in calculus, where students are asked to solve some nasty looking integral problem, which is much simpler if you know and use properties regarding symmetry and stuff like that.
I'd say for most people, there's no free lunch when learning math. You'll need to understand it, and you'll have to practice.
There's always going to be some extremely high-IQ individuals that can do pretty advanced math purely by logical deduction - but for the vast majority, it comes down to hard work.
> A lot of people hate rote learning - but elementary math classes, meaning all the way up to calculus (or prior to more rigorous proof based classes), do require a bunch of memorization
A lot of people still think reading and re-reading textbooks is the way to memorise theorems but recall (attempting to remember them without the book) is much better.
I cannot recommend the book "A Mind for Numbers" by Barbara Oakley enough - it put me on a journey to re-learning Maths as an adult.
I have still not graduated and this is my sixth year at university (in Europe). I find math too difficult and I still have calculus, linear algebra and probability. Discrete structures is the only math class I managed to pass and it took me too long to realize it's because the other classes have a long list of prerequisites. I have passed all of my software/IT classes aside from the maths and it's because they virtually are built from the ground up. On the other hand, strong math foundations are required even for the introductory math in my college. What I did was get great algebra and precalculus textbooks and went through them with great detail. After that I found the classes were not that hard to grasp.
The fact that the push primary school students through to higher and higher levels of math, when they 'fail', is a sacrilege to the student and a stain on the education system. I see so many capable students that don't have the prerequisites for the current math level, and are now just completely lost with NO chance of finding the path in the current system. And completely demoralized and disenfranchised.... all for what? To meet some BS metrics? This really needs to change.
I learned this as a teenager when I went from great at math to terrible because I got stuck with crappy teachers. Then, in 11th grade, I got put in algebra 2 with a great teacher and was tutoring other kids.
Math is completely different than other subjects. You can't catch up by cramming or reading a book over the weekend. You have to consistently learn and use it over the years. And have competent teachers to teach it to you.
Once you get placed in the remedial math, where they are just corralling misbehaving teenagers, and slapping out worksheets so kids can pass, you are basically screwed, unless you can get out of that situation.
Reminded me of the Feynman’s technique. I relate completely. One of my biggest challenges in returning to university after several years of work was exactly having lost the grasp of prerequisite knowledge. Unfortunately, from experience, more often than not lecturers just play the “you should know this from previous courses/high school” card and you are pretty much left alone in your struggles. Gets even worse if an exam problem relies on some borderline trick that wasn’t practiced throughout the course. You could probably tell I haven’t let go of some grudges.
This was my takeaway from college. In HS, you don’t have a lot of prerequisites, excepting the “II” level classes. I quickly found out how unstable my math and physics foundations were. Luckily, few realize that Bs are closer to failing than they are to mastery.
Even the college gut courses have hidden dependencies. I still feel for the business majors in my entry level stats class when the prof, bragging about learning calculus at 10, required calculus proofs for all the things.
Much like Civilization and Diablo games, and @godber’s comment, those tech trees should be required for all course syllabuses.
Late to the thread, but my view on being "dumb" in general, even if you have the pre-req's knowledge and ability are different things. Most people, in my opinion, are smart enough to understand and apply just about any complex subject or topic. Being smarter just means you can "compute" and understand or apply the subject in question faster than others.
In the end, what we prioritize and how much time is available for us to tackle different subjects is the biggest limitation, not genetics or luck. Art and entertainment heavily influence these things.
> A cage of 5 mice costs ~$1k upfront and ~$5k/yr recurring
You can get mice a lot cheaper than that, I'm not sure what kind of mice he's referring to but the prices depend on the vendor and mouse type.
Where I work it's about $2 a day to house a cage of 5 mice. It's about $30 a mouse if you get C57BL/6NJ's from Jackson: https://www.jax.org/strain/005304
So more like $150 for 5 mice and $800 to house for a year.
Another good one to know if the size of antibodies (10-12 nm).
I discovered this myself when I attended a data science summer school, a 2-day bootcamp. I knew Python and Jupyter. I know basics of Operations Research, though I barely passed the class back in time. I took classes on optimization classes thanks to industrial engineering classes. But at the end of the boot camp, I was as illiterate on data science as I was at the beginning. I was just more confused. Then, I understood that I was missing the mathematical prerequisites for the understanding. I still felt dumb though.
I don’t even have to read the entire article for the title to resonate with me.
But when you are poor and really need a leg up in society, you will do anything to push yourself forward - including going into student loan debt.
I certainly wasn’t equipped nor ready for computer science. Well let’s say my computer science classes I did well. It was the Calculus and Physics that I struggled because I didn’t have a good background from High School.
I didn’t have the necessary pre requisites.
When I recently completed my Masters in Systems Engineering, getting a 4.0 GPA was no problem.
Assessments are getting better in education and they help find missed skills. It's possible the author was smart enough to copy but not understand why they were doing what they were doing. Whether your local district focuses on skill acquisition versus graduation rate may determine the students success long term. I know little, but what I've seen from reading programs they're pretty much 'there' in discovering these blind spots. I don't know if there is comparable assessments or programs for mathematics.
Lately I've begun thinking that the way maths is taught, with each new concept following on the previous and no real way to revisit older content, just might be why people think there is a sharp divide between people who seem to understand all maths immediately and people who don't get it at all. Of course it might be the case that maths people exist, but maybe it's mostly survivor bias.
One thing I noticed going through school is that math concepts are usually taught first before physics and other subjects —- precisely because the math is viewed as a prerequisite for the other material. But this always seemed entirely backward to me, because much of the math was invented for and motivated by people trying to solve actual problems in these other disciplines. I think we should teach people in the same order of operations, rather than treating math as an abstraction to be learned by itself.
It's absolutely true. At work I keep a notepad open where I write down questions to get to the bottom of things. What occasionally stops me is shame that I might be asking my peers too silly questions as expected from my level. It's a daily struggle but there's no other way for me. Life's about learning and growing so you better get the right mindset for it.
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.
I did have friends lol. I was just a bit of an outsider from then through the end of high school. I didn’t correct it until midway through uni and I think I missed out on a lot
To generalize. When people fail they can blame themselves or can examine how to change. I saw one person try open a gate, fail and say they were no good with mechanical things. I said "Its not you, the gates broken" and the person immediately opened the gate. This is pervasive. "I'll see it when I believe it" as the saying goes.
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.
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.
Math/physics/engineering/programming rely on similar skills but need somewhat different attitudes. I know people who are great at advanced math (especially earlier generations) but don't think about it computationally at all, and their code is very messy. On the other hand I know great programmers who just don't like to think in terms of proofs and dislike the ambiguous and vague nature of math notation (yes, math is much more vague and handwavy in notation compared to its reputation, especially compared to programming). And there are people who love programming as a kind of puzzle and like prodding it for its own sake, pursuing elegance and will get nerdsniped for weeks if you teach them about quines, while others just love building things and whatever gets them there doesn't matter too much.
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.
It's a silly sort of comparison. Is reaching the programming competence necessary to hold an average software engineering job easier than attaining a PhD in mathematics? Yes, 100%. Is becoming a well-regarded software engineer by building large, complex systems or contributing significantly to projects like the Linux Kernel easier than getting to the top of your field in academic mathematics (tenured professor, high-impact papers)? I don't think so, or at least, it's not obvious to me that one is more "difficult" than the other. They take different skills and personalities, and I don't think that talent in one would necessarily translate to talent in the other.
> average software engineering job easier than attaining a PhD in mathematics
This isn't what I'm trying to say, I'm just saying that someone that can become good at programming could become good at advanced mathematics. If your criteria for good in advanced math is a phd then we disagree.
> They take different skills and personalities
I disagree, I think the skills are largely the same. Programming is literally encoding logic using a programming language which requires mathematical reasoning ability. Programming is more immediately practical, accessible and requires fewer credentials and that's why I think more people become good at it than math.
I have absolutely no problem being a 10x developer. While I might be a better than average developer I am not more than 10x more talented than my peers.
I have no problem achieving 10x status because I ask better questions and loathe doing repetitive work, especially out of social conformance. It completely blows my mind that most people strive first for emotional comfort, especially amongst a social reference group. Any effort moving in the opposite direction of that emotional comfort results in fear and possibly anxiety.
That is why I abhor software as a career. As a dumb soldier I feel like I am the least educated in the room and my delivery is inversely proportional to that. That is because, as a dumb soldier, I focus first on delivery. Focusing on delivery first means knowing the end state and cutting out all the bullshit in the middle. Military people think like that because they are highly assertive. The average software developer is meek.
Can you see the friction that follows? You have this dumb guy with less education that is a 10x developer but not because they are better at writing software. Nonetheless the output executes much faster with greater durability written in a fraction of the time only because of a difference in value system. That leaves the dumb soldier believing they are surrounded by a bunch of cowards.
learned to code pretty easily. the rapid feedback was key vs math where you do a problem set and it might be days before the graded assignment is returned. I found mathacademy 2 months ago an its been a game changer for me. I just completed the first of their foundations course and it filled in a bunch of cracks in my foundational knowledge I didn't realize I had. the big advantage is that you do the problem and you get feedback immediately. They figured out how to close the loop on feedback to make learning it efficient.
I can relate - I was quite bad at math through high school.
I eventually hit a wall in college then, like the author, decided to start from the complete basics: positive and negative numbers, fractions, arithmetic, algebra, then calculus.
Khan academy made this possible for me (in 2010), I don’t know where I would be without it.
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.
That is what I dislike about Computer Science course ( BSc ) and much prefer Computer Engineering ( BEng ). There are far too many abstraction involves that most people just remember or know the skills set of the abstraction layer without ever understanding how that abstraction comes into the field in the first place.
Over the past 10 years the media has have popularised the term First Principle often spoke about by Elon Musk ( He didn't invent the term but media help to spread it ). And this is precisely it.
And this isn't just computer but literally every single subject taught are now about the grade and not the "WHY". We just dont know how most things are derived from. We just memorise it and society will reward you with Certificate and a "Smart" status.
In Maths Richard Feynman [1] explaining mathematics in 4 pages from algebra to calculus. As the saying goes, I dont have time to write you a short letter, So I wrote a long one. Getting something simple and concise in 4 pages is the work of genius and takes a lot of time. I only wish something like this exists for all other subject with video course, completely free of charge in dozens of languages to kids all around the world.
When I think of "Computer Science" I definitely think that it respects "build great things from a small set of first principles" as much as any field. In fact, I might argue that CS is just a subset of pure math. It's those pesky Computer Engineering concerns, warts like memory locality and branch prediction, that make CS "less pure" than it could have been :) [no offense to CE, I personally love those warts]
> most people just remember or know the skills set of the abstraction layer without ever understanding how that abstraction comes into the field in the first place
This sounds like you're describing pragmatic software development, or "software engineering" if you insist on sounding fancy. My degree was in SE and in retrospect I would have enjoyed CS more, but it really didn't matter in the long run. But I digress... the point is: a course that focuses on using an abstraction is a programming/SE course, whereas a course that focuses on the principles one might use to build such an abstraction is a CS course.
I think intelligence (however defined) is important. But the problem is often people misuse this metric to predict a binary conclusion of whether they can acquire a skill or not instead of just considering it as one variable involved in the learning curve.
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.
This post biases way too hard into the nurture side of the equation. The difference between the author and someone who is genuinely smart is that the genuinely smart don't need to spend months carefully working through all those prerequisites in carefully arranged order. Until you meet somebody like this, it's easy to delude yourself into assigning yourself more brilliance than you posses and think that everybody struggles the same way. It turns out some people really are smarter than you, prerequisites or not.
It’s heartwarming to read comments from clever people, focusing on their struggles.
Too often I interact with people who lock conversations into their own sphere of competence with the outcome being that I feel incompetent.
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.
“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).”
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Many branches of math aren't focused on numbers. One cool branch that has almost no prerequisites and isn't focused on numbers is graph theory. It can be a great subject to cut your teeth on, regarding formulating theorems and proving them.
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 ;) ).
I am not sure I entirely agree with the premise: eg. you maybe are "dumb" (lack mathematical talent, really), but with proper instruction, you can learn a lot of math.
Let me dive deeper.
Our school system teaches math in a pretty inflexible way: "this is how everyone can get it". But even math talents don't learn it that way: as one, I was usually ahead of the school with my own reasoning (sometimes by a couple of grades) and could backtrack to the school method to understand it and apply it.
Second, if you are good at maths naturally, everything else at school becomes easier: people simply treat you as "smart" in whatever you do just because you have a natural leaning to mathematics (both if they do or don't themselves). Even rote memorization subjects like history and geography become easier since, well, you are "smart": teachers simply do not ask much of you.
And finally, I've met many an extremelly intelligent mathematician (uni professors and math competitors) who simply are outright dumb: they could not process a simple logic statement in human language, even if they were regularly working on advanced research calculus.
So, anyone can learn a lot of math, and doing so requires internalizing the foundations. However, people talented for mathematics find it easy to internalize them in various ways (not always the textbook way), so it's not hard work for them (eg. I could coast through the entire undergrad math and CS program too, cramming for a weekend for all but a couple of exams: memorizing all the axioms and theorems was the struggle, operating with them and proving them once I knew them was comparatively easy and I finished with a GPA equivalent of 3.4 or so).
But math instruction is hard because math is a formal language representing a very specific mindset that not everybody can naturally get. And instruction is usually performed by people not having attained that internalized knowledge of the foundations, thus not being able to look at it and describe it from numerous viewpoints required for individual students.
Finally, we need to fix the society not to equate "good at maths" with "being smart": plenty of smart people who have a hard time with maths, and plenty of math wizards who are outright dumb.
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.
To me there are either two ways: when you are trying to learn the thing XYZ you are seeking, drill down to the first thing you don’t understand and consult a lower level resource. Continue until you reach a level you understand.
And the second way is: Re-learning “all” of essential math and then going back to XYZ.
I don’t think the second step is feasible, as you cannot possibly learn everything in a breadth-first kind of way until you are deep enough to learn the (now level-adjacent) topic XYZ.
But for strategy 1, the question is 1) how to identify the problem that you are lacking (e.g. how to isolate math gibberish into a concrete concept) and 2) how to find a good resource to learn and practice this concept at this level?
I do struggle with this and sometimes randomly learn some lower concept again but notice later it did not help me in the end and just left me with a million untied knots that were infeasible for me to entangle.
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
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.
I’m interested in data structures and algorithms, particularly those related to mathematics. If you have any relevant materials or resources, I would greatly appreciate them.
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)"
I wish we didn’t think of understanding math as being smart or dumb. Like, it was a singular ability. Math is a subject deep, wide, and rich of ideas as well as results. For me, learning and understanding new concepts and results is like discovering new mountains and trying to ascend them. I won’t be able to ascend most, but I can certainly sit back and appreciate and enjoy their magnificence.
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.
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.
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.
Then my main point is that wealth and/or income is likely more studied than IQ, because it is/they are easier information to acquire. Even in psychology, I suspect it's quicker to assess someone's "socioeconomic status" than IQ.
The article you linked me to is paywalled. By "wealth" I mean family wealth even when one is 12 years old and have no income. The abstract says of another line of research: "Sibling comparisons ensure factors like the resources available during childhood, the impact of growing up in particular neighborhoods and genetic predispositions are all controlled, without explicitly adding these variables to the research."
If they normalize for parental wealth, then the paper you linked to concerns a different topic.
Yeah but good luck going through the entirety of khan academy as an adult, it's not impossible but it's also a task as hard as learning a new (or multiple) languages...
If you can dol it though and complete everything up to precalc you can most definitely do well in university.
I often tell people they're not dumb (and therefore I'm not smart), they just lack the prerequisites. But they never go and acquire said prerequisites. I do. Maybe they are dumb.
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.
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.