What really stuck out to me was how R failed in a bunch of other subjects except math because he wasn’t interested in them.
I know society and norms expect students to learn all these other subjects.
But what if those just aren’t interesting to someone?
I wonder how many geniuses we skip on because doing the chores of homework and getting through boring classes is busywork and memorization for the sake of getting an A.
Meanwhile, hardly anyone actually remembers anything about those topics and even the best students mostly go on to achieve only above average things.
My class valedictorian went on to become a doctor and while that is certainly impressive to me, there are many doctors and he practices (like almost every other doctor) and isn’t pushing the boundaries of medical science. I feel terrible writing that because I’m certainly not as smart as him, but R is just so impressive and I’m glad he got his lucky break.
People like R would be lost in the sea of averages because their genius would be kept shut by norms.
Almost every extraordinary person I read about seems like they were 1 step away from being forgotten, and got some huge universal break that boosted them.
It's good that public school exposes children to many subjects - hopefully most of them. So that they can discover if they click with one of them. The real danger is that someone never gets exposed to a subject at all. College is the place to specialize in a subject.
Exposure is good, but success in school requires you to be successful in these subjects as well. I have had similar thoughts recently as OP thinking about what I want for my young kids and what my experiences in school were. Being _really_ good in one thing should allow you to make up for being subpar in other areas, but it doesn't. You can only get an A (or A+) in math for example, even if your a genius. But maybe you should be able to get an A++++ that makes up for D's or F's in English for example and still get accepted into top universities. We need a system that accommodates spiky people better.
Specialization does begin earlier than that. Most high schools in the US have advanced classes that students can opt in to, and there is the AP program.
Personally I think that we could do better by tailoring every student’s education to their abilities. Put in simplest possible terms, we could arrange classes by complexity rather than by year. Have one class for addition and subtraction, another for multiplication and division, then geometry, algebra, etc, etc. Then let students graduate from one to the next based on proven ability rather than by age. Do the same for language, history, etc. Let every student proceed through the courses at their own speed.
I'm honestly no genius but I can relate to R in that one way.
As a child I used to get all As and even got into a Stanford pre-collegiate program as a kid where I learned C++ and geometry.
Unfortunately after a surgery in 9th grade that left me unable to attend school for 3-4 months and just terrible QOL for about a year my grades slipped (went from A+ studen to C grade student) and I basically became average. I lost all interest in most subjects at school due to depression and other things.
My goal as a child was to get a Stanford JD/MD MBA (lol I know..), and today I have only a bachelors from a low ranked state college in business.
I enjoyed programming so much as a kid that one summer, so later in life I ended up going back to it. Taught myself enough in a month to get on some projects as a swe. Later I got lucky working at a unicorn company that IPO'd.
Now I am trying to build my own company and see how far I can get as a solo founder. Sometimes I wonder how my life would have turned out if it wasn't for that injury, but oh well. Shit happens, right?
Jeez sorry for the sob story but it feels good to get it off.
The miss, which in my state has changed recently, was kicking kids out eventually who failed one subject but were succeeding in others. Sure, that kid isn't going to an Ivy League school, but there's value in finding ways to make up for what they aren't getting versus producing a high school dropout with bleak prospects in today's society.
To quote an artist friend: exposure is good until you die from it.
Being forced to do subjects that you hate is not exposure, it is being forced to do things which you are completely unsuited for.
I would go so far as saying that being forced to take music until 7th grade put me off any musical pursuits for the next 20 years. The less said about the torture disguised as education that is PE the better.
The problem is that there is too much of this general exposure and specialization happens way too late once the brain is not as good at learning anymore. People peak in competitive fields when they are in early 20s or even before that. In our education system we are not even allowed to do the job before we are like 25 or later. There is only so much you can learn sitting in a chair listening to a professor. It's a completely backwards system that limits potential of about anyone with average+ intelligence.
> It's good that public school exposes children to many subjects - hopefully most of them. So that they can discover if they click with one of them. The real danger is that someone never gets exposed to a subject at all. College is the place to specialize in a subject.
While some exposure is probably better on average than none, in some instances bad experiences can trip the fuse on developing an interest.
The rote nature of canned education, bad teachers, bad parents, or bullies can turn kids off of subjects they might otherwise come to love.
It's good that schools expose children to many subjects.
It's not good at all that school reward being OK in every subject more than being really good in a handful of them. The school system is a mediocrity factory.
Agreed. Given that most people are not exceptional at anything it makes more sense to give them a well rounded education by exposing them to a lot of subjects to see what takes. And all the exceptional people, well, they're just gonna have to use their gifts to push through the monotony of the things they don't like while still pursing their passion.
Unless they are born wealthy or find a benefactor so they can go to private school or a gifted person school, that caters to their specific needs, it's the best we can do to serve everyone.
> College is the place to specialize in a subject.
In Europe maybe, but in America a lot of students receive their general purpose liberal arts education in College, and will then specialize later with a post graduate degree.
It's good that public school exposes children to many subjects - hopefully most of them. So that they can discover if they click with one of them. The real danger is that someone never gets exposed to a subject at all. College is the place to specialize in a subject.
then why does this 'discovery process' have to continue into college? That was the OP's point. When money and time is on the line, let adults decide what they want to study. An 18-year-old is no longer a child.
Is it "good", certainly, but I don't think most of the stuff is worth it in the technological world. Let kids follow their interests, keep the general stuff to a minimum and you will have a lot more of happy kids with who excel more.
To overemphasize in a complex topic as good | bad is overly simplistic and doesn't help at all. Hardly anything is complete good or completely bad, it's meaningless to make a black or white point.
"Exposure" would be spending much much less time on those subjects, especially the free home time. Then the number of subjects is practically infinite, so expecting "most" is just as unrealistic (and colleges also continue this "exposure")
Ramanujan was, what, a 1 in 100,000,000 level genius? The gap between Ramanujan and the average class valedictorian is much wider than the gap between the class valedictorian and the average student. I think if we optimize schooling for people like him, we are probably not going to do as well for the other 99,999,999 students.
We probably aren’t going to get it quite right for that 1, either. Extreme outlier geniuses are extreme outliers and there’s no easily generalizable pattern around them. The ideal education for a young Ramanujan is probably different from the ideal education for a young Von Neumann. Of course in an ideal world we would give an extremely individualized education to every child, but that’s much easier said than done. Failing that maybe we could identify and then invest in the extreme geniuses, but that’s what we already try to do.
> People like R would be lost in the sea of averages because their genius would be kept shut by norms.
Well norms were in place when R did his work. Even the most strict systems have made concessions for extraordinary people. It is just that mostly average people go around claiming they'd be genius, had system not smothered their creativity.
> I wonder how many geniuses we skip on because doing the chores of homework..
I think from not many to hardly any as I can't believe if kids who are really genius can just go on for more than a decade of primary schooling without ever finding outlet for their creativity.
If you are optimizing for finding geniuses like R, you may be right. Many probably fall through the cracks of the educational system. But I don't think this is what we are or should be optimizing for. The vast majority of people would end up unemployable if they weren't "forced" to study things they don't enjoy because some skills are just more employable than others. You're lucky if you enjoy engineering/science, but not so lucky if you only care about art literature.
Instead of giving kids grades with a ceiling, each subject would have (unlimited number of) levels of proficiency, and to attain a level, kids would have to pass a test (demonstrate certain skill). The choice of subjects and levels to attain would be up to each kid, but they would have to choose to do something (working at getting next level of something would be mandatory). (Although perhaps they should be encouraged to explore different subjects and attain some minimum of levels.)
Also, I would group kids by subject, and not by age. So kids of slightly different levels would train together, and the higher level kids would be obligated to help kids on lower level to learn, while lower level kids were taught to be respectful of higher level kids.
> My class valedictorian went on to become a doctor and while that is certainly impressive to me, there are many doctors and he practices (like almost every other doctor) and isn’t pushing the boundaries of medical science.
I think in this example you are vastly overestimating the “average genius”, in comparison to those rare few who truly push the boundaries. We tend to do this because of the way our brains estimate things of significant scale, like the fact that keeps floating around social media about how bad most of us are at having a concept of the difference between thousands, millions, and billions.
There are many valedictorians (of the order of some per thousand) but few Ramanujans (of the order of tens per billion?), and gearing an overall education system specifically for those few could do a disservice to a great many others at every level below. Ramanujan was not the result of an effective education system anyway: like a lot of other world changing minds he was largely self-taught. Perhaps there is room to encourage more investigation to the side of the curriculum a lot more than we currently do, but the problems that stop other "Ramanujan"s, that could so easily have scuppered Ramanujan himself, are usually not caused by the education system but by other societal problems (death & disease, racism, sexism, caste or class biases, etc.) not giving them a chance to explore & self-learn or have useful access to education & other resources at all. Addressing those problems will help a much wider chunk of the population, as well as reducing constraints for the truly exceptional geniuses¹ amongst them.
----
[1] And of course those non-geniuses who luck out and have that one brilliant idea. Their contribution is often vital for progress too, and I'd wager that there are orders of magnitude more of them than there are true geniuses!
You're applying the logic of "that someone" is "Ramanujan", but the system isn't designed around students at the extremes.
Generally I think: "Unless you're Ramanujan, then you should probably have some breadth to your knowledge rather than pure depth" is not a terrible policy.
I have always been only interested in the 'exact' sciences since I was a little kid; I did not do other things even if I had to. I just didn't turn up; I was doing 'more important stuff'. I graduated with a special letter from the queen; all aces for exact sciences and the rest massive fail. It turned out that this made me a good programmer and employer, so I made a shitload of money (in eu terms; pocket change compared to what usa peers did). But it was a big mistake; now I really want to learn languages and history, but I never had the basics as a kid so I struggle far more than my peers. My ability to memorize things is not very good as I never needed to in school; formulas and code is not really memorizing as such I found. It is a massive regret. There are no do overs, but I guess even if I could time travel, I would've not be able to explain this to myself enough for me to listen.
Now think about the college admission process in the US, where kids are expected to take arbitrary number of AP courses and get 5 in all of them, and write world class essays about passion and solving world hunger while excelling at several extra curricular activities and showing leadership and so on and on…..
> I wonder how many geniuses we skip on because doing the chores of homework and getting through boring classes is busywork and memorization for the sake of getting an A.
Zero? If you qualify as a prodigy, it is apparent from a young age. Maths prodigies are especially easy to distinguish. Given a little time, they will self-learn, grok and innovate on anything you throw at them and will likely attend higher education early unlike "the brilliant kid"s who will struggle with advanced concepts all their lives.
It seems you are assuming that all child prodigies are polymaths (no pun intended).
There are children with exceptional mental abilities in a limited range of topics or even a single topic, such as math, music or drawing, They may struggle to various degrees with subjects or life skills that don't correspond closely with their specific topics of interest and ability.
The best way to learn is to play and come up with stuff yourself. But playing doesn't get you anywhere specific. People who play around a lot, clearly know much more and in depth than everyone else, but when you hand them a random checklist, chances are they won't know a few.
Standardized tests are screwing everything up. People who learn on their own might stumble upon the entire alphabet except for the letter "B," but standardized tests want only the first 5 letters. Hence the incredible efficiency of knowing the entire alphabet is thrown under the bus in favor of making sure none of the 5 are missing.
You can't teach someone to play, and there is no way to play systematically, at scale, and with guaranteed results. All the incredible people I know have some hole in "basic" knowledge, and if it is revealed nobody cares about them being miles ahead elsewhere. "Their basics seem lacking, in the name of stability and norm, throw them back to square one."
Following standards never produces something new, but the world is so afraid of failure and lack of definitions in "messing around" that they are willing to trade their souls for it.
Take any hacker here on HN, and ask how much they learned in CS class vs. how much they learned messing around with Perl on a weekend.
Standardized tests are tool for systems to be able to compare and work toward a uniformity of outcome. Expecting it to help anything beyond that is a foolish errand. Public schools need to educate million of people each years with differing deposition and life circumstances and do so with relative competency.
Excellence requires individual attention and cannot be so readily mass produced.
I was this person, also from India. I am interested in most topics under the sun, but was never interested in being an excellent student, apart from math where I was ranked in top 30 in my national olympiads.
I got mediocre marks in high school but thankfully did well in JEE. Now I have a decent PhD in math with an extremely mediocre school record.
Unfortunately, the purpose of the education system at this stage of human civilization development is not the realization of individual potential or the creation of geniuses. Geniuses, however cool they may be, are not necessarily the most effective increase in unit production efficiency per person that you can create. The point of the education system is essentially to create workers that make the economy productive.
I don’t think you can say, even all these years later, that Ramanujan, that mathematician, made the economy more productive, but he certainly increased the high watermark of human civilization and created an inspiring story for individual achievement, creative realization, and artistic and mathematical expression. There’s something sublime and transcendental—no pun intended—in the kind of truths that he was able to tease out and the unique, idiosyncratic way that he expressed them. Sort of like a Basquiat of mathematics, I suppose. Or probably better than that.
That aside, I think it’s unfortunate that he died of cholera or something, isn’t it? I mean, he apparently didn’t think it was unfortunate that he was going to die. And certainly, the formal education system didn’t necessarily fail him, in that a professor at a university recognized his genius and sponsored him to the UK.
But I think, in a sense that you identify, there is this general failure of the education systems in the human civilizations on this planet to foster perhaps the best thing that they could be fostering. They’re more like a manufacturing assembly line to produce cogs as part of the economic machine.
Not that there’s anything necessarily wrong with that. I think it’s good that people can have a role to play in the larger economy and that there are pathways to bring people to the level of capability where they can contribute like that. But the lack of pathways that these systems provide—those that could contribute to the creation of the full realization and expression of individual potential—I think is sad. And I think that’s what you’re kind of identifying.
> People like R would be lost in the sea of averages because their genius would be kept shut by norms.
I have a family member who scored perfectly in math on the final exam of middle school but failed other subjects so badly that he couldn't advance to high school. What frustrated us even more was that the next year, he achieved a perfect score in math again but still failed overall, leaving him no choice but to drop out of school.
This story has stuck with me for years, highlighting how our education system focuses solely on maintaining an overall average score above a certain threshold without considering how individual subjects are mastered.
I can’t help but think that if he had been allowed to pursue mathematics, he might have accomplished something great. It’s really disheartening.
> But what if those just aren’t interesting to someone?
That doesn't matter. A student is unable to evaluate the value of unknown subjects and the synergy between subjects. People are often unmotivated to do important things, because we are just lazy s**.
> I wonder how many geniuses we skip on because doing the chores of homework and getting through boring classes is busywork and memorization for the sake of getting an A.
And how many lunatics making wrong decisions did we prevented by forcing them to learn necessary things?
> Meanwhile, hardly anyone actually remembers anything about those topics
They are engraved in your mind. Even when you don't remember every little detail, they still helped you in forming your understanding of the world.
> My class valedictorian went on to become a doctor and while that is certainly impressive to me, there are many doctors and he practices (like almost every other doctor) and isn’t pushing the boundaries of medical science.
You seem like someone who thinks things through, so I suspect you’ll know what I’m about to say, but given the sentiment of your comment, I think it’s worth explicitly sharing this:
The fact that your class valedictorian went on to be a doctor is great. Not everyone needs to push the boundaries. Your classmate may end up saving/helping countless lives.
There is an Isaiah Berlin essay called The Hedgehog and the Fox. In it he divides artists/philosophers into two categories: foxes and hedgehogs. It is inspired by a fragment of ancient Greek poetry which reads, "the fox knows many things, but the hedgehog knows one great thing." In other words, some great thinkers looked at the world through the lens of one big idea, and others can't be easily reduced.
Two major caveats. The first is that Berlin never wanted the essay to be taken all that seriously. To him, it was an intellectually parlor game. But if TvTropes is any indication, people are addicted to classifying things into simplistic categories, so the fox and hedgehog game took off.
Second, the rest of the essay actually challenges the binary by focusing on Leo Tolstoy, an unimaginably creative thinker and writer who, according to Berlin, was a fox by nature but wanted to be a hedgehog.
Why did I share all of that? Well, without taking the categories too seriously I do think some people are more foxy by nature and others are more hedgehog-esque. I've always thought of myself as a fox (which to be honest, might limit me), and I had no interest in focusing on a single subject in school. At worst, I'm a dilettante who can have a surface level conversation about any intellectual topic at a party. At best, I can muster up a real enthusiasm about trying something entirely new.
I say all this not just because I'm running on 4 hours of sleep, but also because I think foxes get a bad rap. Not everybody should strive to be Ramanujan, because it isn't in everybody's temperament.
You’re absolutely right. The education system was meant to be a factory of production line workers, at the cost of missing out on geniuses who don’t care about other subjects.
This system is meant for a worker that no longer exists. If someone wants to specialize at the age of 13 and has shown reasonable aptitude in that subject, then let them do it. Sure, give them a well rounded education, but don’t weight those grades anywhere close to equally.
What better system do you propose to identify and nurture the one-in-a-million genius (the productive genius, not the crazy genius) while also serving tens of thousands of smart strivers, as well as the vast majority of the rest?
I don't think we can or should design a system optimized for that one-in-a-million genius. It would be worse for everyone else, and ultimately for society.
I'm pretty sure I only have a successful career because I deeply enjoy programming and have a slightly neurotic obsession with code quality and ergonomics. I can't fathom giving enough of a shit about anything I don't enjoy for long enough to be successful otherwise.
These people are still out there. When I was in high school we had the normal people, then people who took advanced placement stuff, then the "super nerds" who were at the top of all the advanced placement stuff with perfect grades, and then there was this one guy who was most of the way through all the advanced math classes at the nearby university. Same guy was in one of my English classes, and was failing. More or less he couldn't be bothered.
Sadly the later part of your comment may hold - I don't remember what ended up happening with him, whether he graduated high school or what. Hopefully at that level you just disappear into academia and not off the face of the earth in general.
I agree. That's also why I just don't believe at all when people say we have a shortage of talent (as in we need stuff like H1B) there is a ton of talent wasted. Everyone know that smart person who is working a menial job.
In my experience it’s not talent but opportunity that is in short supply. One of the smartest people I know is an electrician, simply because he grew up in a rural area and couldn’t afford to leave for college.
Exposing students to other subjects is what make them discover what they like. I think the optimal balance would be studying a wide number of subjects, but being able to select for an even larger set.
"...they weren't interesting to him". That’s how I felt about mathematics. History, literature and language were the areas I was interested in, for example. It seems to me that an individual’s strengths aren’t necessarily indicative of an all-encompassing genius. Rather, they are often more focused, tied to specific subjects or fields in which that person naturally excels. That's not to say an individual can't have a wide range of strengths.
Meanwhile, I'd be working a backhoe in the middle of nowhere Louisiana if someone hadn't put books in my hands and forced me to read them under penalty of failure. Standards aren't standards because they're fun. They're standards because you need them to have fully functioning, working societies.
The US political system is currently in a state of utter dysfunction because people have decided that facts "just aren't interesting."
Education is optimized for average citizen who must work through boring tasks every day. I feel like geniuses probably more like survive in school rather than being supported.
> But what if those just aren’t interesting to someone?
The school I went to grades 1--12
tried to be especially good
so taught Latin, French. Some of the girls were in ballet. MIT came
recruiting. The year before me two guys went to Princeton and ran against each other for President of the Freshman Class (whatever that meant!). In my class, one guy (did nearly as well on the SAT Math as I did!!!) went to MIT.
In one of the early grades, I got dumped on (adenoids, couldn't hear well until that got fixed). Apparently the teachers talked to each other and had me with a dunce cap until I proved otherwise. In 1st algebra, discovered math: I liked it, was good at it, was the best in the class, proved myself, got sent to a math tournament, couldn't get dumped on, etc. Continued that way: Was so good at math that I got an unspoken but powerful by in any subject, e.g., English literature, I didn't like.
Got sent to summer math/physics enrichment programs.
So, for that example, for
> But what if those just aren’t interesting to someone?
some schools will let a student who is good at some one subject get a by in other subjects.
Really, schools, K-Ph.D., have a tough time finding any students really good in even just one subject, are thrilled when they find one that is, and don't want to block him/her because
he viewed fictional literature as a not very credible presentation of common reality?
That by pattern continued: In grad school, they insisted that I take their computer science course. My background in computing was already nicely above that course, and I'd already taught a similar course at Georgetown. Soooo, mostly laughed at the course: E.g., they had a test question about Quicksort (very common topic then), and I answered with material they didn't know.
The best case of by: Took a reading course; decided to address a question in the pure math of optimization; two weeks later had a surprising theorem and from that an answer to the question. The work, clearly publishable, was instant news all over the department, some profs angry that I had done well, others pleased. Angry/pleased, the work got me a general purpose by, a gold crown, immunity from any criticism, and an unspoken, implicit, easy path to the rest of the Ph.D.
It's the crazy ones that push humanity forward. We lose far more than we can imagine by not enabling even just one of them. This is one of the most important problems for us to fix.
You're assuming it's luck. But maybe we're actually good at identifying boundary-pushing geniuses? There are huge, huge incentives for being good at that.
Honestly I suspect a lot of potentially able people get left by the wayside in our lowest cost educational system. It's particularly tough on neurodiverse people, which one has to suspect included Ramanujan.
This thread is a good example of one of the perhaps overlooked reasons it can be hard to discuss education in our societies - any attempt at making a general point, or a meta-observation of some kind, quickly gets swallowed in a massive influx of personal anecdotes about things that happened to people during their education.
Maybe there are other topics where the same phemonen exists. I'm not sure if off the top of my head I can think of one where it gets so intense so quickly - long, detailed, emotionally charged anecdotes immediately when school comes up! [I love a good yarn, and am not judging or belittling people for expressing themselves here, I must stress].
I have speculated in my comfy armchair as to why our education structures seem to make venting such a need for people. Anecdotally, there does seem to be a suggestion of some sort of lingering and powerful malaise around the whole thing, in any case.
I wonder if it's like an abusive relationship - the emotional work required to move on and have better relationships (i.e., different educational structures) gets so large at some stage that all you can focus on is "coping". And you're sort of stuck, and the cycle more or less continues.
EDIT: may I just add - I read that whole article, and am very fond of Ramanujan. He made university much harder for me when I learned about his existence, as my maths classes seemed so far removed from whatever it was he was doing.
I think there is another force at play here - the assumption that there is a platonic ideal good education system that works at every scale of population.
Attempting to scale something that large forces people into boxes that fit inside the framework - one must necessarily ignore little detailed differences between humans.
But from the perspective of those individual humans, ignoring those little details doesn't really work. Those things are important, and tend to ping our egos, so we react accordingly, and are eager for opportunities to give voice to this frustration.
There's another potentially missed point. There is no education system, nor any amount of effort which could have granted me the genius of Ramanjuan. He had something I never will. It would be like dream to run faster than Usain Bolt.
I think it's just because every single person here has at least a decade of personal experience in education. One of the things many people like about HN is that when an article about some obscure subject is posted, there is often someone in the comments with personal experience who can share a story. Education is one of the few subjects where we can all be that person.
> it can be hard to discuss education in our societies
I found it quite ironical how the whole topic has been hijacked into this side discussion. For the record, Ramanujan is NOT a product of the Indian education system. In fact, that system was quite cruel & off-putting to him. He was a self-taught math prodigy. Apart from the well known 2 big biopics, there are numerous TV soaps telecasted in India in various languages that regularly emphasize this point. He taught himself mostly using Inequalities by GH Hardy, & several other books, all of which are freely available right now at the click of a button. Nobody is stopping anybody from studying math, and education or the lack of it has nothing to do with any of this.
> I found it quite ironical how the whole topic has been hijacked into this side discussion
Indeed. I submitted this post, because I found the article's discussion of the math involved fascinating, along with biographical details of Ramanujan I hadn't heard about before. I'm surprised to see how popular the post is, but also that there's maybe 5 or so posts that engage with the actual mathematics.
I probably shouldn't be surprised - generating functions pointing out unexpected connections between different restricted integer partitions is pretty niche.
> as to why our education structures seem to make venting such a need for people. Anecdotally, there does seem to be a suggestion of some sort of lingering and powerful malaise around the whole thing, in any case.
Teaching has not incorporated morality or ethics as strongly as some other professions where the need is more obvious. This is combined with the way teaching "quality" is measured. The result is that your average school teacher often uses tactics and methodologies that border on straight up public bullying of their students in order to "achieve results."
> I wonder if it's like an abusive relationship
Neither the teacher or the student engage in a process which selects each other nor is there any process to manage particularly bad combinations of such. They are assigned to and stuck with each other. The ethical failures in teaching abound.
[...]the day I started reading educational research was the day my life changed.
It all began with Daniel Willingham’s Why Don’t Students Like School, which led me onto his ridiculously good series of Ask the Cognitive Scientist articles for the American Federation of Teachers and the associated research papers he quoted. I devoured them all with eyes wide open and jaw firmly on the floor. Next came Cognitive Load Theory, which I was introduced to via the incredible blog of Greg Ashman and our subsequent conversation on my podcast. I was familiar with the work of Dylan Wiliam through the development of my Diagnostic Questions website, but I read and re-read everything I could get my hands on from the great man. I didn’t think life could get any better — and then I came across the work of Robert and Elizabeth Bjork. Good God.
One thing led to another, and before I knew it I had read well over 200 hundreds books and research papers. I would wake up in the middle of the night, my head buzzing with ideas[...]
> ... in a massive influx of personal anecdotes ...
Not sure if you're in the US.
Here, ALL non-technical topics, e.g. nutrition, healthcare, immigration are just a few of the big ones, devolve into a discussion of anecdotes and "feelings."
My guess would be it's because it's an experience (most of us) have in common in years full of first times. It's also an experience that took as least as much space in our life than work now does. It's also a place where we had to deal authority and groups. So whenever you make any point about education, if the point you are making seems to negate/go against the grain of my own experience, I will probably tell you :)
(Venting is a need that everyone has when under pressure not sure it has anything to do with our education structures)
a.k.a. sampling from the distribution of interest. Once you have enough anecdotes, you actually have a representative distribution of the real thing. Selection may be biased but you can weight those data points as you like, it's still good data.
I mean everyone on this thread seems to be pointing to the negative. I would like to point out that what we have achieved from a civilization perspective is literally through education and the structures that we have have gotten us this far. It is a broad tool and could be better. However lets not throw the baby out with the bath water because of all these anecdotes. It feels a little like everyone complaining about capitalism for all the woes in the world.
In the Ramanujan story, a true MVP is G.H. Hardy. He read letters from some random unknown guy (a savage "native" no less!) half the world away, and took them seriously. And then organized resources to have that guy travel to England. A true MVP. All the others Ramanujan wrote to ignored him (understandably so). Such a tragedy that he died so young.
If you want to understand how human potential was wasted in old world, Ramanujan belongs to a caste in India that is only caste that is supposed to be educated (Representing probably < 5% of population) in those days.
Ramanujan short life itself is a loss to the world, Imagine how many Ramanujan's were ignored where there is no G.H. Hardy and what about Ramanujans in the other 95%?
In the New World, the Ramanujans' lives are spent optimizing high-frequency trading, ads or video recommendation algorithms. This is arguably even worse for society than them not being discovered altogethrt.
Ramanujan himself survived childhood smallpox and died at just 32 from what's thought to be complications from an earlier bout with dysentery. But he had lucked out in that he was born male in an urban setting in a high caste, with access to education, textbooks, and the language of the imperial core, and managed to make it to adulthood at all.
“I am, somehow, less interested in the weight and convolutions of Einstein’s brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops.” ~ Stephen Jay Gould
This is why education and opportunities should be available for all, regardless of social / economic status. I reaped the benefits from that in the Netherlands where the government paid for most of my education (bachelor equivalent) and consequent professional life (white collar / middle class), instead of staying in my parents' / ancestors' "class" (blue collar / working class).
But people in the upper classes don't like that, so they're telling working class people that people even worse off are after their jobs.
caste system in india is an evil that persists even today, and is a complex topic.
majority of the indian-origin users on hackernews or people in the IT community are very likely upper castes. see [1]
If you are interested to learn more about the caste system in india it is recommended to get your facts from multiple sources. start from [2]
This is such a common and dangerous rhetoric, which has been repeated millions of times in India today, that people are willing to believe that one community has wickedly tried to subjugate everyone by denying them the right to education. But this is completely false. The early government records of British India themselves attempt to record detailed demographic records of the communities that studied in the early 'modern' schools. And no, the Brahmins didn't have a monopoly.
This is similar to another popular narrative that no girl received an education until some sympathetic Englishman in partnership with a local woman allowed girls to go to school. What a load of BS.
Agreed. It is interesting to contrast G.H.Hardy's treatment of Ramanujan (nurturing) with Arthur Eddington's treatment of Subrahmanyan Chandrasekhar (petty and back-stabbing) decades later. A discussion with lots of links can be found at https://news.ycombinator.com/item?id=41284239
I think he was referring to the way colonizers commonly referred to the subjects of their colonies pejoratively as "savage natives". This applies to Native Americans, Africans, South Asians, etc. Racism was the norm in the old days.
"The statements had been proved 20 years earlier by a little-known English mathematician named L.J. Rogers... Rogers was content to do his research in relative obscurity, play piano, garden and apply his spare time to a variety of other pursuits"
The stories of mathematicians like Srinivasa Ramanujan, who claimed to have derived complex partitions and identities in dreams, have always captivated me. It's as if their minds were tapping into some hidden reservoir of knowledge. I'm curious what drives these intuitive leaps. Was Ramanujan's brain quietly processing patterns during sleep, leveraging its default mode network in ways we're still struggling to understand? Or was it something more fundamental – an emergent property of complex neural networks, perhaps, or even a glimpse into Jung's collective unconscious?
I'm curious to hear how others think about this phenomenon. Do recent advances in neuroscience, AI, or cognitive psychology offer any clues about how innovators like Ramanujan access these hidden sources of insight? Or are we still stuck in the realm of "genius is mysterious"?
Starting from the basics, Ramanujan was known to spend huge amounts of time in the library pouring over mathematical texts. He was also personally and spiritually obsessed with mathematics, thinking it was an expression of divinity. So its quite probable a significant chunk of his memories were already mathematical and random accesses to it were the same.
As the article mentions ,we was familiar with the literature. He communicated with other mathematicians, read papers, and submitted in journals while in India. he was not some hermit in a cave or something. I think this claim that he just dreamed the results part of mythology that has been built around him. From what I read, he he did a lot of the grunt work deriving these formulas but only published the final results, so it only appears that he conjured them out of nothing. It's not like he could have sent Hardy a book-sized letter of all the steps to derive those results.
This myth of him just coming up with results without all the hard studying he did is often repeated. His results are made to be seen as impossibly hard and his impact is overstated. Galois solved a problem that was open for 350 years, he made significant contributions to new branches of math and he died at 20. I don’t like the whole ranking thing, but with all the myths surrounding Ramanujan I feel compelled to say Ramanujan is overrated and Galois is by far the more gifted and accomplished to just name one person.
There’s a relevant quote from Swami Vivekananda on this
From Karma Yoga, Chapter: I, Karma in its effect on character—
What we say a man “knows”, should, in strict psychological language, be what he “discovers” or “unveils”; what a man “learns” is really what he “discovers”, by taking the cover off his own soul, which is a mine of infinite knowledge.
We say Newton discovered gravitation. Was it sitting anywhere in a corner waiting for him? It was in his own mind; the time came and he found it out. All knowledge that the world has ever received comes from the mind; the infinite library of the universe is in your own mind. The external world is simply the suggestion, the occasion, which sets you to study your own mind, but the object of your study is always your own mind. The falling of an apple gave the suggestion to Newton, and he studied his own mind. He rearranged all the previous links of thought in his mind and discovered a new link among them, which we call the law of gravitation. It was not in the apple nor in anything in the centre of the earth.
Whenever I read about Ramanujan having divine revealing formulas in his dreams, I remember Swami Vivekananda’s quoute on consciousness and mind.
edit: found another relevant quote from Upanishads on tapping the infinite knowledge:
Translation: “The Self hidden in all beings does not shine forth, but it is seen by subtle seers through their one-pointed and subtle intellect.”
Explanation:
The ultimate knowledge or truth is hidden within all beings and is revealed through subtle inner perception. The idea is knowledge is latent within the mind, and it is discovered, not externally found.
Not sure I follow the explanation. And apologies cause I’m not that familiar with these texts, but afaik it talks about “soul” here, not information? Words like Brahman, Atman come to mind.
In case I mistook it, ignore the above.
All knowledge that the world has ever received comes from the mind
Doesn’t it come from the mind’s interaction with the world? I don’t think that any important knowledge comes from doing nothing (apart from psychology due to its nature). It all sounds like a typical religious salad, tbh. “You” discover from your “mind”, but then some seers understand that “you” and “all” and “mind” are the same, etc. Well, you don’t have to be a seer to see it. “You discover from your mind” is a useless semantic loop that creates all this unnecessary complexity, imo.
That’s beautiful, thanks for sharing. I’ve seen that firsthand growing up. It’s like when you hear a song as a kid, and it just sounds cool. But then, later in life, you hear the same song, and it hits you on a whole different level—like the meaning was always there, but you had to live through stuff to really get it.
> Translation: “The Self hidden in all beings does not shine forth, but it is seen by subtle seers through their one-pointed and subtle intellect.”
That doesn't look sound to me. If the "seers" are seeing "The Self", are they beyond and separate from "The Self"? If they do so with their "subtle intellect", is that intellect outside of "The Self"?
If affirmative, then "The Self" is something external to the seer, making the term a misnomer. And furthermore, there is something outside of The Self (that which is seeing The Self), which remains to be explicated.
If you spend an enormous amount of time (like, too much) working on a single piece of software, you'll come up with solutions in your dreams and wake up and write them down. It's not that rare. Now... the solutions might be for loops so I'm not comparing such situations to Ramanujan, but it's not some extremely rare phenomenon.
I am also intrigued by this question: What was different for guys like Ramanujan, and how were they able to tap in to this hidden reservoir of knowledge. And how can we replicate it
One guy able to tap into this knowledge in dreams is an indication that it is possible. Now, how do we make this the default for everyone is the question I wonder about
The way we found one variant of wheat in Mexico that was resistant to bacteria, and replicate that to the whole world -- can we do something like that for humans
( even I don't like the sound of it, but I hope you get the feeling )
> how were they able to tap in to this hidden reservoir of knowledge.
My guess would be by filling the reservoir first. After filling it, then you can stir it and extract new knowledge. In other words: you get better at math by studying, reading, and thinking a lot about math. I don't think there has been a brilliant mathematician who has not immersed himself in math at first.
We all already do solve problems in our dreams. Even if you haven't awoken with a Eureka, surely you've occasionally found that something that was difficult to solve yesterday is obvious with a pair of fresh eyes.
how do we make this the default for everyone is the question I wonder about
You delete stupid prejudices from society, then it allows itself to do that. One group of people forcibly practicing it for idiotic reasons now plagued the whole field with its label, amplified by religious ideas. The irony is, it’s that same limitation that disallows to address itself and continues to propagate.
As others say - just study a lot of math. You will not be Ramanujan, but you will be better at math.
That's not to say there isn't something spiritual and mystical in knowledge. I think the fact that he was from a different background gave him a different perspective. But yeah you cannot skip all the math studying.
The recent book Mathematica by David Bessis attempts to describe this exact topic. One core theme of the book is that math is done by building mental models and using our intuitions that come from. Formalism is then used to shape and refine these models. By spending time developing these models, insights eventually become obvious. These mental models are the essence of mathematics, not theorems. The author explicitly uses neural networks as an example to describe how this negative feedback could work to make these changes to our mental models happen in our brains.
The core premise of the book is to describe how mathematicians work and think and to show that this is a process that everyone can do (although some will be better at it than others). It includes interesting accounts of Grothendieck, Bill Thurston, and Descartes as well as from the author's own research career at Yale and École normale supérieure. The book is targeted at the general reader and at times reads a little like a self-help book, especially in the first third or so. However, I found it to be an enjoyable and fascinating read. It provoked a lot of interesting questions about the nature of learning and provided a framework to begin to answer them (e.g. "How can I have proved something and yet feel no understanding of it?", "How can some people solve problems orders of magnitude faster than other smart people, as if they don't even have to think about it?", "Why do I sometimes watch a presentation on a new topic, follow every step, and come away feeling like I've learned nothing?"(* see except below)). I don't think I'm doing it justice here, so I'll stop by saying I highly recommend it based on your comment.
_______
* I'll use this as an excuse to provide a related excerpt featuring Fields Medalist and Abel Prize winner Jean-Pierre Serre:
One day, I had to give a lecture at the Chevalley Seminar, a group theory seminar in Paris. I didn't have substantial new results to announce, but it was an opportunity to make a presentation even simpler than usual. [...] A couple of minutes before the talk was to start, Serre came in and sat in the second row. I was honored to have him in the audience, but I let him know right off that the presentation might not be very interesting to him. It was intended for a general audience and I was going to be explaining very basic things.
What I didn't tell him, of course, was that his presence was intimidating. Still, I didn't want to raise the level of my talk only to keep him interested. I just kept an eye out to see if he'd taken off his glasses, which would mean he was getting bored and had stopped listening. No worries there—he kept his glasses on till the end.
I gave my presentation as I would have without him there, speaking to the entire audience, especially the students seated in the back, whom I was pleased to see listening and looking like they understood.
It was a normal presentation, fairly successful, not very deep but well prepared, clear, and intelligible. At the end of the seminar, Serre came up to me and said—and here I quote verbatim: "You'll have to explain that to me again, because I didn't understand anything."
That's a true story, and it plunged me into a state of profound perplexity.
Apparently, Serre wasn't using the verb to understand the way most people use it. The concepts and reasonings of my talk couldn't really have caused him any difficulty. I'm sure he wanted to say that he understood what I had explained, but he hadn't understood why what I had explained was true.
There are two levels of understanding. The first level consists of following the reasoning step by step and accepting that it's correct. Accepting is not the same as understanding. The second level is real understanding. It requires seeing where the reasoning comes from and why it's natural.
In thinking again about Serre's comment, I realized that my presentation had too many “miracles,” too many arbitrary choices, too many things that worked without my really knowing why. Serre was right; it was incomprehensible. His feedback helped me become aware of a number of very big holes in my understanding of the objects and situations I was working on at the time. In the years that followed, research into explanations for these various miracles allowed me to fill in some of the holes and achieve some of the most important results of my career. (However, some of the miracles remain unexplained to this day.)
But the most troubling aspect was the abruptness, the frankness with which Serre had overplayed his own incomprehension.
My intuition is that some people are able to develop abstractions on top of abstractions (compression is a key part of intelligence) that allow them to traverse the search space much, much faster than without these, and with enough repetition this becomes routine, even faster in the brain.
I don't think we have a good theory of how this works, or at least I haven't come across it.
Neural network guided search is somewhat similar but I think we are missing several key pieces
For people interested in learning more about Ramanujan and his Works;
1) Mathematics Wizard Srinivasa Ramanujan : Some glimpses into his Life and Work by two Indian Mathematicians Narendra Kumar Govil and Bhu Dev Sharma is a good biography with an introduction to his Mathematics and links to further resources. Good complement to Robert Kanigel's book The Man Who Knew Infinity.
2) In order to understand the fascination that Mathematicians have for Ramanujan see this and other lectures by Prof. Ken Ono who credits Ramanujan as his inspiration in becoming a Mathematician; Why Does Ramanujan, "The Man Who Knew Infinity," Matter? - https://www.youtube.com/watch?v=7ynhiZJUMzA
The article talks about a recent paper[1] by one of the interviewees that uses McMahon partition functions for primality testing. I wonder how its running time of compares to that of the AKS primality test, and something more practical like BPSW.[2] Could it find an application in practical cryptography?
Ramanujan’s story is very interesting but I would love more Indian mathematicians and scientists to become household names. Mathematicians like Harish Chandra, C. R. Rao, Manjul Bhargava, Narendra Karmakar etc. Physicists like C. V. Raman, Satyendra Nath Bose, Meghnad Saha. Others like Har Gobind Khorana and Venkatraman Ramakrishnan too.
You're right, some Indians don't have the recognition they deserve, but if it makes you feel better, few "western" mathematicians or scientists are household names either.
Fwiw Chandra, Rao, and Bose are instantly recognizable to me. I’m not a mathematician or physicist and don’t know the other folks. That said I am very aware that Indians have made significant contributions to math, physics and I imagine other disciplines.
This is entirely the fault of the Indian Education System and Popular Media. The current generation knows almost nothing about these Indian Greats.
In order to rectify the status quo;
1) Everybody should get the monthly magazine Science Reporter published by National Institute of Science Communication and Policy Research (NIScPR), Council of Scientific and Industrial Research (CSIR), New Delhi, India. which gives a overview into Indian Science - https://sciencereporter.niscpr.res.in/
2) The two-volume The Mind of an Engineer by Purnendu Ghosh et al. published by Springer contains essays from many of our recent Scientists/Researchers/Engineers etc. - https://link.springer.com/book/10.1007/978-981-10-0119-2
3) Books on Indian Science/Scientists by various authors are available on Amazon India and are worth getting.
Universities press in India had bought out a series of books by G Venkataraman called Vignettes in Physics it also had books on Saha, Bhabha, Bose, Chandra and Raman. https://universitiespress.com/books?id=0&sid=161
National Book Trust also has several books on Indian scientists.
What about Madhava of Sangamagrama[1], who (among other things) described various Taylor expansions for trigonometric functions (including an estimate of the error term) in the 14th century? (ie 300 years before James Gregory discovered the method for the Taylor expansion in the West).
Part of the answer is that research funding in India is predominantly from the public sector, and investments in pure science research have been low for a long time (not that applied sciences are doing much better). Thus many researchers lack the resources for experimental science whereas theoretical study is more accessible.
Chandras(h?)ekhar is already there, at least if you're the kind of nerd who knows about physics at all. Probably even more so than Ramanujan, but that could just be my science bias as a kid.
Readit News
What really stuck out to me was how R failed in a bunch of other subjects except math because he wasn’t interested in them.
I know society and norms expect students to learn all these other subjects.
But what if those just aren’t interesting to someone?
I wonder how many geniuses we skip on because doing the chores of homework and getting through boring classes is busywork and memorization for the sake of getting an A.
Meanwhile, hardly anyone actually remembers anything about those topics and even the best students mostly go on to achieve only above average things.
My class valedictorian went on to become a doctor and while that is certainly impressive to me, there are many doctors and he practices (like almost every other doctor) and isn’t pushing the boundaries of medical science. I feel terrible writing that because I’m certainly not as smart as him, but R is just so impressive and I’m glad he got his lucky break.
People like R would be lost in the sea of averages because their genius would be kept shut by norms.
Almost every extraordinary person I read about seems like they were 1 step away from being forgotten, and got some huge universal break that boosted them.
It's good that public school exposes children to many subjects - hopefully most of them. So that they can discover if they click with one of them. The real danger is that someone never gets exposed to a subject at all. College is the place to specialize in a subject.
Personally I think that we could do better by tailoring every student’s education to their abilities. Put in simplest possible terms, we could arrange classes by complexity rather than by year. Have one class for addition and subtraction, another for multiplication and division, then geometry, algebra, etc, etc. Then let students graduate from one to the next based on proven ability rather than by age. Do the same for language, history, etc. Let every student proceed through the courses at their own speed.
As a child I used to get all As and even got into a Stanford pre-collegiate program as a kid where I learned C++ and geometry.
Unfortunately after a surgery in 9th grade that left me unable to attend school for 3-4 months and just terrible QOL for about a year my grades slipped (went from A+ studen to C grade student) and I basically became average. I lost all interest in most subjects at school due to depression and other things.
My goal as a child was to get a Stanford JD/MD MBA (lol I know..), and today I have only a bachelors from a low ranked state college in business.
I enjoyed programming so much as a kid that one summer, so later in life I ended up going back to it. Taught myself enough in a month to get on some projects as a swe. Later I got lucky working at a unicorn company that IPO'd.
Now I am trying to build my own company and see how far I can get as a solo founder. Sometimes I wonder how my life would have turned out if it wasn't for that injury, but oh well. Shit happens, right?
Jeez sorry for the sob story but it feels good to get it off.
Being forced to do subjects that you hate is not exposure, it is being forced to do things which you are completely unsuited for.
I would go so far as saying that being forced to take music until 7th grade put me off any musical pursuits for the next 20 years. The less said about the torture disguised as education that is PE the better.
I found that I have a certain knack for it and really enjoyed performing.
While some exposure is probably better on average than none, in some instances bad experiences can trip the fuse on developing an interest.
The rote nature of canned education, bad teachers, bad parents, or bullies can turn kids off of subjects they might otherwise come to love.
It's not good at all that school reward being OK in every subject more than being really good in a handful of them. The school system is a mediocrity factory.
Unless they are born wealthy or find a benefactor so they can go to private school or a gifted person school, that caters to their specific needs, it's the best we can do to serve everyone.
In Europe maybe, but in America a lot of students receive their general purpose liberal arts education in College, and will then specialize later with a post graduate degree.
then why does this 'discovery process' have to continue into college? That was the OP's point. When money and time is on the line, let adults decide what they want to study. An 18-year-old is no longer a child.
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We probably aren’t going to get it quite right for that 1, either. Extreme outlier geniuses are extreme outliers and there’s no easily generalizable pattern around them. The ideal education for a young Ramanujan is probably different from the ideal education for a young Von Neumann. Of course in an ideal world we would give an extremely individualized education to every child, but that’s much easier said than done. Failing that maybe we could identify and then invest in the extreme geniuses, but that’s what we already try to do.
Well norms were in place when R did his work. Even the most strict systems have made concessions for extraordinary people. It is just that mostly average people go around claiming they'd be genius, had system not smothered their creativity.
> I wonder how many geniuses we skip on because doing the chores of homework..
I think from not many to hardly any as I can't believe if kids who are really genius can just go on for more than a decade of primary schooling without ever finding outlet for their creativity.
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Instead of giving kids grades with a ceiling, each subject would have (unlimited number of) levels of proficiency, and to attain a level, kids would have to pass a test (demonstrate certain skill). The choice of subjects and levels to attain would be up to each kid, but they would have to choose to do something (working at getting next level of something would be mandatory). (Although perhaps they should be encouraged to explore different subjects and attain some minimum of levels.)
Also, I would group kids by subject, and not by age. So kids of slightly different levels would train together, and the higher level kids would be obligated to help kids on lower level to learn, while lower level kids were taught to be respectful of higher level kids.
I would be cautious with this, they may have the same academic ability but a large gap in social skills.
I think in this example you are vastly overestimating the “average genius”, in comparison to those rare few who truly push the boundaries. We tend to do this because of the way our brains estimate things of significant scale, like the fact that keeps floating around social media about how bad most of us are at having a concept of the difference between thousands, millions, and billions.
There are many valedictorians (of the order of some per thousand) but few Ramanujans (of the order of tens per billion?), and gearing an overall education system specifically for those few could do a disservice to a great many others at every level below. Ramanujan was not the result of an effective education system anyway: like a lot of other world changing minds he was largely self-taught. Perhaps there is room to encourage more investigation to the side of the curriculum a lot more than we currently do, but the problems that stop other "Ramanujan"s, that could so easily have scuppered Ramanujan himself, are usually not caused by the education system but by other societal problems (death & disease, racism, sexism, caste or class biases, etc.) not giving them a chance to explore & self-learn or have useful access to education & other resources at all. Addressing those problems will help a much wider chunk of the population, as well as reducing constraints for the truly exceptional geniuses¹ amongst them.
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[1] And of course those non-geniuses who luck out and have that one brilliant idea. Their contribution is often vital for progress too, and I'd wager that there are orders of magnitude more of them than there are true geniuses!
Generally I think: "Unless you're Ramanujan, then you should probably have some breadth to your knowledge rather than pure depth" is not a terrible policy.
It may now be possible with Videos, Games and VR.
Zero? If you qualify as a prodigy, it is apparent from a young age. Maths prodigies are especially easy to distinguish. Given a little time, they will self-learn, grok and innovate on anything you throw at them and will likely attend higher education early unlike "the brilliant kid"s who will struggle with advanced concepts all their lives.
There are children with exceptional mental abilities in a limited range of topics or even a single topic, such as math, music or drawing, They may struggle to various degrees with subjects or life skills that don't correspond closely with their specific topics of interest and ability.
Standardized tests are screwing everything up. People who learn on their own might stumble upon the entire alphabet except for the letter "B," but standardized tests want only the first 5 letters. Hence the incredible efficiency of knowing the entire alphabet is thrown under the bus in favor of making sure none of the 5 are missing.
You can't teach someone to play, and there is no way to play systematically, at scale, and with guaranteed results. All the incredible people I know have some hole in "basic" knowledge, and if it is revealed nobody cares about them being miles ahead elsewhere. "Their basics seem lacking, in the name of stability and norm, throw them back to square one."
Following standards never produces something new, but the world is so afraid of failure and lack of definitions in "messing around" that they are willing to trade their souls for it.
Take any hacker here on HN, and ask how much they learned in CS class vs. how much they learned messing around with Perl on a weekend.
Excellence requires individual attention and cannot be so readily mass produced.
I got mediocre marks in high school but thankfully did well in JEE. Now I have a decent PhD in math with an extremely mediocre school record.
I don’t think you can say, even all these years later, that Ramanujan, that mathematician, made the economy more productive, but he certainly increased the high watermark of human civilization and created an inspiring story for individual achievement, creative realization, and artistic and mathematical expression. There’s something sublime and transcendental—no pun intended—in the kind of truths that he was able to tease out and the unique, idiosyncratic way that he expressed them. Sort of like a Basquiat of mathematics, I suppose. Or probably better than that.
That aside, I think it’s unfortunate that he died of cholera or something, isn’t it? I mean, he apparently didn’t think it was unfortunate that he was going to die. And certainly, the formal education system didn’t necessarily fail him, in that a professor at a university recognized his genius and sponsored him to the UK.
But I think, in a sense that you identify, there is this general failure of the education systems in the human civilizations on this planet to foster perhaps the best thing that they could be fostering. They’re more like a manufacturing assembly line to produce cogs as part of the economic machine.
Not that there’s anything necessarily wrong with that. I think it’s good that people can have a role to play in the larger economy and that there are pathways to bring people to the level of capability where they can contribute like that. But the lack of pathways that these systems provide—those that could contribute to the creation of the full realization and expression of individual potential—I think is sad. And I think that’s what you’re kind of identifying.
Even within academic stem fields you have people who know how to promote and speak and they have the most influence.
I guess what I’m trying to say is the system is mostly selecting for what it wants.
I have a family member who scored perfectly in math on the final exam of middle school but failed other subjects so badly that he couldn't advance to high school. What frustrated us even more was that the next year, he achieved a perfect score in math again but still failed overall, leaving him no choice but to drop out of school.
This story has stuck with me for years, highlighting how our education system focuses solely on maintaining an overall average score above a certain threshold without considering how individual subjects are mastered.
I can’t help but think that if he had been allowed to pursue mathematics, he might have accomplished something great. It’s really disheartening.
That doesn't matter. A student is unable to evaluate the value of unknown subjects and the synergy between subjects. People are often unmotivated to do important things, because we are just lazy s**.
> I wonder how many geniuses we skip on because doing the chores of homework and getting through boring classes is busywork and memorization for the sake of getting an A.
And how many lunatics making wrong decisions did we prevented by forcing them to learn necessary things?
> Meanwhile, hardly anyone actually remembers anything about those topics
They are engraved in your mind. Even when you don't remember every little detail, they still helped you in forming your understanding of the world.
You seem like someone who thinks things through, so I suspect you’ll know what I’m about to say, but given the sentiment of your comment, I think it’s worth explicitly sharing this:
The fact that your class valedictorian went on to be a doctor is great. Not everyone needs to push the boundaries. Your classmate may end up saving/helping countless lives.
He is saying that the people who add most value to science isn't always the ones who are at the top of the hierarchy in the school system.
Performing well in school is like a F1 racing car: very fast, but can only go on paths very well trodden already, i.e. paved road.
Two major caveats. The first is that Berlin never wanted the essay to be taken all that seriously. To him, it was an intellectually parlor game. But if TvTropes is any indication, people are addicted to classifying things into simplistic categories, so the fox and hedgehog game took off.
Second, the rest of the essay actually challenges the binary by focusing on Leo Tolstoy, an unimaginably creative thinker and writer who, according to Berlin, was a fox by nature but wanted to be a hedgehog.
Why did I share all of that? Well, without taking the categories too seriously I do think some people are more foxy by nature and others are more hedgehog-esque. I've always thought of myself as a fox (which to be honest, might limit me), and I had no interest in focusing on a single subject in school. At worst, I'm a dilettante who can have a surface level conversation about any intellectual topic at a party. At best, I can muster up a real enthusiasm about trying something entirely new.
I say all this not just because I'm running on 4 hours of sleep, but also because I think foxes get a bad rap. Not everybody should strive to be Ramanujan, because it isn't in everybody's temperament.
This system is meant for a worker that no longer exists. If someone wants to specialize at the age of 13 and has shown reasonable aptitude in that subject, then let them do it. Sure, give them a well rounded education, but don’t weight those grades anywhere close to equally.
I don't think we can or should design a system optimized for that one-in-a-million genius. It would be worse for everyone else, and ultimately for society.
Sadly the later part of your comment may hold - I don't remember what ended up happening with him, whether he graduated high school or what. Hopefully at that level you just disappear into academia and not off the face of the earth in general.
Being a genius is a phenomenon of the individual, but being able to pursue one's talent is a social aspect of the phenomenon of civilisation.
The US political system is currently in a state of utter dysfunction because people have decided that facts "just aren't interesting."
The school I went to grades 1--12 tried to be especially good so taught Latin, French. Some of the girls were in ballet. MIT came recruiting. The year before me two guys went to Princeton and ran against each other for President of the Freshman Class (whatever that meant!). In my class, one guy (did nearly as well on the SAT Math as I did!!!) went to MIT.
In one of the early grades, I got dumped on (adenoids, couldn't hear well until that got fixed). Apparently the teachers talked to each other and had me with a dunce cap until I proved otherwise. In 1st algebra, discovered math: I liked it, was good at it, was the best in the class, proved myself, got sent to a math tournament, couldn't get dumped on, etc. Continued that way: Was so good at math that I got an unspoken but powerful by in any subject, e.g., English literature, I didn't like.
Got sent to summer math/physics enrichment programs.
So, for that example, for
> But what if those just aren’t interesting to someone?
some schools will let a student who is good at some one subject get a by in other subjects.
Really, schools, K-Ph.D., have a tough time finding any students really good in even just one subject, are thrilled when they find one that is, and don't want to block him/her because he viewed fictional literature as a not very credible presentation of common reality?
That by pattern continued: In grad school, they insisted that I take their computer science course. My background in computing was already nicely above that course, and I'd already taught a similar course at Georgetown. Soooo, mostly laughed at the course: E.g., they had a test question about Quicksort (very common topic then), and I answered with material they didn't know.
The best case of by: Took a reading course; decided to address a question in the pure math of optimization; two weeks later had a surprising theorem and from that an answer to the question. The work, clearly publishable, was instant news all over the department, some profs angry that I had done well, others pleased. Angry/pleased, the work got me a general purpose by, a gold crown, immunity from any criticism, and an unspoken, implicit, easy path to the rest of the Ph.D.
https://www.rfcafe.com/miscellany/factoids/Jean-Shepherd-Cla...
on youtube as well
But I always wondered how polymath like Leonard davinci and Isaac newton that are excellent in many area are possible.
It's the crazy ones that push humanity forward. We lose far more than we can imagine by not enabling even just one of them. This is one of the most important problems for us to fix.
People who are "weird" and yet are entirely functional are the best of both world and a much rarer combination.
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How would you characterize R's master and the "normie" master?
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Maybe there are other topics where the same phemonen exists. I'm not sure if off the top of my head I can think of one where it gets so intense so quickly - long, detailed, emotionally charged anecdotes immediately when school comes up! [I love a good yarn, and am not judging or belittling people for expressing themselves here, I must stress].
I have speculated in my comfy armchair as to why our education structures seem to make venting such a need for people. Anecdotally, there does seem to be a suggestion of some sort of lingering and powerful malaise around the whole thing, in any case.
I wonder if it's like an abusive relationship - the emotional work required to move on and have better relationships (i.e., different educational structures) gets so large at some stage that all you can focus on is "coping". And you're sort of stuck, and the cycle more or less continues.
EDIT: may I just add - I read that whole article, and am very fond of Ramanujan. He made university much harder for me when I learned about his existence, as my maths classes seemed so far removed from whatever it was he was doing.
Attempting to scale something that large forces people into boxes that fit inside the framework - one must necessarily ignore little detailed differences between humans.
But from the perspective of those individual humans, ignoring those little details doesn't really work. Those things are important, and tend to ping our egos, so we react accordingly, and are eager for opportunities to give voice to this frustration.
I found it quite ironical how the whole topic has been hijacked into this side discussion. For the record, Ramanujan is NOT a product of the Indian education system. In fact, that system was quite cruel & off-putting to him. He was a self-taught math prodigy. Apart from the well known 2 big biopics, there are numerous TV soaps telecasted in India in various languages that regularly emphasize this point. He taught himself mostly using Inequalities by GH Hardy, & several other books, all of which are freely available right now at the click of a button. Nobody is stopping anybody from studying math, and education or the lack of it has nothing to do with any of this.
Indeed. I submitted this post, because I found the article's discussion of the math involved fascinating, along with biographical details of Ramanujan I hadn't heard about before. I'm surprised to see how popular the post is, but also that there's maybe 5 or so posts that engage with the actual mathematics.
I probably shouldn't be surprised - generating functions pointing out unexpected connections between different restricted integer partitions is pretty niche.
Teaching has not incorporated morality or ethics as strongly as some other professions where the need is more obvious. This is combined with the way teaching "quality" is measured. The result is that your average school teacher often uses tactics and methodologies that border on straight up public bullying of their students in order to "achieve results."
> I wonder if it's like an abusive relationship
Neither the teacher or the student engage in a process which selects each other nor is there any process to manage particularly bad combinations of such. They are assigned to and stuck with each other. The ethical failures in teaching abound.
[...]the day I started reading educational research was the day my life changed.
It all began with Daniel Willingham’s Why Don’t Students Like School, which led me onto his ridiculously good series of Ask the Cognitive Scientist articles for the American Federation of Teachers and the associated research papers he quoted. I devoured them all with eyes wide open and jaw firmly on the floor. Next came Cognitive Load Theory, which I was introduced to via the incredible blog of Greg Ashman and our subsequent conversation on my podcast. I was familiar with the work of Dylan Wiliam through the development of my Diagnostic Questions website, but I read and re-read everything I could get my hands on from the great man. I didn’t think life could get any better — and then I came across the work of Robert and Elizabeth Bjork. Good God.
One thing led to another, and before I knew it I had read well over 200 hundreds books and research papers. I would wake up in the middle of the night, my head buzzing with ideas[...]
Not sure if you're in the US.
Here, ALL non-technical topics, e.g. nutrition, healthcare, immigration are just a few of the big ones, devolve into a discussion of anecdotes and "feelings."
(Venting is a need that everyone has when under pressure not sure it has anything to do with our education structures)
a.k.a. sampling from the distribution of interest. Once you have enough anecdotes, you actually have a representative distribution of the real thing. Selection may be biased but you can weight those data points as you like, it's still good data.
Maybe it's worth a bit of perspective.
Edit: or happy, one-parent homes. I just have an inkling it is good for children to see a contrast of parents, not just one parent.
Ramanujan short life itself is a loss to the world, Imagine how many Ramanujan's were ignored where there is no G.H. Hardy and what about Ramanujans in the other 95%?
“I am, somehow, less interested in the weight and convolutions of Einstein’s brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops.” ~ Stephen Jay Gould
But people in the upper classes don't like that, so they're telling working class people that people even worse off are after their jobs.
[1] https://www.hup.harvard.edu/books/9780674987883 [2] https://en.wikipedia.org/wiki/Caste_system_in_India
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This is similar to another popular narrative that no girl received an education until some sympathetic Englishman in partnership with a local woman allowed girls to go to school. What a load of BS.
Tall claim.
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divinely inspiring
I'm curious to hear how others think about this phenomenon. Do recent advances in neuroscience, AI, or cognitive psychology offer any clues about how innovators like Ramanujan access these hidden sources of insight? Or are we still stuck in the realm of "genius is mysterious"?
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From Karma Yoga, Chapter: I, Karma in its effect on character—
What we say a man “knows”, should, in strict psychological language, be what he “discovers” or “unveils”; what a man “learns” is really what he “discovers”, by taking the cover off his own soul, which is a mine of infinite knowledge.
We say Newton discovered gravitation. Was it sitting anywhere in a corner waiting for him? It was in his own mind; the time came and he found it out. All knowledge that the world has ever received comes from the mind; the infinite library of the universe is in your own mind. The external world is simply the suggestion, the occasion, which sets you to study your own mind, but the object of your study is always your own mind. The falling of an apple gave the suggestion to Newton, and he studied his own mind. He rearranged all the previous links of thought in his mind and discovered a new link among them, which we call the law of gravitation. It was not in the apple nor in anything in the centre of the earth.
Whenever I read about Ramanujan having divine revealing formulas in his dreams, I remember Swami Vivekananda’s quoute on consciousness and mind.
edit: found another relevant quote from Upanishads on tapping the infinite knowledge:
Mundaka Upanishad 2.2.9:
“Eṣa sarveṣu bhūteṣu gūḍhātmanā prakāśate, dṛśyate tvagryayā buddhyā sūkṣmayā sūkṣmadarśibhiḥ”
Translation: “The Self hidden in all beings does not shine forth, but it is seen by subtle seers through their one-pointed and subtle intellect.”
Explanation: The ultimate knowledge or truth is hidden within all beings and is revealed through subtle inner perception. The idea is knowledge is latent within the mind, and it is discovered, not externally found.
In case I mistook it, ignore the above.
All knowledge that the world has ever received comes from the mind
Doesn’t it come from the mind’s interaction with the world? I don’t think that any important knowledge comes from doing nothing (apart from psychology due to its nature). It all sounds like a typical religious salad, tbh. “You” discover from your “mind”, but then some seers understand that “you” and “all” and “mind” are the same, etc. Well, you don’t have to be a seer to see it. “You discover from your mind” is a useless semantic loop that creates all this unnecessary complexity, imo.
That doesn't look sound to me. If the "seers" are seeing "The Self", are they beyond and separate from "The Self"? If they do so with their "subtle intellect", is that intellect outside of "The Self"?
If affirmative, then "The Self" is something external to the seer, making the term a misnomer. And furthermore, there is something outside of The Self (that which is seeing The Self), which remains to be explicated.
One guy able to tap into this knowledge in dreams is an indication that it is possible. Now, how do we make this the default for everyone is the question I wonder about
The way we found one variant of wheat in Mexico that was resistant to bacteria, and replicate that to the whole world -- can we do something like that for humans ( even I don't like the sound of it, but I hope you get the feeling )
Great analogy.
Borlaug's famous Mexican dwarves.
My guess would be by filling the reservoir first. After filling it, then you can stir it and extract new knowledge. In other words: you get better at math by studying, reading, and thinking a lot about math. I don't think there has been a brilliant mathematician who has not immersed himself in math at first.
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You delete stupid prejudices from society, then it allows itself to do that. One group of people forcibly practicing it for idiotic reasons now plagued the whole field with its label, amplified by religious ideas. The irony is, it’s that same limitation that disallows to address itself and continues to propagate.
That's not to say there isn't something spiritual and mystical in knowledge. I think the fact that he was from a different background gave him a different perspective. But yeah you cannot skip all the math studying.
The core premise of the book is to describe how mathematicians work and think and to show that this is a process that everyone can do (although some will be better at it than others). It includes interesting accounts of Grothendieck, Bill Thurston, and Descartes as well as from the author's own research career at Yale and École normale supérieure. The book is targeted at the general reader and at times reads a little like a self-help book, especially in the first third or so. However, I found it to be an enjoyable and fascinating read. It provoked a lot of interesting questions about the nature of learning and provided a framework to begin to answer them (e.g. "How can I have proved something and yet feel no understanding of it?", "How can some people solve problems orders of magnitude faster than other smart people, as if they don't even have to think about it?", "Why do I sometimes watch a presentation on a new topic, follow every step, and come away feeling like I've learned nothing?"(* see except below)). I don't think I'm doing it justice here, so I'll stop by saying I highly recommend it based on your comment.
_______
* I'll use this as an excuse to provide a related excerpt featuring Fields Medalist and Abel Prize winner Jean-Pierre Serre:
One day, I had to give a lecture at the Chevalley Seminar, a group theory seminar in Paris. I didn't have substantial new results to announce, but it was an opportunity to make a presentation even simpler than usual. [...] A couple of minutes before the talk was to start, Serre came in and sat in the second row. I was honored to have him in the audience, but I let him know right off that the presentation might not be very interesting to him. It was intended for a general audience and I was going to be explaining very basic things.
What I didn't tell him, of course, was that his presence was intimidating. Still, I didn't want to raise the level of my talk only to keep him interested. I just kept an eye out to see if he'd taken off his glasses, which would mean he was getting bored and had stopped listening. No worries there—he kept his glasses on till the end.
I gave my presentation as I would have without him there, speaking to the entire audience, especially the students seated in the back, whom I was pleased to see listening and looking like they understood. It was a normal presentation, fairly successful, not very deep but well prepared, clear, and intelligible. At the end of the seminar, Serre came up to me and said—and here I quote verbatim: "You'll have to explain that to me again, because I didn't understand anything."
That's a true story, and it plunged me into a state of profound perplexity.
Apparently, Serre wasn't using the verb to understand the way most people use it. The concepts and reasonings of my talk couldn't really have caused him any difficulty. I'm sure he wanted to say that he understood what I had explained, but he hadn't understood why what I had explained was true.
There are two levels of understanding. The first level consists of following the reasoning step by step and accepting that it's correct. Accepting is not the same as understanding. The second level is real understanding. It requires seeing where the reasoning comes from and why it's natural.
In thinking again about Serre's comment, I realized that my presentation had too many “miracles,” too many arbitrary choices, too many things that worked without my really knowing why. Serre was right; it was incomprehensible. His feedback helped me become aware of a number of very big holes in my understanding of the objects and situations I was working on at the time. In the years that followed, research into explanations for these various miracles allowed me to fill in some of the holes and achieve some of the most important results of my career. (However, some of the miracles remain unexplained to this day.)
But the most troubling aspect was the abruptness, the frankness with which Serre had overplayed his own incomprehension.
My intuition is that some people are able to develop abstractions on top of abstractions (compression is a key part of intelligence) that allow them to traverse the search space much, much faster than without these, and with enough repetition this becomes routine, even faster in the brain.
I don't think we have a good theory of how this works, or at least I haven't come across it.
Neural network guided search is somewhat similar but I think we are missing several key pieces
1) Mathematics Wizard Srinivasa Ramanujan : Some glimpses into his Life and Work by two Indian Mathematicians Narendra Kumar Govil and Bhu Dev Sharma is a good biography with an introduction to his Mathematics and links to further resources. Good complement to Robert Kanigel's book The Man Who Knew Infinity.
2) In order to understand the fascination that Mathematicians have for Ramanujan see this and other lectures by Prof. Ken Ono who credits Ramanujan as his inspiration in becoming a Mathematician; Why Does Ramanujan, "The Man Who Knew Infinity," Matter? - https://www.youtube.com/watch?v=7ynhiZJUMzA
3) Mathologer on Youtube has good walkthroughs of some of Ramanujan's most famous identities (eg. 1+2+3+... = -1/12) - https://www.youtube.com/results?search_query=mathologer+rama...
4) All of Ramanujan's published papers and unpublished notebooks can be found online at - http://ramanujan.sirinudi.org/
PS: In the submitted article, George Andrews is wearing a Ramanujan tie :-)
https://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinity
The Man Who Loved Numbers - Srinivasa Ramanujan documentary (1988) - https://www.youtube.com/watch?v=PqP2c5xNaTU
Srinivasa Ramanujan: The Mathematician and His Legacy (2016) - https://www.youtube.com/watch?v=K5jsgBvJMUc
1. https://arxiv.org/abs/2405.06451v2
2. https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_...
Well, at least to western people. Are Indians more familiar with Indian scientists?
In order to rectify the status quo;
1) Everybody should get the monthly magazine Science Reporter published by National Institute of Science Communication and Policy Research (NIScPR), Council of Scientific and Industrial Research (CSIR), New Delhi, India. which gives a overview into Indian Science - https://sciencereporter.niscpr.res.in/
2) The two-volume The Mind of an Engineer by Purnendu Ghosh et al. published by Springer contains essays from many of our recent Scientists/Researchers/Engineers etc. - https://link.springer.com/book/10.1007/978-981-10-0119-2
3) Books on Indian Science/Scientists by various authors are available on Amazon India and are worth getting.
4) Also see the books by the great astrophysicist/cosmologist Jayant Narlikar (https://en.wikipedia.org/wiki/Jayant_Narlikar), specifically; The Scientific Edge: The Indian Scientist From Vedic To Modern Times. - https://www.penguin.co.in/book/the-scientific-edge/ and Science and Mathematics: From Primitive to Modern Times - https://www.routledge.com/Science-and-Mathematics-From-Primi...
I imagine most people won't recognise the name. But everyone's heard of a boson. So he's somewhat immortalised — more than most.
National Book Trust also has several books on Indian scientists.
[1] https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama
And Mani Chandy for computer science.