That was a fun read.
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.
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.
On the other hand, it's likely that you have the potential to be really good at something that is totally outside your awareness right now, and having a broad education can be beneficial to expanding your exposure to such possibility.
Also, there is ADD and the likes which make mastering a subject very difficult or even impossible for many people, regardless of intellect.
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.
From: https://medium.com/eedi/how-i-wish-id-taught-maths-8ec9b0578...
[...]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)
> massive influx of personal anecdotes
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.
Except for the fact that anecdotes aren't a random sample, they're selected because they're interesting.
Trama mostly.
Trauma-drama? :)
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.
Maybe it's worth a bit of perspective.
Parenting.
This is the long and short of it. Education doesn’t replace happy, 2-parent homes.
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.
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.
People like Ramanujan are on a different planet compared to your everyday PhD working stiffs. There's smart (PhD), there's genius (Einstein, Feynman, von Neumann), and then there are freaks of nature like Ramanujan.
I couldn't comment on the impact of his work, which anyways is evidentially still unfolding, but the WAY he seems to have worked is just bizarre - coming up with these deep baffling series formulae offered without proof. His mind seems to have worked in an extraordinary way.
Do you know the saying "comparison is the thief of joy"?
It was intended to apply to oneself, but seems just as true when appreciating the genius of others. No need to diminish von Neumann, etc., AND doing so just diminishes your own credibility.
Well, he's dead, so there's that ...
I'm not sure there's much you can do to help someone at this level, other than perhaps take away outside distractions.
It's not remotely clear how Ramanujan's mind worked, or whether he really knew himself. He seems to have been a lot more highly functioning than an idiot savant, but still operating at a level that is hard to comprehend. Was this an expression of humility or devotion by Ramanujan, or was he just expressing his own lack of understanding of how his mind worked?
Arguably. Assuming that's actually where they end up, they earn lots of money and they get to choose what they want to do with it. We should force them to be mathematicians if they didn't want to be?
The other version in the West is he moves to rural Montana and sends bombs through the mail.
I think they're talking about specifically the Unabomber, who was known to be a math genius too.
You mean a world where someone like Terence Tao gets to be a professor of mathematics at UCLA, where he holds the James and Carol Collins Chair in the College of Letters and Sciences?
That brilliant geniuses have chosen corporate America and to not go into academia is yours and mine and the world's loss, but a defining feature of the Western world is that they get to make that choice for themselves.
Those bombs might have done more good than his potential math career. Considering we just had our first AI millionaire. Perhaps there is hope for humanity to wake up.
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
If we individual makeup was so irrelevant we wouldn't have just had me Ramanujan so far. Although I agree that Einstein is probably very unremarkable and was mostly the right (genius) person in the right place at the right time in history of science to marry together things other people figured out and sprinkle some unique framing onto that.
It's not that individual makeup is so irrelevant, it's that if we assume (potential) Einsteins and Ramanujans follow a normal distribution (or really any random distribution), then the fact that so many people in the world live in situations where any genius will simply never be noticed means we are necessarily missing out on the vast majority of them.
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]
[1] https://www.hup.harvard.edu/books/9780674987883 [2] https://en.wikipedia.org/wiki/Caste_system_in_India
Unfortunately, India's poorest still don't get govt benefits due for them: A few gobble up all the caste-based reservation from the govt. I have seen many people who have made a career out of talking about caste all the time.
I know more than 3 people in the US, who talk about caste all the time (as in that's their job now) but grew up in very privileged families.
Many Indians who talk about caste all the time, do it in ways that aren't immediately obvious. "What part of India are you from?" and "What's your surname?" are not just innocent questions.
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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.
> ...only caste that is supposed to be educated...
Tall claim.
People in the West love to attribute everything in India to caste.
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The cited book has no information on how the surveys themselves were conducted. The book is an overall romanticization of pre-colonial india, and frequently indulges in nostalgia.
The following books have a good overview of education&caste in india.
A. L. Basham, The Wonder That Was India
Gail Omvedt, Understanding Caste
> The cited book has no information on how the surveys themselves were conducted.
These are not surveys. These are actual full data from the erstwhile British Administration’s revenue records with the data collected by each Provincial head and the District administration revenue collectors and officials at that time. The modus operandi of the data collection is also detailed in the book.
Do these books fix the day issue? How do they do it?
Not to mention Ramanujan grew up in a colonial British era, and to attribute his genius and success (instead of the oppression and difficulties) him and his environment struggled with due to British colonialism is utterly rich!
Ramanujan succeeded in spite of it, not because of the favors of the British. Imagine how many Ramanujans were ignored due to 200 years of British colonial rule in India, wherein caste system developed and thrived*
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Can you share any links which can help me read up more on this?
Also, a cursory search on internet doesn't brings anything conclusive about Aryabhata being a Shudra.
Unsuprisingly, avowed Nazis like this ... like in Ukraine are great friends of the Western establishment.
Who are you referring to, specifically?
Except the groups the article refers to are quite marginal, have no influence in government, are a certainly not "great friends of the Western establishment".
Basically a propaganda piece, designed to push your buttons. And not surprisingly, published originally in RT.
What a piece of Russian-funded propaganda.
That is funny how russians claim everyone of their enemy is a Nazi.
While real white supremacists with swastika tattoos are fighting in the Russian side, just to have an opportunity to make money while doing atrocities.
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
If it had not been for Hardy, we would know a fraction of what R did.
What do you mean by savage "native" here ? He is coming from a culture of long and rich intellectual history.
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.
It's the norm now as well.
Only the target has changed. The land grabs and slurs and dehumanization are essentially the same, if sometimes with a modern twist.
exposure and opportunity means so much in professional life. There's a lot of value out there but someone has to find and promote it.
What a kind heart!
"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"
divinely inspiring
Indeed. Also is the retirement dream for many working software engineers.
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.
This is the thing. You think about something with compulsion for long enough, it shows up in your dreams.
People are impressed by the seemingly spontaneous origin of genius ideas, but forget all the analytical labor that went into preparing the mind for conceiving such thoughts. Genius doesn't come out of nowhere. The 'stroke of the genius' required a lot of hard work and experience.
And, it works the same with us 'normal' people, even though the results are less spectacular. Want to have a good idea? You need to feed your brain everything you can and obsess over it, then let it stew for a bit. Again, and again. Only then do you stand a chance for a really good idea to just pop out.
Please forgive, it's *poring
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:
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.
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.
>> All knowledge that the world has ever received comes from the mind
> Doesn’t it come from the mind’s interaction with the world?
There is no difference between these concepts, is there? In both cases knowledge comes from the mind.
Also, you can come up with ideas without interacting with the outside world, can you not? This feels like nitpicking.
I think you're too hung up on the mind part. I don't understand this as saying the mind synthesizes knowledge from nothing. That's why I said, there is no fundamental difference for me - whether the mind conceives of something new, or synthesizes knowledge from outside information is not very interesting a difference, since a mind is required to grasp this knowledge.
Maybe a better term would be information vs. knowledge.
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.
The master appears when the student is ready.
> 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.
I'll try to slightly dive into Advaita philosophy here. Both the Upanishads and the Yoga Vasistha affirm that the distinction between the seer and the Self is an illusion created by the mind. When this illusion is dispelled, the oneness of the Self is realized. The intellect, which seems to function as a separate tool, is ultimately part of the same illusion. True knowledge is realizing that there is no separation—everything is the Self.
Quoting one from Katha Upanishad 1.3.10:
"Indriyebhyaḥ parā hy artha, arthebhyas ca param manah, manasas tu parā buddhir buddher ātmā mahān parah"
Translation: "Beyond the senses are the objects, beyond the objects is the mind, beyond the mind is the intellect, beyond the intellect is the Self."
This quote emphasizes that the intellect is still a part of the illusion. Beyond even the intellect lies the Self, which is one and undivided.
Stupid question but why didn't Newton come up with Relativity instead?
Likely because it's far more complicated than Newtonian gravity and he wouldn't have been aware of the shortcomings of his theory e.g. Mercury's orbit. Also, the study of light was only just beginning in Newton's era - Ole Rømer's measurement of the speed of light was in 1676, but Maxwell's theories didn't come out until 1865.
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 )
> The way we found one variant of wheat in Mexico that was resistant to bacteria, and replicate that to the whole world
Great analogy.
Borlaug's famous Mexican dwarves.
> > ... found one variant of wheat in Mexico ...
According to the Wikipedia page of Norman Borlaug, he _developed_ that variety of wheat.
> 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.
That's a very interesting article. Thank you!
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.
thanks, will be checking the book!
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
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 :-)
I enjoyed the movie on The Man Who Knew Infinity.
Also see some of Ramanujan biographic documentaries on Youtube, specifically;
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
Thank you! Really enjoyed the 2nd one. Haven't had the time for the 1st one yet.
Thank you. I'll watch them. It's incredible that Hardy gave Ramanujan a perfect 100, no reserves at all. I guess that only comes due to the close interactions with him.
Another marvelous mathematician story is that of Yitang Zhang. The Newyorker had a good article about him if you haven't seen it yet.
https://web.archive.org/web/20220527042904/https://www.newyo...
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?
1. https://arxiv.org/abs/2405.06451v2
2. https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_...
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.
However, pretty much all of the ones who are household names, are western.
Well, at least to western people. Are Indians more familiar with Indian scientists?
Amoung the people who are interested in science? Yes. But to the general public? No.
I don't think a non STEM guy would know Ramanujan or C V Raman.
C V Raman's name is known via school text books; though granted that few would remember it unless they carry on with STEM.
Household names (in the west) are maybe Aristotle, Galileo, Newton and Einstein. If you are lucky. For a lot of people it is just Einstein, that's it.
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.
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...
To this excellent list, I would humbly add Bhavana: https://bhavana.org.in/
Two More;
a) Resonance : Journal of Science Education published by Indian Academy of Sciences - https://www.ias.ac.in/Journals/Resonance_Journal_of_Science_...
b) Science India magazine published by Vijnana Bharati- https://scienceindiamag.in/
Great; I didn't know of this Mathematics magazine.
> Satyendra Nath Bose
I imagine most people won't recognise the name. But everyone's heard of a boson. So he's somewhat immortalised — more than most.
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.
When I first encountered the Mahalanobis distance, I thought it sounded strangely Indian. Turned out it was!
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).
I've noticed India seems to be full of ring theorists/algebraic geometers. I wonder if that's actually true and, if so, why.
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.
Ah. I was only comparing with other 'pure' disciplines like analysis and topology/geometry, but I guess these are still more strongly connected to applied things.
Physicists like Subrahmanyam Chandrashekhar and George Sudarshan. Also Mahalanobis for statistics.
And Mani Chandy for computer science.
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.
Manjul Bhargava is not Indian.
You’re right, I can’t edit my comment anymore though.
Ramanujan responsible for inspiring generations of mathematicians throughout the world. His life was a beautiful tragedy. One that leaves me in awe and also great inner sadness. If you come from a hardcore traditional br*hmin family, just to cross over the ocean by boat would risk you getting excommunicated. The culture which he came from makes the entire story all the more legendary. Just cutting off your topknot and forgoing the dhoti to wear a western suit. We don’t understand what he went through and what he gave up to give us his mathematics. What he had to sacrifice to practice his art.. to be.
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Let's not be so aggressive.
Comments on HN are meant to be for civil discussion, even on contentious topics.
No hate here. And you’re very much not quoting anything I said so who knows where you got that from. Don’t be so insecure.
I cannot recommend reading A Mathematician's Apology enough. It was written by GH Hardy and I think it's one of the best non-math texts out there to understand how a mathematician's brain works.
https://en.wikipedia.org/wiki/A_Mathematician%27s_Apology
Fairly short and beautifully written.
It boggles my mind what could have been if he had lived to a ripe old age.
He attributed his success to his family Goddess, claiming he had dreams of scrolls unfurling theorems against a bloodied wall.
"An equation for me has no meaning, unless it represents a thought of God" - Ramanujan
> Hardy, a mathematical giant in his own right, is said to have quipped that his greatest contribution to mathematics was the discovery of Ramanujan.
I love the Humility and Kindness in Hardy's statement.
There is a movie covering his story, it's incredibly good!
The Man Who Knew Infinity [1]
I often think about Ramanujan in the context of AI, if reasoning is type 2 thinking how did he (apparently) come across a lot of his ideas as intuitions
I wonder if you can train a neuronetwork to have the kind of intuition Ramanujan had. How incredible would it be for math discoveries. Then separate AIs to try and prove or disprove the insights.
probably but we don't know how to build that type of intuition. in humans or in machines.
AlphaProof does do some kind of neural network guided search and automated theorem proving to validate it https://deepmind.google/discover/blog/ai-solves-imo-problems...
But it's still fairly brute force and inefficient
When we get to cloning people he's probably the most worthy person to get cloned first. Best chance of getting something usefult from a specific genetic makeup. You'd still probably need to make hundreds of clones to get one anywhere close to his level due to random epigenetics, even if raised in optimal environment but it's the best shot.
My education prepared me to make millions of dollars by the time I was in my mid-forties. I learned that if I take every task seriously and strive to do the absolute best I can on that task, no exceptions, regardless of how I personally felt about a subject or an assignment, I would end up being an employee that saw to it that anything I did, I did to the absolute best of my ability.
Not every task ended up being worth doing, but some sure did. And regardless, more important than any one individual tasks, I earned a reputation with management. This in turn resulted in a steady stream of bonuses, stock grants, and promotions. I suppose I was doubly-fortunate to have focused my efforts in the tech industry from the early 90's onward. You have to be really unlucky for nose-to-the-grindstone efforts in FAANG companies to not work out pretty well for you from 2003-ish until today.
With continued good health I can look forward to another 20 or 30 years of life, where I get to do whatever I want with my time because I earned so much money being content to conform to whatever my employers required of me. Nobody will remember my name, but that's okay with me, because I won't either.
This is like looking at the kids who collect Pokemon cards out of the sheer love of the Pokemon and playing the game compared to the scalpers and “investors” / “speculators”. Yes they may end up with a lot of money but they will always lack that sweetness, purity, and wholesome joy that you see when a kid finds and enjoys something.
That's cool!
Mourtada was a professor of mine during undergrad. Really happy to see him in this article ^_^
> “This is a very weird fact. It’s mysterious,” said Shashank Kanade of the University of Denver. “I mean, where did the 5 come from?”
This is how I felt in 90% of my theoretical math classes in university.
I wish we lived in the multiverse where he didn’t die. Imagine what he could have done. Cool article though.
At least we don't live in the multiverse where he was never discovered and all his work was thrown away.
True very true.
I don’t understand why people praise Ramanuja so much. Why not this praise for Euler or Gauss?
I have no idea how you could possibly have picked up the notion that Euler and Gauss are not praised enough. They are literally the top two names when you google "greatest mathematicians all time"
There is such praise, just not on this post, but what's powerful about the story of Ramanuja is where he came from. The almost-miss that the world would not have his genius, but for a couple of lucky turns, is a powerful driving story about a diamond in the rough. I'm no such a genius, but the idea of being discovered is very alluring; to idly fantasize that I have some gift that could lead me to fame and riches. It's that nice feeling to hold while reading and thinking about his story.
Interesting. My experience with maths education is pretty much limited to watching Numberphile and other YT videos these days, but Euler and Gauss get plenty of praise.
There is something cult like, mystic, almost magical about Ramanujan that almost transcends the raw intellectual horsepower (which he certainly had).
Is there a movie about it ?
The Man Who Knew Infinity
Why aren't we working on drugs to make people smart?
I used to think like you do. But the real place where we could make tons of progress is in relationships. Many stories of great thinkers involve one or a few crucial mentoring/pedagogical relationships. Without those, a person could forever find themselves trying to fit their square peg into the round hole of what "normal" society around them seems to expect. I can easily see how my life could have ended up like that.
As someone who benefited greatly from a few mentors in childhood and adolescence, my goal is to be able to give the same to at least a few other people in my lifetime.
Was just discussing various parenting stories with my colleagues over lunch and, like relationships, childhood environment and parental boundaries (or otherwise) would likely be the greatest influence to any individual.
But then, if 'mentors' applies to parents, then I guess I'm saying the same thing.
Having said that, my teenage son wastes a lot of his life playing online games with his friends, but having said that, I heard him say mid-game to a friend of his "Brazilian isn't a language you idiot!". So, I mean it's trivia, but he knows there's no language called Brazilian. He's a smart kid (that's not the only data point).
(I had to look it up: Portugese is the official language of Brazil).
>> As someone who benefited greatly from a few mentors in childhood and adolescence, my goal is to be able to give the same to at least a few other people in my lifetime.
+1. Same for me.
Relationships has been more important than what I could consider ideal or optimal (in terms of how many missed opportunities are there).
Once I was reading a technical paper from a colleague and could not follow much. Spoke to him and he could explain the crux of it in two-three sentences. I was left wondering why that was not there in the paper itself. But that happens repeatedly. People are able to explain much better than the literature they produce*.
This perhaps has a contribution as to why good relationships and mentors end up being so important. But then, this ultimately comes in the way of scalability.
* Of course, it may possibly also be that in a verbal conversation when the author personally knows the reader, they can explain to that reader better. However, I think this effect is less important than people being able to write better.
Lots of people are, they're called nootropics.
Whether they are successful and whether they are mostly a bunch of snake oil is another question...
I feel way more productive since going sxe.
I am naturally so tired around 9pm when I shut the lid of my laptop that I fall asleep within minutes of getting in bed.
On a side note.. Somehow my dreams have been insane and I’ve low key enjoyed the vivid worlds I find myself in over the past few months.
Wake up around 5 or 6, go for a stroll and then eat some breakfast.
Then I can work taking only breaks for lunch and dinner. Sometimes a 30 min nap in the afternoon in the park.
For those unfamiliar with the abbreviation sxe:
https://en.wikipedia.org/wiki/Straight_edge
> Straight edge (sometimes abbreviated as sXe or signified by XXX or simply X) is a subculture of hardcore punk whose adherents refrain from using alcohol, tobacco, and recreational drugs in reaction to the punk subculture's excesses. Some adherents refrain from engaging in promiscuous or casual sex, follow a vegetarian or vegan diet and do not consume caffeine or prescription drugs. The term "straight edge" was adopted from the 1981 song "Straight Edge" by the hardcore punk band Minor Threat.
> sxe
Weird. First time I've ever seen that (abbreviation?) For straight-egde. Thought you were talking about some supplement at first.
I thought the same and instantly googled, and immediately remembered I had seen it before. It's like a play off HxC
I've always been pretty much straight-edge. Don't drink much alcohol, and recently went off caffeine.
I'm naturally tired at around 2am. I barely dream, or can't remember my dreams. I struggle to wake up at 6:30 even after 7+ hours of sleep.
I do like an afternoon nap, but Sunday's are almost the only opportunity. As a bad consolation prize, I involuntarily micro-sleep at my desk, working from home or in the office, a handful of times most days, generally at peak afternoon nap times (1:30pm - 3:30pm).
I wouldn't say I'm productive, but I would say that the work I produce is generally of a high quality.
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Next he'll grow fangs and start drinking blood.
How so?
'Source'.
Interesting, I have seen some pretty scary interviews with Elon. If I had noticed a family member acting the same way I would have taken them to the ER.
I wonder what effects do you expect from that on a societal scale in the long term (at least 3,4 decades) ?
For instance we banned meth and other drugs that have tremendous productivity effects at the expense of the individual and how we had to deal with them, so it's not a rhetoric question.
We didn’t fully ban them. We just prescribe them to anyone a doctor decides has ADHD.
Not quite, but if one does not have ADHD or something similar, things like adderall have a very different effect than they do to someone who has ADHD.
Your apparent disbelief in ADHD doesn’t make it imaginary, by the way. Consider yourself lucky that you do not have it; I am unemployable without medication.
People without ADHD take adderall &etc for focus/performance enhancing reasons. Some get it from a friend, some are incorrectly diagnosed. I don't know if you disagree that this is the case, but I don't think it implies anything about ADHD.
I have ADHD. It doesn’t change the fact that people feign ADHD in order to get access to amphetamines.
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Don't forget the military. Go pills.
You will be surprised to learn that methamphetamine is not banned and that it is currently prescribed for refractary ADHD under the name desoxyn!
I'm aware we never "ban" any specific substance, as we say the dose makes the poison. And any substance that has any effect is also a potential cure for the disease that has the opposite effect.
I should have been clear I saw it in the "make people smart" light, as doping an already acceptable situation, instead of correcting something perceived as a pathology.
Meth was widely available over the counter at some point, and we made it legally disappear outside of strict medical settings.
Anything in Schedule I is almost fully banned, though people will try hard enough to get around it that it doesn't matter.
Caffeine is the most used drug in academia.
Surely the most used because of its affordablity and easy access. Coffee, energy drinks, tea, caffiene pills, etc.
I wonder what academia would look like if adderall, vyvanse, modafinil were just as accessible, or even less controlled substances that are considered to enhance mental performance like L-tyrosine, alpha-GPC, Lion's Mane mushroom, Bacopa, or Ginko.
Alpha-GPC is just choline, so you can get it by eating eggs. Amino acids and mushrooms are also quite accessible.
Modafinil is straight up better than caffeine though, which is a crappy and addictive stimulant.
"just" choline is a little reductionist, it's much more bioavailable.
I'm not sure where the distinction actually lies, but it is also considered a (generally recognized as safe) drug.
Caffeine also has a track record of several centuries.
We really, really know the long term effects.
What are the long term effects? How harmful is caffeine addiction?
no, you mean Ritalin. caffeine is a joke compared to actual stimulants.
IIRC nicotine is the most effective nootropic by far, the problem being that it's super addictive.
But none of them work as well as sleep and exercise.
And eating right! Gotta complete that trifecta, each one compliments the others.
IIRC nicotine itself isn’t super addictive when used solely in a less addictive form, as in gum or patch.
Have you ever tried taking a vape away from a teenager?
...or me?
In the past, when I've heard the 'less additive form', it was in contrast to nicotine from tobacco, where there are other chemicals and compounds at play. While it's true ROA can be a big part of habit formation, I'd argue it's a stretch to call it different form when it's the same chemical in either case.
tbf tho, I am somewhat contradicting another comment I made about bioavailability on this very gp thread so suffice to say, I do understand your point.
Careful what you wish for: https://web.archive.org/web/20140527121332/http://www.infini...
I think gene therapy type stuff seems more interesting at this point:
https://www.psypost.org/new-intranasal-rna-therapy-shows-pro...
I bet if such an effective drug were created, that just like Ozempic, we'd end up in a situation where the rich buy it up and use it to increase their advantage thus only widening the wealth gap. Like some dystopian Monkey's paw effect.
I think modafinil is a wonderful drug, far more potent than caffeine or nicotine. It's also very easy to get with no long term ill effects. There is a whole subreddit dedicated to this.
Wonder why more people aren't using it.
> no long term ill effects
That is not totally true for at least three reasons:
1. Modafinil can interact badly with some other stimulants. One must remain careful.
2. Modafinil by itself can cause SJS and similar serious problems in rare cases.
3. Chronic use of modafinil can easily produce anxiety, so much anxiety that it makes using modafinil impossible. This is even in a very low dose.
> Wonder why more people aren't using it.
Because it's classified as an amphetamine in large parts of the western world and would be illegal obtain without a prescription.
We did and still are. You can only safely push hardware so much.
@HN: Downvoting questions is stupid.
You're perpetuating "be ashamed of not knowing" instead of encouraging "learn by asking questions".
If you agree, consider upvoting downvoted questions.
Positive eugenics would result in more smart people with zero drugs or genetic engineering involved. And it requires zero oppression or violence. Just pay couples with high IQs to have babies.
It's called nicotine.
for people who are interested in Ramanujan's biography https://en.wikipedia.org/wiki/The_Man_Who_Knew_Infinity
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> he came up with thousands of elegant and surprising results, often without proof. He was fond of saying that his equations had been bestowed on him by the gods
And somehow this guy is remembered as a 'genius'...
I wish Ramanujan's gods gave him a formula to calculate ignorance of people!
Obviously someone's burning
IMO, you're thinking about this backwards.
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.
The admissions process to universities in the province of Ontario in Canada has a direct solution for this, which applies to well-known universities in the global technology industry, such as the University of Toronto and the University of Waterloo.
Most of these universities look at an applicant's grades for just six courses. After looking at the courses required for certain programs (such as calculus and physics for certain programs), the remainder of the six courses comprise the student's top grades for any courses at the Grade 12 (final year) level.
So, a high school student aiming for a top engineering or mathematics program will not be hamstrung by a poor grade in Grade 12 English, nor will a student aiming for a top international relations program be hamstrung by a poor grade in Calculus. At the same time, the student going into a STEM program will have an exposure to Shakespeare, which can provide inspiration and a rich set of works to explore later in life. The student going into international relations may later be inspired some years later to study mathematics for its beauty as a hobby, some years later.
I remember the feeling that I was wasting time with many of my courses in those years, despite having good teachers for many of them—I thought my time spent on mandatory humanities courses like music took time away from more practical subjects, and I wish I took a programming course (though I did love my English classes). Perhaps this remains true for many students, but I personally took an interest in music performance as a hobby years later in life, and the years-old lessons in music theory came back to me. My English classes also introduced me to literature, which has remained a very important part of my life that has guided me through highly consequential life decisions for the better. It is unlikely that I would have taken an interest in literary works without my exposure to English in school.
This is the same reason why STEM admission became so competitive in UCs in California.
UCs historically admitted using a mix of class rank, GPA, and test scores, but the number of seats at UCs didn't really increase in the past decade+ despite a small baby boom in the 2000s, and the growing prominence of STEM in the 2010s, so the average GPAs and SAT scores for UC admissions skyrocketed.
Plenty of Californians have anecdotes of getting rejected from mid-tier UCs but getting into MIT or Stanford. It's had a downstream impact out-of-state as well, as plenty of Californians now attend out-of-state STEM programs for that reason (played a major role in upleveling UT Austin/UW/UIUC/GT/UW Madison's reputations among STEM-targeting HSers ime) and make STEM admissions harder in out-of-state colleges as well.
That said, education quality for STEM majors is consistent across all UCs so the UC you go to doesn't matter as much academic quality wise.
Imo school grades should be disregarded entirely. We should bring back standardized testing, but make the ceiling much higher. Most people should score fairly low on it. This way, you can really see how good people are. At this point, people just get weeded out in university.
This would also help the case of people like Ramanujan, he might score perfectly on the math portion of the standardized tests, and despite poor scores on everything else, he'd be distinguished.
Are there not standardized exams in Ontario high school? When I grew up in Alberta standardized exams took up half your grade and those could not be fudged by your teacher.
The exams were made by a small panel of teachers and nobody else knew the exam contents until the day of the exam. Everyone across the province takes the exam on the same day and time. Exams were proctored by people external to the school. Exams are again marked by a panel of teachers outside of your school.
Cheating is nearly impossible outside of someone leaking the exam early.
And it should always be included as one of the important courses in Canada. If you can't communicate your thoughts clearly then your thoughts never really translate into the real world.
A mathematics-focused student in Ontario could take Calculus & Vectors, Advanced Functions [1], Computer Science, and Physics—the first two courses should be straightforward for this student.
Though Computer Science and Physics are distinctly different from the mathematics courses, these are still directly useful for a mathematics student to learn—the problem-solving skills should also carry over. Key mathematical discoveries have been inspired by problems in computer science and physics, and many rigorous university-level mathematics books still draw from problems in these fields to motivate certain problems. At the least, they are less laboratory-heavy than Biology and Chemistry (the student could still attempt these subjects, though, and choose to omit the grades for university admissions).
That leaves a couple of other classes—or just one if English is required, as noted by another commenter. My school offered subjects like Grade 12 Drama, Visual Arts, and Music, where much of the grading was effort-based. In my school, most students in my classes saw these courses as a break from other intensive courses, with grades not being as much of a concern. This would allow the student to avoid using a grade for History, Economics, French (or another foreign language), or another subject.
The English requirement would then be a difficult challenge for the mathematics-focused student. I wish I could speak more about what it was like for most of my classmates who went on to study engineering, as many of them took the standard English course (I took a more demanding version of the course, due to personal interest). My classmates at the time did not seem to have an issue with university admissions to competitive programs despite not enjoying the subject at the time, but the other commenter makes a good point that minimum grades for admission standards have increased greatly since then.
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[1] As an aside: a past classmate—who was brilliant at mathematics and also great with people—later poked fun some years later about the Ontario government's naming for math courses. He said, "there's Grade 11 Functions... and then in Grade 12, there's Advanced (!) Functions." The last I heard, he went on to work as an investment banker at a top hedge fund by profitability in the United States.
You can get tested into qualification, by most colleges. Colleges (which includes Universities, State schools, etc) aren't robotic. This is part of the responsibility of the administration. You may get delayed a semester or a year, with a little community college gatekeeping. Dedicated students can always get into a University.
I like the first reply immediately try to scoff you :) Maybe oop is right, but the real problem is people always try to do this.
Note here, in the current years, grades / competition has exploded so for the more competitive programs it is nearly impossible to get in without high 90s in all the required courses (English is required for all programs).
So responding to OP, you indeed must be an expert in all subjects to have a chance to study in your field of expertise.
I don't know how things work across the ocean, but here in Greece essay grading is a veritable mystery. The highest grade I ever got in a school essay was 16/20 and I had thought that was my masterpiece. Feedback always seemed cryptic (your essay contains platitudes, you are not developing your point enough etc.) and when I tried asking for specifics I got shrugs and you-still-don't-get-it groans. There are people who get 20/20 on essays consistently, so there must be some method to the grading madness; but I could not crack that method in my school years :(
A few points on this:
- How and when would you know a subject doesn't click? In my case German (my first language) and English sucked for roughly 8 years. I ended school with A in both and was the only student in my school without a single mistake in my exam.
- School gives you good grades if you managed to learn the topics at hand. Being able to learn what is needed or what you are bad at — not what you want and are good at is a skill in itself that can be important in life. I'd argue unless you are an exceptional genius (unlike 99,999% of pupils) you gain more from pulling through than you would if you called yourself a genius and focused on a single topic. Schools goal is to educate the majority
- pupils (and often also their parents) are utterly unable to judge which bits of school will be essential to their later life. I had many collegues who utterly hated every second of a multitude of subjects, only to years later tell me how glad they are now to have had been subjected to it (which brings me back to my first point)
We could (and should!) argue on how school works as a system of grades, teachers and pupils — ideally teachers would motivate students to become curious about and proficient in subjects without the motivation of good or the threat of bad grades. But if my experience as an educator at the university levels (without grades in mh case) shows one thing it is that those first semester students who are really able to judge what will be useful to them later on are not many. Many of them are more like the dog in the meme: "Only stick, no take" — they want to be able to do the cool thing without knowing what is needed to do the cool thing.
This is also part of my point, e.g. having learned Italian for years was exhausting and mostly pointless when I did it, and nowadays I can somewhat understand it whenever I am in Italy, which is maybe every two years. Was that worth it for the level of skill I gained: probably not.
But I wouldn't be able to tell if some of the mental agility I have today (beyond languages) could be attributed to the fact that I was exposed to this language which differs from my first languages in multiple ways. The lessons you may take away from any given subject is not necessarily limited to its raw content.
This is a quite known topic in educational sciences, e.g. when answering the question why it still makes sense to teach kids how to do maths without a calculator in an age where you have it always in your pocket — that is because it (A) forms an intuition about mathematical correctness and (B) logical and numerical thinking gets trained as an side effect.
Ofc all of that doesn't happen when you are unlucky enough to have had bad teachers, but that wasn't the argument.
Personally I loved history, social science and geography, but none of that has been useful in working life. Being good at mathematics is prerequisite for nearly all well-paying jobs, humanities are not of much use except for being able to write and read well.
Hindsight bias?
Yeah I mean, I can come up with 1000 lessons that would be utterly usless to 999 out of 1000 students in practise — but that wouldn't stop me from teaching those 1000 lessons in a way that really hooks those students to the topics discussed.
On the flipside you can take the most essential, practical and important topic and deliver it with such dullness that everybody falls asleep.
The first step in teaching (IMO) is to make people interested to hear what is being said. If your subject is useless and boring to most people you have to find a way to make it useful and interesting to them. This requires rethorical and social skills, good presentation skills and a deep understandingnand interest for the subjects to be thought.
In reality most subjects have interesting bits, but is the task of the teacher to make them relate to the world. If your prof thougth you about the OSI protocol stack so long everybody forgot its practical applications and how it binds together with higher level stuff, that is certainly knowledge, but it doesn't help you making it part of your understanding of the world.
Yeah, these are definitely points worth noting. Especially as in many cases, the way something is taught can have a huge effect on how well you do in it/understand it, and students will end up not liking subjects that they could have liked in university or the workplace.
So you also have to wonder how many potential math prodigies we've missed out on, simply because they had a bad maths education at school.
> Exposure is good, but success in school requires you to be successful in these subjects as well.
Being specific, it's not school, it's what school grants you i.e. a paying job. The higher paying, thusly more coveted jobs, generally filter against good grades which then the requirement pushes downwards into schools because, at scale, it's a decent system; leveraging the schools to help decide who is good.
> Being _really_ good in one thing should allow you to make up for being subpar in other areas, but it doesn't.
I counter with, if you are "_really_" good, it shows because you truly are a genius and you get fast tracked on that subject, but I think your "_really_" is actually just "_pretty_" and you're trying to include more than the 1 in 100 million.
To directly answer your point, for the "slightly smarter than everyone else" my middle school allowed kids to attend highschool in specific subjects and then highschool into the nearby community college and considered "harder/more prestigious" than the AP programs - admittedly only in math for this latter part. The school was in a more affluent neighborhood so I recognize the privilege.
I think that's the point - "pretty good" isn't good enough for a top school to want to admit you. Second year college/CS is pretty good but I went to a great high school where there were 50+ kids at that level. That's not enough to stand out in a meaningful way.
Whereas if someone was Ramanujan-level, their raw talent would be so apparent they wouldn't have this issue and would clearly stand out.
So in grade 5 you maths is A++++ (like college entry level), you're excused from English, Civics and what not, and when by the time of graduation you've fizzled out (which most prodigies do) you're just an unemployable nerd.
School education standards are the barest minimum and anyone of IQ > 85 can make them.
When you're a scientist, you'd assume the calculus is the most useful thing you learn in high school, but actually its the 10 page essay on deadline for grant writing.
It's weird to say that about English when good communication is essential for success in any engineering field. Even if all English does is force you to read more, it's probably a win in this regard.
I learned anything professionally useful in English
Even as someone who went full 'STEM' for both my education and career, English is probably among the most professionally useful courses I did in high school. A surprisingly large part of my job involves reading things, understanding them, and then writing a clear and reasoned response to them, all skills I first learned and really got to practice in English class.
The school should form decent individuals before than useful workers, and for that it's necessary to have a passing level of culture. For example, everyone should have a basic grasp of ethics (and know a bit of history), even those of us working in technology or science. Geniuses like Ramanunjan or John von Neumann are such a rare occurrence that the school system can not and should not optimize for them (and my very personal view is also that we'll have even less geniuses in the future, as our distraction-based society is not conducive anymore to cognitive development).
That is an advantage of the British system. You have a wide education until 16 and sit a broad range of exams at that age (GCSEs) and then specialise in a few (most often three) subjects from 16 to 18.
You usually need to pass English and maths GCSEs, but universities mostly care about the subjects you do in the last two years (except for very competitive courses).
This can be a problem for those who want to keep options open until they are 18 (like my younger daughter).
Even the system up to 16 is pretty flexible. There is a huge range of available subjects - although most schools offer only a limited selection (my kids were out of school by secondary school age so we had a huge choice and did some less usual subjects like astronomy and Latin).
I did IGCSE's (the international version as I lived in Indonesia) and AS/A levels myself, while a lot of my friends who went to a different school did IB and I generally disagree. My most hated school years were the IGCSE times, exactly because I was forced into learning about crap I didn't care about in the slightest, like English Literature when all I wanted to do was the various sciences and especially mathematics.
I feel like the IGCSE times were more than enough exposure to those other subjects to give me a reasonably well-rounded image of those subjects. Now, a decade later, I'm definitely glad I went the STEM route rather than ever touching on any humanities subjects in any amount of detail.
Same in the Netherlands, it's become more flexible even since I went through school. It sucks until about age 16 (or 17/18 if you take a higher level education that includes e.g. Latin / Greek), then you get to go to vocational education and either do a work/school combo (1 day a week of school), or continue fulltime school for 3-10 years (3 years for most associate's degrees, 4 for bachelor's, more for a master's, and often you can go from the one to a higher level if you choose to).
But yeah, until that age it sucks and a lot of people struggle because they have to do classes they aren't at all interested in.
> but success in school requires you to be successful in these subjects as well.
It requires a very small success on a very basic level. It is not good to be a super math genius and know nothing about geography and history.
If you’re sufficiently good or accomplished at one thing then you’ll be fine. But most students aren’t. Most students who are good at math are just good at math. But if you win two golds at Math Olympiad while in high school or publish a groundbreaking result then your mediocre grades in history will be ignored.
But a 780 on the math SAT isn’t likely to put you in the company of people whose math is sufficient to ignore poor grades otherwise.
I was put into the "advanced" class in grade school, only to hate the increased homework load, and I missed my friends in the non-advanced class, so I intentionally refused to do the work so that I would be put back in the "regular" grade school classes. I was great at math, I taught myself fractions and algebra at a young age. I just didn't like spending so much time on the other subjects, they seemed too easy or too boring and took up too much school time. They wouldn't let me into the computer programming class in High School because I hadn't passed geometry, and the teacher was a total asshole - I failed his class, but I got an A in the make-up class with a different teacher. I got kicked out of electronics class because some other students stole stuff and blamed me for it - I had nothing to do with the theft, I was the teacher's favorite and had access to everything, I didn't need to steal anything, he gave me anything I wanted. God I hated High School so much.
After graduating High School, I couldn't really see myself sitting through 4 more years of basically the same subject material - history, English, social studies, etc... when all I really wanted to do is use my talent in electronics and computers that I'd been accumulating since I was 5 years old. So I went to a trade school for electronics and computers instead of a college, this was back in the late 80's.
I can't say that not going to college hurt my job prospects at all, I've been highly paid for many years for doing something I love. Jobs hired me based on experience, not because I went to a college. That may have changed a bit now, since some tech companies favor college graduates, but I've worked with some college graduates that are just dumb as shit and couldn't get anything done - not all are like that, but going to college isn't an indicator of success at all.
In my country there is a system where a kid applying to high school gets accepted based on a standard test, or can skip the line by performing well on a contest organized by the ministry of education. In my class something like 80% of kids were admitted through contests. When applying to the university there's a similar thing, except it's much harder to skip the line, but universities are free to set up their own admission rules as long as the rules are based on the national standard test. In my case, the final admission score was calculated something like "90% maths 10% everything else"
In my experience, people who are really good at something usually are at least average on many others. Thus they can do the minimum in other courses to pass, it's just a matter of working enough.
In my country, there are several students (12-18 years old) who can mix sport at high level (national championship) and a lighter school activity. They work like mad but they do it. But they have to prove there are good enough to get it, which is OK to me.
Being really good is not something you appreciate yourself, it's the others that notice.
> maybe you should be able to get an A++++
What does an A++++ mean? I guess it's equivalent of publishing a novel result/idea, solving a long standing conjecture sort of ability? If yes then I'd say the system does accommodate them. Terrance Tao for example was fast tracked and didn't get lost in the system.
Though it probably requires some effort from parents to figure out the right path for their prodigies.
>but success in school requires you to be successful in these subjects as well
It doesnt, at least in my country.
Here you have a few school leaving exams that you need to perform at
Math, native lang and english.
And one at advanced level, but its your choice when it comes to subject.
What you are talking about is GPA and GPA is meaningless tbf
That’s irrelevant to 99.99% of people. An A in math is not especially difficult to achieve.
To actually be meaningfully better at math you need to go far, far further than the coursework presented in schools, at which point it becomes an extra curricular which is already considered. If you’re a math savant proving novel theorems, they will notice that.
But nobody gives a shit if you’re just really, really good at factoring polynomials from Algebra 2 tests.
Something about this sniffs as elitist to me. A person who’s intelligent is curious and a person who’s curious should be curious about all things, not just some limited set.
Now, that’s not to say the only issue is someone’s curiosity. Traditional teaching methods make it very hard to be interested in some topics (history and language comes to mind), but barring that, I’m not sure I accept “it’s not interesting” as a reason not to explore a subject.
The only intelligence on earth capable of being equally curious about all things, of not needing to discriminate and focus only on preferred subjects, is the artificial kind of intelligence.
The rest of us don't live long enough compared to our reading speed.
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.
One such school:
https://en.m.wikipedia.org/wiki/AME_School
The kids that went there … some succeeded, some really struggled to adjust to other schools and environments.
> In 1992, there was a proposal to fully integrate the AME school with the ACT Government school system. However the idea floundered after the ACT Teachers Union amongst other things, knocked the idea on the head, due to compatibility issues. It felt the idea of having a school principal having selection over teaching staff was outside the scope of the Unions control. It came out fighting against the proposal.
Now there’s a depressing thing to read. Teacher’s unions are the worst.
A bit too late if you are comparing to football level of specialization to produce people like messi
Most colleges now inundate students with painful core classes that go on into senior year. It's getting ridiculous.
College is not trade school. College exists not to generate people who are masters of Framework v3.0, but to generate people who can quickly learn to use whatever tool they're given and who can connect the dots to solve generic problems. Part of that is exposure to a broad range of ideas. Part of that is showing that you can learn about and deliver results on things you're not necessarily excited about.
You can get exceptions. My social science degree had math requirements, but the lowest math class I took was too high level to count for them. I had to fill out a sheet of paper and submit it to the dean, but other than an hour split between talking to my academic advisor, filling out a form, and talking to the dean it wasn't a problem.
I still find it funny and informative that multivariable calculus, linear algebra, and high level math classes did not automatically fulfill the math requirement because no one ever tried to use them to satisfy it.
I feel your pain. What a dumb system.
I don't think college teaches people how to learn, and if they do it's only by accident. There's a body of knowledge on how to teach and how to self educate and it takes a long time for systems to incorporate these knowledge.
Most of school is primarily baby sitting these days, if were being really real.
That is what high school is, for families where parents want their kids to learn to learn and are well-resourced enough to teach the kids themselves or pay someone to teach.
High school has been dumbed down and is mostly a waste of time.
I wasn't particularly excited to learn philosophy or economics, but that gave me the foundation to understand Marx and Engels. Learning "things you're not necessarily excited about" makes you well-rounded. That's a _good thing_.
Let's use another example. Communicating with other people is a very important part of my job as a software engineer. I'd say that at this point in my career, writing good proposals, documentation, project charters, task orders, change requests, etc. is vital to my individual and collective success, so I'm glad I was taught lots of rhetoric and literature and writing and all that artsy stuff my younger self spent too much time disdaining.
Or another example. I have written precisely zero lines of Lisp or ML for work, and I flunked the first computer science class I took that used Lisp (good old _Essentials of Programming Languages_). And yet I use the concepts I learned in those classes _all the time_. Heck, I was just talking about side effects the other day in the context of 21 CFR Part 11 computer system validation.
My last metaphor. Brushing one's teeth or taking a shower isn't particularly exciting, but that kind of routine maintenance work is a vital part of your physical health. So it goes for mental exercise. Reading something challenging, learning the work's historical context, and writing a critical response well all helps keep you mentally healthy. That's good no matter what socioeconomic system you live in.
It's probably because secondary school has become mostly worthless in the US, so college is taking its place.
Specific citation needed.
At Big US Engineering School, many people are done with their prerequisites in a year.
Unless you're talking about painful core classes like "compiler design" and "networking", which I would say is a different conversation.
Prerequisites are different from core stuff, like say you study computer science but hey take this English class as well for your W credits. I gamed my university on these, taking easy courses that I wouldn’t have bothered with, I think it was one or two quarters of BS (dual acronym meaning) classes.
I used to shit on English class to anyone who would listen, but lately (at age 40) I realize (and grow frustrated at) how poorly most people around me communicate. But then again, I recall my English classes throughout highschool and college and very little time was spent on how to be clear, how to be precise, or how to revise. That definitely wasn't the parameter the curriculum was optimizing for. Sure, there was writing but the grade you got back was simply how much the instructor liked what you had to say.
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.
Another student at a martial arts gym unexpectedly gave me some advice that is somewhat common, but had an impact because the words came at the right time and from the right person: he kindly told me to never judge myself based on the person that I might have been, and to instead compare myself now to how I was a month ago.
I believe that any person here with an inkling of relatable technical experience can greatly appreciate the work you've been doing. Software development can be complicated and frustrating, especially when things don't work as you expect them to (but then, you learn and become better). Leading a business is very difficult, often due to sources of problems you don't expect (such as regulatory and legal requirements, accounting, and publicity).
Some people cruise on to great careers without facing many barriers. But many others face unexpected setbacks and have to manage them. A close friend of mine was living an overall good life until it was profoundly disrupted by a civil war in his home country. But he made it to my country where he began his undergraduate degree at a great university that he loves. A past colleague of mine spent much of her early twenties managing physical disability, but successfully received treatment and went on to graduate with an engineering degree. She has since landed a position at a top aerospace company that she really wanted to work at.
You are setting up a good life for yourself. Many people lack that kind of drive or struggle with executing ideas; several people I personally know would be very proud to one day experience just a small part of your successes so far.
Thank you for your kind response. I am happy for your friends and also the message you received at your gym.
Life is truly a journey unique to each one of us.
I’ve found peace and I carry a signed index card in my wallet on which I’ve written my “ethos”. It took me a long time to come up with it and I’m sure it’s common, but those 5 points are something I try to remain true towards.
Cheers mate, best.
yep ... we are all running our own individual races ... instead of reminiscing and thinking what coulda been ... better to look forward and look at what we can do now.
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.
Yeah that's a common result of forced learning by lets be polite mediocre folks. Utter hate of the whole topic for easily 2 decades too. Then finding slowly my own personal way back to them, despite school.
If anything, current (and this is valid globally) school system is not designed at excellence at its core, its about raising an army of obedient but not too stupid citizens. And its not missed expectation, just look at what type of work they expect to fill in. Don't expect massive changes unless society changes itself.
Some narrow excellence is not what our society at large values, those few that made it through made it despite their environment.
I missed out on theater and improvisational comedy only because I pigeonholed myself as a computer nerd and engineering type and almost nothing else.
I found that I have a certain knack for it and really enjoyed performing.
The exposure to subjects isn't the problem, this is the problem: https://paulgraham.com/lesson.html
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.
So what is high school for?
Also does the above apply also to the most selective and renowned institutions, or only to community colleges?
In Europe high school is when you get a generic education, and in college you choose your topic. So if you are a languages major you'll have no more math, if you're a STEM major you'll have no more history, if you're a philosophy major you might have logic and some history but no physics or linear algebra, and so on.
I studied computer science (M.Eng.) and I had to provide an English certification as a foreign language, but that was it. The only non-computer science, non-math, non-electronics courses were one semester of chemistry, one of economics and three of physics (which you could say is related to electronics though). So basically 15-20%.
Having to study more history or biology would have been a huge waste of time in a CS university, and I say that as someone who did more Latin and Greek than math and physics in high school.
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.
But if their overall SAT scores aren't good enough to get into the elite colleges, won't we just be denying the eccentric geniuses?
I like to look at the backgrounds of the people who win the Nobel prize. Everyone is interdisciplinary.
For college and life in general, I think main skill needed is emotional regulation. Everything else flows from that.
Well, the eccentric geniuses have already left the system... Frankly, if you go through the last 30 years, how many such geniuses you can find in American universities? I only see a little bit higher than average, so it seems that the system has already eliminated the geniuses.
> I only see a little bit higher than average
Could you share your source for statistics on "eccentric geniuses"?
The eccentric geniuses at these elite schools will end up doing stupid shit for a bank.
It is good to expose them. But that doesn't mean the previous point is backwards.
A subject "clicking" is strongly dependent on the teacher.
"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.
Well, my solution would be this:
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.
>Also, I would group kids by subject, and not by age.
I would be cautious with this, they may have the same academic ability but a large gap in social skills.
Why is that worse than having large gaps in academic ability and social skills?
> 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.
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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!
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.
What we need to do is, make the subjects interesting to learn with tech.
It may now be possible with Videos, Games and VR.
Tech is not necessary. You just have to make the subjects interesting to learn. Good teachers are way more value than they are valued, and will make the classes interesting to learn.
No, the subjects are already interesting. What we need is to pay teachers a good, above-average wage, and to continuously invest in their professional development throughout their careers. Only a good, well-trained teacher can make a subject interesting and engaging.
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.
> and do so with relative competency
I dispute this on the grounds that students are going through American schools and many of them don't even know how to read.
Right there you have 8%. But it is worse because it says they're able to "read to varying level", which includes 2nd grade level, or barely at all.
Over 20% of American adults have a literacy proficiency at or below Level 1. This is defined as "difficulty using or understanding print materials" and "may struggle to understand texts beyond filling out basic forms." So 1 out of every 5 Americans is functionally illiterate. I would say that counts as "many of them don't even know how to read."
In terms of not being able to read, I would count 8% as "many"? Select a group of ten random Americans and you'd expect one of them to be illiterate - that's appalling.
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.
a very clear analysis, that's also very well written
Thank you, kindly! I dictated it to ChatGPT which paragraphed but kept verbatim. I was on a roll! Hehehe :)
But if you notice the people who are in administrative positions are the people who are “well rounded” not those who are good at one thing.
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.
> 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.
He isn't saying that being a doctor isn't great.
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.
Please excuse my misinterpreting OP’s comment, and thank you for clarifying.
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.
> every extraordinary person [was] 1 step away from being forgotten, and got some huge universal break
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.
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.
Everyone here seems to have missed the significance of L.J. Rogers in this story.
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.
your thought reminded me of the radio program about Jean Shepherd getting his Class A radio license.
https://www.rfcafe.com/miscellany/factoids/Jean-Shepherd-Cla...
on youtube as well
Thanks a lot for writing that. I agree 100% with you.
But I always wondered how polymath like Leonard davinci and Isaac newton that are excellent in many area are possible.
Agreed.
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.
We shouldn't need crazy people to push the boundary. Rather, the crazier you are, the more likely you will flame out.
People who are "weird" and yet are entirely functional are the best of both world and a much rarer combination.
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.
School is for training good test-takers, not finding geniuses.
you can't serve two masters.
How would you characterize R's master and the "normie" master?
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.
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