I think about this often. Having studied mathematics, it's a field where there is a huge range of talents, even at 18 (and younger). There are genuinely gifted people who stride ahead, but the sad reality is that even at top institutions most of the undergraduate cohort will be merely averagely clever people who have received great (often private) tuition.

In a field where finding every genius really matters because of the difficulty of expanding the frontiers it's heartbreaking to realise most just never get the opportunity of good mathematical education in the first place.

I don't think this is the whole story. Of course, it is a tragedy when people who had undeniable raw talent are neglected or passed over and don't get to achieve as they might. It's also sad and disappointing when we observe structural injustices in our society.

However, isn't it possible that giving great teaching resources to people who aren't already at the top level in terms of raw talent, allows them to succeed at a very high level? If so, it's not a zero sum game - the 'averagely clever people who have received great tuition' and who go on to succeed might be a sign of the education system working, not its failure. We might focus more on making sure a wider group of 'averagely clever people' get such good tuition, rather than on the finding and promotion of undiscovered geniuses.

Two examples that I often think about in this context: firstly the Budapest school of mathematics in the early 20th century. A hugely disproportionate number of the world-class mathematicians from that era were directly taught by Fejer and/or part of his seminar and mentee group. (He stands out as a supervisor to a far greater extent then any mathematicians stands out from the pack for his own achievements.) Is it possible that while one or two of this group might have been 'born geniuses', it also includes several who might otherwise have been second-rate, but who became world-class because of the influence both of Fejer and of their high-performing colleagues?

Secondly, George Harrison of the Beatles. In the second half of the Beatles' recording career, he wrote some of their best-loved songs, several of which are regarded as absolute classics. Did the Beatles happen to contain three inherently gifted songwriters, by sheer coincidence? Or is it more likely that working for a decade alongside two extremely talented and successful songwriters nurtured and elevated Harrison far beyond what might otherwise have been expected?

I think the Beatles included three inherently gifted songwriters by sheer coincidence. No doubt the apprenticeship helped, but Harrison was writing songs like Taxman and experimenting with Indian classical music on Love You Too in 1966.

It's not either/or, it's both. They were gifted, but they also had the time, money and motivation to work on their skills and hone their craft. It's not by coincidence that in literally every community that talks about how to level up their skills, the first response is always "practice practice practice".

By 1966 George Harrison had been in a group (with various names) with Lennon and McCartney for 8 years, their songwriting partnership had been actively writing songs for 5 or 6 years, they had written almost all the songs for 6 or 7 albums, several of which are regarded as among the best albums of all time.

> rather than on the finding and promotion of undiscovered geniuses.

There's a catch in it as well, which is that you cannot really identify a genius up-front. A genius is, by definition, someone who thinks differently. A lot of people think differently. But the thing with the genius is that he thinks differently and correctly. The latter part won't be obvious until after the fact.

When Alex Ferguson started out as a Manchester United manager he probably already was a genius, but it wasn't obvious until many years later.

Same with Sergey Brin and Larry Page. They probably had a genius vision for the company right from the start. But it's only in retrospect that we recognize that it was, in fact, a genius vision.

Finding a genius is, I think, almost by definition, impossible.

Alex Ferguson might have been a good football manager, but he wasn't a genius - and I say that as a Man United supporter. He didn't advance football in any significant way, and even towards the end of his career he was somewhat naive from a tactical perspective.

He was just a good man-manager in what is a sea of managerial mediocrity - football management is largely restricted to ex-pro-footballers who can't do anything else. Rinus Michels was a genius, Arrigo Sacchi was a genius - they actually advanced the sport.

On a similar note, Page and Brin were obviously smart and driven and all, but applying linear algebra to the search problem was arguably one of the things that was "in the air". It is hard to judge, because what seems obvious ex post was (obviously) not obvious ex ante. But, again, eigenvalue decomposition/SVD (and linear algebra more generally) - you throw it at the Netflix problem, you throw it at image compression, you throw it at anything really, something's gonna stick.

It's an interesting counterfactual: without Page, when would Page Rank have come around? The idea that the stationary distribution of a Markov Chain (under certain conditions) is given by the eigenvector to the (largest) eigenvalue 1 is certainly decades old, if not a century.

You start to loose correlation with IQ really, really quickly as you move away from feats of solving established math problems to football management and entrepreneurship. Solving established math problems is more like taking an IQ test than any other activity that I can think of.

Yes, and also don't forget that Rajeev Motwani's (Ph.D. supervisor and FFF investor) research field was randomized algorithms. So thinking about random walks in his group was a bread and butter thing, probably, not something that required genius. Rajeev is co-author to several seminal paper on the search engine and worked intensively with Google and joined their board. According to Prabhakar Raghavan, Sergey Brin acknowledged in a BBC Radio interview in 2009 that Google might not have been possible without Rajeev. (Tragically, he passed away in 2009 by falling into his own pool with alcohol in his blood, and he could not swim.)

> Same with Sergey Brin and Larry Page. They probably had a genius vision for the company right from the start.

That seems inconsistent with reports that they tried to sell early to Yahoo! for $2m.

> If so, it's not a zero sum game - the 'averagely clever people who have received great tuition' and who go on to succeed might be a sign of the education system working, not its failure.

The failure is that mediocre students which come from an environment which helps them (good school, private tutors, stable home life) can be successful, while "naturally gifted" yet disadvantaged students don't.

The basic fact that parental income is still the best predictor of educational achievement should tell you everything you need to know.

> The failure is that mediocre students which come from an environment which helps them (good school, private tutors, stable home life) can be successful, while "naturally gifted" yet disadvantaged students don't.

I think you have rephrased what I said to again push the idea that it's a zero sum game. If mediocre students can be successful, given the right educational inputs (for some definition of 'successful' and 'mediocre'), isn't this a huge opportunity? Perhaps we should improve society as well as social justice by trying to promote mediocre students from poorer backgrounds to this same level of success, rather than by working on the problem of undiscovered genius.

> The basic fact that parental income is still the best predictor of educational achievement should tell you everything you need to know.

My whole point is that this does not tell you everything you need to know. If low-parental-income and high-parental-income students had exactly the same potential to succeed, the fact that parental income is a good predictor of achievement might be telling you either (or any combination) of two things: that wealthier students are achieving beyond what should be expected (and so we are ending up with incompetent people in very senior positions because they did not get them on merit), or that poorer students are achieving below where they should be (and so we are wasting the potential of talented people). These are really different problems! And probably have quite different solutions.

I certainly agree that talented students from disadvantaged backgrounds ought to be helped. But thinking parental income is the causal factor behind the observed correlation of student success with income may be mistaken.

Two other hypotheses are that the money itself is not important, but rather the familial transmission of certain cultural traits (which also often lead to high incomes), and/or that, similarly, the causal factor is genetic transmission of some cognitive or personality traits. Note that a factor being genetic does not mean it is immutable - for instance, there could be a genetic factor that leads to students doing poorly in school because they are intolerant of sitting still for hours in desks, which would become irrelevant if one stopped requiring them to sit still for hours.

A mistaken idea of the causal explanation for the correlation with income would lead to interventions that don't actually work.

> parental income is still the best predictor of educational achievement

Is that controlled for genetic influence though? Suppose smart people 1) have high income, 2) have smart kids.

Disentangling the various causal factors is substantially harder than your quote above suggests.

I read a study that specifically addressed this by looking separately at adopted kids. I'll try to look for it and post a link when I'm at my computer.

Given that even our first-tier colleges and universities still have no real idea how to teach math in a way that ensures that students will internalize and understand it, this isn't surprising. I think one of the most curious things about the modern world is that centuries into working with higher math, we still do not have good ways to teach it.

Most students with creative insights that could help advance the field tend to wind up one of two ways: (We need to create a viable, replicable third way - today it seems to require exceptional personalities for both student and teacher...) 1) They slog through the existing math education system, but it crushes their creativity and insight, leaving them mostly useless husks (this pit catches most mathemeticians, IMO), or, 2) They become disgusted by the incomprehensibility of higher mathematics and abandon the field entirely (this is a huge chasm of a pit, that prevents most all engineers and scientists from ever becoming more than moderately competent in mathematics and methods.)

This is the problem right here. The solution is that Maths teaching needs to be brought online Duolingo/Brilliant style (with in person teacher support obviously). We can completely standardise the curriculum and do A/B testing to find out what is the most efficient way to get the information into students’ heads.

The data could be analysed further to see if there were certain mathematical learning styles (I suspect there are). One cohort of students might be better off on the visual/geometry heavy track, one might be better on the abstract track etc.

It should be initiated at the national government level and mandated to use in schools but the actual building of the application should be contracted out to a company from Silicon Valley/Roundabout/$COUNTRY_TECH_HUB.

Until this happens or something similar we’re just going to continue having this grossly unequal and unfair state of affairs where access to good mathematics education is complete geographical potluck for anyone whose parents can’t afford tuition. The amount of wasted potential is staggering.

I think you mean tutelage instead of tuition there (I could be wrong)

But what about the sweatshop owners?

Very nice quote.

I also like to think that intelligence is everywhere, but the persistent pursuit of truth is difficult. Zhang embodies a man determined to work on his particular pursuit.

I hope I can be as persistent over time.

This is EXACTLY what I'm thinking about. Tons of talents were wasted. To fix this we have to change the global political climate.

I mean Alan Turing proved that even a demonstration of your abilities isn't enough to avoid persecution.

We detached this subthread from https://news.ycombinator.com/item?id=33516152.