[g00dn3ss]

Recent evidence tends to indicate that 'not smart enough' is probably a myth. Almost everything can be attributed to exposure and effort at some point rather than some innate smartness.

A lot of advanced math takes some serious concentration to understand. For some non-practical aspects, I found that I lacked the motivation rather than ability to understand it . One particular class where I seemed to hit my tolerance was a theoretical linear algebra class. I could understand the practical applications of most of the topics but some of the theory seemed just out of reach. The book was extremely dry and I think the professor may have been taking lessons from Ben Stein.

Give me a private tutor, a theoretical linear algebra for dummies book, and a pending disaster for which this is the solution, and I bet the outcome would be a little different.

[mlinsey]

I don't think this is true - or at minimum, higher math/theory certainly comes easier to some people than others, and there probably comes a level of intelligence where you were progressing slowly enough that it really wouldn't make a good career option for you.

I hit my brick wall in theoretical classes when I felt like I would understand everything I had been taught up to that point, but to actually construct a proof to solve certain problems I had to come up with some flash of insight which just wouldn't come to me. I simply wouldn't know where to start.

I don't think that this is uncommon. When many people write about famous mathematicians and how they solve extremely difficult problems, it usually happens that they have a certain intuition or insight which didn't really follow logically from the problem as stated up until that point. Perhaps anybody could put enough effort into these problems that these insights would come to them as well, but I think this is doubtful considering just how brilliant these insights are.

Put another way, when you reach a certain level of math it starts requiring a large degree of a particular kind of creativity, which simply not everyone has.

[g00dn3ss]

Actually, there was a recent article on YC news about the large number of discoveries that were made independently at almost the same time.

Our made-for-tv histories usually show a lone scientist toiling away for years, making huge discoveries. But in reality, most discoveries are incremental. The stories we are taught about these discoveries are usually mythologized.

It is true that there is some level of creativity needed. But it's probably way more common than our histories would lead you to believe. The recurring theme you will hear from scientists is that "chance favors the prepared mind."

[nsrivast]

To summarize: intelligence, motivation, and creativity are important.

[bkovitz]

Yup, motivation is the blocker.

I have repeatedly gotten tired of math when it seemed like meaningless puzzle-solving. Why do I really care, for example, how many nonabelian groups of order 36 exist? When math seems to me like it's providing a vocabulary for framing and answering deep questions about the world, or making it computationally feasible to find the stamp of causal influence in data or design real things that are impossible to make without out, then the motivation is there.

This might just be an artifact of the way math is often taught. For example, real analysis is often taught as meaningless proof-finding (e.g. proving things that seem either pointless or obvious). But there's a fantastic book, _A Radical Approach to Real Analysis_ by David Bressoud, that teaches the exact same subject matter as the fruit of deep, pressing, and non-obvious questions that stirred debate among mathematicians for around 100 years.

[manvsmachine]

Many branches of math start out as being "meaningless". Number theory was once widely considered meaningless until it was given importance through cryptography. The current head of Microsoft's Cambridge research location literally said to Bill Gates as she was being hired that her work was completely unrelated to computer science; her work was in combinatorial optimization. I think that may be a part of the problem: not enough math professors teach their subjects with any regards to what they are used for. Sometimes, they don't even know themselves.

[doubleplus]

Would you happen to have any links or book references to this "recent evidence"? While most of success comes from effort and discipline, I've always been under the impression that only the naturally gifted (who also have discipline and drive) can/have achieve/achieved certain things. The dwarf down the street from me will never be a professional football player, and someone with a 100 IQ will never be an astronaut. I've always felt the you-can-always-achieve-whatever-you-want-no-matter-what-if-you-want-it-bad-enough approach to be more than a little cliched and misguided. If recent scientific evidence has disproved this stance of mine, I'm open to changing it.

[elai]

To be honest, I believe that all intelligence comes from innate motivations and practice. You suddenly become 10x more "intelligent" when you actually give a shit or get some innate joy or obsession relieved when you do it. If that is something that is constant within you, you become more and more practiced and a virtuious cycle results. Nothing is too hard to explain, it's just mathematicians are really horrible at communicating.

[byrneseyeview]

For whom, though? Brains are complicated, but if yours is severely damaged (through a stroke or head injury) motivation is not necessarily going to fix that. And as far as I know, there aren't any permanent ways to raise IQ. Some environmental factors lead to higher childhood IQ, but these vanish by adulthood. And there are tricks people can use to bump up their scores on IQ tests, but I haven't seen a situation in which practicing for test A raises your score, years later, on test B. This stuff is just more static than that.

The danger with what you're saying now is that someone with an IQ of 90 -- someone who could be a fine contributor to society in lots of practical, necessary fields that don't require lots of abstract thinking -- could be inspired to throw away lots of time and risk lots of frustration trying to be a mathematician. We should deal with the fact that wasting education on someone who can't use it is as much a tragedy as failing to educate someone who can use it. By pretending that 'smart' just means 'trying hard', you're doing more harm than good.

[DaniFong]

The problem may be that our predictors are terrible. There are a lot of Nobel prize winners who later discover their 'dismal' performance on IQ tests. I know grad students at Princeton who've confided sub-100 IQ scores, bad SAT's, horrible performance on one math test or another -- it's like this big, shameful secret for otherwise brilliant people.

Everyone carries around the absurd burdens of judgments and measurements, but they don't always mean what people think they mean.

[byrneseyeview]

Are there any other quantifiable variables with such high predictive power? I'd be really interested if there were some other test that could better estimate the odds that someone would get a degree, earn lots of money, stay out of jail, vote often, delay having kids, etc.

It would be interesting to know more details, which I'm sure you can't divulge without violating someone's privacy. But it would be neat to find out if those people had other skills that correlate strongly with IQ, like the various digit-memorization/recitation tests, or reaction time. Were these people autistic? Were they taking a test in a language they didn't know too well?

[DaniFong]

They weren't autistic, and there didn't seem to be any strange behaviors socially, and they were quite inquisitive and swift on the uptake.

I once heard IQ defined as "ability to navigate bureaucracy, getting the answers others think correct in a manner testwriters imagined, and color in the lines" (paraphrased). This may have something to do with the phenomenon here.

The odds that someone would 'get a degree, earn lots of money, stay out of jail, vote often, delay having kids, etc.' seem to have more to do with successfully conforming to certain values of society's upper-middle class.

I think I'm capable of inventive thought, but I don't particularly want to get another degree, or earn much money, or delay having kids, or vote often, and the sort of things that one has to do to stay out of jail, are, honestly, quite often absurd, and I often rail against them.

And if you start to observe, closely, just how these things are tested, you'll start to get the impression that maybe bright people will find their ways through the cracks more than anticipated. There's an enormous weight given to quick answers -- time directly influences scoring. Linear answers are expected, and alternative interpretations are docked. There's often insufficient information in the questions, or it's based on a model of the world that's wrong. Domain knowledge like mathematics or vocabulary is brought into it. Analogies are made to hone in on one relationship from the many that could exist. Scorers of essays give insufficent weight to substance and too much to form -- despite their lipservice, they are indeed swayed by big words.

Even tests like GRE physics are bad. They claim to be testing rapid physical intuition, but in practice what divides good from poor scores is prior experience with 100 simple systems and the ability to get the factors of 2 and pi right with three minutes per question.

[aswanson]

someone who could be a fine contributor to society in lots of practical, necessary fields that don't require lots of abstract thinking -- could be inspired to throw away lots of time and risk lots of frustration

But the bottom line though, is that it's none of your business to determine how one should spend ones effort and life. Nobody asked for your take on parameterizing their lifes limits. Thats what living is for. I could tell you what you should do with your life after a test or battery of tests and send you on your way. Like that idea? Even if you do, let it apply to you and you alone.

[byrneseyeview]

How do you go from "Nobody should prevent someone from aspiring to do what they want to do" to "It is wrong to point out that people with low IQs are very unlikely to succeed at tasks which require high IQs, and telling them that these real constraints are not real at all is cruel and insensible, since it's factually incorrect advice that will lead them to make poorer decisions."?

I mean, I could tell you that if you are really super-passionate and very very outraged, you can convince me to abandon my factual beliefs for some kind of feel-good claptrap. But that might lead you to make another comment of similar tone and content to the one to which I responded, leaving us both worse off. Do you see where I'm coming from?

I am in a business that most batteries of tests would predict I'd fail at, so I clearly don't follow your parody of my thought processes.

[aswanson]

since it's factually incorrect advice that will lead them to make poorer decisions

The problem with this assertion is the assumption that you have all the facts, facts meaning absolute certainty what a person is capable of given a scalar score from a test.

But that might lead you to make another comment of similar tone and content to the one to which I responded, leaving us both worse off. Do you see where I'm coming from?

I'm guessing not from a vantage point of omniscience.

I am in a business that most batteries of tests would predict I'd fail at, so I clearly don't follow your parody of my thought processes.

Not a parody, just an observation that your aggregates mean nothing to a given instantiation. And congratulations on defying what the stats say you would fail at. Hard worker, indeed.

[g00dn3ss]

Here is a pop-sci version of some recent research:

http://www.sciam.com/article.cfm?id=the-secret-to-raising-sm...

However, this isn't particularly surprising. Research on "grandmasters" in many activities, like chess or pilots, usually shows they've put in 25K hours of practice in their field.

I'm certainly not in favor of saying anyone can do anything. Especially in physical activity. There are definitely physiological advantages.

The same is also true in mental activities to some extent. A kid with down syndrome probably isn't going to become a math professor. However, I think the point is that most normally developed folks probably haven't hit a ceiling in their math aptitude. They just stopped caring or putting in the effort.

[aneesh]

Very true. I really enjoy math, but after abstract algebra and real analysis, I slowly lost the motivation to go deeper into pure math. Interestingly the most enjoyable areas of math for me are those arguably most applicable to CS: graph theory and number theory.

[manvsmachine]

Haven't gotten to that point yet, but I can definitely attest to the significance of good materials and enthusiastic teaching. Because I started out as a engineering major, I had taken the entire Calc sequence up through Diff. Eq's, but I ended up having to take Prob/Stats and Discrete when I Switched to CS. I was sick of taking math classes at that time, but my Discrete prof was awesome and motivated me to see how far I could go with it. I went into fall semester never wanting to take another math class again; now I'm taking more so I can get a minor in it. I'll be taking combinatorics in the fall and topology(?) the following spring to close it out.

[flashgordon]

Totally agree. Case in point. Up until highschool I was very good at math. Mainly because I would actively seek out books and try to solve problems where ever I could find them. As soon as I started Uni, my interest dwindled (was replaced by love for all things CS). People who I had dominated back in highschool were now getting a good 10-20% higher than me. All in practise I think.