Terry Tao about his own childhood as child prodigy (1985, at ten)(terrytao.files.wordpress.com)
terrytao.files.wordpress.com
Terry Tao about his own childhood as child prodigy (1985, at ten)
http://terrytao.files.wordpress.com/2008/12/terrys-my-recollections.doc
26 comments
Interesting.
I absolutely suck at math, and I blame the computer. Here's why :) :
When I was a kid I was good at 'arithmetic' in school (grade school, I really wouldn't call it math). Anything up to say raising stuff to arbitrary powers and doing lots of multiplications, long division up to fairly high numbers without pencil and paper.
Then high school came along. In the first year I was probably the best of all the kids at math, it didn't cost me any effort either, trigonometry and so on was a breeze. I liked doing it, the teacher was great. Then in the second year of high school we got a fairly lousy teacher (sorry, it's the truth, especially when compared to the guy from the first year), and I got access to a computer.
That changed everything. Computers were so much more fun than math yet somehow related that my interest in mathematics dropped like a stone and my interest in computers went through the roof. After that mathematics never managed to wrest my attention away from coding long enough to make it count.
Even today I have an almost untouched calculus course sitting on my bookshelves that I've been planning to work my way through 'one of these days' for the last decade or so.
Programming is just too much fun.
I know I'd be a better programmer though, with more math under my belt, so maybe, one of these days...
I absolutely suck at math, and I blame the computer. Here's why :) :
When I was a kid I was good at 'arithmetic' in school (grade school, I really wouldn't call it math). Anything up to say raising stuff to arbitrary powers and doing lots of multiplications, long division up to fairly high numbers without pencil and paper.
Then high school came along. In the first year I was probably the best of all the kids at math, it didn't cost me any effort either, trigonometry and so on was a breeze. I liked doing it, the teacher was great. Then in the second year of high school we got a fairly lousy teacher (sorry, it's the truth, especially when compared to the guy from the first year), and I got access to a computer.
That changed everything. Computers were so much more fun than math yet somehow related that my interest in mathematics dropped like a stone and my interest in computers went through the roof. After that mathematics never managed to wrest my attention away from coding long enough to make it count.
Even today I have an almost untouched calculus course sitting on my bookshelves that I've been planning to work my way through 'one of these days' for the last decade or so.
Programming is just too much fun.
I know I'd be a better programmer though, with more math under my belt, so maybe, one of these days...
" And yet whenever I read stories of "real" prodigies, I never see anything like this--they always seem to continue their trend and prove to be brilliant mathematicians."
No, it's very, very rare that child prodigies go on to have significant impact in the field they were brilliant at as a child.
See http://en.wikipedia.org/wiki/Study_of_Mathematically_Precoci... for some details. (I read a very interesting article about Tao previously which addressed this - trying to find it now)
No, it's very, very rare that child prodigies go on to have significant impact in the field they were brilliant at as a child.
See http://en.wikipedia.org/wiki/Study_of_Mathematically_Precoci... for some details. (I read a very interesting article about Tao previously which addressed this - trying to find it now)
Hmm. Reading the papers from that study group seems to prove me 100% wrong. I suspect my selective referencing skills need improving.
Note: I read the abstract for the 35 year study and glanced over the paper.
The researchers findings seem to be that these prodigies are for most part well adjusted/succesful, but do not necessarily go/stay into math/science fields.
Are those the findings, or is there something more?
The researchers findings seem to be that these prodigies are for most part well adjusted/succesful, but do not necessarily go/stay into math/science fields.
Are those the findings, or is there something more?
They split the prodigies into science/math people and humanities people. I skim read the paper, and it showed a correlation between the field they were talented in and achievement in that field.
There is a famous story about this: Wunderkind, by Carson McCullers. (You can find it online if you search around.)
Your skill stopped developing because you lost interest. Courses beyond basic calculus are not inherently more difficult. They are only more intimidating, and perhaps require a little bit more study time, and spaced repetition. There are more rules, problem types, and proof techniques to remember. If you can do logic, you can do college-level advanced mathematics.
Speaking from personal experience, everyone I went to grade school with who showed exceptional mathematical promise was bested by me in college. Indeed, one of my friends who was put in the "slow" math section is now a math major. Math is doable, and simply becomes something that you have to study for; you no longer get a freebie because you're smart.
Speaking from personal experience, everyone I went to grade school with who showed exceptional mathematical promise was bested by me in college. Indeed, one of my friends who was put in the "slow" math section is now a math major. Math is doable, and simply becomes something that you have to study for; you no longer get a freebie because you're smart.
Math is doable, and simply becomes something that you have to study for; you no longer get a freebie because you're smart.
I found it was the exact reverse. Many concepts, in my experience, could not be learned well how hard I studied. And yet people who did significantly less work than I did found them extremely easy. One example is natural deduction proofs; I slogged my way through them with extreme difficulty while some friends of mine did them nearly effortlessly by comparison.
Overall, at least my own experience suggests that studying and hard work can never fully replace innate ability.
I found it was the exact reverse. Many concepts, in my experience, could not be learned well how hard I studied. And yet people who did significantly less work than I did found them extremely easy. One example is natural deduction proofs; I slogged my way through them with extreme difficulty while some friends of mine did them nearly effortlessly by comparison.
Overall, at least my own experience suggests that studying and hard work can never fully replace innate ability.
There is a cheesy quote that says genius is 1% inspiration 99% perspiration. In college, I found this to be especially true, at least in the math department.
While most people would like you to believe that they breeze through the subject, getting to know people in the department (undergrad) who were very successful, I learned that, while of course mathematically inclined, the amount of effort put forth was quite astonoshing. It was astonoshing to me at least, coming from a non-competitive high school.
It is not unusual for someone on any given week to study 25+ hours for a single class. This was the case for some of the smartest people I met in the program. I found 25+ hours a week turned out to just not seem like a lot to someone who was a) very interested in the subject and b) used to spending that kind of time on things. I struggled to put 10 hours a week into it and would barely pass some classes.
Now that I am older I notice it a lot more. Dedication is a common thread in all stories of success. From Einstein to the kids in my grade that made it. The ability to work hard and push through obstacles is a skill. Some people realize this at a young age and seem to have it very developed by adulthood. I find more and more that being smart is necessary but nowhere sufficient for success, be it in business or academia. All things being equal, I would weigh hardwork/diligence more than intelligence.
There's a funny thing about dedication in that it doesn't really feel like dedication if you're interested enough in what you're doing. I can recall stealing my father's old college calculus book out of the the top cabinet when I was in third grade and reading through it at night when I was supposed to be asleep, purely out of boredom with the math we were taught at that level. By then, I barely had any concept of a variable, and yet I somehow managed to teach myself a decent portion of calculus in the next three years.
These days, I struggle to apply that level of effort to anything I'm required to do.
These days, I struggle to apply that level of effort to anything I'm required to do.
I find that some concepts are slippery to some people, and others are slippery to others, but when you approach a slippery subject and an excellent teacher explains in the right way to you (or a bright peer), it becomes easier to understand. That combined with practice is good enough to fill you in and hard work does indeed compare with prodigy. I don't know if I believe in prodigies -- any piano prodigy I've known has played piano from a young age. Same with math. Practice is everything. Exercise your brain muscle and it gets bigger.
True, in piano though (as well as in string instruments) where finger dexterity is very important it really helps if you get this ingrained in your fine motor control at an early age. Simply put you can distinguish someone that has spent 5 years learning to play the piano from age 6 to 11 from someone that learned to do the same thing from age 30 to 35 with ease. Even if you were to stop at age 11 and restart when you're 30 you will probably still be ahead of the pack.
And then there is the 10,000 hour rule to consider:
http://www.gladwell.com/outliers/outliers_excerpt1.html
And then there is the 10,000 hour rule to consider:
http://www.gladwell.com/outliers/outliers_excerpt1.html
[deleted]
Very interesting. I've had experienced like this. So I guess it's some sort of learning strategy. Perhaps you blitz the field from certain approaches and are slower at others. Do you have any theories about what your strong approach is?
In general I've found I'm best when I'm self-motivated, which really shouldn't be surprising to anyone on HN.
All the math courses I took up to Calculus BC were through the Stanford distance learning program (EPGY), which really helped enforce this. I also took three courses on C programming from them in 2nd-3rd grade; I was pretty terrible at the time but something from must have them stuck with me really well, because when I finally came back to C 10 years later it was nearly effortless.
Similarly, almost all the skills I would say I've learned really well in recent years (e.g. SIMD assembly) are things that I've learned on my own time outside of class.
Another related observation is that I tend to learn on an S curve ( http://en.wikipedia.org/wiki/Sigmoid_function ). For example, take the process of learning x86 SIMD assembly: it took me days to write my first assembly function (from scratch with no knowledge whatsoever other than an example for syntax and documentation pages). Then it took me mere hours to get acquainted with SIMD basics and start writing basic SIMD functions. In just a few more days I had written dozens of optimized SSE versions of MMX functions...
...and yet it probably took me a whole year after that to gain enough experience to feel comfortable writing extremely complicated functions from scratch.
In other words, initial learning is slow, then learning builds on itself to vastly accelerate speed, but then honing ones' abilities is slow again.
All the math courses I took up to Calculus BC were through the Stanford distance learning program (EPGY), which really helped enforce this. I also took three courses on C programming from them in 2nd-3rd grade; I was pretty terrible at the time but something from must have them stuck with me really well, because when I finally came back to C 10 years later it was nearly effortless.
Similarly, almost all the skills I would say I've learned really well in recent years (e.g. SIMD assembly) are things that I've learned on my own time outside of class.
Another related observation is that I tend to learn on an S curve ( http://en.wikipedia.org/wiki/Sigmoid_function ). For example, take the process of learning x86 SIMD assembly: it took me days to write my first assembly function (from scratch with no knowledge whatsoever other than an example for syntax and documentation pages). Then it took me mere hours to get acquainted with SIMD basics and start writing basic SIMD functions. In just a few more days I had written dozens of optimized SSE versions of MMX functions...
...and yet it probably took me a whole year after that to gain enough experience to feel comfortable writing extremely complicated functions from scratch.
In other words, initial learning is slow, then learning builds on itself to vastly accelerate speed, but then honing ones' abilities is slow again.
[deleted]
The article in text format (rather than MS Word):
http://gist.github.com/raw/190081/6731e974f7175b26e351b40dff...
http://gist.github.com/raw/190081/6731e974f7175b26e351b40dff...
This is a link to a Word document. WTF? Do people still write those things?
apparently mathematicians use them. extremely smart ones.
And here I thought that LaTEX is the standard for mathematicians.
On a tangent, bragging about intelligence you can't apply is like bragging about a car you can't drive. So you see, can't overvalue innate intelligence. Although it may just be a sign that he has no reason to be in touch with today's trends.
Huh? I don't think I've ever known him brag about his intelligence (note: I don't know him personally, but I do read his blog), and he has about as much intelligence to brag about as any person alive. And he applies it all the time; that's his job, and he is very very good at it.
Your comments look like they're meant as a criticism of Tao, but I don't see how to apply any of them to anything about him. Would you like to clarify?
(Note: the document linked from here does include the word "intelligent" -- in some such phrase as "I may be considered an intelligent child". It may be worth bearing in mind that at this point he was a 10-year-old who had been asked, on account of his prodigious mathematical skills, to talk to a bunch of very eminent adults. It may also be worth bearing in mind that he immediately goes on to say something like "but I still have a whole lot to learn before I have any hope of being as wise as any of you in this room".)
Your comments look like they're meant as a criticism of Tao, but I don't see how to apply any of them to anything about him. Would you like to clarify?
(Note: the document linked from here does include the word "intelligent" -- in some such phrase as "I may be considered an intelligent child". It may be worth bearing in mind that at this point he was a 10-year-old who had been asked, on account of his prodigious mathematical skills, to talk to a bunch of very eminent adults. It may also be worth bearing in mind that he immediately goes on to say something like "but I still have a whole lot to learn before I have any hope of being as wise as any of you in this room".)
They aren't a criticism of Tao. It was a tangent. I'm saying that intelligence that can only be applied to a few esoteric fields is overrated. I thought a thread about child prodigies would be a good opportunity to make that comment.
He seems to be quite up-to-date otherwise: blogging about new results:
http://terrytao.wordpress.com/
It's absolutely true -- the best way to learn something is to have to teach it to someone else. You must get organized and this focuses you.
I got a 5 on the Calculus AB AP exam when I was ~10 years old and around the same time got a ~1490 (I don't remember exactly) on the SAT. 6 years later, I did no better on the SAT than I did when I was a preteen and I didn't feel as if I was any better at math than I had been 6 years earlier.
Despite the fact that I had done so well at math as a child, I was never competitive in the higher-level math competitions like the AMC and AIME, where I consistently scored mediocre (~110 on the AMC, ~0-1 on the AIME). I went to school with plenty of classmates who scored near-perfect on both competitions and yet had not been nearly as much of a supposed "prodigy" as I was.
Today, I doubt I am even above average at my college; I am relieved to have simply passed the math courses required for my major with barely-tolerable grades. And yet whenever I read stories of "real" prodigies, I never see anything like this--they always seem to continue their trend and prove to be brilliant mathematicians. Or is this selection bias?
I still don't fully understand what happened. Does mathematical ability plateau at an early age? Is there something special about courses beyond basic calculus that are inherently more difficult? Did my skill simply stop developing because I lost interest?