http://en.wikipedia.org/wiki/List_of_child_prodigies I've been looking over this and also reading on different famous people like von Neumann and it seems that all of them demonstrated the ability to apply complex math at a young age.How did they learn that?I mean how does a 14 year old get access to books on harmonic analysis?Also how did they have the discipline to follow trough?I mean aren't people impulsive and inattentive in their teen years?I mean most kids wouldn't even find math interesting, let alone study advanced math with pleasure.I only started to get a slight interest in mathematics around age 20, before that I had never heard about stuff like Fourier series for instance.I feel like I wasted my youth, the math I learned in high school was basically nothing,, when I look at the volume we learned in high school, it looks like something you can learn in 1-3 weeks.12 years down the drain.

I taught myself algebra out of my father's college books when I was in first grade. I was fascinated by the stuff. I have very clear memories of this world of numbers just sort of opening up in front of me. I've lost a lot of that since then. While I was far ahead of my peers in mathematics in the early years, as I started branching out into other interests and stopped focusing so much on the math, I slowed down a bit. If my father had pushed me more (he didn't, at all; I was self driven) then I might have kept with it for longer and done greater things. I think that is a key part of these prodigies: parents who take part. After all, it's the parents that provide the guidance and nurturing for any child; if that guidance includes an emphasis on something, especially if that is something the child finds interesting, then you can keep it going for awhile.

In a way, children are far more intelligent than adults. In a manner of speaking, anyway. Their developing brains are hard-wired from the ground up to empower learning. As the brain matures, it tends to lose a great deal of that expansive capacity and flexibility. It is interesting to note that these childhood prodigies tend to have an advantage only in their formative years. By the time they hit university, many of their peers have caught up and they tend to be more on an equal footing.

I don't have a scientific explanation for you, but I would imagine it has to do with brain function or chemistry. Things just make sense and come easy to some people. Just like some people excel at sports or are great artists. When things come easy to you, I'd imagine you'd get bored and want to apply/challenge yourself to the next logical step (i.e. college courses in high school, post-grad research when you're 16, etc.) instead of being bored out of your skull learning things you could do in your sleep.

Quite often people who are extremely successful have obsessive personalities and are driven to succeed (with the help of their natural abilities).

A great place to pickup such books? The library. Also, old used bookstores. Lots of great math texts to be had in those places.

If you're in the USA then... The schooling system is designed around the idea of going at the pace of the stupidest or slowest child in the room. Note that slowest doesn't mean stupidest, just means you might not pick up things as fast as others do. There are a few problems with this approach:

• It causes extreme boredom in people who AREN'T that slow, which generally leads to them acting out in various ways. The end result, at least in america, is that you get diagnosed with "ADD" or "ADHD" and doped up on drugs, rather than the core issue being dealt with.
• The amount of material covered is minuscule compared to the amount of material you actually need to cover. This is why most american high school graduates are completely unprepared for college. This is also why you should attend a JC after high school (in the USA at least) to prep yourself for college. Classes are harder, they cover the material faster, and your assignments match college standards significantly closer than those of a high school essay.
• Modern teaching methods are terrible at encouraging interest in disciplines like mathematics and physics. They rely on rote memorization of material to pass some standardized tests, rather than focusing on application and derivation of those theorems/identities/etc. This promotes boredom in intelligent students, and promotes the idea that these subjects are "hard" simply because most students have no idea HOW to apply some random identity they've memorized. Memorization is not learning.

All that being said, if you leave your education in the hands of others, you will only learn what they want to teach you. Which is a poor way to learn. People who found themselves interested in topics (such as mathematics) often overcame many boundaries in their way. They did this because they enjoyed the material and the study of it, parents would buy them books.. or they would work jobs and buy the books themselves. They would find mentors and write letters. Etc.

I mean how does a 14 year old get access to books on harmonic analysis?Also how did they have the discipline to follow trough?I mean aren't people impulsive and inattentive in their teen years?I mean most kids wouldn't even find math interesting, let alone study advanced math with pleasure.

We are talking about a few in a thousand.

I think anyone at any age can get into math, usually when people say "I don't understand math", what they mean is "I can work it out but where the hell am I going to use that?" or at least that's what I always thought, especially since 'we' are taught math in a "do this when you see this sum / question" way. It really is a shit way to teach for many reasons and the applications that are almost always engineering / finance are minimum at best, leave nothing to the imagination and are very specific that at 12 year old or whatever age you are, are never going to need. For example, you will spend weeks on working out sums in some major topic, then get presented with one or two real life scenarios at the end talking about some parabola shaped landmark or area under a sail and you think to yourself "so I learnt all this so I can understand a that?". It doesn't exactly make you want to go home and scream "I want to be a mathematician when I grow up because its exciting and fun"

I got really into math when I was younger simply by asking a lot of questions that I wanted an answer for and they always lead to math regardless of topic, this demand for answers resulted in a obsessive relationship with math and I guess at a later age programming too. The true power of math is never exposed well and I am yet to come across a math teacher who makes math exciting.

I personally believe 'through math, all dreams can become reality', as for these mathematician you speak of, read some of the stories on how certain theorems came about, it is almost like reading some fantasy fiction where some hero is in pursuit of some magical item.

Just a quick question: how long did it take you guys to finish calculus 3?I know it's relative, I mean some people "finish" by just passing the exams, while others study 5 times more in their free time and gain deeper knowledge, but I'm looking for a raw estimate.Also it's a good idea to share at what intensity you learned.I mean some say "took calc 3 in 50 hours", but was it divided over a long period of time or just a week...or maybe even 50 hours straight?(some people can probably can do that).

Modern teaching methods are terrible at encouraging interest in disciplines like mathematics and physics. They rely on rote memorization of material to pass some standardized tests, rather than focusing on application and derivation of those theorems/identities/etc. This promotes boredom in intelligent students, and promotes the idea that these subjects are "hard" simply because most students have no idea HOW to apply some random identity they've memorized. Memorization is not learning.

But what you may have left out is it's always been this way. Teachers can't figure it out, they memorized their numbers and it improved their memory, but the average students they witness in their college classrooms seem to have trouble.

Back to the prodigy topic.

They were taught the math at a young age of course, some could even have been tutored, because child labor laws be damned their parents are going to make money off them somehow. Cynical intonation aside, it's a nature vs nurture issue.

Math may have just happened to be a really big thing in their family. TV was a really big thing in most households not too long ago. Maybe one generation of prodigies became drama and talk-show celebrities.

I taught myself calculus quite early and while the rest of the class were a good bit behind, but I'd never consider myself a prodigy and have since lost a lot of the interest I had - I'd now struggle with many things that I once found natural and simple (part of that is growing old, part is broadening your interests and not having time any more). But while I can only speak for myself, I feel that it's similar for many others - it just tickles a part of the brain that gets you curious about stuff, and if you're sufficiently tickled, you're going to go there. From that point it's just a matter of having access to the necessary text books and other references.

One thing that helps is not to have parents who hold you back.

Many parents purposely (and misguidedly) prevent exposing their children to too much information in order to not “burn them out”. This concept is utter rubbish. You couldn’t burn out a normal child’s mind if you tried.

For example, I often claim that if I had a kid, she would learn C++, piano, at least 2 languages, and chess starting from a very young age (starting at birth for languages and no later than 5 for the rest).

Someone told me that that would be ridiculous and overload the child’s mind. “The idea that a child’s mind can be overloaded is what is ridiculous. They learn best the younger they are, so that is the time to be teaching for best (and prodigal) results.”

A perfect example is a kid I “met” on the plane leaving America when I first left. The woman behind me was a Japanese who married a South Korean and they lived in America. Their 4-year-old son spoke fluently and with no accent in all 3 languages.

Potential prodigies are all over the place.

Actualized prodigies are only rare because they were not exposed early enough. You don’t become a 5-year-old prodigy on piano if you starting playing piano at 8.

L. Spiro