There's quite a nice quote on the subject from Terence Tao (Livingston, Sir Ken; The Element; pp 100-101):
"I think the most important thing for developing an interest in mathematics is to have the ability and the freedom to play with mathematics -- to set little challenges for oneself, to devise little games, and so on. Having good mentors was very important for me, because it gave me the chance to discuss these sorts of mathematical recreations; the formal classroom environment is of course best for learning theory and applications, and for appreciating the subject as a whole, but it isn't a good place to learn how to experiment.
Perhaps one character trait which does help is the ability to focus, and perhaps to be a little stubborn. If I learned something in class that I only partly understood, I wasn't satisfied until I was able to work the whole thing out; it would bother me that the explanation wasn't clicking together like it should. So I'd often spend a lot of time on very simple things until I could understand them backwards and forwards, which really helps when one then moves on to more advanced parts of the subject.
I don't have any magical ability, I look at a problem, and it looks something like one I've already done; I think maybe the idea that worked before will work here. When nothing's working out then I think of a small trick that makes it a little better, but still is not quite right. I play with the problem, and after a while, I figure out what is going on. If I experiment enough, I get a deeper understanding. It's not about being smart or even fast. It's like climbing a cliff -- if you're very strong and quick and have a lot of rope, it helps, but you need to devise a good route to get up there. Doing calculations quickly and knowing a lot of facts are like a rock climber with strength, quickness, and good tools; you still need a plan -- that's the hard part -- and you have to see the bigger picture."
"I think the most important thing for developing an interest in mathematics is to have the ability and the freedom to play with mathematics -- to set little challenges for oneself, to devise little games, and so on. Having good mentors was very important for me, because it gave me the chance to discuss these sorts of mathematical recreations; the formal classroom environment is of course best for learning theory and applications, and for appreciating the subject as a whole, but it isn't a good place to learn how to experiment.
Perhaps one character trait which does help is the ability to focus, and perhaps to be a little stubborn. If I learned something in class that I only partly understood, I wasn't satisfied until I was able to work the whole thing out; it would bother me that the explanation wasn't clicking together like it should. So I'd often spend a lot of time on very simple things until I could understand them backwards and forwards, which really helps when one then moves on to more advanced parts of the subject.
I don't have any magical ability, I look at a problem, and it looks something like one I've already done; I think maybe the idea that worked before will work here. When nothing's working out then I think of a small trick that makes it a little better, but still is not quite right. I play with the problem, and after a while, I figure out what is going on. If I experiment enough, I get a deeper understanding. It's not about being smart or even fast. It's like climbing a cliff -- if you're very strong and quick and have a lot of rope, it helps, but you need to devise a good route to get up there. Doing calculations quickly and knowing a lot of facts are like a rock climber with strength, quickness, and good tools; you still need a plan -- that's the hard part -- and you have to see the bigger picture."