@ homeomorphic: you are my role model! I wanted to be a fantasy author most of my life but now I want to switch to math. Out of curiousity, how lacking was your math background entering college compared to most math majors?
I wanted to be a fantasy illustrator. I took calculus in high school and passed the AP exam, so I did have a head start, but not like one of those guys who learns calculus at age 12. I didn't pay much attention in class. My performance was uneven because I didn't take it very seriously. I considered studying for tests to be bad luck. I was pretty good about doing the homework, though. So, I sometimes managed to set the curve on tests.
In college, I spent a lot of time as an electrical engineering major before switching to math. Studying electrical engineering delayed me in some ways, but it also contributed something to my intuition and mathematical knowledge.
I have a friend who is also trying to finish his PhD who started math at a very late age. He got into programming in college and then math. He had to start with trig, I think, but his progress was very rapid and in a few years ended up in grad school. I knew another guy in grad school with a similar story, but he ended up not passing his quals and now he's probably riding his bike a lot or being some kind of outdoor guide. Still, he got into grad school with a fellowship, and those are hard to get, so he must have at least done really well in undergrad.
Some famous mathematicians seem to have gotten into serious math relatively late. Raoul Bott was an electrical engineering major, I think. Atiyah was a chemistry major for a while (although I think he may have done some math earlier, too). Stephen Smale apparently had mixed grades and was more interested in chemistry in high school. But you don't have to be famous. There are all different levels of mathematicians.
I never took advanced math classes K-12 and I did not pay one lick of attention in any of my classes (unfortunate) so I fear that I missed out on the crucial building of math intuition that takes place during formative years.
I don't think paying attention in your classes would have done much for your intuition, unless you had better teachers than is the norm. I will say, that, in my case, I always was inspired by how Einstein said he thought in pictures. I liked to think that way, too. And I did get a little bit of practice with it in high school and maybe earlier.
With piano, one who doesn't start getting lessons around age 6 or earlier has basically zero chance of making it as a concert pianist.
I play piano, myself, and I'm not sure how true that is. I started at age 12, but I suspect if I just decided to quit math and just play piano, if I practiced 10 hours a day over several years, I could be a concert pianist. Maybe I'm wrong. You might not be the best because the best ones head such a huge head start, but I think you can do it if you are willing to put that kind of effort into it.
Those early years are very pivotal for how we develop so I find it amazing that you were able to switch your focus from the arts to math so successfully.
Well, I may never be the greatest mathematician. But it looks like I will at least be a math prof next year or more likely in 3 or 4 more after a postdoc. Math research is really overwhelming. I'm still not sure how good I will turn out to be at it. My thesis has gone slowly, but recently, I have been picking up the pace in order to graduate this year. If nothing else, I know I could be a great expository writer and help a lot of my students to succeed.
Please give me some tips and insight into your journey, because like I said: you are a living example of what I wish to achieve!
Well, it's a long story. In high school, before I got interested in math, I was starting to get interested in a lot of different intellectual things, like philosophical questions, psychology, and eventually physics. With my new found intellectual curiosity, I just wanted to learn as much about everything as I possibly could. So, it was very natural for the question of learning how to learn to come up. Actually, to a great extent, both the intellectual curiosity and the idea of learning how to learning and becoming smarter both came from my best friend at that time. So, I devoted a considerable amount of thought into the best ways to learn as much as I possibly could. I researched it on the internet a bit. Memory techniques, creativity techniques, etc.
One of the things I took away from it was the importance of doing a lot of review. I toyed around with the idea of trying never to forget anything that I learned. After all, what was the point of studying it if you just forget everything you worked so hard to learn?
One technique I had was to keep reviewing the subject in my mind, over and over again, making sure I knew all the main points. Each day, I would try to summarize everything I had learned in the classes I was taking, and I spent some time reviewing previous classes as well. I've gotten worse about doing this lately because grad school is so overwhelming.
Actually, I think having a foggy memory of a subject is good enough for some purposes, and you probably shouldn't try to hard to retain things you don't need. By the time you get to grad school, the maintenance problem becomes difficult because you just know so much, it's hard to keep track of whether you are remembering everything. I still don't know the best solution, but it's safe to say it isn't just letting everything you learn slide away. In math, there is also the idea of just remembering how to derive everything, rather than remembering the stuff itself.
Anyway, the approach you take can make a big difference.
