Should I go back to my math basics?

[RickSilver]

So I'm currently a senior in high school in the U.S. and I'm hugely interested in mathematics. I was accepted to the University of Chicago (my top choice) and I'll be matriculating there in the fall. My background in math is a bit of an odd one. I used to hate it as a kid, but I always read novels and wanted to be a writer. In my freshman year, however, I came upon a senior who was the president of the mathematics club. He showed me Euclid's proof of the infinitude of primes and I was so stricken by the beauty of the proof that I basically took to math like nothing else existed, and I've spent most of the time in the intervening years exploring mathematics on my own, taking courses at the local uni in real analysis (currently in a course developing the Lebesgue integral and it's awesome), linear algebra, geometry, and discrete math.

My question though involves math competitions. While I seem to definitely have some facility for proof based mathematics (I've done pretty well in the proof based courses at my local uni and I seem to manage to do most of the exercises in the math books I read) I seem to have a terrible time with problem solving especially at math competitions. I've never managed to move onto AIME from the AMC for example. The time constraints make me feel uncomfortable and while I'm interested in the topics covered on these tests (I'm hugely interested in geometry, less so in elementary number theory) I seem to not be able to solve these problems in the time constraints. Should I go back to basics and try to cultivate more skill in problem solving? is it really all that important a skill for a mathematician to cultivate?

 

[Quantum Defect]

So I'm currently a senior in high school in the U.S. and I'm hugely interested in mathematics. I was accepted to the University of Chicago (my top choice) and I'll be matriculating there in the fall. My background in math is a bit of an odd one. I used to hate it as a kid, but I always read novels and wanted to be a writer. In my freshman year, however, I came upon a senior who was the president of the mathematics club. He showed me Euclid's proof of the infinitude of primes and I was so stricken by the beauty of the proof that I basically took to math like nothing else existed, and I've spent most of the time in the intervening years exploring mathematics on my own, taking courses at the local uni in real analysis (currently in a course developing the Lebesgue integral and it's awesome), linear algebra, geometry, and discrete math.

My question though involves math competitions. While I seem to definitely have some facility for proof based mathematics (I've done pretty well in the proof based courses at my local uni and I seem to manage to do most of the exercises in the math books I read) I seem to have a terrible time with problem solving especially at math competitions. I've never managed to move onto AIME from the AMC for example. The time constraints make me feel uncomfortable and while I'm interested in the topics covered on these tests (I'm hugely interested in geometry, less so in elementary number theory) I seem to not be able to solve these problems in the time constraints. Should I go back to basics and try to cultivate more skill in problem solving? is it really all that important a skill for a mathematician to cultivate?

I believe that you should ask yourself if the "real life" of a mathematician is like these competitions. What do you think?

From talking to my mathematician friends, I understand that when they receive a grant from a funding agency to do work, they use the money to go to conferences, talk to other mathematicians, listen to other mathematicians talk about how they have solved their own problems, etc. They do all of this to help them develop their ideas for the problems that they are working on. This does not sound to me like what goes on in a mathematical competition.

Sure, you have to be able to demonstrate competence in your student life, and tests represent probably the easiest way to test for competence. Does this kind of test demonstrate the kind of creative intelligence needed to solve unsolved problems? [all questions on any test I have ever taken have been ones for which the solution is known]

Also, your 18 year old self is not such a good predictor of where you will be at your most creative time in life.

It sounds to me like you will do well. Good luck at U. Chicago!

 

[IGU]

Should I go back to basics and try to cultivate more skill in problem solving? is it really all that important a skill for a mathematician to cultivate?

No. Solving artificial problems within small time constraints is pretty much unimportant. Do the things you enjoy. If you want to go back to basics, read Measurement or learn mathematical logic.

 

[homeomorphic]

Some successful mathematicians appear not to have done very well on that type of thing from what I've heard. I think especially the time constraints aren't such a big issue. Doing hard problems from textbooks is good enough.