Academic Math Advice: Geometry Problem Solving for Success

[Broccoli21]

Hey All,
I am an student at an elite math/sci college, and am planning on majoring in math and theoretical physics.
However, my high school preparation is a bit different than most. I went to a pretty mediocre high school, and coasted through the super-easy math classes there. I fell in love with real math later on (I spend about 4 hours a day reading random math books), but I have never really been exposed to good problem-solving. However, I have read both volumes of Apostol, and Axler's Linear Algebra. So I feel pretty confident in my abilities to learn higher-level math.

However, whenever I try to do AMC-level problems, I get a few algebra ones, but can never figure out the trickier geometry ones! I feel that everyone else (that is good at math at my school) got a superb education in geometry and elementary math. Also, most successful mathematicians (and physicists) seem to have done fairly well on math competitions.

For those too lazy to read: I want to study higher math but never got good at elementary math problem solving (esp geometry). Are such skills necessary for success in math? Am I too far behind to do well in the Putnam? Is there something that I can do to remedy my lack of skills in this area?

Thanks in advance!

Broccoli

 

[Mépris]

Get a book on Geometry, get the concepts down and do the problems? :)

 

[Broccoli21]

@Mepris:
Well, that's obviously good for just learning the material. But I doubt that i'll be able to learn enough to do reasonably well on the putnam, right?
So are the problem-solving skills that you get from these competitions necessary or that useful for being a professional mathematician?

 

[micromass]

So are the problem-solving skills that you get from these competitions necessary or that useful for being a professional mathematician?

No. Not at all.

Competitions are very different from research. During competitions you have a limited time to solve a number of problems. During research, you get much time to solve one problem.

I know many professional mathematicians who would never do good in competitions, but who are fabulous in research!

Competitions math does not require you to do new things. It wants you to know a lot of stuff (like inequalities) and apply it.

Professional mathematicians must know a lot of stuff (but not necessarily the competition sutff), must be good with abstractions and must love what they do.

You don't need to be good in competitions to be a good mathematicians. There is little correlation.

 

[Broccoli21]

You don't need to be good in competitions to be a good mathematicians. There is little correlation.

This is very reassuring!
I was worried that my lack of practice over the years in problem solving would hurt me in research. Even though its not needed, would it still help to practice elementary problem solving? Or is that useless for upper-level study?
I mainly care about learning the mathematics. Doing well in competition is secondary. That said, would studying for the Putnam make me a better mathematician in any way? Or would I be better off studying more advanced math?

 

[micromass]

This is very reassuring!
I was worried that my lack of practice over the years in problem solving would hurt me in research. Even though its not needed, would it still help to practice elementary problem solving? Or is that useless for upper-level study?
I mainly care about learning the mathematics. Doing well in competition is secondary. That said, would studying for the Putnam make me a better mathematician in any way? Or would I be better off studying more advanced math?

Doing things like the putnam doesn't hurt. You'll learn how to think logically and how to approach a problem, this isn't bad. You might even learn some new math.

Then again, if you have the choice between studying for the Putnam or doing undergrad research, then I would do research.

It's a condsideration you need to make. Putnam will help you progress, but so will research or reading advanced math books. Do whatever you like best!

 

[Broccoli21]

It's a condsideration you need to make. Putnam will help you progress, but so will research or reading advanced math books. Do whatever you like best!

Thanks for the advice, micromass!

 

[Functor97]

No. Not at all.

Competitions are very different from research. During competitions you have a limited time to solve a number of problems. During research, you get much time to solve one problem.

I know many professional mathematicians who would never do good in competitions, but who are fabulous in research!

Competitions math does not require you to do new things. It wants you to know a lot of stuff (like inequalities) and apply it.

Professional mathematicians must know a lot of stuff (but not necessarily the competition sutff), must be good with abstractions and must love what they do.

You don't need to be good in competitions to be a good mathematicians. There is little correlation.

I realize that speed and timing are different in research but the ability to solve competition style problems is still probably a good indicator of mathematical talent. I mean the problems in the putnam are real mathematics, much more so than many undergraduate problem sets, which often require using the ideas your professor has provided you with.

 

[micromass]

I realize that speed and timing are different in research but the ability to solve competition style problems is still probably a good indicator of mathematical talent. I mean the problems in the putnam are real mathematics, much more so than many undergraduate problem sets, which often require using the ideas your professor has provided you with.

Let me put it like this:
If you do good in putnam, then you'll likely do good in research.
If you do bad in putnam, then you might still do good in research.

 

[Functor97]

Let me put it like this:
If you do good in putnam, then you'll likely do good in research.
If you do bad in putnam, then you might still do good in research.

yes, but the question is how good? 3 of the 4 fields medalists last year were imo medal winners. So competitions are probably a good indicator of some degree of talent, especially at the highest levels.

 

[micromass]

yes, but the question is how good? 3 of the 4 fields medalists last year were imo medal winners. So competitions are probably a good indicator of some degree of talent, especially at the highest levels.

Let's not compare ourselves to fields medalists...

Many mathematicians I know personally are really bad in competitions. But they are very good researchers, but no fields medalists.

 

[Functor97]

Let's not compare ourselves to fields medalists...

Many mathematicians I know personally are really bad in competitions. But they are very good researchers, but no fields medalists.

Yes, i just thought it was interesting to note the correlation between competition winners and field medalists.

 

[Broccoli21]

I think you definitely need mathematical talent to be good at upper-level competitions.
The kinds of minds that produce fields medals seem to start training pretty early. Or maybe the causation is reversed (it takes early training to become a fields medalist)?
However, I really don't know if I'm good at competitions when I've never been to one before. And if I do, won't it will be hard to separate failure due to inexpedience from failure due to inability?

 

[Functor97]

I think you definitely need mathematical talent to be good at upper-level competitions.
The kinds of minds that produce fields medals seem to start training pretty early. Or maybe the causation is reversed (it takes early training to become a fields medalist)?
However, I really don't know if I'm good at competitions when I've never been to one before. And if I do, won't it will be hard to separate failure due to inexpedience from failure due to inability?

Yeah you need mathematical talent, but mathematical talent isn't something easily quantified. As far as i am aware, there is no gene which we can pinpoint and say "golly gosh he has a lot of mathematical talent". I don't buy the old i.q. argument either.

Its seems the only way to demonstrate mathematical talent, is by doing well at research mathematics!

 

[Sankaku]

My take:
- Mathematical competitions are like sprinting.
- Mathematical research is like mountain climbing.

Is there crossover? Yes. Not really as much as people think, though.

 

[chiro]

Hey All,
I am an student at an elite math/sci college, and am planning on majoring in math and theoretical physics.
However, my high school preparation is a bit different than most. I went to a pretty mediocre high school, and coasted through the super-easy math classes there. I fell in love with real math later on (I spend about 4 hours a day reading random math books), but I have never really been exposed to good problem-solving. However, I have read both volumes of Apostol, and Axler's Linear Algebra. So I feel pretty confident in my abilities to learn higher-level math.

However, whenever I try to do AMC-level problems, I get a few algebra ones, but can never figure out the trickier geometry ones! I feel that everyone else (that is good at math at my school) got a superb education in geometry and elementary math. Also, most successful mathematicians (and physicists) seem to have done fairly well on math competitions.

For those too lazy to read: I want to study higher math but never got good at elementary math problem solving (esp geometry). Are such skills necessary for success in math? Am I too far behind to do well in the Putnam? Is there something that I can do to remedy my lack of skills in this area?

Thanks in advance!

Broccoli

Hey Broccoli21 and welcome to the forums.

I do small research of my own in the holidays and I will graduate next year with a degree in math.

The most important thing for research is persistence. You kind of have to have a (many) leap(s) of faith. There really is no guarantee for research. You can get a level of confidence based on what is already out there, but in the end you are going into the unknown and you don't really know what will happen, what you will learn, what results you will find, and what the progress will be like. If you don't like this kind of uncertainty, I suggest you stay away from research.

Also I want to point out that if you do research, find some colleagues to bounce your ideas off and tell people what you are working on. It doesn't matter if you think your research is trivial or not, most research isn't trivial: if it was it would have been figured out already.

Just be aware it's not like textbook problems you are probably familiar with. With my research I have been banging my head against brick walls for a very long time, and it was only till last semester that I got a few ideas from a course on wavelets to pursue. So in that vein, take as much mathematics as you can, because even if you aware that something exists, you can always look up suitable references to use that tool in your toolkit.

When you are in university/college take the opportunity to explore different math and also in different contexts. Math is a very broad area and the context in its applications can be very different even in the most subtle ways. You might find that you don't like the pure way of thinking, and in taking an applied math course you find that you are interested in developing better numerical calculus algorithms. You might find statistics interesting enough to be a statistician even though you had preconceived ideas about it being "boring". So take the opportunity to explore as much as you can to make a more informed decision.

Good luck to you and I hope it all goes well, and that you learn something and have fun along the way.

 

[Broccoli21]

Thanks for your advice everyone. Its a lot of help to me.

The most important thing for research is persistence. You kind of have to have a (many) leap(s) of faith.

I am going to do math only because I enjoy it so much, and I can't see myself doing anything else. I understand that faith/risk is required, but was mostly wondering if my slightly disadvantaged background would have much of an impact on my future.
I will heed your advice about taking some breadth classes (i have some applied math and control theory classes in mind), and will do some research this summer. Maybe I'll learn more about what's right for me.

p.s. i just realized that physicsforums is super awesome

 

[chiro]

Thanks for your advice everyone. Its a lot of help to me.
Maybe I'll learn more about what's right for me.

Thats a good idea. We all have preconcieved notions about things, but sometimes the reality is a lot different and that happens to all of us some point in our lives.

 

[Mépris]

On the note of stats being perceived as "boring"...

I know of a guy who could do really well if he wanted to (in math) and I was telling him about my stats lessons. As I was telling him about how I enjoyed doing it, he shot back with "I really found it annoying/boring as I just couldn't see the point in figuring out the order in which to arrange things."

At the time, I only had exposure to permutations but anyway, we don't get to cover much stats in high school, but I was trying to get him to "see" the coolness of it and I'm not so certain how to go about that. My initial argument was that starting with the simple is a good way to help build an intuition for one to do more complicated tasks later on but that didn't really convince him. Any ideas?

 

[Broccoli21]

On the note of stats being perceived as "boring"...

I have always found statistics terribly boring. Easy stuff can lead to boredom. However I have enjoyed my theoretical probability theory class and my combinatorics class.

If he wants to do math later, maybe he would enjoy learning some more advanced combinatorics (from a book or something). But from my experiance, it takes more interesting problems/facts to get people interested in a certain area. Its really tough to get people to see beauty and coolness in math a lot of times.