Advice for a mathlete

Posted: 2017-08-07 , Modified: 2017-08-07

Tags: math, advice

Parent: Math


Back to Math. (Note: I’d be happy to talk if you still have questions; just shoot me an email.)

1 Advice for high-schoolers (including, but not limited to, mathletes)

First, check out General advice.

Q: Why should I care about other stuff besides math competitions? I just want to focus on math.

A: At this point it may be too early to know exactly what you want to do for a living (especially if you think that you want to do math for a living because you like math competitions – a mathematician’s job is very different from the picture of “doing math” that comes with math competitions). Interests can change, and if you’re not sure, you don’t want to be in a position where you only know how to do math and not anything else. Especially don’t just do math if you have other interests too.

High school is a good time to explore a lot of different interests (college is good for that too), so don’t feel like you have to spend all your time on math. (Paul Graham’s is a good read.) It’s good to figure out why different things are interesting, and what makes people want to do those things for a living.

Every subject has a reason for being interesting; the problem is that most of the time classes don’t move you into thinking that, and you’re not exposed to the interesting stuff. Like many other people, I went into math competitions because it was the only challenging thing where I found very interesting and offered a road forward - and I’m glad for it. Find ways to get exposed to stuff.

(It’s fine also to just focus on math in high school - but it’s important not to inertially have the attitude that “there’s nothing else worth doing.”)

2 Math competition advice

Q: I want to do well on math competitions (ex. USAMO). What do I do?

A: In addition to stuff most people learn in high school classes there is some more material you need to know (number theory, inequalities, geometry (e.g. read Geometry Revisited by Coxeter), combinatorics). What matters is your ability to tackle challenging math problems, and the ONLY way is to actually do challenging problems. The best way to practice for math competitions is by working through past tests (under Olympiad conditions). You can find them online on Art of Problem Solving or the AMC website. Regularly practice under test conditions, but also think more about the problems you didn’t solve. Do NOT passively read solutions - you do not learn anything this way. When you read solutions, asking yourself how you could have come up with them yourself - and then make sure you are able to! - perhaps some time later.

Maintain a set of problems that you can think of at any one time. When you get stuck on one, jump to another. When you solve it, reflect a little bit (though not in the middle of competition). Compare your solution and the official solution if there is one. Ask yourself how you could have come up with it faster. Is there a simpler or more natural solution? How could you apply what you learned to other problems?

As in learning anything you should organize your work. Your efforts should be additive. Keep track of problems you’ve solved so you can find them again. Write or at least outline solutions nicely so you can remind yourself of the solution anytime; note particular techniques that work, pitfalls, etc. The same goes for learning the theory. If you find yourself constantly forgetting math you learn (ex. you keep forgetting a proof of something), then slow down and learn it more completely (ex. trying to derive things for yourself). Leave behind bread crumbs.

Q: What books do you recommend?

A: I’d recommend

(There are a lot more books out there now, these are just the books I personally found most useful when I was in high school.)

It’s less important which book you choose as that you spend the time working on the problems. You can do some from one book and some from another, and/or choose one book to focus on at a time. Take a look at some of them, and then just choose one or a few that’s compatible with your current level. Do identify what areas of math you need to work on the most, and find resources to target those.

How comfortable are you with proof writing? If it’s something you feel you need work on, it could be useful to enroll in a program where you can get feedback on your work, for instance, WOOT or ~AwesomeMath Year-Round. (These programs are also nice if you don’t know where to start on your own).

Q: How much time should I spend? How much time did you spend?

A: This is up to you. It depends on what you want to get out of math competitions, and what other things you want to do. Please don’t feel like you have to spend a certain amount of time on it. It’s possible to get a lot of the benefits of math competitions in terms of general problem-solving ability without going all-out on it - if you don’t insist on winning (too much focus on winning is, in any case, dangerous). If on the other hand you care about doing well very much, then you will probably have to spend more time. It’s good to push yourself to do more than you naturally would (ex. you may want to give up on a hard problem, but you should build up some perseverance). Wanting to get better at math or liking it are good reasons for spending more time; wanting to win is not by itself a good reason.

Personally, I spent the vast majority outside schoolwork on math contest preparation (I had few other commitments - I would recommend not having many commitments if you want to spend more time on math.) While it’s easy to spin this to say that I’m “hardworking”; really, a large reason for this was that I was not exposed to other valuable things to be doing. There is unfortunately some idolization of people who “just do math” - certainly they are to be respected, but not to be imitated unless you are that sort of person.

There’s really two questions: how much of time to spend being “productive” (for a general definition of productive - ex. art is productive, not just science) and how much of productive time to spend training for math competitions. The first ratio is good to increase in any case; the second is really up to you. Don’t chase after the bonus points you get (ex. for college apps) for putting a lot of eggs in the math competition basket.

3 Beyond math competitions

’’Q: I want to do well on math competitions. It’s my only goal in the world right now. Is that healthy?**

**A:’’ It’s good to ask yourself what you want to get out of competition success. Is it the recognition? Is it the math community? Looking for recognition is unsustainable in the long run; your desire to do math should come from something deeper than just wanting to do well on competitions (though that is a perfectly okay way to start; I got interested in math a large part because of competitions, and so did a lot of other people I know). As to community, the quality of a math community isn’t all about how smart the people are; perhaps you can find others in your school, and if not, there are plenty of other math programs and camps (for me AwesomeMath was a great experience). It’s good to be around other people who like math for the sake of math, and it tends to rub off. I did not make it to MOP myself, and at this point, it matters little to my happiness or sense of self-worth. What really matters are not the results but what you get out of the training and the experience. You learn a lot of problem-solving and hard work, and those things are useful.

There’s no clear “antidote” for thinking in terms of achievement, but I think being exposed to interesting problems/projects and being around people who really care about what they do (and don’t think in terms of achievement) is the most helpful.

Q: What else should I be doing besides math competitions?

  1. On the math side:
    • Learn some collegiate math (which is quite different from competition math - you go a lot deeper).
    • Read math blogs and articles.
    • Understand how math can be applied to various subjects.
    • Teach other people math!
    • Work on a research project.
    • Find a community of math people to join.
  2. On the non-math side:
    • If you have other strong interests, pursue them!
    • Explore.
    • If you have things that you want to do but don’t know how to do - learn how to do them! High school and college are good times to do something like this, be it music, acting, philanthropy, etc. - there are classes, clubs, programs. After college, you can still get into new things, but the activation energy is a lot higher.
    • Gain general transferable skills. Math teaches a specific skill - that of problem-solving. If you don’t go into pure math research, then you will need other skills. Leadership/organization, people skills, communication, business, experimentation, experience abroad, performance, etc.

Q: What is there to math beyond math competitions?

’’Q: I didn’t do well on the competition:( What now?**

A: Performance on USAMO does not correlate in how good of a mathematician you will be (or whatever other profession you pursue).

**Q: How can I get into a good college like MIT?’’

Checked out the MIT admissions blogs. They give you a good idea of life at MIT, and what they look for in students. (Also see In addition to the standard admission stuff (test scores, grades), MIT likes people who have a good sense of “fun,” the motivation/interest/passion to pursue their own projects.