Escape Velocity

First published April 24, 2020.

The current state of mathematics competitions is incredibly unwelcoming to beginners, especially high schoolers. Here are my thoughts on why this is the case, and how to alleviate this problem.

Starting Costs

It takes an incredible amount of energy for high school students to actually get into the competitive mathematics scene. This problem doesn't exist for middle schoolers for a couple of reasons.

1) The competitions are more simple so theory is minimal. The problems, though usually boring, are approachable.

2) The lack of a good way to train for MATHCOUNTS makes it so that the only way to practice is to do problems.

3) There's usually an actively involved coach and a larger peer group (since entry costs are low).

But when you're in high school none of these are true anymore. Local competitions are not a reliable introduction to competition math, so the AMC is the only introduction to competition math that's cheap, convenient, and reliable. But the AMC is a very harsh introduction to competition math, especially for juniors and seniors. The introductory questions are either too boring, too weird, or too hard. 2019 was a particularly egregious year - here you have beginners hoping that maybe they'll be able to do a couple of questions and get around 80 points even though pretty much two thirds of the test is inaccessible to them. Then two weeks later score reports come back and they learn that they scored 60-70 because of stuff like this.

This is very discouraging and can very quickly kill interest. Most people say that in the long run, learning the strength of their competition encouraged them. But I suspect survivorship bias is at play here. [1] There are a lot of times when this kind of stuff gives off the undesirable message "you suck" and completely kills off a kid's interest. This actually happens surprisingly often, and the most common way to counteract this is to be dragged through it by force until you're past escape velocity. Often this role is played by the parents and it ends up being a balancing act where the parents have to make their kids try something, but if they push too hard any interest is completely killed.

It turns out that parents are actually really bad at fulfilling this role. But it helps tremendously to have a friend dragging you along to do something that matters. [2] If you hang out with smart people and want to do a smart person thing, chances are one of your friends are involved in it. So force them to take you along for a ride. It'll be helpful for them too. And if your friend insists on dragging you along - seize the opportunity. This will help you significantly lower the starting costs of getting into something meaningful. [3]

One of the biggest issues is that few people have externalized the way they learn math. And very few of them know how to recreate the experience of getting into math competitions. It's very easy for math competition kids to get into physics, chemistry, computer science, and so on is because USACO has a very smooth learning curve by nature, and physics/chemistry competitions have their material covered at school. But it's very hard for those physics kids to get into math competitions because there is no good introduction to math competitions.

I think the only long-term solution to this problem is to create a low-stakes introduction for high schoolers that's thrilling enough to get beginners to come to the scene, similar to MATHCOUNTS. A previous version of the draft had "that feels important enough," but I think this is a misleading way to think about it. I think deep down middle schoolers know that they'll have to start contributing to the community in some way eventually. The realization hits them just late enough so it doesn't scare them away from the scene, and I think this is the best of all worlds. And the thing about MATHCOUNTS is it's also cheap (on your side) and convenient - you make the tryouts for your club (if there are any to begin with) and your coach will take care of all the logistics for you.

The biggest problem is for those without a community of other smart kids in their geographical area. For these kids all of these problems have always been there, and as an added bonus, they get to deal with them without guidance. For these kids I highly recommend going to a summer program - you'll experience what it's like to have a group of really smart kids and you'll be able to stay connected to them to some degree. But I live in a big competition math area, so I can't really help much in this regard.

The reason math competition isn't really able to spread out of its hearths is because it's really easy to lose people along the way. We need to do a better job of reaching out to people and helping them find their way.

Introductory Texts

I've tried to get into several competitive academics, including debate, physics, chemistry, and computer science (basically the entire list). I think maybe one of the reasons I had trouble doing this is because the introductions are very abstract and philosophical. For example, Learn C++ takes this up to eleven. [4] Perhaps something similar is happening with math for a lot of kids; they're having trouble getting into it because there's no action. There's no excitement. It's just a bunch of explanations about a topic, and I think increasingly often kids just get bored and stop reading at some point, and they have to actually force themselves to continue reading until it gets interesting.

In fact I think this is a large reason there's such a huge gap in mathematics competitions; most people who've made AIME can honestly say "I felt like it was just going to happen," while many people just fall behind and lose interest. It isn't a gap in intelligence - it's a gap in interest. The people who're drawn to math by some combination of curiosity and enjoyment are far likely to do better than the people that are doing it because they feel like they should for a useful reason. The qualifier "for a useful reason" is very important - speaking personally, there are times when I haven't worked on math for a while and feel like I should do math because not having done it for a while feels wrong. Not wrong because "I'm not spending my time productively," because I get this same feeling when I work on homework for too long, much to the chagrin of my grades. Wrong because I haven't done math. [5] The hardest part about getting into a competitive academic is usually having enough interest to force yourself through the beginning - the momentum will carry you from there.

But here's the issue with just "getting rid of the fluff" - it isn't fluff. This type of philosophical stuff is actually important. It's also general, which is why the best place for it to be is in the front. In fact the same issue exists with the theory for a handout, and this definitely isn't "fluff." So what can textbook designers do, and what can students do?

For textbook designers, make the philosophy and theory feel out of the way. The problems are what matters. Theory is just incidentally a toolbox you can refer to, and philosophy is just a rough idea of "how and when should I use these tools?" Make your design reflect this. Publishing companies: take care to physically design these things to be out of the way - they can be referenced at will but do not take the focus away from the problems. Student teachers: Make your theory feel out of the way by making it as concise as possible.

And make your text naked. What that means is you don't need to add filler sentences or whatever to inflate page count, or to make the text not feel naked. If what you're writing is worth reading, there's no need to dress it up. In fact it's much easier to understand when it's not dressed up. Present your conclusions the same way you thought of them, but streamlined. In this way what you write will reflect reality. As an example, consider Projective Geometry in math olympiads. Every text will explain the projective plane, cross ratios, and harmonic bundles. Then you get what feels like a "configuration dump," where you have a bunch of configurations that imply harmonic bundles. This is the part where inexperienced writers would try to dress it up. [6] But in fact, laying it bare as a configuration dump is not only more concise, it signals to the reader that it is a configuration dump. When you're writing, just make sure what you're writing about is significant - being interesting will naturally come with being useful.

In particular, put "skip" tags on skippable stuff. (This includes useful stuff that can be returned to later when the interest barrier is cleared.) If there's something you honestly think someone who knows it already can skip, communicate this. Either do this implicitly by putting this in an "Extra" section, or a Chapter 0, or explicitly say "If you already know this, skip it."

For students, I recommend the following four step process.

1) Skim the theory so you have some minimal idea what's going on.

2) Do problems (and inevitably fail).

3) Go back and read the theory while looking for something that solves the problem(s) that you failed on.

4) Repeat until the theory becomes second nature to you.

I must emphasize that this will not work unless the problems are significant. So if you're reading a textbook filled with boring and routine problems, you're reading a bad textbook and you need to put it down. A good first order approximation of interesting problems is if they're from a reputable competition, and a good second order approximation of interesting problems is if they feel like they could be from a reputable competition.

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[1] In particular, the type of people who become notable enough to speak about math competitions are not representative of the general population of kids who become interested in math.

[2] Note that things that matter don't have to be "important." Even if admission officers and your parents don't care, I'd argue that your friends asking you to hang out/bake cookies/water the school garden/etc are much more real and matter more than stuff like homework or filling your life with empty extracurricular activities. (I wonder what would happen if the best way to get into college was to just live your life fully.)

[3] This is one of the many reasons smart people like to hang around other smart people. I'd go as far to claim that if you had to predict someone's intelligence with only one metric, that metric would be how smart their friends are.

[4] I only succeeded in getting the motivation to start learning C++ when I decided I would pretty much ignore all of the philosophy. This is not to say Learn C++ isn't good - on the contrary it is the best resource I've seen. (It's also free.)

[5] I think this is maybe why a lot of people also don't breach the computational to olympiad barrier, because they perceive it as separate subjects and philosophy is really prevalent during olympiad.

[6] This doesn't happen because in this specific case because inexperienced writers will generally not write about projective geometry.