Stop Paying Me Per Hour

Occasionally I am approached by parents who ask me if I am available to teach their child in olympiad math. This is flattering enough that I’ve even said yes a few times, but I’m always confused why the question is “can you tutor my child?” instead of “do you think tutoring would help, and if so, can you tutor my child?”.

Here are my thoughts on the latter question.

Charging by Salt

I’m going to start by clearing up the big misconception which inspired the title of this post.

The way tutoring works is very roughly like the following: I meet with the student once every week, with custom-made materials. Then I give them some practice problems to work on (“homework”), which I also grade. I throw in some mock olympiads. I strongly encourage my students to email me with questions as they come up. Rinse and repeat.

The actual logistics vary; for example, for small in-person groups I prefer to do every other week for 3 hours. But the thing that never changes is how the parents pay me. It’s always the same: I get N \gg 0 dollars per hour for the actual in-person meeting, and 0 dollars per hour for preparing materials, grading homework, responding to questions, and writing the mock olympiads.

Now I’m not complaining because N is embarrassingly large. But one day I realized that this pricing system is giving parents the wrong impression. They now think the bulk of the work is from me taking the time to meet with their child, and that the homework is to help reinforce what I talk about in class. After all, this is what high school does, right?

I’m here to tell you that this is completely wrong.

It’s the other way around: the class is meant to supplement the homework. Think salt: for most dishes you can’t get away with having zero salt, but you still don’t price a dish based on how much salt is in it. Similarly, you can’t excise the in-person meeting altogether, but the dirty secret is that the classtime isn’t the core component.

I mean, here’s the thing.

  • When you take the IMO, you are alone with a sheet of paper that says “Problem 1”, “Problem 2”, “Problem 3”.
  • When you do my homework, you are alone with a sheet of paper that says “Problem 1”, “Problem 2”, “Problem 3”.
  • When you’re in my class, you get to see my beautiful smiling face plus a sheet of paper that says “Theorem 1”, “Example 2”, “Example 3”.

Which of these is not like the other?

Active Ingredients

So we’ve established that the main active ingredient is actually you working on problems alone in your room. If so, why do you need a teacher at all?

The answer depends on what the word “need” means. No USA IMO contestant in my recent memory has had a coach, so you don’t need a coach. But there are some good reasons why one might be helpful.

Some obvious reasons are social:

  • Forces you to work regularly; though most top students don’t really have a problem with self-motivation
  • You have a person to talk to. This can be nice if you are relatively isolated from the rest of the math community (e.g. due to geography).
  • You have someone who will answer your questions. (I can’t tell you how jealous I am right now.)
  • Feedback on solutions to problems. This includes student’s written solutions (stylistic remarks, or things like “this lemma you proved in your solution is actually just a special case of X” and so on) as well as explaining solutions to problems the student fails to solve.

In short, it’s much more engaging to study math with a real person.

Those reasons don’t depend so much on the instructor’s actual ability. Here are some reasons which do:

  • Guidance. An instructor can tell you what things to learn or work on based on their own experience in the past, and can often point you to things that you didn’t know existed.
  • It’s a big plus if the instructor has a good taste in problems. Some problems are bad and don’t teach you anything; some (old) problems don’t resemble the flavor of problems that actually appear on olympiads. On the flip side, some problems are very instructive or very pretty, and it’s great if your teacher knows what these are.
  • Ideally, also a good taste in topics. For example, I strongly object to classes titled “collinearity and concurrence” because this may as well be called “geometry”, and I think that such global classes are too broad to do anything useful. Conversely, examples of topics I think should be classes but aren’t: “looking at equality cases”, “explicit constructions”, “Hall’s marriage theorem”, “greedy algorithms”. I make this point a lot more explicitly in Section 2 of this blog post of mine.

In short, you’re also paying for the material and expertise. Past IMO medalists know how the contest scene works. Parents and (beginning) students less so.

Lastly, the reason which I personally think is most important:

  • Conveys strong intuition/heuristics, both globally and for specific problems. It’s hard to give concrete examples of this, but a few global ones I know were particularly helpful for me: “look at maximal things” (Po-Shen Loh on greedy algorithms), “DURR WE WANT STUFF TO CANCEL” (David Yang on FE’s), “use obvious inequalities” (Gabriel Dospinescu on analytic NT), which are take-aways that have gotten me a lot of points. This is also my biggest criteria for evaluating my own written exposition.

You guys know this feeling, I’m sure: when your English teacher assigned you an passage to read, the fastest way to understand it is to not read the passage but to ask the person sitting next to you what it’s saying. I think this is in part because most people are awful at writing and don’t even know how to write for other human beings.

The situation in olympiads is the same. I estimate listening to me explain a solution is maybe 4 to 10 times faster than reading the official solution. Turns out that writing up official solutions for contests is a huge chore, so most people just throw a sequence of steps at the reader without even bothering to identify the main ideas. (As a contest organizer, I’m certainly guilty of this laziness too!)

Aside: I think this is a large part of why my olympiad handouts and other writings have been so well-received. Disclaimer: this was supposed to be a list of what makes a good instructor, but due to narcissism it ended up being a list of things I focus on when teaching.

Caveat Emptor

And now I explain why the top IMO candidates still got by without teachers.

It turns out that the amount of math preparation time that students put in doesn’t seem to be a normal distribution. It’s a log normal distribution. And the reason is this: it’s hard to do a really good job on anything you don’t think about in the shower.

Officially, when I was a contestant I spent maybe 20 hours a week doing math contest preparation. But the actual amount of time is higher. The reason is that I would think about math contests more like 24/7. During English class, I would often be daydreaming about the inequality I worked on last night. On the car ride home, I would idly think about what I was going to teach my middle school students the next week. To say nothing of showers: during my showers I would draw geometry diagrams on the wall with water on my finger.

So spiritually, I maybe spent 10 times as much time on math olympiads compared to an average USA(J)MO qualifier.

And that factor of 10 is enormous. Even if I as a coach can cause you to learn two or three or four times more efficiently, you will still lose to that factor of 10. I’d guess my actual multiplier is somewhere between 2 and 3, so there you go. (Edit: this used to say 3 to 4, I think that’s too high now.)

The best I can do is hope that, in addition to making my student’s training more efficient, I also cause my students to like math more.

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