In the last week of December I got a position as the morning instructor for the A* USAMO winter camp. Having long lost interest in coaching for short-answer contests, I’d been looking forward to an opportunity to teach an olympiad class for ages, and so I was absolutely psyched for that week. In this post I’ll talk about some of the thoughts I had while teaching, in no particular order.

**1. Class Format **

Here were the constraints I was working with. After removing guest lectures, exams, and so on I had four days of teaching time, one for each of the four olympiad subjects (algebra, geometry, combinatorics, number theory). I taught the morning session, meaning I had a three-hour block each day (with a 15-minute break). I had a wonderfully small class — just five students.

Here’s the format I used for the class, which seemed to work reasonably well (as in, if I were to teach the class again I would probably not change it very much.)

- (0:00-0:10) I usually started the class with a quick warm-up problem (something pretty easy), just to soak up time from latecomers and give students a chance to get ready and glance through the handout. (If you give smart students a pretty handout, the first thing they will do is look through it, regardless of what you tell them to actually do.)
- (0:10-1:30) Afterwards I would go through the lecture, both theory and examples, up until the break. On average this got split up with about half the time for the theory and half the time for the examples. I typically let students try the examples themselves for five minutes (again, smart students will automatically start on the problems regardless of whether you tell them to or not) before I discuss the solution, just so they at least have a feeling for what it is — I consider it immoral to start talking about a solution before students have had a chance to try a problem.
- (1:30-2:40) After a break, I would give the students a long period (a little over an hour) to try the practice problems in the last section of the handout. Since the class was so small, I would prepare about 5-7 practice problems and then let each of them pick a different problem to start working on. (Once they solved their own problem, they would go on and try other ones.)During this time, I was able to take advantage of the small class size in a pretty great way: throughout the hour I would walk around the room talking to each of the students about the problem they were working on. In particular, I tried to make sure every student at least solved the problem they started with.
- (2:40-3:00) In the last 20 minutes of class or so, I had each student present the solution to the problem which they worked on. I think the main utility of this is that it forces the present-er to know clearly in their head what their solution is. This was actually possibly more useful for my feedback than for the students: if a student could present the solution to their problem to the class then I knew they understood at least each of the individual steps.Overall I think this format did more or less what I intended it to do, and will definitely be re-using it if I ever teach an olympiad class in this style again. Though I don’t know how well the second half might work in a bigger room: I actually had to do a bit of running to keep up with questions and ideas that the students came up with while working, and of course the presentation time is proportional to the number of students you have (maybe 3-5 minutes each). So if I had, say, 10 students, I would probably re-think how to run the end.

**2. Picking Topics **

I think it’s general kind of useless to teach a class where you do a mix of unrelated problems. For example, I never really liked “functional equations” as a class. And don’t even get me started on the typical “divisibility” class. That’s what the IMO Shortlist is for, and the students already have that. Anyone competing at this level already knows how to pick up a collection of problems and practice against it. Class needs to something more than that.

My idea is that problems in an olympiad class should be linked by some underlying, specific theme. It doesn’t have to be a specific technique, but it can be. The reason is that this way, you can see the theme re-appear over and over again. By the time you see it the fifth time, hopefully things start to click.

Let me phrase this another way. Suppose I gave wrote down the following:

You can tell what each of the ‘s, ‘s, ‘s have in common by looking for a few moments. But what happens if I intertwine them?

This is the same information, but now you have to work much harder to notice the association between the letters and the numbers they’re next to.

I think the class is kind of the same idea. If you want to draw out the idea of orders, pick a bunch of problems that involve orders in spirit. They don’t have to be exactly the same problem, but they should be reasonably related.

So to produce a good olympiad handout, you need to have something to say. I think my Chinese Remainder Theorem handout is a good example (it was actually something I was considering using for the NT session, but I decided on something else eventually). I want to point out how CRT is used in constructions, so the examples and practice problems are all designed to illustrate this point. There’s a large degree of micro-control throughout the entire thing.

Honestly, I think it’s really easy to teach olympiad math badly: just pick a bunch of unrelated problems, go through the solutions one by one, then give some more unrelated problems for practice. The students will still get better, because they are practicing. But is that all you can do as a teacher?

For the record, here’s the topics I ended up using for the camp.

- Orders / Lifting the Exponent
- Irreducibility of Polynomials
- Projective Geometry
- Double Summation

**3. Narrowing Problems **

Something new I tried for this lecture was trimming a lot.

At MOP, I’d often get a handout for a MOP class with something like 30 problems on it. We’d get to pick which ones we worked on, and then we’d see or present some of the solutions in class. The issue is that, well, a class isn’t that long, so I would only be able to work on two or three problems, and these wouldn’t be the same as the two or three problems other people worked on or presented.

I think the hope was that when we went home we’d still have like 20 various problems to work on. The problem is that I couldn’t possibly have worked on the left-over 20 problems from every class even if I wanted to — there were just too many.

I fought this issue at A* by trimming down the practice problems a lot. My handouts essentially had only 5-7 problems to work on. This way, more people had looked at the same problems when it came time for presentations.

The reason I picked 5-7 was so that every student could work on (and hence present) a different problem. In retrospect I’m not sure this was a good idea. If I were to teach again, I might even cut it down to fewer than that, maybe four problems or so. That way, everyone really works on the same problems, and presentations of solutions are infinitely more useful. I would just have to work around the fact that on any given day, not all students would have a chance to present.

**4. Things I Did Badly **

Finally, here’s a couple things I wish I had fixed.

First, I made a lot of assumptions about what people knew and didn’t knew. I thought I had made the NT lecture too hard because the room was very quiet, but it in fact turned out that it was because the students had actually seen most of the order material before.

The only reason I found out was because after I had finished presenting all the order material, I asked out of curiosity whether anyone had seen this already, not actually expecting anyone to raise their hands. Instead, the entire class did. Students really are too polite — I must have bored them to tears for those first 45 minutes.

The solution to this is really simple: just ask the students if they’ve seen it before. Any teacher knows that students are shy to admitting they don’t understand what you’re talking about, but if you just ask “have any of you seen this before?” the students will in general be pretty honest. (If you phrase it as “have any of you *not* seen this before” the results are less accurate.) So that’s something I will remember to do much more of later on.

The other thing is that I likely made the practice problems too hard. I felt like I had to give too many hints: at least the students understand the solution, but I’m not sure how helpful it is to only solve the problem because the hints given amounted to an outline of the solution. In my defense, I was guessing in the dark as to the abilities of the students, and erred on the side of hard. (Any of you who do math olympiads know how useless and boring a too-easy class is; in contrast, classes which are a tad too hard can often still be beneficial.) But the point stands that my estimate was wrong.

Finally, I think I wore the students out a bit too much. I was happy with their performance in my class, but apparently they were all pretty tired during afternoon. But I think that might just be because of the way the camp is set up — six hours of class a day is really a lot, even for the very hardcore.

**5. Closings **

Overall I was quite happy with how the classes turned out, and I think the students were too (either that or they were very generous with my instructor ratings). I can’t wait until I get an opportunity like this again, but that might be a long time in coming — there really aren’t that many USAMO-level students out there as I would like!