On Tuesday 3 August 2010, I had an opportunity to observe the Awesome Math Camp at UCSC.

The camp is for high school and middle school students (grades 6–11) who are very good at math and looking for some real challenge. It is a 3-week residential camp, with about 6 hours of math a day.

The students I saw there were mostly male and mostly Asian (more East Asian than South Asian, I think). I think that this is driven mainly by which parents are willing to pay a fairly substantial summer camp fee for just mathematics. Many of the students there are hoping to improve their performance on various math contests (such as AMC-10 and AMC-12) or improve their chances of getting onto the US Math Olympiad team.

The snippets of classes I observed were not contest math prep, though, but a variety of different topics in math (strong induction, graph theory, game theory, … ). They were preparing students more for being math majors in college than for contests, though any sort of math problem-solving practice can help with contests. Personally, I think the broader approach they take is better for the kids and more fun.

Here are examples of the problems the kids were working on while I was visiting that afternoon (less than 1/30th of what the kids were getting).

- . The professor was showing the students how to prove that was a polynomial in $n$ with rational coefficients and that its leading term was . Although the math is beautiful, I was not greatly impressed with the presentation. It reminded me too much of my grad student days in math, when a professor would step through a proof without motivating any of the steps or letting students know where he was headed. I saw the students in the class with their heads down furiously scribbling notes, in the hope that they’d be able to reconstruct the argument and make sense of it when the professor was finished.
- Another problem was a game in which there are 5 boxes initially with 0,10,20,30,40 pieces. On their turn each player can choose one box and move any number of pieces from that box one position left (the leftmost box being a black hole that swallows all pieces moved into it). Whoever moves the last piece wins. The task here is to figure out whether the first player or the second player has a forced win, and what the winning strategy is. The kids were making some progress on figuring it out, but the professor was running out of time and ended up giving the answer. The kids were clearly engaged in the lesson, but didn’t quite have the tools or the time needed to guess the solution. They had no trouble understanding why the solution was correct, once the trick was shown to them. After the class, the professor told me that the class was usually working on more advanced topics, but that the game exercise had been done to give them a little break and catch their attention again.
- Another class involved attempting to prove a theorem about bipartite graphs: if every subset of k nodes on the left is connected to at least k nodes on the right, then there exists a pairing in which each node on the left is paired with a different node on the right. In this class the students were very actively involved in trying to find the solution. Even the background chitchat in the room seemed to be about the proof. Whenever someone had an idea, the professor would invite them up to the board to show their idea, then the professor would try to find counterexamples or other gentle ways to point out the flaws, without either stifling the kids’ desire to participate or tipping his hand too much for the desired solution. This was masterful teaching, and the kids in this class seemed almost hyper with excitement about the math. I stopped observing that classroom before they got to a solution, so I don’t know if he coaxed a correct proof out of them, nor how much hinting he had to use.
- The lowest level class I saw was one in which the kids were presenting solutions to problems that had worked on earlier. One kid was presenting his (correct) solution to the problem of measuring 5 liters of water given only a 7-liter jug, an 11-liter jug, a water tap, and a drain. A few of the other kids were paying attention, but several seemed disconnected: I couldn’t tell from my few minutes there whether they were bored because they already knew all this, or bored because they didn’t understand most of what was going on: the two are often hard to tell apart. As I was leaving, the teacher was segueing into a general discussion of what integer amounts you could measure with two integer-sized jugs, probably leading to a proof based on properties of gcd. The students seemed to be more alert during this lecture portion, so perhaps they just didn’t like “call a student to the front to present his solution” segment of the class.

Overall, I was impressed by the level of the math the middle school and high school students were doing and by some of the math teaching. Although the classes were fairly large, the kids all seemed to want to be there. Of course, this was the first week of the camp—maybe some of them burn out after 3 weeks of intensive math.

I would recommend the camp to my son, who would have enjoyed 2 or 3 of the classes I observed. I doubt that I can get him to go, though, as the camp conflicts with the teen theater conservatory with the actors from Shakespeare Santa Cruz, and he’d much rather do the theater camp than the math camp. Frankly, I think that is a wise choice, as he can learn a lot of the math on his own, but theater is inherently a group activity, and the teen conservatory one of the best theater experiences he can have at his current level.

I would also recommend the math camp (in a year or two) for the 6th grader who did well on the math team I coached last year. I would **not** recommend it for students who were not at least 2–3 years ahead in math, though, as the pace was fairly fast. Students at the camp do have to choose their classes carefully,though, as the teaching styles varied enormously, and a mismatch between the student’s learning style and the professor’s presentations could result in a pretty miserable experience.