The Information for Counselors page at Promys is not very sapecific: what does one need to do to apply? On which basis are candidates selected? Does anyone have any information from past years (I'm willing to apply next year)?
There will be an application form on the website during the application season. It gets removed during the rest of the year to make sure nobody wastes time filling it out when there's no hiring going on, but this ought to be clarified on the website. (I'll look into fixing this. Full disclosure: I've been involved with PROMYS in different ways over the years, but I've never played any role in hiring counselors. This answer is not official or on behalf of PROMYS.)
The counselors are chosen based primarily on their mathematical knowledge, talent, experience, and enthusiasm. This is judged partly from the application form and partly from letters of recommendation. Of course other factors can play a role. For example, if you come across as a jerk, you'll be less likely to be hired. However, you can't make up for a weak mathematics background by being really good at working with teenagers.
Many counselors are former PROMYS students, but many others had no connection with the program before applying. (I don't have statistics handy.) Alum applications are often easier to judge since they already have track records at the program. Whether this is an advantage or not depends on the track record, of course, and PROMYS wouldn't hire a less qualified alum in preference to other applicants.
$\begingroup$ Dear Dr. Cohn, like the OP, I'm interested in PROMYS. Can you clarify how the application form ascertains "mathematical knowledge, talent, experience, and enthusiasm"(e.g. are there problems to solve? an interview?) and which mathematical knowledge is typically seen as adequate (e.g. elementary number theory, abstract algebra, etc)? Thank you. $\endgroup$– user2015Aug 18, 2014 at 9:07
2$\begingroup$ @Nick: The last I saw, the application form was relatively brief (no problems to solve or the like), and mainly asked general questions about background, experience, and interests. Two letters of recommendation are required, and they carry a lot of weight. I don't think any specific background is absolutely mandatory, but some knowledge of number theory and abstract algebra is certainly very helpful, and it's important to have enough mathematical maturity and perspective that you can guide the high school students in problem solving and rigor. $\endgroup$ Aug 21, 2014 at 2:46
$\begingroup$ P.S. See also promys.org/program/counselors-faq for more information about the role of counselors in the program. $\endgroup$ Sep 27, 2014 at 18:44
See the FAQ: "Are counselors always former PROMYS students? Not always, but often."
If they're like most American summer programs, they prefer to hire alumni. When there aren't enough good alums, they'll hire people that some alum or staff member knows personally. Strangers are usually a last resort.
So if you're asking here, you're not in a great position....but you have plenty of time before next summer to make friends with someone from the program.
$\begingroup$ It's probably true that some summer camps strongly prefer to hire alums or other people already personally known, but that's not the case for PROMYS. $\endgroup$ Sep 27, 2014 at 18:43