[L]et’s just say that you’re a prospective undergraduate who wants to study one subject more than any other… [H]ere’s how I think a prospective who self-identifies as highly interested in one topic or subject ought to work through the questions involved.
First, are you sure that you’re really that interested in a single topic or issue, so sure that you want to make that a primary axis of your decision about where to go to college? Why are you that sure? Do you just like the topic or are you thinking already of a profession narrowly based on it? Are you sure based on an understanding of what a likely undergraduate-level curriculum around that topic looks like, or based on what you know about it from your high school experience? Are you making that choice with a wider awareness of the subjects that even a small college will offer to you that virtually no high school curriculum can focus on?
Second, are you SURE? Really? Then you’re a really unusual applicant. Most of what prospectives think they’re interested in is not the same as what those subjects turn out to be, and most of their interests are based on a very incomplete understanding of the range of academic subjects even within a particular discipline.
Third, if you’re really that kind of unusual person, absolutely certain that your first, second and last priority is to comprehensively study a single subject area while you’re an undergraduate and that this priority is unlikely to change, then: a) don’t apply to any small undergraduate institution; b) pick a place with as few general education requirements as possible; c) find a program in your preferred subject at a large institution that is stuffed to the gills with faculty and courses and make sure undergraduates with a dedicated interest get access to the most prestigious or high-powered faculty in your area of subject interest. The relative difference between one small college and the next doesn’t really matter to you if you’re that driven, because in either case, they’re going to have a relative paucity of resources in comparison to a large institution. You don’t really care about any of the other resources at an institution if you’re that focused: just your area of study and whatever direct supporting skill areas you need (say, language or quantitative training). An undergraduate applicant who is this specifically focused is really more like a proto-graduate student, and should use selection rules much closer to what a graduate student might employ.
This sort of student may be extremely unusual in history, but I would say that they are only somewhat unusual in mathematics. By time I got to Harvard, I knew I wanted to be a mathematician and was struggling to learn as much as I could, as fast as I could. I was not at all the most sophisticated student in my year, and I think that the top undergraduates and high school students are noticeably better now then we were then.
So, I thought it might be interesting to see which of our readers identify with Burke’s hypothetical student, and what they think of his advice.
I did what Burke suggested. The schools I seriously considered were Harvard and MIT, which are large schools in the sense Burke is talking about. (Although in absolute numbers they are only medium sized.) Harvard had very weak core requirements, and I exploited them: I took 2-3 math courses a term, and did basically the minimum required of me in other subjects.
Did it work out well for me? Sure! I had a great time in college, learned a lot, and am now a successful mathematician. Would it have been better to be more rounded? Maybe. I often regret my gaps in history and economics, and I sometimes wish I could read Hebrew. But at the time, I couldn’t have imagined taking the time out of math to gain those skills, whereas now I am interested and am making the time to do so on the first two fronts.
Did I win up much better off than students who took a more balanced approach to their studies? I don’t think so. In particular, Ben Webster went to Simon’s Rock, a tiny liberal arts college, and wound up an outstanding mathematician. (He’s not the only example, but he’s a striking one.) I’d be curious to hear from people who went to small liberal arts schools, or who went to large schools but pursued a well rounded education, and wound up mathematicians.
Finally, a piece of advice of my own. If you are certain that you want to work in a particular field, you should begin meeting other people in that field; both people older than you and contemporaries. I spent my high school summers at MOP, and benefitted enormously from meeting both fellow mathematical high school students and older mathematicians in this way. I think that the PROMYS and Ross programs probably do an even better job than MOP of bringing high school students into the community of mathematicians. If these sort of programs are not available to you, then walk into the math department of your local college. Modern mathematics is really different from the picture you will get from books or contests, and the sooner you meet its practitioners, the sooner you will know whether it is for you.