Advice for specialized undergraduates

Timothy Burke, a professor of History at Swarthmore, writes

[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.

21 thoughts on “Advice for specialized undergraduates

  1. As someone already ancient (to lift Jim Milne’s helpful
    description), I can’t help responding from the viewpoint
    of an entering student at Swarthmore in 1957. Having
    coasted through a mediocre public high school but also
    having gotten a National Merit Scholarship which then
    covered almost all costs for four years, I had to decide what I was most interested in majoring in. Since my
    high school math experiences, and most others, had been
    totally negative, I gravitated at first to an English major.
    Since childhood I had devoured everything in print, so
    why not English? Little did I know that it would kill all
    interest in reading. (Now I read mainly Proust anyway,
    in self-taught French.)

    In truth, I did find bits of mathematics exciting but had
    a major problem there: I knew I would never be as good
    as the geniuses in that field, so why bother? I still have
    that problem and am still bothered, but haven’t regretted
    the belated choice.

    One never really knows where life will lead at an adult
    age. One of my nieces insisted on studying Russian
    (her late father having coincidentally gotten a Ph.D. in Russian history), but has wound up being a mother and Fed Reserve analyst and married to a philosopher.

  2. David’s description of his focus on mathematics applies pretty well to me too.

    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.

    For me, in math, Caltech was a serious exception to that rule. Definitely they didn’t have faculty covering all subject areas, in a way that would probably bug me as a grad student or faculty member, but I wasn’t specializing enough as an undergrad to notice that then.

    Mainly, though, Caltech was special for having tenured faculty teach calculus, and having visiting faculty teach one-shot courses. There was always way more on offer than I could take, and I did try. It was a real disappointment going to Princeton, supposedly the #1 math grad school at the time, and finding so much less being taught despite their being so many more professors.

    (Warning to prospective undergrads/grads: this information is 15+ years old, and not a good way to choose one institution over the other!)

  3. I was one of those very specialized students, and I did go to a large institution with a great math program (UMCP), but I’d still give even more caution. If absolutely nothing else, take a strong supporting sequence in a related field. You might be dedicated to studying math (or whatever other subject you come across) but life may have other plans. When the one ship you’ve put all your hopes in runs ashore, you’ll need to be able to do something else.

  4. I went to UNC Asheville, a public liberal arts college, to be a high school English teacher, and I am now a mathematician. The suggestion to pick a school with as few general education requirements as possible sounds terrible to me. Core curricula in the humanities aren’t things to be avoided.

  5. I’ve heard that university study outside the US is often more specialized than it is here. In particular, it might be good to suggest that high school students in the US who have decided on their field of study should at least think about applying abroad.

  6. I can confirm that Scott’s description of university study is at least true of Australian universities. As a student, I was of the type that Burke’s advice is aimed at, and (after dabbling a little in English as well, which was only
    possible due to an unusual arrangement which I dispensed with after a year or so) I took just maths and physics courses. Having no general education requirement (and also having no homework sheets or midterms, just one final per subject on which all of ones grade rode) gave plenty of time to learn mathematics, which is what I wanted.

    Without being able to see into the future, there is always the possibility that advice, however well intentioned and thought out, will turn out to have been bad. This is not in itself a reason not to give or take advice. Similary, I don’t think a student’s decision should be dominated by the consideration of unpleasant possible contingencies
    in the future. Four years is not a short time, and so it makes sense to
    do what will make them happiest in college. If they think that they want to study mathematics in a very serious fashion, and a big university is the best way to do this, then do it. After all, if they change their mind along the way, at a big school there will be other options (even if they are not compulsory). And a degree in mathematics and the physical sciences is not the worst thing to end up with, even if the original plans of being a mathematician don’t work out.

  7. In France too studies are much more specialized, for instance there isn’t a major-minor system: if you’re into science then your first two years are a mix scientific topics only (no economics, no history…), then in the third and fourth year one studies only a single subject (even sensible dual curricula like math+physics during those last two years, although possible for a small additional fee, are only allowed to those with very high grades).

    Bigger universities there have larger libraries, which does make a difference in allowing one to learn a subject properly (larger choice of recent books, more students to talk to, etc.). So to answer Scott’s comment: if you’re sure you want to study say math or biology then it’s true that going to a large french university is a good way to focus on your subject early during your degree and meet like-minded folks.

  8. Comments posted so far reinforce my own conclusion
    that advice to younger people should be tempered by
    uncertainty about the range of real outcomes they may
    experience. There is no broad scientific study of the
    origins of research mathematicians and probably will
    never be, but the anecdotal evidence suggests caution
    about generalizing too much from limited evidence.
    If you want to prove that selective liberal arts colleges
    are the best place to start, you will note that Robert
    MacPherson and Eric Friedlander (a current AMS president
    nominee) graduated from Swarthmore in the mid-1960s,
    while Craig Huneke and other established people were
    Oberlin graduates. At the time I transferred from
    Swarthmore to Oberlin in 1959, to get distance from my
    unsolvable sexual orientation problem but failing to do
    so, Oberlin was actually the better choice for mathematics.
    I did encounter there good people with doctorates at MIT,
    Princeton, Yale, etc. And my senior honors work was
    vetted by a junior Wisconsin faculty member Charles
    Curtis who later became a colleague at Oregon. But I had
    no idea what I was doing with my life and was late
    deciding. My middle class junior high school friend
    Dick Beals went with a National Merit Scholarship to
    Yale, struggled between economics and mathematics,
    wound up with a solid career at Yale.

    Still, people who do decide early risk disappointment.
    One I recall was the son of a recent AMS president, who
    entered Cornell in 1962 when I switched to math from
    philosophy there and who had an enriched grad fellowship
    which we others envied. He didn’t make it in academia
    and apparently became a corporate lawyer. The only
    moral is to narrow your advice to students somewhat and
    hope for the best. There are roles for family background,
    intrinsic aptitude, sheer determination, sheer luck, karma.

    My own family teaches me lessons. My parents grew up
    in poverty and had almost no education, but somehow
    their three sons and daughter all got overeducated
    and never took over the small family business. My
    oldest brother started college when I did, graduated
    with junior Phi Beta Kappa from Allegheny, got a Ph.D.
    in philosophy of science at Yale (all with several kids),
    taught at New College (where Bill Thurston later went),
    and was a founding faculty member at Evergreen State
    in Olympia. My middle brother followed his high
    school girlfriend to a state teachers college, learned
    nothing but met his future wife, learned much in one
    year at Union Seminary, got a Ph.D. at Penn in Russian
    history but couldn’t make a real academic career work
    and instead raised wonderful kids. My sister is harder
    to summarize but manages to survive in the East Bay area.
    You never know. Advice comes without a warranty.

  9. Since I grew up in Brazil, I had to make my choices before even applying to the university. All my courses were mathematics, with the exception of a few pro forma requirements. It was actually really hard to get permission to attend a course outside your specialty, though I managed once or twice.

    It worked pretty well for me, but there are things I regret. For example, not learning a language (German or ancient Greek would’ve been nice). It’s very hard to learn a language later; as an undergraduate I had more time and was far more willing to be patient with quizzes and other pesky tools for pressuring students.

    I think it’s correct to say that if you are certain of where you’re going, you should avoid small colleges and go for as many advanced courses as you could possibly take at a large university. On the other hand, most of us are not certain, and for most of us life won’t be located at “research I” institutions. In that case, a broader education is better, if only because it offers other interests, broader knowledge, and so on. Liberal arts colleges offer things that help make life interesting.

    Would I be happier if I could read Greek (or German)? No, but certain possibilities would be open that are now much harder to realize.

  10. I think it’s definitely much more common for people to be focused on math early on than it is for people to be focused on another subject. (Music, and maybe art or some kind of engineering being the only other things I can think of this happening in.) But even out of these people, I think a lot of them end up switching to something else. Fortunately, math is a pretty good background to have if you want to switch into philosophy or physics or economics or a bunch of other fields (especially as these are often the ones that someone with a math background is most likely to be interested in).

    Also, I have to say that you should mention the Canada/USA Mathcamp as well. Ross and PROMYS probably give a better impression of what it’s actually like to study mathematics, but the Canada/USA Mathcamp gives exposure to a wider variety of areas of mathematics. It’s not at all clear to me which is more important in terms of figuring out whether one is interested in doing math.

  11. The suggestion to pick a school with as few general education requirements as possible sounds terrible to me. Core curricula in the humanities aren’t things to be avoided.

    Basically every other country in the world seems to manage to avoid them without suffering any terrible consequence. On the other hand, the lack of breadth requirements has not led to the creation of a race of super-mathematicians that overshadow all those educated in the US, so you can probably use this fact to argue things either way you want.

  12. I feel strongly that a more balanced undergraduate education is at least no impediment to a later career in a highly technical, specialized, competitive field like mathematics. In other words, I don’t think it matters much in the end how many math classes you take as an undergraduate (I mean, above and beyond what is reasonably standard for a math major and that you could learn at hundreds of American colleges and universities, no matter the size).

    As others have said, in many other parts of the world, university education is far more specialized. When I began as a graduate student (at Harvard), it was remarkable to me how much farther along in their studies most of the foreign students were than almost any of the American students: in some cases their reputations preceded them (and were in turn preceded by their publications). Still, American students can “catch up”, in the sense that their thesis work is not necessarily inferior to that of foreign students. A few years past the PhD is enough to rectify virtually any inequity of prior knowledge if you are serious about doing so.

    Having said all that, as a matter of personal taste I believe in the well-rounded liberal arts education. I went to the University of Chicago, which is antithetical to Harvard in terms of its distribution requirements (when I was there, half of all your classes were to be taken in in the “common core”, although about 1/3 of that half could be bypassed by a well-prepared student). I wonder how many other mathematicians can say that as undergraduates they read works of Weber and Eisenstein? (Yes, there is a trick here and a test of your non-mathematical education.) As to what extent this reading changed/improved my life since then and to what extent it is simply a reflection of the interests I’ve always had, I’m not sure, but I think that my undergraduate experiences did deepen my already extant love of language and writing.

    Nevertheless, most European mathematicians that I’ve met are “better cultured” (don’t ask for a precise definition!) than most American mathematicians, despite having ostensibly studied nothing but mathematics since the age of 19 or so. So I have cultivated quite a fantasy of what the high school education must be like in, say, France or Germany. Not to even mention that their second language skills are infinitely superior to ours — that is the one deficiency of my “specialized education” that I have not yet been able to overcome.

  13. I agree with those who believe even highly focused students should choose a less specialized education. For one, there is a pretty big gulf between what constitutes math for a high school student, and what a professional researcher does. In particular, the latter tends to be much more abstract. Not to mention the job market forcing people out. Also there are some fields which it’s very rare for high school students to know much about, and which might be a good choice for them. You see many high school math whizzes, but not too many say high school meteorology or statistics whizzes, and clearly there are those students who would be good for those fields and many might consider themselves math whizzes in high school.

    Another thing… it may not really even be an option nowadays for a highly focused student to go to a top liberal arts place, like Swarthmore which someone mentioned. Those schools are increasingly hard to get into, and their admissions processes emphasize well-roundedness to a great degree. I think a lot of the people who end out going to Caltech, to choose an extreme example, would have a hard time getting into there. If such a student had to end out choosing between Caltech and a second-rate liberal arts school then I think the choice would be determined by the quality issue.

  14. Because most of the important stuff has been said, just a collection of random comments:

    1) My next intended non-mathematical project is to get some familiarity with the ideas of John Dewey, specifically because he ties (well-rounded) education with his (peculiarly American) understanding of democracy.

    2) I do think that my broad education (I read Weber but not Eisenstein as an undergrad) has made me a better person, whatever that means. But then again I have a hard time imagining myself having had a different education.

    3) I distinctly disagree with the idea that European mathematicians are more “cultured” than American mathematicians (or at least, the subset of Americans educated at liberal arts colleges (and for the purposes of this sentence I am defining Harvard as a liberal arts college but not MIT)). Part of this might be national cultural markers; my instinctive notion of “cultured” includes familiarity with Shakespeare, but of course that’s completely unreasonable for a French or German person (who might instinctively substitute Moliere or Goethe, neither of whom I know anything about). But I seem to find it much more common for an American mathematician to have a nuanced understanding of the difficulties and complexities of, for example, the notion of “historical fact”.

    4) The specifics of what one learns as an undergrad is pretty irrelevant for most jobs anyways. What matters is that one hopefully gets a lot smarter from all that education, and most courses of study seem to accomplish this.

  15. I don’t really have an answer to the original question, except for the obvious “people should do what they want to do and avoid attending colleges whose requirements are likely to irk them.” But maybe I can at least satisfy David by standing up as one of the people “who went to large schools but pursued a well rounded education, and wound up mathematicians.”

    I went to college at a big research school (Harvard). I think I took just one math course over the minimum required to graduate, and did only one semester of graduate coursework, and spent the rest of the time doing distribution requirements and electives, mostly in English. After college I spent a year doing no mathematics at all.

    When I started my Ph.D., I certainly felt I was a bit behind many of my classmates. But, as Pete describes, it evens out very fast. And I’ve never once felt I would have been a better mathematician had I choked down a lot of grad courses when I was 20. (But again: that’s because I didn’t want to do that, and obviously lots of people do want that experience, and benefit from it greatly.)

    On the other hand, I don’t see myself as any more cultured or educated than my friends who focused more on math as undergraduates. I’ve started to think that people become cultured and educated by reading a great quantity and variety of books, and in no other way.

  16. Assuming Burke’s context— we have a student who genuinely knows that he would benefit by specializing early, and we have to advise him— I feel that little is possible in the way of general advice. I don’t see how people in this tiny and unusual segment of the population could be so alike as to merit Burke’s specific recommendations, or any specific recommendations. They could do no worse, and may well do better, by following a mix of their own impulses and the advice of people who know them personally.

    And assuming Burke’s context ignores the most important point: how can a young person know— that is, _really_ know— whether or not they would benefit from specializing early? My guess is that although almost everyone who would benefit from specializing early knows it, the majority of the people who are certain that they would benefit are wrong. (The way Burke leads into his advice, it wouldn’t surprise me if that’s his guess, too.) If my guess is right it is a bad idea to give out general advice. You might as well just say “if you will benefit by doing X, then do X.” It would do less harm.

  17. My impression is that in the U.S. system, there is pressure to have a diverse course of study, and not to specialize in one subject too early.
    (Certainly not in high school, and not in the first year or two of university as well.) Given that this is the the ambient pressure, I think of Burke’s advice more as giving permission to a very ambitious, somewhat single-minded, student to push back against this pressure and devote themselves to their passion. As Ben pointed out in his comment, in many other parts of the world, *all* university students (and even junior and senior high school students) are forced to make such choices of specialization in any event, and don’t come to any harm, as a general rule.

    If the pressure were in the other direction (as it is in those other parts of the world), this post would be discussing a different question: what should the student who is passionate about both maths and english do to pursue both subjects at university? And there would be advice of a different kind.

  18. Interestingly, my husband and I have arrived at very similar questions in the same week, without discussing it together. My version touches off from Stanley Fish’s weblog post on the NYTimes, “What Should Colleges Teach?”:

    @Alex (#14), point 4: I agree that in later life one’s undergraduate curriculum is unlikely to matter in its specifics. I wouldn’t phrase what does matter as that one is supposed to “get smarter from all that education,” but rather that undergrad should teach one critical thinking and contextualization skills–both of which aims are well served by a broader liberal arts eduation.

    @Ben (#11): It’s highly doubtful (and, from my experience at the Dutch public secondary schools, categorically false) that basically every other country AVOIDS the humanities. I had 11 required subjects in 7th grade in the Netherlands, including 3 foreign languages. Had I stayed through 8th grade, the subject count would have been 16, including both Latin and classical Greek, as well as drawing. I think *that* is what the “cultured” comment (Pete, #12) points towards.

  19. Lark,

    I agree with you about secondary school, but if you had gone to university in the Netherlands, my understanding at least is that you would have have essentially only studied one subject; that’s all I was saying. I certainly believe that European countries have a more rigorous and well-rounded secondary school curriculum that makes distribution in university less necessary. I’m just saying, they seem to do OK with a very focused university experience.

  20. When I was an undergrad at Princeton I cleverly gamed the system to avoid most courses in subjects other than math and physics – for example by placing out of the English writing requirement, taking mathematical economics to satisfy the social sciences requirement, and studying mathematical logic to satisfy the humanities requirement. In retrospect this seems a bit extreme, but after four boring years in high school I was eager to plunge into my favorite subjects, so it seemed great at the time. And luckily, education doesn’t need to stop after college, so I’ve been able to catch up on various subjects in the ensuing decades.

    I guess my only big educational regrets are not having been forced to take piano lessons and become fluent in some other languages back when I was a kid – but if I *seriously* regretted these things, I would have done something about it by now.

    I’m not sure it makes sense to give youngsters advice unless you know them personally and have a feeling for what they’re like and what might make them happy. People are so different. But I guess it’s good to have a bunch of well-argued, contradictory advice out there for youngsters to choose from, and hope that they will choose the advice they need.

  21. I second John’s comments above, but this topic also reminded me of Edward Witten’s own story. It seems he graduated from Brandeis with a history degree and a minor in linguistics, worked in a political campaign, attended the University of Wisconsin-Madison as an economics graduate student, enrolled in Princeton to study applied mathematics before switching to theoretical physics and graduating with a PhD in physics.

    None the worse for wear, apparently, for he ended up with a Fields Medal in 1990.

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