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My advisor strongly encouraged me to check out two (highly renowned) American universities for my PhD study.

Her collaborators in these two institutions are good people and great mathematicians, so, assuming I can get in, working there would be very pleasant.

However, I'm confused by the structure of these graduate schools. In the first two years I'm supposed to follow courses and do a comprehensive written exam and an oral one which are mostly about topics (complex analysis, basic functional analysis, ODEs) that we do in the first two years of undergraduate study in my current institution.

The exam questions from the past years appear to be quite difficult, but revising those relatively elementary topics and spending a great amount of time solving difficult problems on them seems like taking a step back after a Bachelor's and a Master's degree (which had a quite significant research component).

So my questions are the following:

  1. what is the rationale behind this structure of graduate school in the US?

  2. why is it effective?

  3. should I be concerned about "wasting time" revising basic topics in my area instead of diving directly into a research program after a Master's degree?


Added context from comments:

"Students who already know the material can take the exams in the first month of their Ph.D. program" is exactly what my advisor's collaborators told me. However, although the core material is well-known to me, it appears that the exam consists of many problems in a short amount of time and that such problems are mostly about clever ways to sum series, evaluate multiple integrals, do contour integration, solve tricky ODEs, and so on. That is, it is about elementary things but requires lots of exercise. That's why I'm concerned that it could be an unnecessary detour.

migrated from mathoverflow.net Jul 17 '17 at 17:17

This question came from our site for professional mathematicians.

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    American universities typically emphasize breadth at both the undergraduate and graduate levels much more than is my impression at European universities. Consider for example the American tradition of the comparatively broad liberal arts education. It is a cultural difference, but with certain advantages. (Which is not to say that specialization is without advantages.) – JDH Jul 17 '17 at 12:37
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    Could you clarify which country you are studying in as an undergraduate? – Yemon Choi Jul 17 '17 at 17:04
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    Note that the type of examinations given to PhD students also vary a lot within the US: across fields, departments, and universities. In the program I was in, for example, there was no coursework-based preliminary examination: the preliminary exam was a thesis proposal and was designed to make sure students had a plan after their 2nd year or so for completing their research. – Bryan Krause Jul 17 '17 at 22:39
  • That sounds like the course of study that the (physics) program I attended laid out for students coming in without a masters; there was an understanding that students with a masters would be evaluated to figure out how much of that classroom prep they needed to do—i think the comprehensive exam was still required. You might contact the department and ask if they have an arrangement for better qualified incoming students. – dmckee Jul 18 '17 at 14:56
  • My two guiding comments are: (1) spending time revisiting important and fundamental mathematics is always beneficial, even if we feel that we've seen it before; (2) a program's comprehensive exams do (or at least should) represent what that program considers important and fundamental mathematics. – Greg Martin Jul 19 '17 at 6:03
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  1. The rationale is because graduate programs in the US are generally structured assuming the incoming student does not have a masters, and got a bachelors in the US, which can sometimes involve an embarrassingly small amount of math (and maybe more importantly, there is a very wide variation in what students learn at different universities, due to the flexibility of the US system). So, you're entering the system at a drastically different level of preparation from most people. Of course, it seems off to you. My experience at quite a number of US schools is that almost all incoming students need some sort of preparation of this type, often even ones with credentials suggesting that they shouldn't. I think some of the logic here is that it's easier to set out somewhat tougher requirements and then make exceptions to let people accelerate than the other way around.

  2. I think it's hard to assess whether it really is. But I don't know anyone who seriously suggests we don't need to do something along these lines. If something is covered in the syllabus of these classes, it is because the students don't consistently come into the program knowing it.

  3. Maybe I'm being too dismissive here, but I am never sympathetic to these complaints. If you really know the subject, the classes will not take much of your time. No one will stop you from talking to professors about research or doing reading on the side while you're taking the classes, and generally the timetables around exams and classes are flexible. That time will only be wasted if you decide you want to waste it.

  • 14
    Thank you. This is a useful and insightful answer. About 3.: my concerns stem from the fact that, although the core material is well-known to me, it appears that the exam consists in solving many problems in a short amount of time that are mostly about clever ways to sum series, evaluate multiple integrals, do contour integration, solve tricky ODEs, and so on. That is, it is about elementary things but requires lots of exercise. That's why I'm concerned that it could be an unnecessary detour. – user76182 Jul 17 '17 at 17:55
  • Complementary to this good answer, for some cases, you may also request to transfer credits from your masters. Although, you would still need to complete qualifying/preliminary exams. Consider the fact that you are taking those classes again to prepare for these exames, to excel at it, practice your skills, learn new ways or improve your way of thinking, and also meet and learn from other brilliant minds. – iled Jul 17 '17 at 21:03
  • Do you mean embarrassingly small amounts of math for math grad school or in general? – Azor Ahai Jul 17 '17 at 23:10
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    @Zyl I can't comment since I can't look at the exam, but at least consider the possibility that if the exam looks too tricky to you, it might be you don't know the field as well as you think you do, and you actually need to take the class. I don't mean to be insulting, but every professor on here has had experiences with students swearing up and down (and presumably believing) that they know an area, and then freezing up when asked to do a relatively simple problem. – Ben Webster Jul 18 '17 at 12:22
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    @BenWebster Thanks for your comment. "If the exam looks too tricky to you, it might be [...] you actually need to take the class." You're absolutely right: I need to revise and do exercise. Indeed, I took the relevant classes and did exams on those topics 3-4 years ago as an undergraduate (and they rarely pop up in my research area). However, I don't know if it is convenient to put research on hold for one or two years to do this revision, especially after that I have completed a research Masters program. – user76182 Jul 28 '17 at 15:45
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Speaking as a long-time graduate program director and adviser in a mid-level US department (at a big public university), I'd emphasize that most of our Ph.D. students in past years have had a less-than-adequate preparation in some basic areas. This is true especially for US students coming from a typical 4-year college math major but is also true for some international students. On the other hand, most of our students aim for a college teaching career, or perhaps a non-academic job, rather than aiming for a research career in some kind of mathematics (or statistics, part of our department). At the most elite departments, it is taken for granted that everybody plans to do research. But my own contemporaries in the 1960s Ph.D program at Yale actually went in many different directions.

Like most other US departments, we have had an evolving procedure involving written (and perhaps oral) qualifying exams. Anyone applying to one of the top few programs should certainly ask questions about how and when such exams are administered, and how they are evaluated. Typically no one wants to hold back a talented student, but sometimes a student overestimates his or her own talent and knowledge (and future job prospects). Even Harvard and MIT Ph.D.'s sometimes end up teaching at small colleges or out-of-the-way universities.

In any case, identifying a potential thesis adviser (or two) is equally important, though obviously it's difficult to predict one's future interests precisely or to predict the future logistics of an active faculty member (sabbaticals and other leaves can upset plans as can personal crises). Good luck navigating the US system, including the evolving immigration rules!

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In US programs I know about, e.g. UC Berkeley, there are preliminary exams (typically taken after the 1st year), on a broad range of topics, and qualifying exams, on the intended area of the PhD. Unless you went to some very, very good MSc programme, this is meant to prepare you for academic career, and broaden your view of the subject.

I wish I went to one of these, instead of diving headfirst into PhD research.

As to "why is it effective", consider excellent products the system has been producing :-)

  • 6
    This system is common at many American universities. And usually, if an incoming student already knows a topic well, it is allowed to skip the class and simply take the exam directly to verify his or her knowledge. Often, incoming students have diverse backgrounds and some students can pass out of one topic or another. In this sense, there is no time wasted, if one already knows a subject well. – JDH Jul 17 '17 at 12:33
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    At CUNY, we have qualifying exams in six topics, and students must pass exams in three of them before continuing with dissertation research. See gc.cuny.edu/Page-Elements/… – JDH Jul 17 '17 at 12:41
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    This description is very similar to what we do at Michigan, except that we reverse the terminology: qualifying exams are about broad areas of math, and preliminary exams are about the intended research area. The qualifying exams are given 3 times per year. Students who already know the material can take the exams in the first month of their Ph.D. program. – Andreas Blass Jul 17 '17 at 16:04
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    @AndreasBlass "Students who already know the material can take the exams in the first month of their Ph.D. program" is exactly what my advisor's collaborators told me. However, although the core material is well-known to me, it appears that the exam consists of many problems in a short amount of time and that such problems are mostly about clever ways to sum series, evaluate multiple integrals, do contour integration, solve tricky ODEs, and so on. That is, it is about elementary things but requires lots of exercise. That's why I'm concerned that it could be an unnecessary detour. – user76182 Jul 17 '17 at 16:35
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Supplementing the other good answers, and echo-ing one of @BenWebster's points: I, too, am unpersuaded (statistically...) by complaints that "prelim exams" are too speed-oriented and/or too low-level/cleverness oriented. True, these are easy accusations to make, and complicated to attempt to refute.

However (without claiming anything about the questioner's situation), almost without exception, the on-the-ground complaints I've heard in this direction were from people who were not nearly as competent or mature as they imagined... and easily stood to profit greatly by further attention to basic things.

In particular (although, yes, there certainly can be "bad" instances of various prelim exams), an important meta-point that I (in writing such exams) try to test for is a higher-level understanding, especially which enables examinees to "see the forest for the trees" (or vice-versa), and realize that the given question is "just another" instance of an iconic, cliched, and useful/important idea... perhaps manifest in a slightly prankish-seeming context.

What I tell my own people is that they "are done" with "algebra, complex, real, ..." or whatever labels these exams are given, when they realize that there are "only eleven (or pick your favorite smallish-number) different issues that will come up". Prior to some point, they cannot see the commonalities (which is not so good), and then after some point, they do see the commonalities, and everything is easy.

In the latter context, almost all the complaints I've received are from people in a state prior to seeing how simple these basic parts of mathematics are (or can be). This is not a moral failing, but it is not a useful professional state (especially as part of a belief system...).

  • 2
    +1 Years ago, a student was furiously complaining about a particular question (not in my department but in computer science). According to the student, the question tested only the ability to do complicated arithmetic, for which reasonable people would use a calculator. (Calculators were not allowed in this exam.) What the question actually tested was the ability to see that complicated arithmetic was entirely unnecessary and the question could be solved almost trivially. – Andreas Blass Jul 24 '17 at 3:44
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I can offer a more cynical point of view, as someone who has studied for and passed exams three times, at three different U.S. grad programs in mathematics: I completely agree with you. If you think preparing for exams is a total waste of time for you, you're probably right.

There might be students who actually learn things in the course of preparing for an exam, but there are also students for whom the whole exam routine actively interferes with meaningful learning. I am solidly in the second category, so I minimize the cognitive resources I allocate to exam preparation because it diverts time and energy from doing something useful. As another reply here points out, the exams at each school are usually recycled variations of the same manageably-sized bag of tricks, so it's probably less time-consuming than it seems at the outset.

From a philosophical/pedagogical point of view, it's unfortunate that programs don't even try to be a little more data-driven about the effectiveness and efficiency of this hidebound process. I could give a very long critique detailing why the coursework-written exam-oral exam gantlet is completely counterproductive, starting with the educational research that high-stakes exams reduce long-term retention. But this is the sort of thing that no one has any interest in hearing because they have no interest in doing anything about it. As you can see from the other replies, most (not all!) professors within this system are under the impression that such exams are a reliable way to determine whether a student "really knows the material," so why fix what (to them) isn't broken?

  • finally. someone with the critical mind... there is whole research book and critic on this subject. it is called Disciplined mind by Jeff Schmidt – SSimon Jul 30 '17 at 17:42

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