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but the exam questions -- and often, homework problems -- come from the classics, e.g., Baby Rudin, Ahlfor's Complex Analysis, Lax's Linear Algebra. This creates a huge discrepancy in one's level of training and practice, compared with the level exams one must take.

The students who know this early on, i.e., the one who go into this dept with their eyes open, are at somewhat of a significant advantage, while those who studied with lower-level / friendlier books, recommended by the dept, struggle on the exams - whether it's for a course, or comprehensive exams that graduate students must pass.

Is this done on purpose to sort of keep the barriers to entry to mathematics as high as possible?

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    I don't recognize the practice you're describing. On first glance it sounds lazy to me: most exam problems should not be taken directly from any small collection of famous texts. I would expect that if a department did that it would swiftly become known among the students and the effect would be mitigated because students would know what to study for the exams. Anyway, this practice may be localized to one specific department, or it may be a feature of some particular subset of academic culture. With the information you've provided, it's hard to know. – Pete L. Clark Nov 29 '15 at 3:51
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    I couldn't make any more out of your proposed explanation than "Because the faculty in my department are mean to undergraduate students." I don't find that very plausible. Two explanations spring to mind: (i) the faculty are not putting time and energy into their teaching, and they are grabbing exam questions too quickly and lazily. (ii) The exams are more appropriate than you claim. In my experience, some students always claim that the exams are much harder than the lectures and the homework, even when one can map each exam problem to a homework problem or an example done in class. – Pete L. Clark Nov 29 '15 at 3:58
  • I've heard the phrase "sink or swim" used to describe a number of practices in some math departments that reflect some professors' disregard for student success. I don't think this lack of consideration is necessarily as ill-intentioned as it sometimes seems. Some math professors are a bit disconnected. Some are snobs. Some wouldn't recognize good pedagogy if it bit them on the nose. // Perhaps the solution to the particular problem you noticed would be for experienced students to create a guide for new students. – aparente001 Nov 30 '15 at 0:36
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Is this done on purpose to sort of keep the barriers to entry to mathematics as high as possible?

Almost certainly not. What makes you think this might be the explanation? Unless you have further evidence, this doesn't seem even remotely plausible. As Pete L. Clark points out in a comment, it wouldn't be a terribly effective policy once the students found out, while there are plenty of other ways to increase standards if that's the goal. Furthermore, it's counterproductive: high standards are one thing, but there's no point to deliberately misleading people regarding what they need to know and then punishing them for not learning more than you asked them to. That's just not rational behavior, so if your department is actually doing this, then something has gone terribly wrong.

I find it difficult to imagine this is the case, though. Instead, I'd bet the department is trying to help students learn. Even graduate students in relatively strong math departments can find these classic books difficult to read, and they often benefit more from studying more accessible books. This may be poorly thought out if the comprehensive exams do not match the level of the suggested reading list, but it's probably well-intentioned. If I had to guess, I'd guess that the faculty assume the students should be able to solve the most difficult problems in these books, while many students assume they are doing fine if they can solve the average problems, which may be quite a bit easier. If this is the explanation, then it's certainly worth clarifying the standards.

Another possibility is that the reading list was set by a committee and is rarely changed, while the comprehensive exams are written each year by faculty members who are guided mainly by tradition and their own sense of what is appropriate, and who don't pay much attention to the details of the reading list. They might be surprised to hear that students actually took the reading list seriously. That would be dysfunctional, but it's a different sort of dysfunction than creating deliberate barriers to entry. A variant of this explanation is that the reading list may have been intended to indicate just the topics to be covered, with no implication that it should be a source of sample problems or indicate the level or difficulty of the exams.

It can't hurt to ask faculty members about this issue and explore ways in which it could be resolved. However, I'd strongly recommend against accusing anyone of doing it deliberately to create difficulties for students.

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    I'd guess that the faculty assume the students should be able to solve the most difficult problems in these books, while many students assume they are doing fine if they can solve the average problems, which may be quite a bit easier. This sounds quite plausible to me. I've witnessed this kind of mismatch causing problems from precalculus classes through PhD qualifying exams. I've learned to tell my students that problems on my exams will be comparable to the most difficult homework problems, and that the easier homework problems should be thought of as warm-ups. – Mark Meckes Nov 29 '15 at 15:43
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As a student, I have experienced this phenomenon before. (Disclaimer: I am only a graduate student (from outside the US), hence can only answer from the student's point of view) Some plausible reasons I can think of are:

1) Faculty may be required to "bell curve" the examinations and hence have to set some difficult questions. If everyone gets 90% in the exam, they might face some trouble grading on a curve in certain institutions. This is also to separate and identify the exceptionally good students.

2) For Pure Mathematics, setting an exam proof question is not just a matter of "changing the numbers", hence it may be safer and more convenient to set a question from the classic books, rather than risk setting a wrong proof question (asking students to prove a wrong statement).

3) The lecturer genuinely thinks the questions from classic books are "easy", they are not out to make things difficult. It is not hard to imagine that someone who has studied the field for decades would think that Rudin/etc are actually easy. You may reach this level yourself in a few years time.

Personally as a student, I would much rather have the opposite: learn from classic books and have exams from the "lower level" books.

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    To counter point 2, there is an option (which I've read that some mathematics instructors take) to ask prove-or-disprove kind of statements. Further, in an ideal scenario, the solutions to such "new" questions would be worked out before asking them in the exam, so such a problem should not arise. – cutculus Nov 29 '15 at 8:01
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    Knowing that you recognize this phenomenon from your schooling "outside the US" is somewhat helpful. (I am "inside the US" and don't recognize it.) Would you be willing to be even more helpful and tell us where (at least, what country) these experiences took place in? – Pete L. Clark Nov 29 '15 at 8:28
  • My undergraduate studies (these experiences) were from a university in Asia. I have to add that not all modules were like that, it depends on the lecturer teaching the course. – yoyostein Nov 29 '15 at 8:47

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