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I noticed while doing practice math PhD exams that the newer exams tend to be a bit easier than the older exams. The online archive goes back about ten years or so, but I was able to dig up some old exams from the mid-80s. I was absolutely shocked. I would get wrecked if they gave me one of these. I don't even think I could finish one if they let me take it home.

I looked around at some other PhD exam records at other universities, and it seems to be an across the board thing, not just my department.

Some people I've asked about this give the explanation that PhD programs are bigger now than they used to be-- in other words, in the past, only geniuses got PhDs, and since then cretins like me have gradually wormed our way into academia and lowered standards for everybody. This would make sense, but I'm not sure if this is just something people say, or something that is actually true. People have a tendency to put the past on a pedestal, and I'm skeptical.

Another explanation I could think of would be drift in material content. Maybe they were studying different stuff back then at the same difficulty, or using different terminology to study the same thing we do now. Maybe it just seems harder to me because I'm not used to how they talk about it. A lot of PhD level material is, after all, reasonably modern.

I also wonder if exam practices used to be different-- open book, take-home, more hand waving tolerated in grading, offered at the end of the 5th year. Something?

All this is speculation, though, which I find unsatisfying. Has this actually been studied with healthy rigor and skepticism? What is the history?

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    I can't speak to a rigorous study of the history, but I do offer another possibility. Many fields have seen substantial advances in the past few decades. As a result, it's likely that PhDs have become much more specialized. Thus, you don't cover as much breadth of material in pursuing your degree and therefore those old exams contain a much larger spectrum of material than you are used to seeing. – marcman Aug 4 '16 at 6:42
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    It's very hard to evaluate the effective "difficulty" of an exam by looking at the questions without knowing the grading standards, and these are rarely made public. (Sometimes they are not even formalized, but are the subjective opinion of that year's graders.) You speak in the first paragraph of not being able to "finish" the exam, but it may be entirely possible to pass the exam without "finishing". – Nate Eldredge Aug 4 '16 at 16:15
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    I would speculate that the most likely reason is the instructors, and also course content/emphasis, has changed. I'd be skeptical about asserting a change in difficulty. I often find reading older treatments of material harder than more modern ones, because the latter are closer to the way I think about things. – Kimball Aug 5 '16 at 10:44
  • When I started programming, it was basic C on a text editor without any sort of syntax highlight. The newer kids get a full-fledged IDE from the start, with more modern stuff like Java and C#. My course was all about understanding how code works and what it really does, understanding memory models, coding patterns, etc. The newer grads can barely write a "Hello World" program. Yes, I'm really salty about that. – T. Sar - Reinstate Monica Aug 5 '16 at 19:09
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(In mathematics:) I'd be willing to believe that pre-college math curricula, and lower-division college (in the U.S.) curricula have become easier to get through, for reasons mentioned in other answers and comments. But I think this becomes less universally so at upper-division level (last two years of undergrad in the U.S.) and even less so with regard to the sort of "prelims" that math grad students typically do/take in the first year or two of their graduate work. This reflects my observations over the last 45 years in math in the U.S.

So then how to account for the phenomenon apparently observed by the questioner? Again in my own direct experience, decades ago there were impulses to ask "interesting/challenging" questions on prelims, that even the other exam-writers might not be able to do. Some contest-like spirit, resembling the old Tripos in the UK? At the same time, again somewhat contest-like, the questions were at a lower mathematical level than nowadays. This, combined with the impulse to ask challenging questions, led to many Baroque, Rube-Goldberg-like questions whose statement alone might be difficult to parse, and whose relation with any known mathematics was unclear. E.g., very complicated questions about iterated operations in point-set topology, with delicate separation assumptions? Difficult questions about proving that a bunch of relations in a group implies another one?

After a decade or so of that kind of thing, it was apparent to me that such exams did not encourage forward-looking study on the part of the students, which was a bad thing. Often, as in prep for contests, some amusing tricks were learned, but basic, standard, useful more-advanced ideas were neglected entirely because they'd never show up on those exams. Thus, here, we deliberately changed the nature of those exams to address less-tricky, higher-level mathematics that would actually be used by people in doing their PhD work.

At least at my own University, it's not at all the case that basic graduate education is being diminished in favor of supposedly jumping into research immediately. As in some comments above, it's not feasible to really start doing modern mathematics research while not knowing anything... But, yes, we do try to encourage a more active version of engagement than merely "fulfilling requirements" (even while those "requirements" are aimed to be useful).

So, in the end, I'd argue (as in other comments) that the exams are not truly easier, but just "modernized", so that some acclimitization to modern mathematics makes them seem easier in our context.

(And, yes, I recognize that in fact some grad programs in math have "thinned" their prelim requirements, apparently motivated by "getting students into research faster", but it is not clear to me that this can truly accomplish the avowed goals... though I am equally confident that programs would be disinclined to candidly discuss such a thing. The ever-increasing commodification of "research" does provide considerable pressure to degrade things...)

  • we deliberately changed the nature of those exams to address less-tricky, higher-level mathematics Would you please clarify when did this happen in your university? In 80's? 90's? After 2000? Or recent years? – scaaahu Aug 6 '16 at 4:04
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    @scaaahu, the basic change was 1989, from lower-level to higher-level topics addressed in the prelim exams. Another attitudinal change has been within the last 10 years, when we have moved completely away from surprise/tricky questions at any level. – paul garrett Aug 6 '16 at 12:16
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This very much depends on the area at hand ("mathematics" is a quite general thing that comprehends many different lines). It is however generally not true that newer exams and exercises are easier: it is instead the converse, as the study, the knowledge and the level of understanding go deeper as science evolves.

Notice however that finishing (or not finishing) a test/exam is by no means indication of the level of difficulty of the topic. As an example you could be asked to manually calculate the determinant of a 1000x1000 matrix in one hour: you will never finish that on time but it is monkey job with no difficulty or understanding required; as such, do not grant deepness of knowledge according to how messy or long or cumbersome the exam tasks may be.

Moreover, the argument of "difficulty" is usually very subjective. There are no easy or hard topics, there are just topics you are not familiar with because you did not study them before. One thing that is however true is that the flavour of research has changed as the time went by and different efforts and emphasis have been put on different topics; as such, you may find yourself to have a more natural understanding and practice on some topics and aspects rather than some others.

Are those old-style-exams that you are mentioning really more difficult and intense or is it any of the above that may come into play?

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    I've spent 8 years in academia. When I started, the maths were so hard just 4 students, out of 30, passed the Basic Calculus class. The exam I took back then was around five pages full of puzzling stuff. The newer, 2016 exam is around a page-and-a-half, with multiple choices, and no need to show real calculations. I felt almost cheated - I almost killed myself studying back then for an 8, and now those "fresh bodies" barely study at all and get a 9.5. Also, the grade you needed for an approval on that college dropped from a 7 to a 5. If this is not "getting easier", I don't know what it is. – T. Sar - Reinstate Monica Aug 5 '16 at 19:05
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    I have never taken a "multiple choices" maths exam in my life, never seen anything of that sort. I've always taken maths exams with theorems to prove or sensible calculations to make. I guess the teaching of mathematics varies from one country to another, at this point. – gented Aug 5 '16 at 19:17
  • Never did I during my college years. Still, it seems it's the norm nowadays in several colleges on my country. There seems to be a trend to mass-produce diplomas, unfortunately, even in the high-profile colleges. – T. Sar - Reinstate Monica Aug 5 '16 at 20:05
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I think that you've discovered a fairly good example of credentialism and educational inflation. It's pretty well established at this point that over time there's a trend towards more jobs demanding higher degrees, jointly with more higher degrees being awarded. It makes sense that, for this to occur, standards are being relaxed over time. Note that this is a pretty solid evidential counterpoint to the de rigueur defense that an older generation always complains about a younger, or that "people have a tendency to put the past on a pedestal".

At a different level, we could demonstrate the same thing at the community college where I work. A senior faculty member has kept final exams for a variety of mathematics and computer science courses for a number of decades, and if they were lined up side-by-side there would be very clear evidence that current students would have nearly no chance at passing prior exams. In some cases what was once a one-semester course is now a two-semester sequence to cover the same material. And in the decade that I've been there, there's been a fairly public process with making general-education exams easier in an attempt to get a majority of students to pass them (maybe 4 iterations of downshifting the university-wide basic math exams by administration).

You might consider whether this discovery itself makes for a worthwhile academic paper. Personally, I'd love to see that documented and published.

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    "standards are being relaxed over time" are you sure this applies to mathematics? – gented Aug 5 '16 at 9:36
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    @GennaroTedesco The second paragraph of this answer pretty much answers your question "are you sure this applies to mathematics?". Furthermore, I myself witness similar phenomenon in my location - Taiwan which was well recognized by its quality math education, but not anymore. – scaaahu Aug 5 '16 at 10:04
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    I further clarified that these observations are especially about mathematics (and computer science) education. – Daniel R. Collins Aug 5 '16 at 13:04
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    I downvoted this answer because I do not believe it applies to PhD qualifying exams in mathematics (the topic of the question). – Tom Church Aug 5 '16 at 19:43
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    I've seen evidence of that in basic chemistry exams (inorganic and organic) where the same pool of questions has been used over several decades and a corresponding change in the requirements for passing. (However, even for undergrad subjects one may argue that school curricula may have been covering fewer subjects more deeply, and that this has consequences for undergrad courses.) – cbeleites supports Monica Aug 5 '16 at 22:13
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I believe the larger trend is that PhD programs are putting much more focus on research over coursework. Not to say that research was ever not an important component of a good PhD programs, but the importance has greatly increased over recent years in many fields.

Students are expected to start writing papers earlier and to write more papers before obtaining their PhD, which necessitates spending less time on coursework. At the same time, as research becomes more and more specialized, it becomes harder to design good courses that will be relevant to the students' future work.

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    I'm not sure about such a trend for pure math--exams in the US are usually in the 1st or 2nd year, before one starts on research. Though somewhat along these lines, I could imagine that some PhD programs in the US have been started demanding more teaching duties from their students, but I don't have any actual knowledge of this. – Kimball Aug 5 '16 at 14:01
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    I did not down-vote this answer (I have not up-voted it yet, neither). But, I am furious about something here. Don't you need to have foundational knowledge before you do research? – scaaahu Aug 5 '16 at 14:10
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    @Kimball I'm in statistics, not math, but many programs I know have students started research during the summer after their first year (or even during their first year) and continuing it through second year. This goes hand-in-hand with many programs making qualifying exams easier or having fewer of them. Though it's entirely possible that my experience does not match the trends in pure math. – Roger Fan Aug 5 '16 at 15:34
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    @RogerFan Students starting research sooner than they should is also the reason behind the skyrocketing of the number of bad papers out there. – T. Sar - Reinstate Monica Aug 5 '16 at 19:12
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    Although I do hear people say "learn X by reading papers", in much of mathematics that would simply be impossible without considerable background, usually obtained by fairly systematic study of advanced textbooks or notes, whether in an official course or not. High-end published papers assume a very substantial body of knowledge, and it's essentially impossible to bluff one's way through such things, since there is also no general tendency to give useful backward pointers to textbooks. – paul garrett Aug 5 '16 at 19:37

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