Say you have college A, B and C that are each ranked in the top 100 in the US News or other similar rankings.

Admissions (SAT scores, GPA, et cetera) standards, quantitatively, rank A>B>C as the hardest and the easiest of the three to get into.

It's clear that college A has the 'best' students according to the metrics the entrance standards measure. It could measure IQ (SAT), effort and communication skills and critical thinking skills (GPA).

Would you say, in mathematics and elsewhere, that the content of the material is markedly different? For instance, could the undergraduate textbooks used be different?

In the end, why do students from prestigious colleges have such a leg up in terms of finding jobs and getting into graduate schools of similar prestige, if the material taught is similar, in terms of the amount of material taught and the difficulty of the exams?

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    The content taught a different schools is different. Even at a given school, variation across instructors is common. Often the professor who is avoided due to a lack of "easiness" is perhaps the best option for those who wish to optimize learning. As to how the elite stay elite, I cannot say. It's outside my sphere :) – James S. Cook Oct 31 at 3:47
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    I think a large amount of the difference is the selection process itself rather than the training being superior. – guest Oct 31 at 6:34

Different universities teach different courses and different content in them, though typically there are similarities in the basic courses. They are likely to use different books. Custom lecture notes are not uncommon, either. (They might be uncommon in USA, but the question does not, at the moment, specify a country).

The specific lecturer often has a fair amount of freedom in deciding the specific content of a course.

I do not know whether the difficulty or learning outcomes of courses are correlated with rankings or other measures of quality of a university. At least the US News rankings also include social factors like sports teams and fraternities, the quantity of alumni donations and access to facilities.

It is a plausible hypothesis that universities with higher rankings teach more or better, but it is also a plausible hypothesis that the quality of incoming students determines the quality of outgoing students. It would be interesting to read studies about this, but I am not an expert on the field.

One should also remember that the university rankings tend to be controversial, and even if such rankings make sense, the quality of mathematics departments (or any other specific departments) need not follow the general rankings.

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    +1 for "the quality of mathematics departments (or any other specific departments) need not follow the general rankings" I was thinking this myself, and I thought I wrote a comment earlier today saying this, but apparently I only thought about writing such a comment. – Dave L Renfro Oct 31 at 16:36
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    It may also be worth noting that the US News rankings are based around quite a bit more than academics. They also include social factors like sports teams and fraternities, the quantity of alumni donations, access to facilities, and so on. The rankings may, in fact, have very little to do with the actual academic quality of an institution, let alone an individual department within the institution. – Xander Henderson Oct 31 at 18:01
  • @XanderHenderson Added. – Tommi Brander Nov 1 at 9:45

The short answer to your question is "of course there is". I think that other responses will probably adequately take care of the many exceptions to this in terms of how questionable university rankings are. But there is absolutely no question that even in a course like calculus which (in the United States) has a fairly well-defined set of topics (see the MAA calculus report) there is considerable difference in how rigorously things are done, how difficult of applications are considered, etc. (An example might be whether one barely talks about implicit differentiation, whether it's connected to parametric curves, whether one then does logarithmic differentiation, or whether one then introduces the implicit function theorem.)

However, I want to focus on a different problem with this, which is that at many universities (most?) there may be different versions of the "same" course taught.

  • In graduate school we had three different varieties of calculus to teach, for instance, ranging from fairly standard to using Spivak.
  • Some institutions will have separate linear algebra (often not in a course by that name) for engineers, for math majors, for computer scientists, and for others - all depending on the application.
  • Even (US) upper-division courses like number theory or geometry might come in several varieties; for instance, particularly if there is a large teacher preparation program, you could have two courses covering non-Euclidean geometry, one focusing on the axiomatic viewpoint and one on the differentiable viewpoint.

In all of these cases the content might be extremely different, certainly with different texts, but in general the "hard" version at University C probably would fall higher under your hierarchy than the "easiest" version at University A. If there were just one calculus or linear algebra (or whatever) at each university it might be easier to make such comparisons, but there isn't.

Yes. MIT and Caltech are notorious for having freshman math and physics classes that go 30 percent faster than corresponding courses at other highly ranked institutions. Sadly, this means that 20% of MIT freshmen do poorly in their freshman math classes, despite being some of the best-prepared college students in the country. MIT's standard freshman physics courses used to have similar rates of Ds and Fs, prior to a major pedagogy change during the first decade of this century.

And no. In California, almost anyone can attend a community college. And either after passing enough remedial classes, or after demonstrating that the remedial classes are not needed, you can take essentially the same calculus classes as are offered by the University of California at Berkeley. If you manage to get in to MIT, you can use those the non-remedial classes to get out of MIT's classes. But because MIT's classes go 30% faster, you will need to have taken 30% more semesters-worth of non-remedial classes at the community college.

Many colleges offer multiple versions of some courses, depending on students' interests and planned majors. These courses can vary widely in difficulty, content, and teaching approach. Some "honors" versions of classes are only offered to students who meet special admissions standards.

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    Without entering into whether your conclusions are correct, I ask where do the claimed 30 and 20 percent numbers come from? What does it even mean that a class goes "30 percent faster" (how is the rate quantified?)? What does "20% ... do poorly" mean concretely, and where does this number come from? (By the way, that 20% do poorly seems very low when compared with what is typical in most engineering programs.) – Dan Fox Nov 2 at 8:18
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    A. This link shows the course description for MIT calculus. It is two semesters covering what is traditionally 3 semesters. I would call that 50% faster. (3/2)/(3/3)=150%. B. I don't find the 20% do poorly to be surprisingly low, given the capabilities of the incoming students. – guest Nov 2 at 10:30
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    The Catech sequence is also 3 semesters in one year and even includes some linear algebra. Maybe (3.5/2)/(3/3) = 175%, this 75% faster. Or even 200%, 100% faster at 4/2 if you figure the linear algebra is that of a full semester. Note they use a term (sometimes called quarters, but really trimesters) system so even though 3 courses are listed,, it is 2 semesters long, not 3. – guest Nov 2 at 10:44
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    At Caltech, the courses are more proof based and difficult (e.g. AP credit not accepted, students expected to have already mastered the traditional course) This can be verified by (1) checking the online course descriptions, and (2) several Q/A answers on the web at Quora or College Confidential or Reddit.. Google for several of them, but see in particular Hsu's answer here: Note, it is debatable if baby real analysis helps physics or engineering students. – guest Nov 2 at 10:50
  • @DanFox -- The standard calculus sequence (such as at Berkeley or California's community colleges) is 4 semester units of Calculus 1, 4 semester units of Calculus 2, and 3 semester units of Calculus 3, where a full time load of 4 courses is 16 units, and an average load of 15 units is needed to graduate in the standard amount of time. (Most American universities grant units based on the number of hours of class time per week.) – Jasper Nov 2 at 17:18

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