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Background: 1) In the book Gems of Geometry, chapter 9 Relativity, the author wrote: "The General Theory concerns gravitation and the mathematics behind it is considered rather difficult ( post- graduate in these dumbed-down days I am sure ).
2) To get an idea of which mathematics books Alan Turing read during his studies I read Enigma by Andrew Hodges but the book did not answer that question. I even contacted Hodges by e-mail. He replied but he couldn't answer.

See also: How much math education was typical in the 18th & 19th century?

Question: How did the teaching of mathematics, at what we now call the Undergraduate Level, changed since the midth 19th century, ( when Riemann was studying ), until today? In topics and in depth and difficulty of the exams. Is it possible to reconstruct what Alan Turing likely had read when he started to write his paper on computable numbers? Is there any truth in John Barnes' remark "these dumbed-down days"?

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I would leave that question about the future of teaching mathematics for another question, it makes the question a bit too broad in my opinion. – Mikko Korhonen Jul 3 '12 at 9:41
Agree with @m.k. - if we knew what was going to happen in the future, we'd be doing it already. – Chris Taylor Jul 3 '12 at 9:45
Also, I suspect that people borin in generation $g$ feel that education is generation $g+1$, $g+2$, ... is "dumbed down" compared to what they experienced, so I wouldn't put too much stock in it. – Chris Taylor Jul 3 '12 at 9:48
I have removed the part of the question related to the future. I don't agree with that if we knew that we would be doing it already. Teaching programs have to be agreed by governments, boards, etc. Unless it is teaching at the frontier of course. – nilo de roock Jul 3 '12 at 10:33
What you are being taught as an undergraduate also depends on where you study, i.e. whether you study in the United States or Germany, say (I'm saying this because you were referring to Riemann; now if you want to compare how mathematics was taught back then to how it is taught now, you may end up comparing how mathematics was presented to German undergraduates in the 19th century with how it is presented to American undergraduates today). – Lennart Jul 3 '12 at 11:06
up vote 5 down vote accepted

A few comments:

  1. Undergraduate education as a whole changed a lot between the 18th and 19th century and now. The modern liberal arts model where the student "majors" in a certain course and takes electives in other subjects were not really developed until around 1850. Prior to that colleges and universities follow the system that is still followed by many modern European technical universities, where upon enrollment your entire course load has been designed and assigned for you.

  2. For the Cantabrigian perspective, Littlewood gave in his Miscellany some description of what 18th and 19th century (a practice that even lasted into early 20th century) Mathematics education is like in University of Cambridge. While the tripos program is rather the exception than the norm, it provides some insight of what math education could be in the 19th century. And in terms of exams, the topics were much broader (physics and astronomy are included) and harder (especially on the computation side). The emphasis was heavy on the applied mathematics side and valued computational abilities compared to the modern Part II/III program where there is more equal balance between theoretical studies.

    One should however keep in mind (when discussing British mathematics) that in part due to the reverence of Isaac Newton, mathematical analysis / advanced calculus came to Britain slower and later compared to the other European countries (France in particular).

  3. Turning our attention to the United States, one can look at Princeton as an example. Here we find that:

    In 1760, entering students were required, "to have an understanding of the rules of arithmetic, and underclassmen learned algebra, trigonometry, geometry, and conic sections."

    To additionally illustrate the time period, we should note that the first mathematician appointed to the Princeton faculty was in 1787. The second was Henry Fine in 1885, for whom the current department building is named. Mathematics just weren't, well, "big" at that time. (Though one should note that the subjects listed above are pretty much state of the art for 1760, except for the calculus.)

    Similarly, at MIT, a technical university founded around the American Civil War, the teaching of Mathematics was originally under Course IX, general studies as "service classes" to engineers. It wasn't until 1933 that a Department of Mathematics was formed in its own right.

  4. Back to Britain: here's a webpage at St Andrews giving an overview of the evolution of British mathematics education through history.

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