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I often find myself debating the content and structure of such courses and I would find it useful to know the basic history.

I don't remember any such offerings during my own undergraduate days in the '70s. I have always supposed these courses appeared as compensation for a decline in high school Euclidean geometry teaching, but I would call my understanding anecdotal.

Some old-timers may have the whole history in their heads, if so thanks. Otherwise it would be useful to me to hear if you had such a course a long time ago (and where, and from who), but please only comment if your date trumps the earliest previously posted date. In any case I feel sure such courses were popular by the 1980s.

Polya's How To Solve It dates to 1945 and roughly addresses these needs, but I have always understood it as a popular book rather than a a text. So I wonder what were the first texts written to support such courses?

Please refrain, of course, from opining about the efficacy or effectiveness of such courses, but feel free to cite any published research addressing the same. Thanks.

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Dear David, could you please explain in more detail what kinds of courses you are talking about? – Alex B. Jan 6 '11 at 7:09
My guess from the comment about high school geometry courses is that the OP is referring to courses introducing students to the mechanics of proofs and abstract mathematics, as opposed to the purely algorithmic courses one generally sees in high school. My understanding is that this began in some secondary schools with the "New Math" movement (now defunct, though the song it inspired is pretty well-known) but I obviously have no personal experience to back up that claim. – Daniel Litt Jan 6 '11 at 7:20
@Alex I'm asking about college and university courses that go by many names, "Introduction to Proof," "Introduction to Abstraction," "Basic Structures of Mathematics." I have in mind courses intended and required of primarily mathematics (and possibly computer science) majors, designed to follow calculus, but to precede one-variable analysis, abstract algebra and topology (ideally so that professors in those more advanced courses can focus on content rather than methodology). – David Feldman Jan 6 '11 at 7:48
@Daniel I don't associate such courses with "New Math." Indeed quite possibly they arose to compensate for its demise. But I'm not sure. – David Feldman Jan 6 '11 at 7:49
My impression ... when mathematics students are combined with engineering and physics students at first, the courses don't do much on proofs, so there needs to be such a course for the mathematics students to take before they can go on. Another impression ... in Europe, mathematics students do not take the same courses as engineering and physics students, so their courses can include proofs from the start. Who knows if my impressions have any validity? – Gerald Edgar Jan 6 '11 at 12:47

I assume that this question is asking about Mathematics Education courses in the United States. A short answer to this question is to read a nice piece on the history of the mathematics major (in the U.S.) by Alan Tucker. The citation is:

Tucker, A. The History of the Undergraduate Program in Mathematics in the United States. The American Mathematical Monthly, 120(8), pp. 689-705.

(I believe a free version is available here.)

Here are a couple relevant pieces to your specific question from the Tucker article. The first one references the sort of class that you mention and which was available prior to the 1970s. The latter one notes that it was primarily the universities themselves, rather than, e.g., recommendations from mathematicians (an example being the CUPM = the MAA's Committee on the Undergraduate Program in Mathematics) that were the driving forces behind the institution of such courses.

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Separately, I was recently reading through a Mathematics Education thesis from Columbia University Teachers College that discusses the history of the mathematics major at liberal arts institutions. You can find a copy through DigitalDissertations; the citation is:

George. M. The history of liberal arts mathematics. Teachers College, Columbia University, ProQuest, UMI Dissertations Publishing, 2007. 3288599.

Besides being of general interest, this thesis also led me to work by a mathematician who worked at the University of Chicago by the name of E.P. Northrop. Already by the 1940s Chicago was instituting a course quite similar to the sort that you have described. More information can be found in a couple of Northrop's papers:

Northrop, E. P. (1945). Mathematics in a liberal education. The American Mathematical Monthly, 52(3), 132-137.

Northrop, E. P. (1948). The Mathematics Program in the College of the University of Chicago. The American Mathematical Monthly, 55(1), 1-7.

Here is a relevant excerpt from the former citation:

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As requested, you can see in the above a reference to a specific textbook (Fundamental Mathematics) that was used for purposes similar to those you outlined, and which dates back to the 1940s.

I am sorry I cannot respond to your question in more detail; a full answer is probably fodder for an entire thesis in the history of Mathematics Education. Hopefully, though, the few references provided here can get you started if you wish to read further.

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I took such a course at Temple University in the spring of 1972. The professor was Ray Coughlin, and the book was Fundamental Concepts of Mathematics by Foulis, which I fell in love with. I still keep a few copies around and occasionally give them away to students at about that stage.

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