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Most differential calculus courses begin with the theory (and analysis) of differentiation, followed by computations, and likewise integral calculus courses. That's a lot for a three credit course, which is why most college courses of this sort carry four credits.

I initially wondered if it were possible to cut back these courses to three credits by removing the analysis portions, but this question goes the other way. I am now wondering what would happen if colleges "supersized" these calculus courses by adding most of an intermediate analysis material and making them six credits. The advantage is that students could learn both theory and practice in one course. Also the average math program requires about 39 credits, so the student will have completed 12 credits by the end of the freshman year, and could complete the major in six or seven semesters instead of eight.

Are there any university programs that offer this kind of supersized option in introductory mathematics? (Some years ago, Yale had an "early concentration" option in Economics, featuring six credits of principles and intermediate microeconomics in the freshman first term, and similarly for macroeconomics in the second term.)

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    $\begingroup$ Harvard is well known for its supersized (but no extra credits) honors honors version of 3rd semester calculus for incoming students with AP credit, MIT is well known for not offering 1st semester calculus at all, and Caltech was the last holdout to teach first year calculus out of Apostol (or a similar textbook with a significant amount of theory). Those are about the only places with the student population for such a course. Here we consider it a success (and we mostly fail) if we can teach someone the intellectual tools necessary to appreciate intermediate analysis by their senior year. $\endgroup$
    – Alexander Woo
    Commented Feb 9, 2021 at 21:01
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    $\begingroup$ The precise numbers of credits suggest this question is about a particular country. Could you add a country tag to that effect so we know which one? $\endgroup$
    – Tommi
    Commented Feb 10, 2021 at 7:45
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    $\begingroup$ In 1974, at University of Michigan, I took their honors honors (higher of two levels of honors) course. It was very theoretical and counted for upper division credit. I have a BA with only 30 units / credits of math. $\endgroup$
    – Sue VanHattum ♦
    Commented Feb 10, 2021 at 20:49
  • $\begingroup$ The honors course @SueVanHattum refers to is Math 185-186 ; the honors honors course is Math 295-296. Unfortunately, the current course webpages are hidden behind a Canvas log in, but here are versions from previous years. dept.math.lsa.umich.edu/~zieve/math185.html math.lsa.umich.edu/~kesmith/Math295.html $\endgroup$
    – David E Speyer
    Commented Feb 12, 2021 at 14:36
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    $\begingroup$ Note that these courses are much smaller than our main calculus sequence, Math 115-116. Most students are not served well by courses like these. $\endgroup$
    – David E Speyer
    Commented Feb 12, 2021 at 14:37

1 Answer 1

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A. CIT is (in)famous for having a freshman calculus course, with substantial analysis content.

http://www.math.caltech.edu/~2014-15/1term/ma001a/

The course is not supersized in terms of credit size though. And it presupposes AP5 level of capability for the incoming students. And the kids hate it.

B. For various reasons, I think the suggestion to mix more real analysis into freshman calculus is a bad idea. (A recurring suggestion though.)

  1. Mixes in content that most students (science, engineering, econ) will never need. This is the bulk of a typical class. Most of these kids, if they validated calc 1/2, would move to calc 3 and ODEs, well, well before a theoretical calc course. Which realistically most will never take...even at grad level.

  2. Furthermore, realize that most math majors don't self designate, at least in the states, freshman year. As a hedgie, you should understand "option value", the whole trope of the Hollywood sequel option validating the negative NPV of a first movie. Well...learning all that shit before, is forfeiting option value.

  3. Moves away from gradual progressive learning and deepening insight to a "learn it all at once" approach, that is not pedagogically optimal. Realize the human brain is imperfect and requires training, in a very different fashion from the way a computer can do everything at once, provided an instruction set.

  4. Taking an extra couple credits has to come from somewhere. If you go from 4 to 6, where are you getting these 2 credits from? As a financial guy, think of this as an investment decision. Not "money from the air", but a trade-off.

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  • $\begingroup$ Great googly moogly, mad respect for that Caltech course. Talk about being counter-cultural in our current world. That is a formidable course, I think I believe that university is actually challenging. Someone who could take that on is someone I would respect, admire, and if I was in the position, hire. No coddling there. $\endgroup$
    – James S. Cook
    Commented Feb 10, 2021 at 5:45
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    $\begingroup$ Yeah, no coddling in the course I took either. And I wasn't well enough prepared, so it was a huge challenge. I agree with our guest that it is (usually) a bad idea to do this. $\endgroup$
    – Sue VanHattum ♦
    Commented Feb 10, 2021 at 20:51
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    $\begingroup$ -1 for numerous reasons. (a) The OP asked for existence, not argument/justification, and this answer is mostly the latter. (b) The linked course is not meant for an introduction to calculus; it says, "We assume that the Caltech freshman has reasonable familiarity with single variable calculus as a computational system". (c) The link is to the 2014-15 semester, what is the current situation? (d) The listed textbook is Apostol, which A. Woo in an earlier comment notes is no longer used there. Looks like a misleading and out-of-date example. $\endgroup$
    – Daniel R. Collins
    Commented Feb 10, 2021 at 21:07
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    $\begingroup$ Calling Caltech “CIT” looks as strange as calling MIT “Masstech”. $\endgroup$
    – KCd
    Commented Feb 11, 2021 at 13:38

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