I have in mind a course taken by liberal-arts students who will probably never take another math course. I would like such a course to convey some of the way mathematical thinking is done (i.e. not a cookbook course) without getting "rigorous" (since such students cannot be assumed to understand that and can learn rather little of it during a one-year calculus course). Apparently I am not the first to think of excluding the mean value theorem, since one of James Stewart's books does that. I would also like to include some of the ways in which differential and integral calculus have played a role in the history of science.

I'm teaching a course in which I began with this and I use that idea repeatedly in exercises. It will of course be used in explaining the fundamental theorem. One of various places where I've used that proposition so far is #3 in this assignment, where I was told by multiple students that no one else who teaches math ever asks students to think through steps like this. They "know" very well that that's not at all how math is done. Hence, they say, it is quite confusing. I do some topics that might normally be done only in "rigorous" course, such as things like #1 in this, but as you see, I don't do it in the way in which rigorous arguments are written.

I'd like to see skills taught in such a course only to the extent to which they aid thinking, and I like to have students write carefully about that thinking. This contrasts with a practice that perhaps few if any mathematicians intend to do, but which is widespread, and that is that students in such courses are taught that mathematics consists entirely of skills. This leaves no place for things like one that I like to include: What is "natural" about the number $e$? (Here is how I begin the treatment of that question.)

It seems as if mathematical thinking is often reserved for advanced courses rather than freshman calculus or the like, despite what is probably overwhelming empirical evidence that it can be done even at the most elementary levels, e.g. teaching graph theory to 4th-graders.

The question here is: Which specific topics should be included in a course consistent with the ideas sketch above and why? In particular, which that are now customarily not included should be there, and vice-versa?

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    $\begingroup$ You might want to mention that you have asked a similar (not duplicate) question before: mathoverflow.net/questions/28695/… . Also I suggest you make this community-wiki. $\endgroup$ – danseetea Nov 10 '10 at 21:21
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    $\begingroup$ I would teach all of the sections in the calculus book by Hughes-Hallett that do not involve the rules of differentiation and integration. Another book that I always enjoyed teaching with was a calculus book by Goldstein, Lay, and Schneider. $\endgroup$ – Deane Yang Nov 10 '10 at 21:28
  • $\begingroup$ To clarify danseetea's comment: that previous question was on what to teach liberal arts students who will take only one math course. This is specifically on calculus for liberal arts students, regardless of how many math courses they take. $\endgroup$ – Michael Hardy Nov 10 '10 at 23:04

Several years ago the liberal-arts-ish university where I was at the time was pushing to have more interaction between the sciences and the humanities. In that spirit I volunteered to give an hour-long seminar entitled, "So You Think You're Educated, But You Don't Know Calculus: A Brief Introduction to One of Humanity's Greatest Inventions." It was aimed at the humanities faculty. My goal was to explain the big ideas behind calculus and place them in their historical and philosophical context for an audience of very smart people with weak math backgrounds. You might be able to use the historical and philosophical context part of the talk for the "ways in which differential and integral calculus have played a role in the history of science" aspect of your question. You are welcome to borrow freely from my presentation.

In retrospect the title may have been a bit too audacious, but the talk went much better than I had expected. A few of the scientists showed up for fun, but most of the audience were folks from the humanities and social sciences. They were engaged, and they peppered me with questions for half an hour after the talk was over. After I left there were still people who stayed behind to discuss the seminar. Later I even got an email from the provost (a religion scholar) who wanted me to meet with him to discuss the ideas in the talk! It was, frankly, the most successful academic talk I've ever given - and much more so than the one I gave three days ago at a math conference that was attended by eight people in a room that could hold hundreds and yielded no questions. :(

One caveat: When I discuss the philosophical implications of calculus, I'm doing so as I think they appeared to people at the time, not today. Clearly, humanity's consensus on these big questions has changed in the last 300 years.

The other thing I would say is to second Deane Yang's recommendation to look at the Hughes-Hallet, et al, calculus texts. I know there are strong opinions on the calculus reform movement, and I don't want to wade into that. But what the Hughes-Hallet texts do well (in my opinion) is to emphasize ideas and mathematical thinking over rote computation. Since you're after the former, looking at what they've done may be helpful.

  • $\begingroup$ That talk sounds awesome: I love the boldness of the title and agree with the idea too. I think it's a shame that the way calculus is taught means that students can't see the forest of ideas for the tree of minor computations. $\endgroup$ – Thierry Zell Nov 10 '10 at 22:37
  • $\begingroup$ I read through the slides. I'd have emphasized instantaneous rates of change rather than tangents. I think the characterization of integral calculus on the second page falls short, since one might want to find an integral that one can't find by using the fundamental theorem. I hadn't realized Newton's book published in the 1680 avoided using the new discoveries. The part about God initially seemed off-topic, but apparently Newton's mathematics did unintentionally lead to Deism. $\endgroup$ – Michael Hardy Nov 12 '10 at 17:04
  • $\begingroup$ ..... Explaining "why nearly everything on earth and in space moved the way it did" seems like a stretch if you apply it to living organisms. Nonetheless calling Newton the greatest of all scientists may well be well justified. Definitely efforts like this to explain the importance of calculus to laypersons are needed. $\endgroup$ – Michael Hardy Nov 12 '10 at 17:04
  • $\begingroup$ @Michael Hardy: Thanks for the feedback. Those are fair criticisms of my characterization of integral calculus and of the comment about motion. I chose the tangent and area problems as prototypes because they would be easy for my audience to grasp and because I still think it is fascinating that they are (in some sense) inverse problems. I'm glad you got something helpful from the talk. $\endgroup$ – Mike Spivey Nov 12 '10 at 21:11
  • $\begingroup$ That the instantaneous-rate-of-change problem and the area problem are inverses is easier to see directly than that the tangent problem and the area problem are inverses: wnk.hamline.edu/~mjhardy/1170/handouts/September.8.pdf $\endgroup$ – Michael Hardy Nov 13 '10 at 4:57

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