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?
Best Answer
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.