Thanks Chris for your comment on my last post. There was a lot there and it stirred up memory of one of the biggest sources of advice for this project: a book review by Underwood Dudley from nearly twenty years ago that is repringed on talldarkandmysterious’s site. Before starting to write last summer I sent Dudley an email asking if his views have changed since 1988 and he said no that his objections are pretty much the same.
So what triggered this memory was Chris’ note about geometry. Dudley’s second conclusion is that Calculus books need more geometry. I think that Calculus can be taught as a nice combination of calculation and intuition and that driving the geometry out of it makes the intuition less – umm – intuitive.
It is Dudley’s first conclusion that is the biggest risk in Calculus books. It is easy for Math practitioners to be dismissive of a book on Calculus that doesn’t do something in a particular way. But, as Dudley notes, Calculus books should be written for students and not for professors. In fact, I think that students should be able to read and learn from the book as a supplement to what the instructor is presenting. If the instructor is merely re-teaching exactly what is in the book (which is done way more often than you would like) then there seems to be little value in the student owning a copy of the book.
This memory was spurred by Chris’ note that we shouldn’t be teaching Calculus for that one in 100,000 that is going on for a PhD in Math. There was a conference at Breadloaf in the 1980s at which the point was made that the PhD requirements for Biology helped drive the science we teach first graders. The arguement was that undergraduate curriculum in Biology was designed by Biologists who, among other things, wanted to ensure that students would be prepared for grad school in Biology. The High School AP program and hence standard Bio courses were therefore designed to adequately prepare students to take Biology in college. These high school teachers told middle and elementary school teachers what science the younger kids would need to come in with. The result was that the science taught in first grade was not necessarilly the science that would be taught without the constraints of beginning the preparation of preparing a small percentage of that population for work as a professional.