Calculus Crank

August 17, 2006

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.


12 Responses to “Calculus Crank”

  1. “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.”

    Or little value in having a genuine teacher, if all they’re going to do is read the book aloud to the students. I had some CS profs like this in college — it made class eminently skippable.

  2. The rehashing of what the book said likely killed most of my interest in upper-level math (that or I started hitting the limits of my abilities to do things easily, which is also a possibility).

    I like the idea of making the class to prepare you for the PhD, though I would also throw in that it’s not just a PhD in Math for which someone needs to prepare, but upper-level Engineering and the like, as well. The question is how to properly deal with if there is a conflict between teaching something for better Math understanding and better Engineering understanding, but I don’t know if that’s a true worry.

  3. SusanJ Says:

    I have a Ph.D. in an area that’s basically computational mathematics so I’m not sure I’ll be adding any non-Ph.D. insights here.

    Also an Oberlin [!] A.B. ’62 from back when they didn’t even offer a course in differential equations.

    Over the years I’ve thought about what I didn’t understand when I took first-year calculus and introductory physics.

    One of the key things that I really didn’t understand for a long time is what a function is. Somehow “independent variable” and “dependent variable” were answers to multiple-guess questions, not something I really grokked.

    And it is very embarrassing to admit that I had no idea why you wanted to know the area under a curve. I’m pretty sure I understood algebraic formulas like d=rt but didn’t understand that d was the area under the curve when d=r(t)t. Also I got mixed up in physics about v=dx/dt and then using v in some other formula.

    My Ph.D. research involved both numerical quadrature and numerical solution of coupled PDE’s so I guess I got it eventually.

    The real problem was that I didn’t know that I didn’t understand these things.

  4. SusanJ Says:

    Just thought of something else.

    Maybe my problem with “dependent” is because in a lot of examples the dependent variable isn’t **really** dependent. A lot of times they try to illustrate this concept with charts where, perhaps, the day of the month is the independent variable with something artificial like the number of cookies sold being the dependent variable.

    Or even automobile speed as a function of clock time. You could actually have varied the speed a lot of different ways so the speed isn’t dependent on the clock time in the way I understood “dependent.”

    This artifactual dependence is different from physical laws where, say, the density of a material really really does depend on its temperature and its pressure.



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  11. Hello – Did you every finish writing what was to be Head First Calculus? If so, what is the name of the book and where can I find it? I’d love to read it!

  12. Your blog is great! I hope you gonna keep writing that way. Come on

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