05.21.07
On books
Last week an Ontario-based programmer named Antonio Cangiano started writing his Math Blog - Mathematics is wonderful! (I agree, BTW). Only two articles so far, but one of them rose up the digg ranks pretty quickly and crashed his server. So maybe he’s doing something right. 
Refresh your High School Math skills is a post containing precalculus math problems. I’d agree with him that these are the kinds of faculties we’d like our students to have going into calculus–algebra, trigonometry, inequalities, familiarity with exponentials and logarithms, etc. I wish we could assume more of the conic sections material was taught but it doesn’t seem that way anymore.
His other post is called “The most enlightening Calculus books” and is about his favorite books. There is massive debate among college math teachers about how best to teach calculus: reform, IBL, “Harvard Calculus” (which I do not teach), the list goes on. And as someone who has perused dozens of free calculus books from publishing companies, I can say that I still haven’t found the perfect book for wide university appeal.
What I want in a freshman calculus book is:
Tell no lies
I don’t insist on epsilons and deltas in a book, but I think we can get within epsilon of it (sorry). The concept that f(x) can be made arbitrarily close to L by taking x sufficiently close to a is precisely the definition without the greek letters, absolute value bars, and the dreaded less-than sign.
I think the derviative should be defined as a limit of difference quotients, and the integral should be defined as a limit of Riemann sums. I don’t think we need to prove that all continuous functions are integrable (that requires uniform continuity, which requires compactness of closed intervals, which I think is a little much), but the Fundamental Theorem of Calculus needs to be proved.
There is a tightrope to walk here. If you get too technical, students’ eyes will glaze over. I just don’t think everyone needs to know about epsilons and deltas. But if you get too hand-wavy, you lose the faculty to speak in any rigorous fashion about any limit, and suddenly every theorem becomes an article of faith.
Relevance
I think today’s students are interested in putting everything together rather than following many subjects down their separate paths. So I’d like a book that includes as many applications as possible. Calculus is the universal language of science, and I want my students to think of it as something that continues to be relevant. Of course there are the myriad physics applications that mathematicians are most familiar with, but the majority of our students are concentrating in (a) economics or (b) some sort of life sciences or pre-med. So give me problems in comparative statics, theory of the firm, population systems, rates of absorbing medicine, etc.
Problems
Many of our students get discouraged about the difference between homework and test problems. I really believe that for a student to demonstrate mastery of calculus, they need to be able to solve “new” problems. I don’t think the students are well served if each exam problem is similar to a homework problem. Again, calculus is not something that has been solved and put in books to be memorized; it is a tool which can be used ad infinitum.
So I also want conceptual problems that are unique, and enough of them to give the impression that this is what calculus “is.” I like drill-type problems for practicing the techniques (after all, the word “calculus” means a set of rules for deriving something), but limiting calculus to that is like saying all you need to know to be a carpenter is how to saw a board in half.
Antonio’s a big fan of Calculus by Michael Spivak. Indeed, it is a beautiful book; it changed my life in my first year of college at the University of Chicago. It has excellent prose, wonderful, challenging problems, and the kind of snarkiness that appeals to smart math students and their teachers. I still pick it up about once a month. Yet, as someone in charge of teaching calculus to hundreds of college students, I can’t imagine using it. I don’t think every single student is going to be receptive to that kind of book.
So the quest continues.
technorati tags:math, books, calculus, spivak
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Bret Benesh said,
May 21, 2007 at 8:23 am
Nice list, Matt. I agree with your three criteria, although I want to add that the applications need to be authentic. I am beginning to believe that most of the “application” problems in calculus books aren’t really economic/biology/medical questions, but rather plain-old calculus questions in disguise. I would like to see serious problems taken from work that an actual biologist/economist/etc. did put into a calculus book. I think that this would be much more useful than by claiming that “A fish’s position is given by the function f(t) = t^2+2t+3. What is it’s velocity when t=2?” is a question that has anything to do with biology.
Of course, I have no idea where to find such questions; I’m going to start looking this summer.
leingang said,
May 21, 2007 at 8:37 am
True, that. Although I’ve looked in actual economics books and they have lots of problems like “Assume a product’s demand function is given by the curve Q(P) = 4700 - 0.0001P - 0.0035 P^2…” I do agree that they should be as realistic as possible, which is why I was thinking about GIS applications.
Sendhil Revuluri said,
June 10, 2007 at 10:34 am
Hi Matt! (Ken pointed me to your blog.) Nineteen years after my first calculus course, I have NO recollection of discussions about why simple results were true (derivatives of powers) or connections made between obviously related ideas (integration by parts and the product rule for derivatives). Overall, I felt my teachers were pretty good, so it’s quite possible that they brought these things up but they didn’t stick (until I “independently” realized them a decade or two later). I’d hope that a good textbook led students to understand such relationships in a lasting way. (I’m not sure the usual proof + exposition style really supports that.)
I agree on the applications. More realistic application would start a stage or two before and demand some non-calculus tools (”the deli tried pricing its sandwiches at $3.50, $4.25, and $5.00, and they sold… in each case. Assuming the demand function is quadratic… find… ” or something like that). With minimal emphasis on modeling in secondary math curricula, this might be rather intimidating to students in a first calculus class. It would be a big favor to them for their success in other classes though!
Finally, one question - not having taught calculus yet, due to the aforementioned fog of memory, I can’t remember why familiarity with conic sections would be particularly helpful. Why?
Summer Quarter Begins « Vlorbik on Math Ed said,
June 25, 2007 at 2:58 pm
[…] Today’s Links Philosophy of Mathematics Education Journal Donald in Math Magic Land on YouTube: here and here (spotted at What the heck [05.22.07]). Also, leingang on calculus books (same source, 05.21.07). […]