I have gotten so many interesting and thoughtful replies to the last newsletter that I decided to issue an addendum. Partly this is because they are interesting, and partly because, as Jan Ray (Seattle Central Community College) put it:
We need to have so many more of these [discussions], so that we might REALLY have a good rationale for what we decide to teach. All too often I think we're on automatic. It takes some serious questionning before we are ready to attend to hard choices that should be made within the curriculum.
First I have a note from Paul Goerss which provides another calculus-specific example with huge non-calculus resonance:
I have only a tiny addendum to your discussion on integrals. In Math 307, in the first homework assigned after the first class, in problem number 1, the students are asked to find the general solution of
y' + y = x.
Any self-respecting symbolic manipulation program can do it, of course; however, if one uses an integrating factor -- very easy even for the 307 students -- one must integrate xe^x. In short, one must know integration by parts. Again a computer can do it. Nonetheless, I make this point: producing the computer for such a problem -- the DE equivalent of 2+2 -- is incredibly distracting, turning a simple problem into a laborious process.
Jerry Folland provided a heartfelt reply to the question that we posed ourselves in the course of the Brown Bag: why, in fact, should we resist the jettisoning of techniques of integration?:
Proponents of calculus reform talk about freeing up time to develop conceptual content by omitting things like techniques of integration. What they seem to fail to realize is that these techniques -- some of them, anyhow -- are also embodiments of important IDEAS. Integration by parts is not just a trick for evaluating some integrals, it's an IDEA that has enormous repercussions in more advanced analysis -- ordinary and partial differential equations, distribution theory, asymptotic expansions, etc. Any time you use the word "adjoint" in the context of differential equations, as in "self-adjoint boundary value problem," you're talking about integration by parts. (Algebraists may prefer the differential form of this IDEA, namely the product rule. Think of all the places derivations turn up in modern algebra.) Integration by substitution is also a fundamental IDEA -- it tells how things transform under a change of coordinates. One of the main reasons for doing techniques of integration is to give students a working introduction to these IDEAS, dammit!
From Seattle University, Andre Yandl came up with a slightly more general version of the same objection:
My blood pressure rises every time I hear educators say we should drop this because the computers can get the answers. I heard the same argument from elementary school teachers. We no longer need to teach kids how to add and multiply because we have calculators. And believe me many have not. So I get students in my classes who get 29 for the answer to the subtraction 15 - 14 and don't even notice that their answer must be wrong since it is larger than 15. It's sad, BUT IT IS TRUE. Now back to techniques of integration. If indeed it is a valid arguement that we don't need to teach them because the computer can get the answers, then we should not have taught them for the last 100 years because we have had tables of integrals that provided the answers to any problem of integration any student ever did!
The real question is not what do we teach the students in our courses, but why do we teach it? The benefit of doing techniques of integration, and mathematics in general, is the developping of the student's ability to make intelligent decisions and choices, and the ability to write precisely and logically. It is for the same reason than an athlete lifts weights. It is not that he/she will have to lift weights during a game, but the process of lifting the weights will make the athlete a better player. I believe that technology must be used. BUT... have the students learn techniques of integration in the second quarter of calculus, and later in the nth quarter where n > 2, use a computer to evaluate complicated integrals, just as we all use calcuators to perform long arithmetic operations.
I am very concerned that if we start dropping some of these topics and do not replace them by topics that accaomplish the same goals (as I stated above) we are going to have a disaster.
And, for a change of pace (and a cliff-hanger) David Pengelley reports from New Mexico State University on a major decision now under way:
I have been chairing a months-long discussion by 11 mathematics and 3 engineering faculty, whose focus was choosing a new calculus text here, but which brought up all the issues of reform, since for the first time we were seriously considering switching to a "reform" text. So far we have been using a traditional text with all our reform efforts tacked on the sides. I will know later today the final result, a ballot of the entire math. faculty. We have options ranging from "lean and lively" Hughes-Hallett et al to thicker reform Ostebee/Zorn to science/engineering oriented quasi-reform Johnston/Mathews to tepid "transitional" brand new book by best-seller Stewart. Whatever the outcome it's been very good for us and our relationship with engineering. By the way, the engineers don't agree with each other. If we choose any of the reform texts, we'll be embarking on an instructor training program also.
As a post-script--I checked with David later on Friday, and the vote was so split they have to do a re-vote. As I said, a cliffhanger!