The Fall of Calculus

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March 2024 TOC icon
March 2024 (Vol. 56, No. 2)

Don’t worry, I’m not suggesting that calculus has gone downhill, or that it’s about to do so. As long as things change smoothly (and the laws of physics, among others, suggest that that’s not about to end) we’ll need derivatives to measure the rate of change, and integrals to total its effects.  Rather, I’m referring to the first semester of calculus, which most science students take in the autumn in which they arrive on campus. 

It’s probably much the same course that you took. Well, maybe you taught yourself from Schaum’s Outline or Prof. E. McSquared’s Calculus Primer, or were admitted into a select class using Apostol… but first calculus courses at most universities are pretty much the same, and the textbooks reflect it.  James Stewart’s excellent “Violin Book”, now in its nth edition, is the type specimen: but most of the others are isomorphic. 

The biggest change since when I started teaching here at Saint Mary’s is that “Math 200” was still a one-year course then, as it was at many universities.  (The number started with 2 because we’d had a post-grade-11 admission stream; 100-level courses had been Grade 12 level.)  Stewart’s text was written back in those days: as everybody was there for the whole year, the important thing was for the order of material to be as logical as possible. Review of Functions, Limits, Derivatives, Theorems and Applications of Derivatives, Integrals.  At the end of Chapter 5,  if all had gone well, the class took their first steps in integration just in time for the Christmas exam, and in January the course began again where it had left off.

The problem with that, of course, was that if anybody fell behind in the first semester,  it would be a long wait until they could retake the course in summer.  So, not long after my arrival, we broke it into two one-semester courses, and offered each one in each semester.  But we were still doing (for most students) limits in September, derivatives in October, integration by parts in January… the same old tune from Stewart’s well-crafted violin.  And why not?  Everybody was still there for the whole concert,  even though some of them were now restarting in January.

But along the way some science departments decided that their students might get more use out of, say, one semester of calculus and one of programming, or maybe linear algebra.  (Engineering, computing science, and other math-heavy fields are of course still requiring “all of the above” and more.)  So their calendar requirements changed… and that’s maybe not a bad thing.  But it does lead to a problem.  Most scientists need differential and integral calculus in approximately equal measure… and the first movement of Stewart’s Violin Concerto in F Prime, on its own, doesn’t provide that.  That’s not his fault: it wasn’t meant to.

So is it possible to rearrange first year calculus in order to make Math 1210 (as we now call it) work better as a standalone course? Maybe some material on limits can be moved into the second semester?  Perhaps there are some integration techniques (trig substitution?)  that many life sciences students don’t need?  It’s not clear yet if we can adapt the content, or how. But… maybe we should fiddle with it. 

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