Tuesday, December 13, 2005

The Troubles With Calculus

Many students could not state the Intermediate Value Theorem. On the first midterm they were asked to state the Intermediate Value Theorem. The review sheet for the final told them that they should be able to state the Intermediate Value Theorem. I told them in class that on the final they would have to state a theorem (but I didn't say which one). This was a fair question. Many of them missed it. I gave some trivial amount of credit for stating the Mean Value Theorem; slightly more credit was awarded for a rote garbling of IVT that included some notion of betweenness being important. Probably many students would find it easier to state the theorems if they understood them. Probably many students would find it easier to think about theorems if they knew the difference between "there exists" and "for all" and the difference between a statement and its converse.

Please note, calculus students, that the amazing synchronies of mathematics do not always hold in the heart-stopping miraculous ways that we want them to. While it is a true thing of wonder that the form of solutions to differential equations parallel those of difference equations, it is, unfortunately for you, not the case that the derivatives of inverse trigonometric functions would resemble those of the regular trig functions. Specifically, the derivative of y = arctan(4x) is not arcsec^2(4x) nor is it 4arcsec^2(4x). Furthermore, it is also not 1/(1 + 4x^2).

Somewhere along the way someone has let my students down, probably by allowing them to bring note cards and formula sheets into exams. (Take the emphasis off memorization.) I once fell victim to the formula sheet, too. At some point, though, knowing the formulas for the volume and the surface area of cylinders goes from "pointless memorization" to "essential information that you should just know." I don't know where that point is except to say that it should have been well before yesterday's exam. There went 10 points (the optimization problem) that they will never get back.

The limit problem that has been on three quizzes and two midterms? I think that almost everyone got it right.