One of the advantages of my job is the incredible scheduling. We finished the fall semester the second week of December and my first class of the spring is next Tuesday! The downside is that this gives me way too much time to plot, scheme, doodle, dabble, think, rethink, and overthink. In the end, I usually settle on a plan that is far too ambitious, pedagogically impossible, philosophically suspect, and utterly indefensible. Thus, I bring you my plan for calculus this semester.
I read the wonderful article, Putting Differentials Back Into Calculus, which argues for using differentials in a way closer to their original creation than the way they are employed in modern textbooks. As a huge fan of Thompson’s Calculus Made Easy, this suggestion didn’t seem half bad. Considering that only a fifth of my students are math majors and that, for the rest, using calculus outside of their physics class is unlikely, why not make things as easy as possible. I’m going to push to teach this same group next fall in calculus II, so the only teacher I can hurt is myself, right? But then I started thinking: the reason the differential approach will work so well with these students is that they will always be using differentiable functions. What reason is there to mention limits and continuity? These are technical issues that won’t help them at all in understanding calculus or how to apply it in their field of inquiry.
Oh dear, so here I am with essentially two months of material (this includes learning to differentiate any elementary function and using this to solve the standard problems). What will I do for the last month and half? I quickly remembered to add Taylor series because I love teaching that in calculus I. Then I added in the obligatory introduction to antiderivatives and integration. I even sprinkled in some partial differentiation at the end so that I could show the students the totally-awesome-implicit-differentiation-trick that would save them five minutes on the final exam. Grr…still two weeks left. These are precisely the two weeks that I usually spend on limits in the beginning. Now I remember why I always do this. It perfectly fills in the semester calendar. And then it hits me.
Any subject can be made repulsive by presenting it bristling with difficulties.
—Silvanus P. Thompson
Limits sure confuse the heck out of students. Why in the world are we leading calculus off with limits, especially to non-math majors? For the purpose of rigor?
You don’t forbid the use of a watch to every person who does not know how to make one? You don’t object to the musician playing on a violin that he has not himself constructed. You don’t teach the rules of syntax to children until they have already become fluent in the use of speech. It would be equally absurd to require rigid demonstrations to be expounded to beginners in the calculus.
—Silvanus P. Thompson
So I decide I just won’t do it. No limits for us. We will just do some extra exploratory work. There is a great article on math in medicine we could read together. Okay, time for sleep.
But sleep does not come.
All right. All right. I’m up.
Why can’t the limits just die? Why do I feel the compulsion to put them back in? They’re like the tell-tale heart beating under my floor. What will become of my math majors if they don’t see limits? No, I can’t do that to them. What a cruel joke to play: send them to real analysis without having used limits. Back in they go. Of course, those biology majors are going to be completely turned off and once you lose them, they’re gone for good. Argh, out they go. On the other hand, if I get audited by the department, they are sure going to ask questions. With my review coming up, I can’t afford that kind of chatter. Put them back in…
…but later. Huh? Put them back in, but later. Yes, of course. Later. The course description says I need to cover limits, but it doesn’t say when! What if we introduced everything in a reasonable way and, only after the students know what is going on and why any of this is important, then showed them those funky limit do-hickies? Hmm. Interesting. And that is my explanation for the following calendar and skills list: