GL(s,R)

January 17, 2011

A multivariable calculus list

In addition to my calculus course this semester, I also get to teach a multivariable calculus course with only six students. I’ll start with the standard list for those interested in that sort of thing.

Spring 2011 Multivariable Calculus Standards List

Let me admit something, here, in between two documents—less likely to read in here—about teaching this course, now for the third time: I’m a fraud. That’s right, I’m a fake, a charlatan, an impostor. I’ve created a counterfeit course and hustle the students with a dash of hocus-pocus and a sprinkle of hoodwinking. It is only through mathematical guile that my misrepresentations, chicanery and flim-flam go unnoticed. In short, and in the passing Christmas spirit, I am a humbug. This is a physics course. It should be taught be someone proficient in physics, someone with honed intuition about the geometry of abstract mathematical notions like div, grad, curl and all that, someone who sees everything as an application of Stokes’ theorem and has strong feelings about whether it should be written Stokes’ theorem or Stokes’s theorem. About the only thing I bring to the table is that I can teach students to remember that:

\mathrm{curl}(\mathbf{F}) = \nabla \times \mathbf{F}

and

\mathrm{div}(\mathbf{F}) = \nabla \cdot \mathbf{F}

because curl and cross both start with c, while div and dot both start with d.

Here is the calendar for the course. After it, I’ll explain a little bit of what I’m trying.

Spring 2011 Multivariable Calculus Calendar

There are several big differences here from how I’ve taught this course in the past. First, I am going to try with all my might to get to Stokes’ theorem before the last week. Part of the way I plan to do this is, similar to my calculus class, to cut out most of the stuff on limits and continuity that I usually get bogged down on in the first couple of weeks—am I the only person who finds interesting the pathological examples that make Clairaut’s theorem necessary? I get to teach an extra hour a week to a subset of the class and that stuff will fit perfectly in there. For the science majors, I’m more interested in helping them figure out how to use this stuff and how to develop intuition. Second, I’m skipping Green’s theorem until the end. Yes, it changes the story I normally tell, one that progresses so nicely up the dimension chart, but the trade-off is that I get more time to show them Stokes’ theorem and more time to focus on the physical interpretation.

Speaking of interpretation, you will notice in the calendar  eleven or so ‘Group Activities’. These are stolen from an excellent guide produced by Dray and Manogue at Oregon State as part of their Bridge Project. To work within their framework, I’ve made another structural change that I’d never considered given how I think about the subject. Immediately after finishing triple integration (which, essentially, finishes the first half of the course), we start with vectors (I never start with vectors as most calculus books do) and then I want to get to line integrals and surface integrals as fast as possible. Normally, I mess around with div and curl before getting to integration of vector fields. Instead, I’m going to push out the Divergence theorem—the theorem I always cover in the last 45 minutes of the course—and use this to motivate the definition of div. Then I’ll push out Stokes’ theorem and use this to help motivate the definition of curl. This ought to give me two solid weeks to explore the physical meaning of these theorems as well as to use them to prove some of the standard cool corollaries (like Green’s theorem).

This class will also be the first of my SBG courses to incorporate a final project. If anyone has good suggestions based on experience about how best to incorporate projects into the SBGrading scheme, I would live to hear them. My current system is quite simplistic. The standards for the course are given a 90% weighting for the overall grade—did I mention that midterms and finals now are simply extended assessments whose grades are treated like an arbitrary quiz, just with a lot more standards tested?—and 10% weighting for the project.

January 12, 2011

A Calculus List

Filed under: High Effort/Low Payoff Ideas,Standard Based Grading — Adam Glesser @ 4:19 pm
Tags: ,

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.

Toss.

Turn.

Toss.

Turn.

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:

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