College Math Teaching

March 9, 2010

The Importance of Integrals and Standards

One of the challenges of teaching lots of “service” courses is that one sometimes comes under heat from client departments if one flunks too many of their prospective students (especially in the engineering/math/science calculus sequence)

Sometimes, we are told that we are too hard on them or teach students what they don’t need to know.
So, it was “art to my eyes” to read the following post at Cosmic Variance:

Having recently slogged through grading an enormous pile of graduate-level problem sets, I am compelled to share one of the most useful tricks I learned in graduate school.

Make your integrals dimensionless.

This probably seems silly to the theoretical physicists in the audience, who have a habit of changing variables and units to the point where everything is dimensionless and equals one. However, in astrophysics, you frequently are integrating over real physical quantities (numbers of photons, masses of stars, luminosities of galaxies, etc) that still have units attached. While students typically do an admirable job of setting up the necessary integrals, they frequently go off the rails when actually evaluating the integrals, as they valiantly try to propagate all those extra factors.

Here’s an example of what I mean. Suppose you want to calculate some sort of rate constant for photoionization, that when multiplied by the density of atoms, will give you the rate of photo-ionizations per volume. These sorts of rates are always density times velocity times cross section: […]

the integral reduces to something that you can start to wrap your brain around:

Basically, they were talking about a change of variables. Of course, the integral is NOT elementary and one would have to use some sort of technique (residue?) to evaluate it.

But the point is that the people in physics EXPECT their students to be able to handle the mathematics.

But about the heat we catch for our flunk-out rates:

to be honest, not everyone is down on us for that:

We make rules we think will help our students–you fail if you don’t do the reading, you fail if your paper isn’t turned in on time, you can rewrite anything you fail, ad infinitum–thinking it will help. Then I come to RYS and see the bodies dropping all over the damned place.

That’s why, this semester, I started to bend instead of break. Kid wants to turn it in late? Okay. Kid can’t be in class. Who cares? I go in every day, try to start a discussion, give an impromptu lecture on days they won’t bite, let them out early once I’ve told them what I guess they probably have to know. If I can make out what the paper is about, I give it at least a B-minus. I mark the hell out of them–I write comments in the margin till there ain’t no margin left, and no ink to write in it with. But the grade is always a B-minus or higher, because if it isn’t, they’ll come to my office requesting a checklist of things they can do to write more effective essays, by which they mean essays that will get better grades. […]

So I inflate grades. So should you, unless you’re teaching your students math or anything related to keeping buildings or airplanes or economic systems from falling apart.

Emphasis mine. What can I say? 🙂

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