Newsletter #15     On Calculus

I was still fretting over my inability to come up with a synopsis of the Brown Bag on Calculus when the conversation at the dinner following David Pengelley's excellent colloquium picked up and took off from right in the midst of it and filled out and extended a whole bunch of the ideas that had come up. Now I can't sort the two out, in the main, but I should be able to offer you a fair collection of insights and comments from the combined pair.

Having said which, I will promptly turn around and report a couple of items that definitely did arise at the Brown Bag, because they were highly specific to 125. One is the source of a number of Neil's recent alterations to the notes: many resulted from comments and suggestions by Jerry Miller in the Physics Department. Jerry did a careful and thorough job of reading the notes and made a number of cogent and useful remarks. The only one that gave Neil trouble was the information that physicists have no truck with negative gravitational constants--they work on a strictly p.y.o.n.s. (provide your own negative sign) basis. So Neil spent a chunk of his summer turning his problems upside down, proving that being a mathematically responsible citizen can throw some distinctly odd spins into one's life!

The other specifically 125 question that arose had to do with articulation. Even in a collection of the world's calculus courses, no two of which are really alike, ours stand out as being really unlike the others. For a student diving in at the midpoint by starting with 125, the results can range from disorientation to disaster. Not, clearly, a totally solvable situation. The possibility that the students might pick up a copy of the 124 notes and learn from them in their spare time went beyond what even the most optimistic among us could envision. One good suggestion did arise, though: we might put out a slim volume of most-often-messed-up topics from 124. Or alternatively, we might require the notes (which cost remarkably little) and then, at well chosen moments, say "You're lost, hey? Well, I suggest that you check out problem 17 on page 41 of the 124 notes." For that the motivation ought to suffice!

The insights that arose from the colloquium dinner were more along compare-and-contrast lines. David, who did his doctorate here and has returned to the fold for a year's sabbatical, is on the faculty at New Mexico State University. The aspect of his career about which lots of people have heard is his Project-Based Calculus (there's an MAA publication out, and there will definitely be a PFF forum on the subject, unless the colloquium chairmen nab him first!) We didn't actually discuss that, but its existence is certainly a tip-off that NMSU is open to interesting ideas. It has, in fact, what David describes as a state of creative anarchy--great levels of independence about what each professor does with his/her calculus section. They have even done away recently with their common final, which David mentioned with regret, because writing it guaranteed several sessions at which serious discussion of the course content and goals went on. This information produced a moment of stunned silence while we all contemplated the question of distinguishing between creative anarchy and total chaos. Then a further question revealed that they do, in fact, have a common textbook (by Simmons, if I recall correctly.) Whew!

David currently teaches totally without in-class tests or numerical grades aside from the semester grades--a process made possible in part by the fact that within a few weeks he knows the working style and ability of each member of the class (right--they do not have classes of 120 students!) Other members of the department cover the range from there to pretty traditional, but they seem to have no problem with connections between sections. We asked what the Engineering School thought of all this--no problem. But there again the situation is not altogether parallel, because most of their math students have entered the university as engineering students--none of this scrabbling for every last hundredth of a point on the GPA in the hope of admittance.

On the calculator and computer front they seem to be about where we are--conscious of their potential, of their dangers and of their bewildering variety. We didn't pursue that one very far--it might be interesting to find out if anyone has cashed in on the independence to run a heavily calculatorified (calculatorated?) section.

You will notice in all of this a singular absence of any conclusions on my part. That's because I don't think that is what this situation offers. What it does offer is a lovely opportunity for two departments both of which are taking their calculus teaching very seriously to exchange ideas and inspirations.

This is getting too long, but I can't end without some mention of the colloquium which gave rise to the dinner, even if I do give it short shrift. David gave us a great talk on teaching math using original sources. He pointed out that use of such sources can run the full gamut from organizing principle for an entire course to enrichment for a quite canonical course. He even showed us some lovely examples right ready to nab. I was particularly struck by Gauss'es eloquent description of the mathematical process and how rarely the first proof to emerge is the best one. Should we perhaps engrave that one in gold to be handed to graduate students as they tackle their theses feeling that the world expects elegantly proved theorems to flow smoothly from their pens (oh, all right, their keyboards!)?

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