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Nov. 19th, 2004
Adventures in Teaching
Nov. 19th, 2004 05:16 pmStoutfellow's First Law of Mathematics: Algebra brings knowledge. Geometry brings understanding.
I'll get back to that.
Last night in my geometry class, I began discussing spherical geometry - in particular, the properties of spherical triangles. (The edges of a spherical triangle are arcs of great circles, like the meridians of longitude or the equator - but not any of the other parallels of latitude.) There were two significant theorems to be covered. The first was no problem; the proof was crystal-clear, easy to present pictorially. The second was something else. The proof in the text was rather terse, leaping from point to point without much elaboration. That's not a big problem if the reader is sophisticated enough to fill in the details, but only a few of my students are at that point. Worse, though, the proof was mostly algebraic in nature. Now, analytic geometry was a brilliant discovery, and I bless Descartes and Fermat for developing it, but a geometric proof that can only be presented in terms of calculations with coordinates is of questionable merit. (See the First Law.) So I spent about half an hour trying to come up with a less calculation-intensive proof, to no avail. When class time rolled around, I still wasn't satisfied with matters. There was a good deal of material to be covered before that proof was to come up, so I decided to stall a little - to go into extra detail on that material, so as to leave the theorem until the next class period and give myself time to come up with a better approach.
So I did. I drew lots of pictures, digressed at significant points, compared spherical geometry to hyperbolic geometry. At one point a student admitted having difficulty visualizing what was going on; I got a globe down from one of the cabinets and traced shapes on its surface until she was satisfied.
All to no avail. There were fifteen minutes left, and no more excuses. I began presenting the second theorem, following the textbook's proof. On the fly, I came up with geometric justifications for the various details, gestured maniacally to indicate where the seemingly off-the-wall constructions actually came from, drew the students along with leading questions, and - with a minute or two to spare - completed the calculations.
I should have allowed myself more time to prepare for the lecture. Nonetheless, given that I didn't, I'm rather pleased that I managed to pull it off.
We get the whole week off for Thanksgiving. For that, I'll certainly give thanks!
I'll get back to that.
Last night in my geometry class, I began discussing spherical geometry - in particular, the properties of spherical triangles. (The edges of a spherical triangle are arcs of great circles, like the meridians of longitude or the equator - but not any of the other parallels of latitude.) There were two significant theorems to be covered. The first was no problem; the proof was crystal-clear, easy to present pictorially. The second was something else. The proof in the text was rather terse, leaping from point to point without much elaboration. That's not a big problem if the reader is sophisticated enough to fill in the details, but only a few of my students are at that point. Worse, though, the proof was mostly algebraic in nature. Now, analytic geometry was a brilliant discovery, and I bless Descartes and Fermat for developing it, but a geometric proof that can only be presented in terms of calculations with coordinates is of questionable merit. (See the First Law.) So I spent about half an hour trying to come up with a less calculation-intensive proof, to no avail. When class time rolled around, I still wasn't satisfied with matters. There was a good deal of material to be covered before that proof was to come up, so I decided to stall a little - to go into extra detail on that material, so as to leave the theorem until the next class period and give myself time to come up with a better approach.
So I did. I drew lots of pictures, digressed at significant points, compared spherical geometry to hyperbolic geometry. At one point a student admitted having difficulty visualizing what was going on; I got a globe down from one of the cabinets and traced shapes on its surface until she was satisfied.
All to no avail. There were fifteen minutes left, and no more excuses. I began presenting the second theorem, following the textbook's proof. On the fly, I came up with geometric justifications for the various details, gestured maniacally to indicate where the seemingly off-the-wall constructions actually came from, drew the students along with leading questions, and - with a minute or two to spare - completed the calculations.
I should have allowed myself more time to prepare for the lecture. Nonetheless, given that I didn't, I'm rather pleased that I managed to pull it off.
We get the whole week off for Thanksgiving. For that, I'll certainly give thanks!
