As I have been meaning to do for several years now, I finally looked up a copy of Euclid’s Elements and got to reading. A complete text – and the one I have been reading – can be found here, in Greek and in a translation a former professor would refer to as a “Bad English Version” done by a physics professor at the University of Texas, Dr. Richard Fitzpatrick.
Compared to the modern text I use at school, I was immediately struck by one thing in particular: Euclid’s proofs rely on a lot of actual drawing, marking off, and otherwise constructing bits and adding pieces to the diagram he starts with. In contrast, the proofs in my current textbook rely mainly on learning a bunch of principles, and then applying them to other diagrams.
The problem with this second approach, I have found, is that it frustrates most students, who are “proving” things which are perfectly obvious – hey, look, the computer-drawn graphic of a rectangle with some lines on in results in some congruent lines! Euclid’s approach says, “Fine, smart-aleck, you draw those equal lines. No, you don’t get a ruler, that’s not how the game works, and anyway how did that ruler get so precise in the first place?”
Off-hand, I can’t remember how my own geometry instruction in high school compared, though vague memories suggest it was somewhere between the two. I am the sort of person who is perfectly content to play with any sort of logical system for any good or bad reason (up to and including general time-wasting and neglect of chores), so I’m not sure I would have attached any importance to the difference if I did not have to teach the subject now.
Of course, the one thing worse than completely abstracted proofs has to be the ridiculously contrived “real life” problems. My own conclusion so far is that for most practical applications – not that I have to deal with any often, so I may be way off the mark here – a few essential facts and a basic understanding of trigonometry will get you much further much faster than messing around trying to understand the “application” of the proof of some triangle theorem to a particular scenario. This is not to write off geometry – the “essential facts” have to come from somewhere.
I am not sure what to make of this – or how exactly to apply it a month and a half into the school year, anyway. To summarize my current thoughts on the subject: geometry should be a hands-on course, essentially a puzzle game, and stick to obvious applications if any – maybe looking at Kepler’s laws, or something like that.