Now that we know something about ZOCS (see 5/30/2009 entry), I’ve been seized with the need to write down why it is that geometry is so valuable to study.
I will list the three things that every geometry book lists, and then I will provide some words to accompany the three other things that I think are even more interesting and beneficial about geometry.
1. Abstract reasoning.
2. Practical applications.
3. General problem solving.
4. Distinguishing between what you know is true and what you can prove to be true.
I maintain that #4 is a crucial skill for success in many white-collar disciplines, especially medicine, the law (much of it), science, engineering, and computer science. A proper geometry course that includes exploration, conjecture, proof (both formal and informal), and opportunities for essay writing furnishes an excellent platform for learning and exercising this skill.
If there is one thing that all white-collar profession require (well, the interesting ones, at least), it is workers who are able to think independently. That means developing new tools on the fly, not simply following cookbook-style instructions. Although there are some well-paid professions that are cookbook-like, the people who go into those either are brain-dead when they start, become brain-dead shortly thereafter, or find some sort of hobby to keep themselves sane. Bootstrapping, by which I mean the ability to start one’s own processes, is something that a good geometry class fosters. Note that this skill goes considerably beyond mere problem-solving. Hirers often list “self-starter” as a desirable attribute for candidates, and the reason is that no white-collar manager wants to spend all of his or her time telling employees to get started, or how to get started.
6. Developing a sense of style/aesthetics.
Geometry is not an art class. Nevertheless, despite the abstract nature of geometry, or perhaps because of it, there is ample opportunity to develop an aesthetic sense, and this should be encouraged. The more that the students have the chance to be creative in both their explorations and their writeups, the more that a personal sense of style can flourish. Geometry is a fact-based course, to be sure, but the facts should be allowed to flourish in an aesthetic environment. This freedom is not unique to geometry among the high-school mathematics classes, but geometry probably gives the greatest opportunity.