“Off-track geometry”

JD2718 is always worth reading. I used to know his real name, but I’ve lost track of it. No matter; he teaches geometry, studies cryptography, and just turned 50, so we know he’s a good guy. A few months ago he wrote an interesting post about his “shaken up” geometry curriculum. As I’ve taught honors and college-prep geometry for most of my 40-year teaching career, I feel compelled to respond — but you should read the original post as well.

First of all, you probably wonder what “off-track” geometry might mean. It turns out that in his New York City school the honors math students take the first semester of geometry as second-semester freshmen, so they are “off-track.” Mr. JD2718 was the only teacher of this “off-track” course that semester, so he had a lot of freedom to shake things up (almost always a good idea). Being the only teacher of a course involves a trade-off: you have a lot more freedom, but you also lose the substantial advantages of having collaborators and colleagues with whom to share work and ideas.

Let’s compare what JD2718 did with what I’m doing this year at Weston — and what I would like to be doing. I’ll discuss his points in his order. First of all, he opened with an extensive logic unit. We do that in honors as well. (Actually, we open with transformations, followed immediately by the logic unit.) He spent four weeks; maybe we should too. Ours was only half that long, and students would have benefitted from four weeks rather than two.

Next, JD2718 offers the following observation:

Have students create their own glossaries/reference sheets. Allow/insist on constant revisions and updates. Allow/insist that the students bring their reference sheets to each quiz and test.

We do almost exactly that. We offer three templates — one each for Definitions, Theorems, and Postulates — which students need to revise and update continually. They use them on quizzes but not on tests, which are announced well in advance so the students can prepare for them without having notes during the assessment.

The next issue is constructions. I rather like his system of doing constructions every Friday, as it allowed for integration of constructions into the content units rather than making them a separate unit by themselves. I want to do that next year.

Then Mr. 2718 makes a particularly interesting observation about proving theorems:

Proving theorems is at the core of what mathematicians do. The students need to be asked to prove theorems. And all the good ones are taken. So I ask students to prove less-known, less-useful theorems. We practice doing the real thing. We talk about the difference between proving a theorem, and doing a proof-exercise from the book. We approach them slightly differently. And we write them differently.

Yay! Let’s do that. Most of my students are fuzzy about this difference — and about how they’re similar.

Read the article for more details, including his students’ reactions to the revised course. It’s good food for thought.



Categories: Math, Teaching & Learning, Weston