# Ends, means, and the content of high-school geometry

Interesting post today in Professor Hirta’s blog. Here are a couple of excerpts:

Got a phone call today from a high school teacher. He was told by his supervisor that he needs to use more manipulatives in class, so he was hoping that I could recommend a manipulative that he could use in geometry class. Nice. More focus by the administration on the means than on the ends.

Also alarming: he described his “basic high school geometry course” as covering topics such as perimeter, area, and volume. When I think “high school geometry” what springs into my mind is Euclidean geometry and proof; I associate perimeter, area, and volume with middle school math.

Let’s look separately at each of these paragraphs:

First, it would be nice to have a little more information about the basis on which the supervisor made his or her comments. Hirta’s response is certainly valid if the supervisor is indeed neglecting the goals. At Weston the administration always tells us — and appropriately so — that we should “begin with the end in mind,” that backward planning is the key to designing both long-term curriculum and one-day lesson plans. Hirta is right to decry “focus by the administration on the means than on the ends” — if that’s really what’s going on. But suppose the supervisor observed the teacher’s class, found out what the objectives were, and only then concluded that using more manipulatives would help achieve those objectives. There would be nothing wrong in that case: an improved means would help achieve the ends. Or maybe the supervisor did no such thing, and I’m just being naive.

The second paragraph deals specifically with the content of high-school geometry. Hirta is right to “associate perimeter, area, and volume with middle school math,” and it hugely bothers me that so many high-school courses merely repeat what was done in middle school. Even in Weston that’s true some of the time, especially in college-prep geometry. The opposing argument, of course, is that 9th-graders don’t remember what they did in 7th grade, so it’s necessary to repeat the material. The high-school course still includes a significant dose of “Euclidean geometry and proof,” but includes review of middle-school math as well. Again, this is at Weston; I have no idea what’s going on at the anonymous teacher’s unspecified high school. Finally, I think it’s unfair to suggest that “covering topics such as perimeter, area, and volume” is automatically a matter of repeating what was done in middle school. It’s entirely possible (and educationally desirable!) to challenge the students with sophisticated area and volume problems that go way beyond what they did in middle school. In that way it’s feasible to sneak in a review along the way, without sacrificing the need to learn something new.

At Weston we’re in a situation where the honors geometry course is a challenge to everyone but the college-prep geometry course is too easy for many of the students. Using differentiated instruction and an appropriate mix of straightforward and challenging problems, the teacher can create a suitable learning experience for both ends of the spectrum: the weaker students can still earn a B if they work hard, and the strongest students can be challenged on their way to an A. Repeating middle-school content is not the answer.

Categories: Math, Teaching & Learning, Weston