Friday, May 27, 2011

The Pentagon is a Pentagon: Against Trivial Application

I teach nothing but geometry. From the start of the day until its end, I'm immersed in it. When I first began to teach, I had little choice but to follow the text. But over time, I've come to doubt the value of the text that was chosen for me, and I ditch it whenever I can. (When I began, the text was Glencoe's Geometry 2007. Today it's the 2011 edition.) I'd like for a moment to focus upon one of its many faults, a fault that it shares with almost all others of its type.

First a bit about geometry. Simply put, geometry is the mathematics of shape, and like all of mathematics, it is abstract. (Hold on to that word. It's central in the argument that follows.) What does this mean? Consider the simplest of polygons, the triangle. (Much of elementary geometry is the geometry of the triangle.) Let us say that we have drawn some conclusion about it, perhaps that the sum of the measures of its angles is 180. This conclusion is not about just this or that triangle. Rather it is about them all. If you like, it is about triangularity as such. Thus we ignore all other properties that a triangular object possesses and consider only its shape, and about that shape we draw our conclusion.

Now of course our conclusion will hold of all triangular objects. But this really is a trivial matter. Of course we must say that if the sum of the measures of the angles of a triangle is 180, then the sum of the measures of the angles of any particular triangular object is 180. But to say this is to say nothing new. It isn't a discovery. It isn't an advance. It is at most trivial application.

Now, what's the relevance of this little foray into the philosophy of mathematics? It's this: the current crop of geometry texts demand near-continual trivial application of elementary geometry.

Here's one of my most despised examples. Glencoe's Geometry gives us a picture of the Pentagon and asks us to determine the sum of the measures of its angles. The task is two part. First we must determine the shape of the given object. Second we must draw a conclusion about that shape. I contend that the first is trivial and hence a waste of time; I contend of the two tasks, only the second is genuinely geometrical. Let me explain.

Here's how we go about our task. First we abstract out the shape, that is we focus upon the shape of the Pentagon and ignore all else. Color, manner of construction (whether brick, stone or concrete), location - all else but its shape is put to the side. After this task is done (and no doubt it is done quickly and effortlessly) one then begins to reason about the shape, and with this one begins to actually do geometry. Nothing is learned in the first step, nothing at all. It requires only that students identify the shape of a thing, and given just how simple the shape is, there's nothing to it.

It's idiocy to have a student already 16 years old pick out pentagons (or triangles, or circles or any other simple geometrical figures). Perhaps we should have them count to ten before we allow them tackle a quadratic, or spell "play" before we have them write an essay about Hamlet. I say that we should dispense with these questions of trivial application and get on to the geometry. That means that we go straight to the shapes themselves. If we wish to reason about pentagons and a picture would be of use, make it as simple as possible. Don't put anything into it that the student will have to immediately abstract away. Don't assume that you've taught students anything that they did not know about the applications of geometry when you present them with pictures of kites, hubcaps, cat whiskers, bobcat tails and all the rest of the clutter you'll find in today's texts. These applications are drop-dead obvious. They're a waste of time.

(A word of caution. I don't mean to say that all applications of elementary geometry are trivial. Some most certainly not. But if it's something that a bright two-year-old could do - and a bright two-year-old could most certainly pick out triangles, circles and other simple geometric shapes - then trivial they are.)

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