Date: Sun, 5 Feb 1995 21:07:06 -0500
Subject: proving the obvious
David Scott Powell rightly points out one of the most widespread methodological errors in teaching geometry: proving the obvious. Indeed, Euclid himself was famous partly for doing just this. It is an historical commonplace, but also quite accurate, to praise the architecture of his Elements, starting as they do with the establishment of the obvious.
Trouble is, our audience is not Euclid's. No matter: the error has received the blessing of History and is difficult to come away from.
So I am intrigued by the various lists of what actually "needs" proof in geometry. In fact, I have often taught a year of geometry, when I could, "backwards," by deriving the non-obvious stuff from a bunch of obvious stuff, then finally setting down axioms and proving the obvious stuff. Euclid would see this as constructing a building from the top down (or at least, from the middle out), but I don't.
In most traditional developments of geometry, it's not until December or so that one proves anything really interesting. Here's a chronicle as I recall it, in the spirit of "1066 And All That":
Back to rectangles, rhombuses, squares. And no one ever draws a rhombus as a diamond (on a pack of playing cards), so that you can see the symmetries. No, you must go by the "properties of parallelograms", and the rules of formal proof.
For this theorem, I usually have the kids trace the outline of their face in a mirror (with soap, not lipstick!) and explain why it's half size. Then you can motivate the proof. I used to ask them to go home and experiment--they never did. They all think they know what will happen. They have some idea that it gets bigger as you get closer, varies with the size of your head, etc. I usually demonstrate it in class, then get a chance to make fun of their smugness (Aristotle thought such and such about falling bodies, everyone knew about spontaneous generation, you're all a bunch of superstitious medieval peasants, you probably don't have a 13th floor in your building, etc). Lots of fun ribbing them, which they love. But I think this indicates that something is non-obvious, and even counter-intuitive, here.
Then we get to areas and coincidence. Now most of the results are not obvious. However, kids have trouble with coincidence. Until they're led to it, they don't understand why it's important to know that three lines meet in a point. It's not a metric result. In fact, it's an existence theorem (although the proof is usually by construction) and so is pretty abstract.
The one about the concurrence of the perpendicular bisectors is a good one to get to early, and you can do it from symmetry of isosceles triangles. It is not at all obvious that any three non-collinear points lie on a circle. I've often had kids, in traditional courses, who doubted this, and even tried to refute it, IMMEDIATELY AFTER having seen a proof. Says a whole lot about the effectiveness of proof at a certain level. This was before I started using the approach from symmetry.
Areas are of course difficult. I don't think that adults really "conserve" area, in Piaget's sense. We are certainly not good at estimating it. And I doubt that anyone "really" conserves volume. Everyone just pretends to, like they pretend to have read "Moby Dick." You math people are all wrong. A baking pan and a ketchup bottle simply cannot hold the same amount of water: water is not a good measure of volume. It seems to know how to spread itself out to cover a baking pan, and squnch itself up inside a ketchup bottle. Volume is a formula. Water is wet.
That is, most results about area and volume are hard to guess, once you get away from the usual formulas, which kids typically have drilled long before a geometry course. We can include in this statement the Pythagorean theorem.
But my favorite milieu by far for introducing chains of reasoning is geometric inequalities. You can start in a number of places (I often take the triangle inequality as an "axiom") and get results that are really hard to guess, but easy to deduce. An example: give a (convex, planar) quadrilateral and ask the kids where on the plane the sum of the distances to the vertices is a minimum. They will do some exploration (take a square, etc.), but then, with only the slightest prodding, will use the triangle inequality to show that this is where the diagonals intersect. They will (probably) go through exactly the kind of proof you want to get out of them, however they write it down, and they will (probably) see this as the most natural way of coming to this conclusion. There are tons of such examples, some of which are really difficult.
They can be done very early in the study of geometry, at various age levels. If and when they see a formal development of geometry, it all fits together.
The idea behind the inequality "proofs", I think, is that they force the kids to think of a "variable" figure. That is, if you ask them about the base angles of an isosceles triangle, they focus on the symmetry of the particular triangle you, or they, draw. If they see another isosceles triangle, they focus on its symmetry. They're not really constructing in their minds a set of figures with a common property. But if you ask where the sum of the distances to the vertices of a quadrilateral is minimal, they are forced to think about all possible types of quadrilaterals (and often spontaneously get into non-convex or skew quadrilaterals, where the problem is more difficult).
It's sort of like the way kids go from arithmetic (individual facts about numbers) to algebra (generalizations about facts about numbers, which become facts about functions).
Just a word about Christina Rose's question (Why teach proof? How is it relevant?). To me, the nature of mathematical truth is the One Main Thing that I want kids to come away with if they do any high school math at all. Despite NCTM propaganda, it seems to me that 70 or 80 percent of my students will not use any math beyond arithmetic--not even algebra. I include in this estimate not just elevator operators and bus drivers, but also doctors,lawyers and teachers of social studies. It's not a class thing at all: a carpenter may use more math than a surgeon.
So why math for all? Easy. (1) We don't know who will be doing what; (2) we don't know how this situation will change with new technologies; (3 and most important) math is GOOD FOR YOU. That is, even if they don't use algebra, geometry, etc., they will know arithmetic better, and use it better, and reason with it better, for having seen algebraic ways of thinking. And if they see mathematical proof in its "pure" form, whether it be connected with geometry or some other subject matter, they will be able to handle logical deduction better in "real life", even if they don't use it on mathematical objects. So I agree heartily with John Benson.
Bertrand Russell once defined mathematics as "the set of all statements of the form 'If A, then B'". Maybe this was said tongue-in-cheek. But there is some truth to it, and if so, the face of mathematics presents itself at every turn in life.
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