If-then-therefore
Doing proofs in Geometry class is hard. Teaching how to do them is harder.
Our tutoring colleague is facing a challenge. It’s the time of year when Geometry students are being required by their teachers to create their own proofs. Now, even when you’ve followed along carefully in the mathematical proofs you’ve been shown over the years, making one of your own is difficult. Most students manage something adequate, under pressure, but few enjoy the experience and only the true mathematicians among them want to continue the practice. Even the other tutors, who have all done reasonably well in Geometry and further math classes, dread this period. Our tutor is still puzzling how to teach it, doing proofs and watching himself while he does them. He has suggestions and hints, but so far no assured process. Maybe there isn’t one.
No doubt many students, past and present, would be happy to see Geometry reduced to a quick study of triangles and squares in an appendix to Algebra I or II, and formal logic dispensed with entirely. One will rarely have to work out the area of a hexagon in real life; even rarer is any rigorous evaluation of a converse. But we believe that is the most important reason (among several) for studying Geometry in the first place. We should all be faced not only with the idea, but with the task of arguing rigorously from a set of premises to an assured conclusion. The deductive proof was one of the most important contributions of the ancient Greeks, quite different from the pragmatic rules of the Egyptians and Babylonians (who were in some ways much better calculators). It remains terribly important.
What makes it so hard? Well, one has to have a firm grasp of the postulates and theorems available to date. They must be memorized in enough detail to know when they can be used and when not, and that is no easy thing. Other bits of algebra can be reduced to automatic procedures; not so the proper application of the parallel postulate.
More difficult, we believe, is the creative application of these tools. If there’s more than a step or two in the proof, there are generally different ways of accomplishing it, and no fixed way to do it. You have to find your way from the givens to the QED without a well-marked path. That’s hard to do, and harder to teach. The most effective way is to do a lot of them.
Most importantly, humans don’t normally think that way. We reason by analogy, by applying rules inconsistently, by gut feeling (which may be completely irrational). Coming to a rigorous, logical conclusion is highly unusual for us. Which is a different problem.
