Introducing the Concept of Proof to Students

This week we saw four more interesting miniature lessons. By a good coincidence, three of them dealt with mathematical proof to differing extents. I want to speak about one lesson in particular, which covered the following geometry problem:

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The object is to determine the sum of the three angles indicated in the figure. I believe this puzzle originated with Martin Gardner (an excellent source for clever math problems and riddles) but I am not positive. We were given Geoboards and elastic bands as tools to help solve the problem. In addition, we we cut up a copy of the figure in order to see that the three angles should sum to a right angle. Although it's a very good problem, this was a rather difficult exercise (perhaps due the fact that we weren't expecting it). It made me wonder how it would be received by grade 10 students. However, if they had been working up to this exercise over the term by doing simpler geometric proofs, perhaps it would not catch them as off-guard. It is probably easier in the beginning to learn proof by proving the obvious, so whether this problem is at-level would depend on how many similar exercises the class had already seen. It is definitely the type of thing that needs to be seen by students more often, and early. If I had a choice, I would have done an algebraic proof, but being forced to do a geometric proof showed me how bad I really am at that kind of thing.

My first real encounter with proof was a second year university algebra course and it completely revolutionized my impressions of mathematics. It was not difficult, it taught me to think, and I really wished that I had seen something like that earlier. Another benefit of proof, particularly of the geometric kind, is that is does not use numbers. It is a purely logical/visual expression of math which I imagine many students may find refreshing, particularly those whose numeracy may be lagging behind their peers. Does the prevalence of calculation, data, word problems etc. perhaps create an implicit bias in the students' minds with respect to what mathematics actually is? Even geometry, when it is taught, is less about structure and more about numerical quantities of area, volume and so on. Which is more fundamental after all, shape or quantity? In schools, quantity seems to be hands-down winner of that debate (whether the debate ever took place I do not know). Perhaps it may be that the math class could do with fewer numbers.