Monday, December 25, 2006

A Reply to Michael Paul Goldenberg re Proof

Apologies to you, Mike...
I was really replying to Anita’s post, however, I was indirectly referring to your comments. Back in the 90’s I bought into the NCTM argument that as long as we ask students to explain their reasoning and continued to ask those ‘Why does it have to be a rectangle?’ types of questions, this would be just fine for conveying the meaning behind those theorems and being able to apply the principles of Euclidean geometry. I watched as more and more geometry teachers moved away from formalism and, even though something bothered me in my gut, I said, ok, fine. But then I noticed some teachers who didn’t compromise and some of these were young teachers who felt the necessity to include some formal proofs, keeping them short (under 8 steps). When I asked why they were doing this even though 2-column proofs were optional, they simply replied, “Are you telling me, Dave, I’m not allowed to!” Of course they were and they argued that it was hypocritical of me to be calling for higher standards while not advocating we continue to include this type of rigor. They felt that, despite the fact that there were opportunities for proof in other subjects, geometry lent itself to developing logical arguments and 2-column proofs provided a structure that paragraph proofs could not. I attempted to refute that by saying some students will find it easier to write a paragraph proof than the structured kind, but they replied that these students were few and far between, since the teaching of writing a coherent paragraph is enough of a challenge for the English teachers!

Since I was trained to be rigorous in math (pronounced, ‘rigormathis’ I believe) it has always been difficult for me to abandon this upbringing, however I did compromise back then. Now, I believe I should have held firmly to my principles. Mathematics is one of the most beautiful structures ever devised, a product of human curiosity and invention but held together in place by the mortar of deductive logic (ok, that metaphor was way overblown, but it’s still XMAS and I felt inspired!). Mathematicians search for patterns and attempt to make beautiful generalizations (inductive reasoning) but they then must prove these conjectures most often using the power of logic and deductive reasoning. Perhaps there is a compromise here but again I will reiterate that famous quote: THERE IS NO ROYAL ROAD TO MATHEMATICS! Oh, and yes, I’ve taught proof in number theory and loved every minute of it, particularly when achieving that aha! reaction. But the students were more mature and they were not ready to venture on their own with these proofs for awhile, although they could begin to imitate Proof by Mathematical Induction (which actually uses deductive methods!). I think you would argue that younger students could explain their reasoning fairly well if given the right models and opportunities, but that isn’t the same as what I’m talking about.

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