Saturday, November 16, 2013

Teaching Proofs

As I mentioned in my first blog post, on Van Hiele levels,  I have some misgivings about the way proofs are presented in the curriculum from which my geometry classes are taught.  The work that we have been doing in Modern Geometry this semester has opened my eyes to the value of conjecturing.  Ideally, proof in my students' geometry classes would develop naturally out of discussion and defense of their own conjectures.  Instruction on how to construct proofs more formally could follow.

As it is, my students face formal geometric proof head on.  They start by writing "algebraic proofs," which basically entails setting up and solving equations with justifications (maybe a theorem or two, mostly properties of equality) for each step.  Then they practice proofs that are more based in geometry concepts.  Early on most of their work is filling in blanks in two-column proofs (which is the format we stick to, for the most part), providing missing statements and reasons.  The next chapter in the course includes parallel and perpendicular lines, and proofs are included again.  The big finish for proofs in Geometry A (although there are more in Geometry B) is triangle congruence proofs.

Proofs are always a struggle, for the majority of my students anyway.  To help them along, I try to point out common patterns in the structure of the proofs we write in class, such as "the final statement is whatever you were asked to prove" or "to change a word into an equation or an equation into a word, use its definition."  I try to remind them to keep the end result in mind, and sometimes we even start at the end of a proof and work backwards a few steps.  A lot of these things that I do to help students "get through" proofs don't really get at the meaning involved, and to a large extent I often feel I'm just teaching them the rules of a game.

It was hard not to be somewhat encouraged though, when by the end of our chapter on triangle congruence, many of my students were able to put together a pretty solid triangle congruence proof on their own.  It made me wonder if teaching formal proof head on might have some value.  The big question, of course, was whether or not my students could transfer any of the skills they were demonstrating to more varied proofs.  I got a partial answer to that question in my class' response to a bonus question on one of their unit tests.

The bonus problem asked them to write a proof of the Triangle Sum Theorem (the sum of the angle measures is 180 degrees).  Not every student attempted the proof.  Among those who did, only two gave fairly conceptually solid proofs, and theirs weren't entirely rock solid as far as our formal standards from class went.  Below is a copy of the problem on the test (I think more could have been said of what is "Given:"), and the two best proofs from my class.

For only two students to come this close took some wind out of my sails again.  But at the same time, it is still the slightest bit encouraging that a couple of them were able to go "off pattern" and prove something fairly different from the triangle congruence cases we had been working with (maybe my hopes for students' proof fluency are way too low).

So I still think that a heavily conjecture-based lead in to proof would serve more of my students better.  But having gone through the usual proof sequence with my students once again, I don't believe what they've done will have no value for them as they continue in mathematics.  If they aren't able to transfer their proof writing to more novel proofs now, might some ground work still have been laid for future proof writing?  It seems like writing proofs formally will always be somewhat awkward the first time, even if it is preceded by lots of practice with informal arguments.

I will have many of the same students again next trimester in Geometry B, and the first unit is on quadrilaterals and their properties.  We are planning to give students a few days to hopefully conjecture some of these properties on their own and then try to prove them with what they already know.  This will be another opportunity to see how much they were able to take away from their education in proofs this trimester.

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