It is all in the proof

Posted: Friday, September 16, 2005

While I was doing some serious spring cleaning last May, I came across a dozen blank books that I used for journals. Each one was found in a different box. Each one was a different color, and each one had two or three entries starting on Jan. 1 of some year. But the rest of the journal was blank. Now given the above information, I could write one of those killer logic problems you find in puzzle books, but really my point is more about why those books were blank.

Up until recently, I thought of writing as something I was supposed to be able to do well enough to get my points across, but nothing more. Now I think of writing more like a logic puzzle to solve. I know what I want to say. I've got the basic pieces. Now I have to put them in a logical order, so the reader can understand what I'm trying to communicate. Several of my peers have commented, "You are a math teacher writing a column for the paper? You should teach English." I don't know the first thing about teaching English, but I do know a thing or two about writing a two-column proof in geometry, and I find writing an essay and reasoning through a deductive proof to be similar processes.

A two-column proof has a drawing and some given information. From only that given information and the ideas you know to be absolutely true about geometry, your job is to arrange, in logical order, a bunch of statements with justifications to lead you to the bottom line, what you are trying to prove. In an essay, your job is to entice the readers with a catchy, little title, and grab them in the introduction so they keep on reading. You then try to knock them dead with the one, two or even three supporting paragraph punch, and cool down with crystal clear conclusion.

For most geometry students, doing a proof is pure torture; as for some is writing an essay. I never felt that way. I was the type of kid who ate two-column proofs up with a spoon. I loved them. The pure flow of logic from one idea to the next was beautiful to me, and it made sense in a world that very rarely seemed to make sense. I'm not going to try to convince you that two-column proofs are fun, because I know from experience that no matter what activities I pull out of my bag of tricks, my students can't wait to finish the unit on logic and reasoning, and hence the introduction to proofs. Little do they know that the unit never really ends, because everything we learn in geometry links to every concept learned prior, and we are essentially building a little castle of ideas.

Some teachers still adhere the very rigid structure of the two-column proof even through it is no longer stressed in the NCTM standards. I don't necessarily think this is a bad idea, because a student who masters this process has essentially achieved a decent level of logical reasoning. The problem is that the percentage of students who really understand the process is incredibly low, even among advanced students. I don't think this success level is due to poor teaching, I simply think that students struggle with the details of the structure. They don't pay enough focused attention to the specificity of definitions, postulates and theorems, just as they may not pay attention to the cleanliness level of the dinner dishes or loudness of their music. I find that the two-column structure often interferes with the real objective of learning logical reasoning. But every year I coach them through the traditional method of setting up a proof, because I think they need to see it and try to understand it. Then I use a variety of other tasks to develop their reasoning skills. Students generally think their methods of communication are naturally clear and logical before this unit. It is challenging to get them to see that in fact their arguments are often missing some simple connecting ideas that are important. Until they see the clarity of a proof, it is hard to get them to consistently distinguish between assumptions and truths. Once they wrestle with those two columns for a little while, I loosen the confines of how they must communicate their answers, and their logic starts to truly flow naturally. I see improvement.

I know something about structures, self-imposed or demanded, that don't jibe with the way I think and learn. It's hard to conform, but those blank books just never got filled. I now have three or four larger blank books floating around my house. I work through project ideas, scribble grocery lists, plan trips, and occasionally write down a funny story or idea. It's interesting to me that it took giving myself permission to lift the structure of the "journal format" from my writing to begin to hear my own voice

I now write to relax. It's a hobby. To the statement, "You should teach English," I reply, "Are you crazy? I'd rather run through a bunch of two-column proofs for accuracy than read and correct that enormous pile of papers English teachers always seem to have chained to their ankles." God bless them.

• Mary-Lou Gervais is a math teacher at Juneau-Douglas High School.

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