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Twocolumn Proofs
Posted:
Feb 17, 1993 2:14 PM


I have little doubt (as well as little direct or indirect knowledge) that teaching twocolumn proofs fails in general to help students to reason better or see the beauty of mathematics. (Given that, I am wondering why I am posting this.)
However, I was taught twocolumn proofs as a freshman in high school and liked it. That you needed a reason for each statement you made seemed selfevident. Arranging the statements and reasons in columns appealed to my own aesthetic sense of organization. I felt a sense of power of being able to prove some proposition, and I even think I had some understanding that these proofs relied on postulates taken for granted. I realize that my experience was exceptional.
A few years ago, I found an NCTM report on proof in the geometry curriculum from the 1950's. There was some study on the effectiveness of teaching proofs. It seemed that the study was not very broad (just a few classes were monitored). I've forgotten the details, but the success of the class was linked to the way the material was presented. If geometry was presented as a corpus of facts (including the proofs), the student failed to understand any of it. I don't clearly remember the more successful way (hah! which way do you think would be more important to remember?). I believe student wasn't told as much as they were led to discover. There was less lecturing and more doing in class. I have the impression that the paucity of data undermined the authority of the report. Perhaps someone recalls the report? I may be able to find it again if anyone is interested.
I personal experience indicates that most individuals are able to understand proofs and produce them until they are 19, 20, 21.... So of all college graduates, only bright math majors are likely to know something about proof. There could be developmental reasons for this, but I am ignorant of any. It seems that 16 year olds might be able to begin to understand proofs, and exposure to proofs might in turn help them later on.
It is arguable whether this is a travesty or not.
My questions are: How prevalent is the beating of the dead horse, the twocolumn proof? How, if at all, does proof enter in to the progressive geometry curriculum?
mike rogers mkr@cs.amherst.edu  



