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Topic: Proofs - My Thoughts
Replies: 11   Last Post: Feb 26, 1993 3:04 PM

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Michael Rogers

Posts: 15
Registered: 12/6/04
Two-column Proofs
Posted: Feb 17, 1993 2:14 PM
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I have little doubt (as well as little direct or indirect
knowledge) that teaching two-column 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 two-column proofs as a freshman in
high school and liked it. That you needed a reason for each
statement you made seemed self-evident. 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 two-column proof? How, if at all, does proof enter
in to the progressive geometry curriculum?

-mike rogers
mkr@cs.amherst.edu
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