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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
Re: Proofs - My Thoughts
Posted: Feb 19, 1993 3:07 PM
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Michelle Manes (manes@park.bu.edu) wrote:
> ...
> High school (maybe even middle school) is a good time to start
> formalizing the argument process, but I would argue that the
> two column proof is not the way to go... It doesn't really lead to
> andy of the 3 positive aspects of proof, and it's not the way
> "real mathematicians" write proofs.
>
> Two column proofs are fairly sterile, they leave little room for
> creativity (and hence the joy that comes from problem solving).
> Often students are simply asked to prove a result -- so they
> already know it's true because they're beings asked to prove it.
> A better way is to have them *find* a result through experimentation,
> and then want to prove it because they're already convinced
> themselves that it's true. And two-column proofs rarely lead
> directly to another result.


I think the two-column proof was created to emphasize the need
to justify (or be prepared to justify) every assertion and show
a logical *train* of thought.

A problem I see with teaching most young people the two-column
method is that they would interpret it too literally to understand
what they are doing. They would think that the most important part
of proving is filling in two columns with stuff. Some would
think that using two-columns is the only valid way of *finding*
a proof. This error could be addressed by the teacher. Still
I would expect some students not to understand. They would miss
that the chief work in finding a proof is rooting around, trying
to find connections via scratchwork and experimentation. They
would not view the two-column write-up as a means of intelligibly
communicating their proof.

As for how matheticians write proofs: Brevity is the soul of
the mathematics' publishing industry. But I think most mathematicians
still operate in a bimodal (assertion/justification) way. If
a justification is too obscure or is important, it is included
although not in a separate column ("By Theorem 3.14, the
circumference...", or "...which can be deduced from Gauss' Lemma").
We also operate in this way when we read mathematics. If a
justification is not supplied in the text, we try to supply it
ourselves. We fill in the second column so to speak.

Finally I would argue that two column proofs, being essentially
the same as proofs, do lead to another result (at least not
rarely). I do believe that they rarely lead young people to
another result. Those students tend to be too worried about
the trees to be able to see their way out of one forest and
into the next.


From archive Thu Feb 25 10:02 PS 1993
In-reply-to: <4@forum.swarthmore.edu>
Subject: Re: Two-column Proofs
From: Keith Grove - Dover Sherborn HS <doversherhs@igc.apc.org>
Date: Feb 25 10:02 PS 1993
Message-ID: <1800700002@igc.apc.org>
Epigone-indent: 1


I am a student of Dover-Sherborn high, and a member
of Mister Groves sophmore geometry class and we have done
equal numbers of two column proofs and paragraph style
proofs, and I have found paragraph style the most usefull.
The paragraph style encourages us to write down all we
know, but the two column proof only needs or asks for a
certain thing. The paragraph lets you write something not so
specific but will help you along in your completion of the
paragraph style proof. Also the paragraph style will make
all those insignifigant (supposedly) in your head, and when
the next proof comes you will remember them, when they
are more important than they were in the last proof. As a
matter of fact the two column proof is not letting me retain
all those "insignifigant" facts. If right now you asked me to
do a proof I did not know how to do, but it involved basics of
triangles, rectangles and what not, and you asked me to do it
in paragraph. If you asked me to do it in two columns I
probably wouldn't even get the first step let alone
completion.








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