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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: Two-column Proofs
Posted: Feb 25, 1993 4:16 PM
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Dover-Sherborn High School (doversherhs@igc.apc.org) wrote:
> 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....


I was delighted to read your response. It seems that you
are learning a lot about mathematics in your geometry class.
However let me address some points about the nature of
proving which your article indirectly raises.

There are (at least) two aspects of proving a theorem.
The first is finding a proof; the second is writing it
down in order to communicate it. To use a weak analogy,
the first part is similar to searching for something you
lost. In the course of your search you come across many
things some of which are insignificant. These insignificant
things you may put back where they belong. Hopefully you will
finally find what you seek. In searching for a proof, you write
down possibly insignificant facts and look for some connection
among them which will lead to a proof. Again hopefully you
succeed.

The second part of proving is to write the proof clearly.
When you write the proof, you should leave out all of the
ideas which you did not use. These will only distract and
confuse the reader. You should communicate only those ideas
which demonstrate why a theorem is true. This second part
if often misunderstood by those just learning to write proofs
(even in college!).

If you will allow me to be a little bit critical, it seems to
me that your conception of a two-column proof is to do the
second part without doing the first part. And your conception of
the paragraph style is to do only the first part. It seems to
me that after you write your first "paragraph proof", you should
rewrite and leave out the insignificant stuff. In fact when my
students include unnecessary statements in their proofs, I point
it out.

Let me clearly say that I do think that you should begin by
writing down everything that seems relevant. What you describe
yourself doing for a "paragraph style" proof is an excellent way
to discover a proof. Every proof requires at least two sheets
of paper---one for finding the proof and one for writing it up.

The two-column "method" focuses one's attention on the second
phase of proving a theorem. It forces one to clearly show why
your assertions are true. One should also be forced to do this
in the paragraph style. One problem with the two-column style
is that it focuses so much attention on the second part of
proving, the write-up, that students do not realize that they
are supposed to still do the first part. And as you said,
without the first part, you are likely to fail to find a proof.







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