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

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Michelle Manes

Posts: 13
Registered: 12/3/04
Re: Proofs - My Thoughts
Posted: Feb 18, 1993 10:54 AM
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Ok, I'll put in my two cents.
Why do mathematicians do proofs?

1. It's fun. (no, really) It's just problem solving taken to another
level. There's the same kick from successfully proving something
rigorously that there is from solving a tricky problem.

2. There are lots of things that *seem* right for hundreds of cases
or more and are in fact false. E.g. (n+1) | (2^n - 1) iff
(n+1) is an odd prime. This is true until. The first counterexample
is at n=340.

3. Proofs lead to new conjectures. Knowing *that* something is true
is all well and good. But if you know *why* it is true, you can use
that to make more predictions about other things that might be true
for similar reasons.

The above are all things we want to instill in our students, no?
The love and excitment of solving problems, the need to be truly
(mathematically) convinced of something before believing it, and the
desire to do experimentation -- search for new mathematical results
from old ones. This is why we have even elementary school students
justify their answers and argue/defend their positions. It's the
start of proof.

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.

So, there you have it.
My opinions are, of course my own and not necessarily those
of EDC or the Connected Geometry project (though we are *not*
including 2 column proofs and we *are* including lots of other
argumentation....), blah, blah, blah


Michelle Manes
Reasearch Assistant, Connected Geometry Project
Education Development Center

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