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

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Rouben Rostamian

Posts: 16
Registered: 12/6/04
Re: Proofs - My Thoughts
Posted: Feb 18, 1993 12:43 PM
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I think it is a skill that anyone can acquire. It's just that often the
students do not have the necessary background skills to think and break
something down in a logical manner. They are plopped right down into the
infamous 'Two Column Proof' and either they get it or they don't. There
is hardly any work done to develop the deductive skills necessary to
really grasp proof.

Part of the problem comes with the reputation, as it were, that geometry
proofs have. They are supposedly something that students have never done
before. True, in a way, but what about in algebra when they were asked
to 'show' that two equations are equal to each other? There they did a
'proof' of sorts, but didn't have to write down the 'whys' of each step.
How about trig identities? Why don't we have them write those out,
giving explanations for each step?

There are a lot of places in the existing curriculum that proofs -
logical, deductive processes which show that something is true - could be
done. And I don't necessarily mean abstract algebra and number theory.
Why is it that geometry has become the showpiece of this supposedly
one-time experience? Is it necessary to terrorize kids (and you must
admit that some of them are) with something that they have done before,
but has never been labelled as such, and consequently they think they
don't, or won't understand it. They use rules and theorems in algebra
and other subjects, but never has so much attention been paid to it as
has in geometry.

Someone (Pamela, I think) mentioned that they use Discovering Geometry by
Michael Serra. Some people have expressed their dissatisfaction with
this book because it doesn't do 'formal' proofs until Chapter 14, and
often a class will never get that far. Yet Chapter 1 (preceeded by
Chapter 0) is on Inductive Reasoning. Exercises throughout the book
encourage and require the kinds of thinking skills that are really
necessary to understand the process of proof. Chapter 13 is about Logic.
If and when students get to Chapter 14, I suspect they are a bit more
capable of actually comprehending proofs than if they had been done in
Chapter 3.

I don't teach geometry. In fact, I don't teach. I have my secondary
certification, student taught, and have tutored many kids in the last
five years in all high school subjects. I currently administrate The
Geometry Forum, a bulletin board about all kinds and levels of geometry.
Previous to that I was involved in writing three sets of workbooks and
videotapes dealing with three-dimsional geometry and teaching it in a
high school setting. I have never used Mike's book. I do own three
copies, though, and might accept a teaching position if I get to teach
geometry with his book. No, it's not perfect, but it's a lot closer to
how I feel geometry ought to be taught than most other books available.

(Disclaimer: the publisher of the workbooks I helped write is the same
one who publishes Mike's book. While this is how I know about
Discovering Geometry, it's not why I like it. It may be one reason why
we like the publisher, though.)

Someone else mentioned the software currently available which allows
manipulation of geometric objects while maintaining established
relationships. While this is not a replacement for formal proof by any
means, it does help the students to understand what they may be asked to
prove. It's one thing to ask a student to prove that the medians of a
triangle intersect. It's quite another to ask them to investigate, using
Sketchpad or one of the other programs, the medians of a triangle. What
do they notice about the medians? Odds are pretty good that they will
discover that they all intersect. _Now_ ask them why. Isn't that a much
more interesting and intriguing way to go about things, than to ask them
to prove things they aren't sure are even true?

Yes, I think proofs are important in mathematics. I was a math major in
college. I also thought proofs were fun in high school, but that's due
not in small part to the fact that I understood them completely the first
time. I think it's more important to develop those skills that are
necessary to comprehend proofs. Sometimes this may need to be done at
the expense of actual rigor, but isn't it the point of education to teach
people to think and reason instead of regurgitate? Any student taking
geometry in ninth or tenth grade is capable of doing proofs _if they are
taught the necessary thinking skills before hand_.


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