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

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David Fowler

Posts: 15
Registered: 12/6/04
Re: Proofs - My Thoughts
Posted: Feb 17, 1993 2:16 PM
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In article <1gkqsbINNmom@forum.swarthmore.edu> Bernie Ivens
<ivens@forum.swarthmore.edu> writes:
> PROOFS- My Thoughts
>
> 1. Students could and should be asked to develop, explore, theorize.
> After that process is it possible to prove seemingly great conclusions?
> Should we rely on insight, totally?

For some students, keeping in mind a goal of the NCTM standards that we
should
try to reach ALL students with SOME valid mathematics, the "Proofs
without
words" of the MAA Mathematics Magazine might be a sufficient
consolidation of
explorations and theorizing. For others, the goal of rigor is worth
striving
for. See "Rigor and Proof in Mathematics: A Historical Report" vol 64 no
5
Mathematics Magazine.
>
> 2. To me the "beauty" of geometry is the development of provable
> theories based on terms(undefined or defined), assumptions (axioms or
> postulates), and previously proven theories(theorems, propositions,etc.)

Yes, axiomatic mathematics is beautiful. The Bourbaki group which arose
partly
in response to the tragic decimation of young French thinkers during WWI,
had a
great impact on world mathematics. The Cambridge Report of 1962, reflects

Bourbakist thinking. The Cambridge Report was an influence on the
curricular
movements of the middle 1960's and 1970's. See the recent book {Pi in the

Sky...}.

You can talk about fractals to an engineer, and she'll be very
interested, but
not when you start to discuss proofs. The litmus test for mathematicians
may
be: Who walks out of the room when you start to talk about proofs. The
ones
left are sieved out, like primes, and they're the mathematicians.

But there is also some beautiful mathematics, and the variant genius
Ramanujan
may have provided some good examples, which just is there. Yes, it can be

proved, but it's the result that's there and amazing, not a gorgeous
axiomatic
structure like you get with Bourbaki Topologie Generale, vol I
>
> 3. Any methodology - discovery, constructions, discussions,
> experimentation which will help students of geometry understand more
> completely is wonderful. Let's follow these eye opening discoveries with
> proving the concepts in an organized way. Let's document, keep records
> carefully, use previously developed ideas to help develop new ideas.

If you want to teach varieties of logic, and that's been a long-time goal
of
education, or at least what is [very approximately] called WESTERN
education,
you could use, Rhetoric, or Formal Logic, or...Geometry. Geometry is a
good
domain for studying principles of logic. However, I'm not sure that logic
is a
good context in which to study geometry.
>
> 4. I wonder if that part of the mathematics educational community that
> wants to eliminate or greatly diminish the use of proofs is succumbing

to
> the frustration of many instructors. (to avoid the word PROOF some texts
> use flowcharts, conjecture- "a rose is a rose")
>

Consider this. Should we teach long division, and then have children
check
their answers with calculators? Should we spend hours graphing
parablolas and
then check the answers with graphing calculators. Should we do lots and
lots of
algebraic factoring and expansions and then check the results with
Mathematica?
Should we spend weeks and weeks doing proofs, when we can check the
results
with automated theorem proving? Now ATP isn't quite here yet, but it's
close.

"Many areas of high school and undergraduate mathematics, by now, are
"trivialized" in the sense that their problems can be solved
algorithmically by
existing mathematical software like Mathematica. Well, if an area of
mathematics is trivialized, why should students bother to study the area?

Rather, shouldn't we just teach the students how to solve the main
problems in
the area by applying, in a reasonable way, the appropriate algorithms in,
say,
Mathematica? There are two dogmatic answers to this question. The
puristic
answer: Ban math software systems from math education! The pragmatic
answer:
Don't spend time in class on any trivialized area of mathematics!" [Bruno

Buchberger, RISC-Linz, Johannes Kepler University. Teaching Math Using
Math
Software: Some Examples for the BlackBox/White Box Principle. Rotterdam
Mathematica Conference, September, 1992]

> 5. SOME QUESTIONS
> a). Are we just interested in students learning a body of information
> or are we also interested in a logical development justifying

conclusions
> that we make? (CONTENT + FORM)
That body of information called geometry is a microworld of a subject
marjorie
senechal and others are calling "shape." Look at the SIGGRAPH notices
from ACM.
People are using simplicial topology and quaternions and other
mathematics to
do very beautiful renderings. Watch the video "Not Knot." This video was

inspired by a proof, but it suggests a future in which you might prove a
result
by flying through a virtual space and checking off the features as you go.
>
> b). Do we think students are incapable of learning how to do "proofs"?


"Imagine then the kind of person coming into our graduate schools, if not

today, then certainly tomorrow. Brought up from early childhood on a diet

involving MTV, Nintendo, graphical calculators packed with algorithms,
Macintosh-style computers, and, in the not-too-distant future, hypermedia

educational tools as well. Such a person is going to enter mathematics
with an
outlook and a range of mental abilities quite different from their
instructors..." [Keith Devlin, Notices of the American Mathematical
Society,
March 91, 38, 3 ].>
> c). Do we need to learn how to teach "proofs" better than we have in
> the past?

differently...
>
> d). A high school in Pennsylvania has a 5 month geometry course- no
> proofs at all. Is that the way we should go?

Back to The Standards again. [The Standards, by the way, are now part of
the
"Conservative Reform Movement."] I think that we need to conceive of
Shape, or
Spatial Cognition, as a vertical theme running through the K-12->
curriculum.
In such a context, I can imagine that there could be a 5 month stretch
when you
aren't doing proofs. But taking a radical point of view, I wouldn't call
that 5
months geometry. Because, I believe Husserl was right. Geometry is no
longer
alive. It's dead, like Latin. You can teach Latin beautifully, and make a
mess
of teaching Japanese. You can teach geometry beautifully, and make a mess
of
teaching fractals. But Latin is dead. So is Euclid. Fractals are alive.
So, and
let's all wish them the best of health, are Mandelbrot, Peitgen, Devaney,

Pickover, and all those other interesting people. That ultimately is
going to
make a difference in most classrooms.
>
>
> I have taught geometry for 30 + years. I have enjoyed combining the
> excitement of students exploring and discovering with the excitement of
> proofs- and I think, very successfully.

I'm sure you have been successful, and I've entered my opinions [as you
asked--opinions] with consideration for your work. The fact that you are
asking
such interesting questions now is [proof (?)] that you must have been
asking
your students interesting questions, for 30 years.

David Fowler, Teachers College, University of Nebraska-Lincoln.
>
>
> ANY OPINIONS OUT THERE?
> Bernie Ivens
> The Geometry Forum
> School Liaison
> Swarthmore College, PA
> ivens@forum.swarthmore.edu







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