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Re: where's the math? so?
Posted:
Apr 20, 1995 6:55 PM
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On an earlier message, Ted Alper wrote:
>Presumably everyone on this list has loved mathematics from a fairly young age.
>How many owe that love to a sense that there were modern mathematicians out
>>there, tackling tough modern problems?
>
Later, Judy Roitman responded: (parts of her response)
>Nope, I didn't. I thought math = calculations (including algebraic) and
>overly formalized proofs of uninteresting statements = uncreative and
>dreary. Except for having run into the proof that the reals were
>uncountable in George Gamow's 123 Infinity back in junior high...
>
(cut...)
>So how to teach kids what math is? I think Ted is right -- you have to
>give kids something they can work with and understand themselves. But
>surely it doesn't hurt to give a glimpse of some of the more understandable
>modern developments, both theoretical and applied. ...(cut...)... Strange and
>>interesting and attractive things were going on in physics. Why didn't we
>know >about strange and interesting and attractive things in mathematics?
>
>
My comments:
Thought that I would ride out this "Where's the math? so?" conversation
without jumping into the fray...but, alas, I can't resist.
Kudos to Judy for trying to strike the practical middle ground...both
approaches can and should be employed within the school mathematics
setting...some educators are better at integrating, or highlighting,
different perspectives of mathematics than others...
However, we need to understand that the ever-evolving mathematics of today
(post WWWII) was/is not developed in isolation of the rich history of
mathematics of previous eras...Conceivably, (and shallowly), I could write
a novel without any deep understanding of the middle age epic poem, but it
is impossible to develop new mathematics without any rich sense of the
mathematics of old...New mathematics does not follow a replacement
theory...it truly is an accumulating evolution...
Let me offer an example: I recently had the opportunity with my student
teachers in a methods class to higlight this interconnection aspect of
mathematics...One of the students engaged the rest of the class in an
activity that developed the idea of using graph theory as a modeling tool
in problem-solving...She used the well-known problem of the lion, the
llama, and the lettuce (or--crossing the river with dogs, or...etc.) where
the owner of the three l's had to cross with one other object at a time,
return back for one other, etc. until he/she had all three--the lion, the
llama, and the lettuce--on the other side of the river...all under certain
prescriptions of which two things could be left alone with each other.
After allowing the other students solve the problem, and having developed
the three-d model of representing the problem with graphs, she then led the
students to the connection to the Tower of Hanoi problem...
What she and the other student teachers did not realize at first was that
this type of modeling that is connected to the Tower of Hanoi problem
was/IS connected to the Sierpinski Triangle---connected to the Cantor
set----connected to Pascal's Triangle---all of which is connected to ideas
of self-similarity, chaos, and fractals--very modern mathematical ideas...
We even explored further (couldn't resist to follow up) that these newer
mathematical topics which had their roots in the Sierpinski Triangle,
Cantor Set, and Pascal's Triangle, which, in turn, have in their roots,
geometric series, binomial distribution, probability, the powers and
divisibility of both 2 and 11, etc. all of which are connected to
Fibonacci sequence numbers, the geometric idea of triangular numbers and
tetrahedral numbers, the Golden Mean, continued fractions, and Farey
fractions, just to name a paltry few!
While slightly (ha!) overwhelming, the excitement and the realization that
students in secondary school can:
(1) study things that we *normally* study in school mathematics;
(2) make connections to the historical mathematics in an
active,
involved, and evolving way;
and (3) examine the development of newer advances in mathematics
both in terms of their origins as well as their application
in the real world...
Well, we all left with a renewed vision of what school mathematics can be...
I dunno about everyone else...but I vote for strange and interesting and
attractive! Moreover, I am tickled pink that the strange, interesting, and
attractive is only so because of its rich undergirding of history--not dead
(date occurred history), but vital (recreating, interpreting)history.
Tim
*****************************************************
Tim Hendrix (hendrix@uxa.cso.uiuc.edu) *
Division of Mathematics Education *
Department of Curriculum & Instruction *
University of Illinois *
*****************************************************
382 Education Building *
1310 South 6th Street *
Champaign, Il 61820 *
*****************************************************
(217) 333-3643 *
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