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Re: where's the math?
Posted:
Apr 21, 1995 9:04 PM


Tim Hendrix (hendrix@uxa.cso.uiuc.edu) writes:
>After allowing the other students solve the problem, and having developed >the threed 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 Triangleconnected to the Cantor >setconnected to Pascal's Triangleall of which is connected to ideas >of selfsimilarity, chaos, and fractalsvery 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!
Wait! What do many of these connections mean? For example, what was the connection made between the Cantor Set and Pascal's Triangle?
(I can see a connection between the Cantor set  as a totally disconnected set  and continued fractions in that continued fractions are a natural representation of the product space N^N, and the cantor set is a natural image of the product space {0,2}^N  but I don't see what this has to do with the subject at hand).
Some of these connections seem a bit tenuous. Does an English teacher get excited by "Less than Zero" because it's written in the same language as Shakespeare?
In any case, the problem with laundry lists of connections, stripped of details, is that it is precisely the substance of the math that is interesting. It's one thing to say "gosh factoring numbers has applications to cryptography"  what do you do with a fact like that? Blink and say "oh"  and quite another to show HOW.
(That's a nice modern application, I suppose  somewhat against my original position  though the mathematics involved is hardly new, only the application to cryptography is. Still, I can see working things like this into the standard curriculum  certainly not because students need to know the math behind public key encryption to compete in the 21st century (or the fermateuler theorem, for that matter), but simply because it is a pretty and accessible modern application of classical math)
Similarly, my complaint with the current invocations of fractals, and particularly chaos theory and such is its lack of content, at least at the level I've read about it in popular articles (this stuff is a little out of my field). I don't find pretty pictures and vague sentiments mathematically enticing. One needs enough details to be able to start at least trying to work out for yourself *some* of the implications... If your regular class work is such a drudge (which seems to be the general consensus here), and all the spicy stuff is vague, what do your students take from this?
As someone who actually DID love math from an early age (am I the only one on this list?), what was so appealing to me was that I could reason things out for myself. When I first saw real mathematical arguments, what was wonderful was that one could FOLLOW them, absorb the reasoning and finally accept the result because one had accepted the argument. So different than history or chemistry, where you had to memorize everything. In math you could work it all out from scratch!
I'm not sure when I first saw any math that was justifiably "modern"  I know I read 1,2,3,... infinity, but I don't think it made much of an impression on me. I also read Martin Gardner's column in Scientific American (and went back through the old issues in the school library's microfilm collection)  but it was the puzzles that absorbed me, not the occasional mention of modern mathematicians. I had won a copy of Herstein's Algebra book at ARML, and leafing through it may have given me some idea of what higherlevel math was like  but it was Spivak's Calculus book (which I saw my senior year of high school when taking honors Calculus at U.Md.) that really revealed to me just how much rigor there could be and how much had been ignored in the school classes I had seen before. But even so, those great theorems are almost all 19th century work, perhaps gliding into the 20th with Peano and Dedekind.
I'm all for mathematics cheerleading when it involves getting students to think about problems that they know enough about to find interesting, and can discover or be taught enough about to make noticable progress in a reasonable amount of time. If along the way, one can work in some famous names, alive or dead, so much the better, but this seems somehow secondary to the activity.
Ted Alper alper@epgy.stanford.edu



