"This Ain't Math!"
"Philosophy is a game with objectives and no rules.
Mathematics is a game with rules and no objectives."--unattributed tagline
What Is Math?
You'd think that mathematics is an easily-defined discipline, especially at the high school level, with well-demarcated boundaries. Guess again.
Case Study #1
"Why the h*** do we have to learn this AND/OR gate stuff? This has nothing to do with the real world."
My response: Oh? And when was the last time you used a computer, hmm?
Case Study #2
"This code stuff ain't math, yo, you're a geometry teacher, so teach geometry."
My response, though this particular student didn't stick around long enough to hear it: matrices can be used for encryption, and as they have a geometrical interpretation as transformations, most certainly there's a connection between cryptology and geometry, though you have to do a little work to get there. Then there are techniques such as steganography (not strictly cryptology, but related!) that can, say, hide messages in an image.
Case Study #3
A friend, in response to my recounting case study #2:
"You'd think they would be relieved to be able to turn their thoughts to something else for a few minutes. I know that's always been my reaction to OT [off-topic] discussions in classes. Silly people."
The student fallacy that my friend is commenting on is the assumption that a rundown on cryptology as a discipline is off-topic in a math classroom. I give you 9-1 odds that there are mathematicians working for the National Security Agency who would beg to differ.
But It Is Math!
I'm no mathematician, but my knee-jerk response tends to be, "What is wrong with you people?" And that's not entirely fair. Societal attitudes toward math, and indeed the emphases in most of the math pedagogy I've seen, are largely to blame. But if you're gonna change things, you gotta start somewhere, and for me, that's here (on this website). (Alas, I am not currently teaching.)
There is an entrenched attitude that if it ain't equations, or lecture and "do exercises 1-20 on p.198," it ain't math. I won't claim to be doing anything greatly progressive in my teaching, but I have run into significant student resistance, both last year and this year, when introducing more "offbeat" (in terms of pedagogy, not necessarily mathematical research) topics.
Literature and math? You bet. Ever read Edwin Abbott's Flatland, which not only introduces the concept of four-dimensional space by gradually building up the necessary frameworks, but satirizes Victorian society? Or Enzenberger's The Number Devil, which is an amusing illustrated tour of basic number theory, with some sly in-jokes (for those familiar with their mathematicians) on Felix Klein and others?
History and math? Absolutely. Just look at Enigma and the codebreakers of World War II. Nowhere is the ethics of mathematics so clear as in wartime cryptology. And math, too, has its history of triumphs and tragedies, its Emmy Noethers and its Turings.
Culture and math? Take a look at the sophisticated quipu of the Mayans; Seki Kowa's early development of calculus in Japan, linear transformations and symmetries in weaving or quilting; base-8, base-10, and base-20 number systems in natural languages; tessellations in Escher's artwork, in Islamic mosque tilings; fractal structures in J.S. Bach's music; ratios and logarithms in musical scales.
As for real-world relevance: you have a bank account (or someone you know does), and you've used an ATM, haven't you? Banks use cryptology for their transactions. Not to mention ISBN numbers, Hill codes in elevators, and CDs, which use non-"secret" applications. You see logarithms every time an earthquake's Richter scale rating is mentioned on the news, statistics in any sociology report (and abuses of it especially in advertisements), probability in casinos, the four-color theorem in maps, arithmetic at the cash register, modulo arithmetic when you read a clock, noncommutativity when you cook (try following your recipes with the steps in reverse), projective geometry when you read a blueprint, Fourier analysis and wavelets (among other things) when you save that digital photo.
And if you can't see the connection between the sciences and math, I don't know what world you live in, but I ain't sharing it with you.
Classroom Applications
My goal is not to end up with all my students adoring math. I'm not sure I adore math. Certainly I adore large chunks of the mathematics I know, but I've never gotten along with combinatorics or probability.
My goal, rather, is to make these connections apparent to students and friends who live under the misconception that math touches on little of their daily existence. Oh, but it does. It does.
And as with many things, you have two major choices: you can become aware of such mathematics and have some power over its effects, or you can be unaware of such mathematics and be at the mercy of those who do, indeed, understand what's going on. Clearly, not everyone can learn everything, and even mathematicians are specialized enough not to know all of what's going on in all of math; but we have choices, and those choices ought to be informed.
Thus I've given a historical and practical overview of cryptology. I've invited in guest speakers (a traffic engineer, an astronomy major, and my husband the physics grad student) to talk about how they use mathematics in their professions. I've emphasized the importance of logic in reasoning clearly, talked about Turing, asked the kids to learn bits of Latin (reductio ad absurdum and quod erat demonstrandum, among others) and deconstructed Greek/Latin etymology, taught coordinates through board-game notation and giving directions (among other things), introduced Boolean arithmetic through the conceit of "voting machines"--oh, and that's not the half of what I'll do. Of course it isn't all I could do, and as the years wend on I'll ask other teachers for ideas, find other guests, think of new relationships between topics...but it's a start.
That's what makes me stifle (most of) my sarcastic responses to student complaints about "this ain't math, yo" or "we're never going to see this in our lives, so who cares?" I don't know if it'll have any effect in the classroom...but if you're reading this, maybe it's a start, too.
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