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Original posted to AMTE listserv, April 20, 2000. Some typos fixed, italics added, hyperlinks activated. For the original thread, and more context, click here. Date: Thu, 20 Apr 2000 10:35:36 -0700 To: amte@esunix.emporia.edu From: Kirby Urner <pdx4d@teleport.com> Subject: More re "ET Math" In a recent post, I introduced the thread of "ET math".[1] I wouldn't claim this idea is original with me. This "imagine a culture in which X is the case" is a time-honored pedagogical technique. Its point is to get students thinking about hypothetical worlds, "parallel universes" as it were, as this helps add perspective and analytical understanding regarding what's actually the case in _this_ world (whichever one that might be). We need "other worlds" for contrast. In the context of math ed, I think mathematics as a discipline can be a disappointing and intimidating subject from the point of view of a newcomer, because: (a) it sometimes appears that all the "easy" discoveries have been made, with the "frontier" being accessible only to those committing to the subject as a career (b) it seems so cut and dried, packaged in a kind of pickle- flavored "this is just how things are" preservative, which many find unsavory, because there's nothing to challenge, question, overturn (in this sense, math, like Latin, is a "dead language" -- at least in K-12). (c) one's accomplishments in mathematics, as well as in many other disciplines, are celebrated only if (a) you add to its ediface (something mostly open to careerists) or (b) challenge its foundations (but how do you challenge self-evident axioms and definitions?) I'm not saying I agree with (a)(b)(c) above, only that these aspects of our subject somewhat account for why it suffers from bad PR and has a difficult time making inroads with many youth. We should be especially sensitive to (a) in that human history is "time asymmetric". There's no "turning back the clock" to where we don't already know about the binomial theorem or Taylor series (unless we unleash the nukes and "restupify" en masse).[2] Kids born today are facing a much larger aggregate of mathematical literature than kids born even as recently as 50 years ago -- and it's not the case that this makes no difference in the early grades. On the contrary, I would claim that this sense of an "ever retreating frontier" is palpable, and affects teacher attitude as well -- even if (as I suggest below), this sensation is somewhat illusory and unsupported ("new frontiers" are a stone's throw away everywhere you look, if you know how to look (and exposure to "ET math" is one way to develop the requisite ways of "seeing")). Many teachers fight (c) by creating an aura of reward and appreciation around: (i) rediscovery (doesn't matter if others before you have "priority" just because they were born earlier in time) and (ii) simple competence (being good at math is like being good at ancient greek -- change and innovation aren't necessarily part of the deal). Some of the NCTM-backed "fuzzy" reforms (I know NCTM doesn't like that word) stem from (i), i.e. are efforts to inject "new life" into a program of dull memorization and jumping through hoops set in stone some centuries ago (the so-called "right ways" to do the basic operations -- which we all agree are important to know; but with insight into their design, an understanding of why they work, if at all possible). I would see a lot of traditionalists as emphasizing (ii). To (i) and (ii) I would add (especially in this AMTE context): (iii) teaching well and showing connections is an art in itself, and is worthy of positive feedback. For example, I personally consider Bill Nye the Science Guy to be an effective teacher of science, because he communicates using the "semiotics" familiar to TV-savvy children (fast takes, fun sounds, creative skits, allusions to culture, and lots of humor). My Videogrammatron Project is much influenced by both Bill Nye and the Childrens Television Workshop (CTW), but its focus is more on spatial geometry and "design science" (for a good approximation of this content, see Jay Kappraff's Connections: the Geometric Bridge Between Art and Science, McGraw-Hill, 1991).[3] Being a very effective communicator of mathematics is itself a major achievement, and worthy of celebration. Innovation in this domain does _not_ require being a career mathematician, but rather requires being a dedicated educator/teacher (perhaps in other subjects as well). However, back to point (b) re "foundations of mathematics", I think "ET math" is a way to open up the whole arena of mathematical innovation, and inspire the sense that this is _not_ a discipline where all new and interesting material requires years and years of background and training. I was talking base 10 positional notation as one example of a "convention" that we may imagine is different in other cultures -- we actually have real life examples of that one. My other example took off from this jargon around "squaring" and "cubing", suggesting that an "ET math" might involve "triangling" and "tetrahedroning" instead -- especially given that a triangle is the polygon with the fewest sides, and a tetrahedron the simplest volume (mathematicians call it a "simplex" for that reason). Then I asked a "leading question" as to whether our ETs might have less interest in "right" angles (90 degrees), considered "normal" in our culture (that's a technical term -- but we shouldn't overlook the connotations). Notice that XYZ is all based on right angles (mutually perpendicular axes) and that we say space is "3D" because we can imagine 3 mutual perpendiculars and not more. Somehow an 89 degree or 91 degree relationship isn't as definitive of dimensionhood -- the 90 degree relationship is critical somehow. Well, can we imagine an "ET coordinate system" that's a lot like XYZ, but seems more "tetrahedral" than "cubic"? As a matter of fact we can, and a lot of work has gone into precisely this topic of late, and one of my websites summarizes the results.[4] We call this the "quadray coordinate system" (I've mentioned this before in other posts, shared an email from a high school student undertaking their study) and it works with four basis vectors from the center to the corners of a regular tetrahedron, and 4-tupe coordinates: A=(1,0,0,0) B=(0,1,0,0) C=(0,0,1,0) D=(0,0,0,1). That's right, 4-tuples, and yet this is ordinary "3-space" (nothing hyperdimensional). And you may notice something else: because 4 vectors to the corners of a regular tetrahedron carve space into 4 volumes, we don't need any "negative numbers" to specify locations. A vector sum of at most 3 positive basis vectors is sufficient to address any point, i.e. P is a linear combination of A,B,C,D. Final note: what is the picture of (3,3,3,3) in quadray coordinates? Four vectors all the same length in the four basis directions net to a zero vector, cancel each other out, so that (a,a,a,a) = (0,0,0,0). This means we "cancel the LCD tetrahedron" out of any (a,b,c,d), by taking the smallest coordinate and subtracting -- which algorithm works even if the smallest coordinate is negative. The result: one coordinate is always 0, with the others positive (what we call the "normalized form" of a quadray). All this is pretty different, pretty ET-like, but is useful from a pedagogical point of view, because it gets students thinking about XYZ, the game of vector addition, the primitive concepts of linear algebra (basis, span, dim...). With the proper graphics and borrowings from Bill Nye's bag of tricks, we're able to get across quadrays (most closely related to simplicial, not barycentric coordinates) without losing a huge percentage of students in the process. They're not that difficult, plus they share an aura associated with all manner of exotica and esoterica which many find intriguing, novel. Toss in that our ET's use a regular tetrahedron as their "unit measuring cup" for volume (cite the "concentric hierarchy" I've been posting about), and you've got a fairly complete picture of an "alternative culture", one fleshed out in the form of websites, applets, applications, artifacts. From a pedagogical point of view, this is exciting, as we can: (iii) encourage teachers to develop their teaching style around new content (fighting (c)) plus (b) show that alternative "foundations" for mathematical investigations are feasible/doable such that (a) likewise falls away: ET math is something students might help evolve even without becoming careerist mathematicians -- and, by extrapolation, math as a whole shares this characteristic (because "ET math", however different, is still just a sub-topic in what we call "math as a whole"). Plus we're training students to be a lot more articulate about philosophical and "meta" issues regarding mathematics. Especially when we're dealing with "foundations", it pays to have some exposure to the discourse going on among the various schools of thought in philosophy these days.[5] The Philosophy for Children program at the Montclair State campus in New Jersey is likewise sensitive to this need for more training and background in philosophy.[6] Otherwise, without philosophy in the picture (as a kind of "glue language"), we'll just develop a culture of narrow-minded specialists, with no ability to "question authority" because everyone else is in their own compartmented sub-sub-discipline (the whole idea of "peer review" breaks down when your "peers" are a coven of like-minded individuals depending on the same sources of funding as yourself). It's amazing what BS passes for "serious-minded scholarship" in such overspecialized environments -- and laypeople are reduced to suckers at every turn because trained to accept the judgement of "experts" or "professionals", even when it comes to the most basic kinds of thinking.[7] I'm a sworn enemy of hyperspecialization in this sense (even if a practicing specialist myself, in some discipline or other). Kirby Curriculum writer Oregon Curriculum Network Notes: [1] http://forum.swarthmore.edu/epigone/amte/twangtimpan [2] http://www.inetarena.com/~pdx4d/graphics/cartoon2.html [3] http://www.teleport.com/~pdx4d/videogrammatron.html [4] http://www.teleport.com/~pdx4d/quadrays.html [5] http://www.teleport.com/~pdx4d/mathphil.html [6] http://www.montclair.edu/Pages/IAPC/IAPC.html [7] http://www.teleport.com/~pdx4d/politics.html For further reading:
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