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 

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, 

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).

Curriculum writer
Oregon Curriculum Network


[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:

Synergetics on the Web
maintained by Kirby Urner