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Topic: [math-learn] My approach to curriculum writing (Urner)
Replies: 2   Last Post: Feb 9, 2001 3:43 AM

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Kirby Urner

Posts: 803
Registered: 12/4/04
[math-learn] My approach to curriculum writing (Urner)
Posted: Feb 5, 2001 1:33 AM
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K. Urner, Oregon Curriculum Network (OCN), Feb 4, 2001

Sometime in the not so distant past, math went through a dark
age -- literally, in that it was "lights out" for
visualizations. Students were supposed to empty their minds of
any naive pictures or diagrams and just get used to operating
with symbols. This was considered the "in" way to think about
matters mathematical -- math, like the rest of culture, is not
immune from fads.

Now the pendulum is swinging back to where visualizations are
no longer considered naive. Semiotics and Wittgenstein have
taught us to see graphical and lexical elements as a part of
the same semantic continuum, neither intrinsically more
"abstract" than the other. Even "visual proofs" are coming
back into vogue.[1]

We also have the "left versus right brain" terminology, which
carries weight regardless of whether neuroscience justifies it
entirely (actually, there's a lot of evidence for hemispheric

Traffic between East and West has only increased in this last
century (1900s) and the right/left brain talk maps fairly well
to yin/yang talk -- there's an isomorphism here, or you can make
it work that way. And this concept of balance, of developing
both hemispheres, of not letting either atrophy, is having an
impact on math education theories, and on pedagogy in general.

So visualizations, the importance of the imagination, is making
a come back under the aegis of the right brain's champions.

As a curriculum writer, I too encourage right brain approaches
to mathematics. This hemisphere I associate not just with
visual/spatial/imaginative abilities, but with the concept of
cardinality (vs. ordinality) -- a link I learned about from
Midhat Gazale.[2]

On the spatial front, I encourage a kind of "mental geometry"
to develop alongside "mental arithmetic", and do so by
exploiting the ancient tradition of nesting polyhedra to form a
"maze" or "matrix". It's something the sacred geometers were
into, down through the theosophists in our own time. Given
this history, our graphics and visuals may tend to raise some
eyebrows.[3] But that's OK. I think it's more interesting to
students as well, to connect to these arts and traditions,
while developing their right brain abilities to consider
geometry in the mind's eye. Opportunities to bring up
historical topics, which offer insight into how we came to
be who we are, should not be ignored.

Where I part company from most polyhedron nesters is I'm not so
fixated on the Platonic Five, thanks to my being a long time
student of the Fuller syllabus, i.e. a reader of books by the
late R. Buckminster Fuller. Rather than the pentagonal
dodecahedron, Bucky, like Kepler, was more into the rhombic
dodecahedron, because the latter is defined by a sphere packing
known variously as the CCP or FCC (or IVM if you're reading
Fuller's 'Synergetics').

The rhombic dodeca isn't even an Archimedean, let alone a
Platonic, and yet chemistry really is a lot about lattices,
which may be referenced to packed spheres. The CCP, like the
XYZ coordinate system, provides a scaffolding, lines of
reference, a grid. It should be taught, starting early. We
need geoboards of both varieties: square-based and hexa-based
in flatland, and cube-based and dodeca-based in space. The
XYZ lattice corresponds to another way of packing spheres
by the way (less dense): the SCP (simple cubic packing).

So yes, I'm into sphere packing and push it as a topic
accessible even to the very young -- that's made very obvious
at my website. And another thing I learned from Bucky:
there's a very streamlined and compact way of both nesting
and scaling a set of inter-related polyhedra so that not
only do they synch with the CCP (with rhombic dodeca = unit
radius sphere domain), but they have whole number volumes
to boot.

I find this intro to spatial geometry far less off-putting to
kids, than the way we phase in non-cubes today. Today, only
cubes get to have simple, whole number volumes, while just
about everything else is irrationally volumed, unless assembled
from cubes, i.e. unless rectilinear, all parallel lines.

Our cubism alienates the other polys, relegates them to the
back of the book, to obscurity. Yet this is really basic stuff
should _not_ be rendered esoteric. It's our obsession with
the cube that makes it so. That was Bucky's critique of our
whole culture and I think there's enough truth in it that I'm
inclined to take decades of practically no discussion of the
issue as more evidence that these valuable diagnostic insights
have been willfully suppressed by those who should know better
than to stand in the way of evolution, yet cling, out of fear
and ignorance, to the status quo.

Here's the new volumes chart:[4]

Shape Volume

Tetrahedron 1
Cube 3
Octahedron 4
Rhombic Dodeca 6
Cubocta 20

Now, how does this work? First of all, we're nesting
polyhedra, ala the ancient tradition of doing so, going back to
Pythagoras and before. We're organizing them concentrically
(they all share the same center). The tetrahedron comprises
the face diagonals of the cube. Another tetrahedron, crossing
the edges of the first at 90 degrees, defines the other 4 of
the cube's 8 corners. These two intersecting tetrahedra are
what Kepler and others called the stella octangula. You can
also see it as an octahedron with tetrahedral tips (a
stellate), these 8 tips defining the corners of our cube. So
we call it a "duo-tet" cube -- because formed from these two
interpenetrating tetrahedra (I use an orange and black duo-tet
as the logo of my company by the way -- 4D Solutions, named
to emphasize continuity with Fuller's own use of '4D' as a
kind of logo).[5]

I mentioned an octahedron, defined by the intersection of the
two tetrahedra (you could use intersection notation at this
point). That octahedron has edges 1/2 those of the tetrahedron,
and therefore 1/8th the volume of an octahedron with edges
the same as the tetrahedron's (double the edges, and volume
goes up by 8 -- it's in the California standard: 7th graders
should all know about the edges:surface area:volume relationship).

I won't go into all the geometric dissections and rearrangements
of pieces (modules, parts), and just say that a series of wordless
geometry cartoons easily communicate a lot of the volume
The students will see, by means of "visual proofs", that the
octahedron has a volume of 4, and inscribes as the long diagonals
of the rhombic dodeca's 12 faces, while the cube's edges inscribe
as the short diagonals. It's a very tight arrangement then:

tetra + dual tetra = cube
cube + dual octa = rhombic dodeca
= domain of unit radius sphere

And 12 such spheres tightly packed around a nuclear sphere, CCP
style, form the vertices of a cuboctahedron of volume 20.

From here, you can keep packing outward, layer after layer of
spheres, always cuboctahedrally conformed, with 10 L^2 + 2
spheres in each layer (L = layer number, starting with 12 when
L=1). There's a way to use simultaneous equations, matrix
algebra, or even Bernoulli numbers, to derive a formula for
the cumulative number of spheres in all layers (plus nuclear).
Let's share that!

So that's what I mean by "mental geometry". There's more to it
of course, but this is the basic framework of what's developed
as a kind of "home base" for the mind's eye. Clearly the
tetrahedron is prominent. This is a divergence from the cubist
standard, but is convergent with other trends and schools of
thought i.e. it may be unfamiliar, but it's certainly not out
of the ballpark, in the sense that tetrahedra are intrinsic to

Tetrahedra are the simplest volumes after all, if limiting
entrants to shapes made from edges, vertices and faces (the
sphere being a limiting shape, a very high frequency polyhedron
of some kind e.g. an icosasphere). The tetrahedron has only 6
edges whereas the cube uses 12. In this sense, the tetrahedron,
or simplex, is the most primitive enclosure. That needs to be
stated directly, in no uncertain terms. If it's not part of
the California standard, it really should be.

So how do we connect to "mental arithmetic" then? Now that
I've outlined the right brain's regimen (including mental
exercises), where do we connect to the other hemisphere?
Answer: through figurate numbers and Pascal's Triangle.

Figurate numbers may be envisioned as sphere packings. The
triangular numbers look like something you'd see in a Pool Hall
-- Pool Hall math. The square and cube-shaped numbers suggest
XY, or the SCP/XYZ lattice. So we're starting to talk about
number series, as the triangular numbers are likewise the sums
of consecutive positive integers. Lots of relationships might
be developed as the students consider and explore, plus we
include a derivation of the aforementioned 10 L^2 + 2, which
applies to icosahedral shells, not just cuboctahedral (links
to viruses, buckyballs, geodesic spheres).

And with Pascal's Triangle, we've got links to the Binomial
Theorem (factorials, probability, Bell Curve), Fibonacci
Numbers, Bernoulli Numbers, Triangular and Tetrahedral
Numbers. That's a very rich set of concepts to go forward
with. Fibonaccis connect to phi, Bernoullis to sums of the
form SIGMA N^c, where c = integer > 0 and N = 1,2,3..., or
to sums of the form SIGMA N^c where c = integer < 0 (Euler's
zeta function -- Riemann's if complex). Throw in continued
fractions, primes vs. composites, and some modulo arithmetic,
and you've set the stage with a lot of important and inter-
related concepts (gcd, lcm, fractions, Fermat's little
theorem, cyphers...) Plus phi gets you back to geometry, to
five-fold symmetry, and to the so-far missing Platonics
(the pentagonal dodecahedron and its dual, the icosahedron).
We also have Pascal's Tetrahedron, and the trinomial theorem,
if we like.[6]

Functions and relations, polynomials, trig, vectors, and much
of what's conventionally covered in math class (plus more
that isn't), all have easy segues or hyperlinks from this
core network of key concepts. We don't lose content, and we
gain a more tightly integrated curriculum, with left and right
brain faculties co-functioning in far greater harmony, producing
more powerful synergies. Sure, that's hype, more pro-OCN
propaganda, but there's plenty of substance to back it up.

The left brain stuff gets developed in tandem with computer
programming (not just calculators), and the right brain stuff
gets developed in tandem with computer graphics -- driven
behind the scenes by the computer programs. Left brain numeric
methods result in right brain artistic renderings. We've got
the bridge between math and art. All we need now is to phase
in music (something computers are good for as well). That's
something I haven't gone into much yet, but some of my colleagues
are exploring in that direction.[7]


[1] James Robert Brown. Philosophy of Mathematics: an
introduction to the world of proofs and pictures
(London: Routledge, 1999)

[2] Re: Cardinality vs Ordinality see:

[3] see L. Gordon Plummer. The Mathematics of the Cosmic
Mind (Wheaton IL: Theosophical Publishing House, 1970)
-- a book both Kiyoshi and I agreed contains a lot of
racist elements, but I'm mentioning it here for the
"maze" depictions.

[4] More re volumes:

[5] 4D Solutions, Porland (PDX):


[7] See Jay Kappraff. Connections: the geometric bridge
between art and science. (New York: McGraw-Hill, 1991)
for more music links. Sir Roger Penrose had an interesting
approach to music using a circle, which he shared with us
at the 1997 Oregon Math Summit

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