Here in Oregon, experiments with alternatives have been
facilitated by the passage of charter school legislation.
This gives resident corporations the "loophole" they need
to design from scratch around their technological wares,
showcasing what a 21st century math/science centric approach
might look like if designed around "engineering math" (as
opposed to investing too early and too exclusively in more
theoretical approaches -- better left for later, and for
students who actually choose mathematics as a specialized
degree path).
As this is a copy of inhouse notes used for internal
consumption, I haven't bothered with a lot of footnotes
or references to specific books or web pages (unlike in
some of my other posts to this list, which have been
more fully annotated). I running low on time for
chronicling all that goes on around me these days, so
please accept my apologies in advance if you send me
emails and I'm slow to get back to you. If you explore
the Mathedcc archives, you may discover I've already
answered your question in any case.
Thanks to those of you who have sent insights and input.
I appreciate all the positive feedback, and the negative
has been challenging and therefore to my lasting benefit
as well (even if I don't see it as such at the time).
Sincerely,
Kirby
Oregon Curriculum Network
http://www.inetarena.com/~pdx4d/ocn/
==================
Notes re our 21st Century high school mathematics curriculum
-- charter school project, Silicon Forest, Oregon. K. Urner
and company, April 11, 1999.
The curriculum streamlining we're brainstorming for our math
sequence for our charter high school (Silicon Forest, high
tech company funding), encourages visualization of spatial
relationships, with a phase-in of computational skills, by
way of coordinate framing (XYZ, polar, quadray etc.) and
Java programming (to get the cool graphics on screen).
The XYZ system, being cube-centric, is too dumbed down by
half to accommodate the natural elegance of natural geometries,
which tend to use triangulation for stability, and hence a lot
more 60 degree angles, as in sphere packing arrangements, where
the intertangency of 3 balls makes an equiangular triangles
(if they share the same radius).
Crystallography is often done with reference to a volumetric
"peg board" known as the isotropic vector matrix (IVM) or,
equivalently, the face-centered cubic lattice. We want to
develop native fluency with this matrix without relaying on
XYZ as a filter (unmediated access vs. awkward processing
through an obsessively rectilinear frame).
To this end, we start in the early grades, or remedially if
working with high schoolers, with a set of polyhedra inter-
proportioned to "nest" in the matrix context i.e. the tetra-
hedron and octahedron share and edge length, because this
is how they space-fill in complement in the ivm (fcc) with
a frequency of 2:1 and relative volumes of 1:4 (tetrahedron
is 1/4th the octahedron's volume). The tetrahedron is its
own dual, so the "stella octangula" or "duo-tet" forms the
definition of our primitive cube -- very standard practice
in the western tradition to nest polys in a "labyrinth"
(theosophic jargon) or "hierarchy".
XYZ maps to fcc sphere centers where x+y+z=only even or only
odd (the two alternatives forming complementary ivms), or
as linear combinations of quadrays {2,1,1,0}. All tetrahedra
have whole number volumes in this peg board, given the "home
base" tetrahedron defined by 4 inter-tangent unit radius
spheres has a primitive (prefrequency) volume of 1. Frequency
growth is introduced as an amplification of edge lengths by
modular increments, e.g. a "two frequency" tetrahedron has
edge lengths double the primitive size, and hence a volume
of F^3 and a surface area of F^2 times the initial values
for volume and surface area.
The rhombic dodecahedron (space-filler) is a combination of
the octahedron and cube (duals of one another) i.e. the
primitive ivm octahedron of volume 4 and cube of volume 3,
interpenetrating. The rhombic dodeca will be studied in
some detail, as a kind of "switch" between 4 alternative
matrices (all fcc), which interpenetrate and in combination
define the lattice points of XYZ (simple cubic) and bcc
(body-centric cubic). The rhombic dodecahedron has a primitive
volume of 6 (also frequency modulated).
Finally, 12 fcc spheres around a nuclear sphere define the
vertices of a cuboctahedron of volume 20, and it's frequency
determines the number of spheres in the outer layer according
to the expression 10F^2+2. This leads to the introduction of
a "jitterbug transformation" from cubocta to icosahedron,
which preserves the sphere count in the outer layer -- useful
for studying the phenomena of metal clusters and viral protein
sheaths, which tend to evidence 5-fold symmetry and icosahedral
counts according to the rule 10F^2+2 (modulated by Kasper-Klug
according to Goldberg's studies to give Class III variants,
as per Coxeter's well-informed contribution to synergetic
geometry back in the 1960s). Of course the buckminsterfullerenes
fit in here as well, with their carbon networks also five-fold
symmetric, with counts modulatable (derivable) from 10F^2+2.
To summarize all of the above in a nutshell:
Shape Volume
=================
Tetrahedron 1
Cube 3
Octahedron 4
Rh Dodeca 6
Cubocta 20 <--> Icosa (~18.51)
<--> = Jitterbug Transform (morph relation)
With the exception of the Icosa, all of the above shapes may
be analyzed in terms of A and B modules, both irregular tetrahedra
of 1/24th. Note that Williams got it wrong about the B mod in
his otherwise excellent book, as his B redundantly includes the
A. Once dissected out, the A and B appear as equivolumed and
complementary. They come in right/left handed pairs and assemble
the most primitive space-filler (a tetrahedron) known as the
MITE (minimum tetrahedron = 2A + B) which is outwardly identical
whether made with a left or right handed B (the As come as a
left/right pair).
With all of this covered early in the curriculum, students will
be ready to move to Java for computational implementation of
various mathematical ideas, including the concepts of translation,
rotation and scaling as simple matrix transformations. Trigonometry
and Euclidean-style omnitriangulation of new theorems, drawing from
principles already know, will come only *after* primitive conceptual
fluency with spatial geometry (unmediated by the XYZ apparatus)
has been established. At the Java level, we will explore the
symbolic apparatus of object oriented programming, as well as
the "metaphysics" of Cartesian coordinates (polar, quadray...),
trigonometry, exponentiation and so on.
For example, exponential relationships will be investigated
spatially using STRUCK, spatial geometry freeware (written in
Java by Beautifulcode, NL) wherein structural members either
expand or contract exponentially towards a user-defined "rest
length", in response to stresses placed on them owing to their
context within the structure as a whole. Understanding the
rules according to which these "elastic intervals" respond
requires familiarity with logarithms -- a good reintroduction
to a subject which caused many math students to disconnect the
first time around.
Vector addition and subtraction will likewise be developed in
the context of the sphere packing matrix (by default this means
the ivm) with quadrays deployed as a pedagogical tool to highlight
various properties of "games with vectors" i.e. we will compare
and contrast using 4 vectors to the corners of a regular tetra-
hedron as a basis, next to the Cartesian apparatus of 6 vectors
(three basic and three derived through negation) to map the same
Euclidean space.
This will set the stage for introducing the concepts of group
theory (closure, identity, inverse, and optional commutativity)
of items (e.g. vectors) with operations (e.g. addition), and
give us a good place to bridge over to other operations with
vectors (cross and dot products) along with their geometric
interpretations. This will set the stage for generalizing to
the Clifford algebras and doing a special case study of
Hamilton's quaternions as an alternative method for
accomplishing rotation (back to our Java implementation). It
will likewise provide concepts necessary to the calculus: those
of div and curl in a vector field (again, geometric
interpretations will be emphasized -- in conjunction with
Maxwell's theory of electromagnetism).
Kirby
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