> So I'm going to try to come up with a set of questions that I think
>could inspire some thinking. As a preliminary inquiry,what are some of the
>major questions each of you has about synergetics or what direction would
>you like to see research go in. What are your goals for the group? Should
>there be any goals?
My own thinking revolves around spark a renaissance of sorts
in K-12 math ed, and I'm looking at synergetics as one of a
few sources of new ideas that might provide necessary leverage
versus the status quo.
I think from a high level math perspective, it may be trivial
to point out the volume ratios as Fuller did, but what's
trivial from a purely mathematical standpoint may yet be a
breakthrough in the pedagogical sense i.e. a way to break
logjams in mental development in the early years, by making
polyhedra and spatial geometry more accessible.
Part of my agenda (in consonance with the above) is to get
a sense of what the status quo actually is. For example,
it seems to me that early introductions to shapes focus
on the following: cube, sphere, cone, cylinder, rectangular
prism, triangular prism. Tetrahedra are mostly introduced
as "three-sided pyramids", enforcing a "base + sides" way
of looking at them (vs the "zero gravity" view, which
doesn't single out a "bottom").
My thought is that the concentric hierarchy is just what
we need to provide teachers and their students with access
to a richer vocabulary of shapes. Topics like space-filling
and sphere packing might be taken up quite early, with
the concentric hierarchy naturally bridging to both
areas of investigation. Later, in the higher grades, we
start doing more with a computer language to build on these
ideas, translating coordinate geometry and algebra into
computer graphics. My view is that numeracy and computer
literacy are convergent goals, and the programming, being
focused on algorithms, needn't be split off from math as
a separate subject (computer science) in K-12 context.
As I write in http://www.inetarena.com/~pdx4d/numeracy0.html:
By supplementing conventional, pre-computer math notations
with a very high level language (VHLL), we give students
access to complementary modes of encoding rules. The hope
is that students will learn to write programs as a means
to test and refine their understanding of math concepts
-- will look at programs as "math poems" (or as attempts
toward this ideal).
These days, the whole analog vs digital thing seems a worthwhile
meditation. Like, the sphere is maybe the ultimate analog
conception, with high frequency geodesics like the n-frequency
icosasphere coming across as "digital samplings" of same.
The tetrahedron is then the most primitive digital sampling
of the sphere, in that it retains the inside vs outside topology
using the minimum possible chordal network.
Which leads me to a concluding thought: I'd like to see
more emphasis on discrete math, and number theory, in
combination with our spatial geometry "backbone". The
calculus could be linked in from discrete math, with the
idea of limits taking us back to a digital vs analog theme,
where real analysis (the calculus) is developed as the
analog abstraction (pure continuity) developed through
an investigation of discrete methods. I think this
approach recapitulates the historical development and
would be a fairly intuitive way to go. With students
already well versed in computational geometry towards
the end of what we call "high school" in the USA
(ages 17-19), I think we'd have a shot at even doing
some "3D" calculus concepts, such as div and curl,
maybe tying in Maxwell's Equations even earlier.
I'd like to do this because I regard K-12ers as
generalists. By the time we get to college, the
specialization has set in, and most students never
get enough calculus to read Maxwell's Equations, while
others wear T-shirts with these equations printed on
them, as tokens of their math-saavy and sophistication
(all symptomatic of the C.P. Snow cultural chasm which
synergetics aims to bridge).