[C] [Wittrs] help the math teachers?
- From: kirby urner <kirby.urner@xxxxxxxxx>
- To: wittrsamr@xxxxxxxxxxxxx
- Date: Fri, 18 Dec 2009 20:15:07 -0800
Here's a bit of applied philosophy of mathematics I could use some help
with:
I've suggested a Wittgensteinian approach, but people don't necessarily know
what I'm talking about.
Picture a simple line drawing like from Remarks on the Foundations of
Mathematics, showing a square with 1 next to each edge and 1 in the middle.
This diagram "shows" that a square of edges 1 has an area of 1. It could be
considered an expression of a rule by some, even though we may as yet have
precious little idea how to follow it.
Now picture another line drawing showing a triangle with a 1 next to each
edge and a 1 in the middle. This would suggest a different rule about area.
Analogous pictures may be placed side by side, of a cube and a tetrahedron,
once again with edges 1, once again with both shapes considered units of
volume, suggesting different rules, different grammars.
We're all familiar, from years of schooling, with square and cubic models
for area and volume. We haven't much experience with any divergent
ethnicity, any contrarian "tribe" or "form of life" that might school its
young differently.
A tetrahedral unit might seem of fleeting interest in an esoteric philosophy
book like RFM, a quick example, no reason to linger.
However, since those early days, a full-fledged philosophy has emerged
making triangular and tetrahedral mensuration a corner stone.
The difference may be regarded as "axiomatic" or "definitional" and in
principle there's nothing mathematically the matter with exploring in this
new direction.
However, the language games involved are somewhat unfamiliar and school
teachers, challenged to cover some of this material, may be confused about
whether any "rules" or "theories" are being disobeyed and/or falsified. I
sense a need for a stronger philosophy of mathematics in the coaching of
these school teachers.
The diagrams described above were in fact published in the 1970s in the
context of a magnum opus dealing with matters geometrical and philosophical.
There's an intro and appendix by an MIT crystallographer and Renaissance
man, Dr. Arthur Loeb. The dust jacket carries endorsements by Arthur C.
Clarke, U Thant. The author won a long list of honors and awards including
a Medal of Freedom from the USA president of that day.
http://www.rwgrayprojects.com/synergetics/s09/figs/f9001.html
Just having more than one way of casting area and volume seems a kind of
monkey wrench to some, an "unfair" plot twist.
Other color diagrams were published showing the slicing and dicing of the
unit volume tetrahedron and assembly from pieces, to give related whole
number volumes for the cube (3), octahedron (4), rhombic triancontahedron
(5), rhombic dodecahedron (6) and cuboctahedron (20).
Of course you might scale any shape, relative to a reference unit, to have
any volume at all. But the above simple numbers correspond to an elegant
and simple "ecosystem" or "sculpture garden" of polyhedra known as the
"concentric hierarchy" in some circles.
Teachers here in Oregon are under some pressure to communicate more about
this material because of its simplicity and memorability. Why not teach
about lattices using simple whole number shapes as basic components? If the
elite schools are doing it, then why not the school down the street?
The rhombic dodecahedron (6), a space-filler and favorite of Kepler's,
contains the octahedron (4) and cube (3) as long and short face diagonals
respectively, with the cube containing said unit volume tetrahedron (1). So
many nice features, means Johnny and Sally, Ahmed and Tag, come home
sounding a lot savvier, more prepared for interesting work. Parents tend to
appreciate these favorable (propitious) signs.
You could expand it a bit, by showing how any m x n area or i x j x k volume
might be consistently rendered using a 60 degree instead of a 90 degree
based visualization.
The way these polyhedra are nested and stacked reminds one of M.C. Escher's
'Flatworms' or the Portland World Trade Center, in turn connecting to
Alexander Graham Bell and his forays into architecture (octet-truss).
It's less a 90-degree world than a 60-degree one, and as such comes across
as more hexagonal-biological than rectilinear-industrial.
Here in the Silicon Forest, a headquarters for nanotechnologies and
bioengineering, having students think grapically, in a right brainy kind of
way, is going to help them with molecular chemistry, chip design and
fabrication, medical imaging, filtration... anything having to do with
lattices.
I'm giving some reasons why there's time pressure. Those stuck in the
status quo seem to be suffering from philosophical difficulties,
insecurities, related to having something new making waves.
Wittgensteinians to the rescue? Or are these more questions for an
empirical science? Which one then? I'm thinking we're needing the services
of bona fide philosophers here. Consider me a recruiter.
Kirby
--
>>> from mars import math
http://www.wikieducator.org/Digital_Math
Other related posts:
- » [C] [Wittrs] help the math teachers? - kirby urner
