Date: Dec 4, 2012 6:12 PM
Author: kirby urner
Subject: Re: Treatment of Negative Numbers

On Tue, Dec 4, 2012 at 2:01 PM, Clyde Greeno @ MALEI <>wrote:

> **
> Stay tuned!

You quoted my whole thing, and Robert's, adding one line!

On some listservs, the moderators would never accept that (Wittgenstein
list comes to mind, where S. Wilson runs a pretty tight ship).

I'm miffed in part because the archived web version has some typos fixed,
whereas my emailed version, which you quoted, has some "my bads".

Anyway, to continue this thread:

In my STEM curriculum, we're interested in notions of convergence and
divergence, oscillation, chaotic sequences.

These may be introduced through numeric series, including those that eat
their previous terms. The On-Line Encyclopedia of Integer Sequences is of great interest in
this respected.

Numeric Sequences (rule driven):
Oscillatory / Periodic
Chaotic / Aperiodic

Periodic / Aperiodic also relates to space-filling and tiling.

Under the heading of Sequences we also have a mini-unification of
rule-based functional expressions e.g. algebra / pre-algebra, with
geometry. How? Through figurate and polyhedral numbers.

Figurate numbers are what they sound like: numbers easily represented as
polygons such as growing squares, triangles, hexagons etc. We can prove
such things as the sum of consecutive triangular numbers is always a square

Polyhedral numbers are what they sound like as well: growing tetrahedra,
cubes, cuboctahedra, octahedra.

A great source: The Book of Numbers by Conway and Guy, and also Gnomon by
Midhat Gazale.

Once you have a growing cuboctahedron (1, 12, 42, 92....) you're looking at
something fairly omni-symmetrical, almost planet-like i.e. the polyhedrons
are a bridge from Geometry to Geography. Start spinning a cuboctahedron
around its axes: those through face centers, triangle centers, mid edges,
vertices. You get 25 axes of spin. With the icosahedron, you get 31.

These grids may be superimposed to create networks with repeating
triangular areas some call LCD triangles (120 lowest common denominator
triangles tile a sphere). We're talking about subdividing a sphere now.
Ed Popko's new book on Divided Spheres is a primer for higher level
students (I'm listed in the bibliography).

One outlet for this material is in San Antonio, Texas where Dan Suttin does
a good job with younger students, helping them get the basics about
Platonic and Archimedian families, space frames, space-filling. I'm not
sure if he teaches about duals (e.g. tetrahedron is its own dual), probably
so. (Dan sharing with younger people about
polyhedrons -- refreshing change from the usual fare).

What I like about this "mini-unification" is it bridges left brain lexical
algebraic computer programming type thinking with right brain graphical
visual type thinking, but without dwelling on any specific coordinate
system at first. It's something we can start spiraling into quite early.

It's when we have a Planet Earth in view (ala Google Earth) that we can
start talking about "number lines" (e.g. equator) and rulers and grids, for
surfaces, for volumes. We don't need to stress "infinite" too much and
we're not scared off by lines that curve, even corkscrew.

Helices are our friends (they're a VPython primitive after all). Helices
are the ultimate chiral phenomenon and as we have seen, this curriculum is
very invested in talking about left and right, mirror images and so on --
much more so than New Math ever was.

Looking down on the north pole, we can see the Earth spinning. Clockwise
or counter-clockwise? Many adults have no idea as this was never a focus
in school. She rolls towards the east. That's counter-clockwise from this
angle. Looking at the south pole, from out further, that's clockwise. You
might need a globe to see this. That's geometry, not just geography.

So you can see how STEM is going: much better integration of solar system
geography (astronomy) with geometry and geometry with numeric sequences.
Numeric sequences become a first opportunity for learning computer
languages in an algebraic / lexical context.

Once the polyhedrons are well developed, we have a much stronger basis for
talking about symmetry and symmetry groups (based on rotation). We also
get into topology with V + F = E + 2 (provable), Descartes' Deficit (720
degrees) and ratios relating to omnitriangulation.

My 'A Bhutanese Mathematics Curriculum' summed up some of this stuff. I
wrote it while in Thimphu in the 1980s.