Wednesday, September 20, 2006

Focal Points

I think I'll swipe the "focal points" lingo and smatter a few for the benefit of our cadres (not a word I like, right down there with comrades -- probably a translation thing).

Addressing my coworker competitors then, I have the following four focal points to propose:
  • Pascal's Triangle, with appended ethnography, tracing it wherever we find it (like in Lost).

  • The Concentric Hierarchy (you knew that'd be here, now didn't ya?). A & B modules in other words. MITEs.

  • The NCLB Polynomial, tying algebra to geometry, with lots of connections to the art world and design science.

  • The Mandelbrot Set, perfect for learning about convergence and divergence in the complex plane (valuable in precalculus in other words).
I realize this seems to leave a lot out, but bear with me, as I'm going to explain why this constellation will help our cause -- and forgive me for choosing Python if that's not your forte (let's get with the cross-training -- I'm always polling for high quality HDTV Rubycasts (not much of a selection these days)).

:: Pascal's Triangle ::

Pascal's Triangle will give us sequences, the starting point for our Python generators segment. Right after we visit Functions, we look at these "state remembering engines" that fire off cyclicly, and remember state between cycles.

Here's a Pythonic way to generate Pascal's Triangle line by line:

generating Pascal's Triangle
(also known to Chinese)

The triangular and tetrahedral numbers are column-wise side by side; our first generators in other words (a little easier than this one).

So here's our link to the cuboctahedral (aka icosahedral) growth curve i.e. to 10 * F * F + 2 (F = Frequency) -- the formula of Fuller's that H.S.M. Coxeter liked so much (provable with high school level math). Sloane's Encyclopedia of Integer Sequences becomes our friend at this point (e.g. cite A005901).

1, 12, 42, 92, 162...

:: The Concentric Hierarchy ::

The concentric hierarchy sets the stage for so much later science that we have to not postpone. This is "without further ado" material that any gnu math teacher must master.

The A and B mods themselves might come later in the intro (I teach 'em in sixth grade), but the basic nomenclature around polyhedra, plus Descartes' Deficit and Euler's V + F = E + 2 need to be there from the beginning (OK to postpone Descartes until you've gone over the Babylonian stuff i.e. 360 degrees, planar triangles having 180, tetrahedron having 720 (the deficit)).

Basically, this is our launch pad for computer graphics careers. You've got XYZ embedded (with optional Hypercross), and our unit volume Coupler right at the origin, at (0,0,0) -- or at (0,0,0,0) if using Chakovians -- thereby giving segue to our four IVMs (R.Z. Chu et al).

Yes, these are abstruse college-level topics, but they all have the Concentric Hierarchy in common (OK to say "Cosmic Hierarchy" at least among hippies). And yes, we're still working on your toon library, stocking it with goodies.

:: The NCLB Polynomial ::
"pentagon math"

The NCLB Polynomial, a quadratic equation with two real roots, is a jump off point into both algebra and geometry. On the algebra side, we meet up with Pascal's Triangle again, and Newton's Binomial Theorem (since rediscovered by other whiz kids his age at the time).

"generating a bell curve"

On the geometry side, we get phi, which also has many interesting algebraic properties. To further tie things together, Pascal's embeds the Fibonacci Sequence (a first Python generator) which sequence converges to 1/phi as follows:

"converging to pow(phi,-1)"

From here, you could jump to any number of connected topics, including to Ramanujan's Pi Engines for more advanced generator drill and practice (e.g. consider using Python's Decimal type for extended precision, for Phi as well).

:: The Mandelbrot Set ::

Kids have a natural curiosity about fractals, as do some adults who still remember the pop art explosion, come across the great artwork on the web.

This is unabashedly eye candy, yes, but exploring the Mandelbrot Set also reinforces key concepts we want our graduates to take with them into their college experience: convergence; divergence; and the complex plane.

This is what'll make 'em ready for Calculus, which, per UCSC's Ralph Abraham's disciplined pioneering, should now probably be taught with a strong dynamical systems bias (there's a similar trend in statistics, undermining the status quo).

Once again, you can use a Python generator for your converging or diverging sequences:

"generator at c"