On Mon, Mar 15, 2010 at 8:40 AM, Joe Niederberger wrote:
> So, the evidence as it stands, lends (strongly) towards the view
> that addition is basic, multiplication derived. It says you can
> always correctly view it that way if you choose to.
> Joe N.
> p.s. Pam, this is a bit related to the little intuition you
> sent me in your last email.
Somehow I thought we were done with this question.
We don't only care about Real Numbers.
Sometimes, we don't care about Real Numbers at all.
Matrices multiply. Adding them may be pretty worthless.
Vectors add, but don't multiply (with each other), although quaternions do.
We preview these meanings of "multiply" with younger students, not by drilling them in all kinds of arcane algorithms, but by giving them a heads up as to the multiplicity of types in the "math objects zoo".
Your nomenclature may vary, but the basic concepts are as traditional and conservative as you need them to be.
At more advanced levels, it pays to show the family resemblances between the different kinds of addition and multiplication.
At these levels a more abstract algebra approach enters in, and we share about closure, inverse, neutral element (identity element), associativity (matrices are associative, not commutative, w/r to that binary operation we call "multiplication" w/r to them).
Using clever little logic games, encouraging free play and discovery with finite groups is not out of reach. I'm not going to make extravagant claims for their pedagogical value, but nor am I going to sneer and jeer like some know-it-all pundit.
Back to my iconoclastic / radical agenda, here are pictures of a robust classroom-ready 4-frequency tetrahedron made of plastic icosahedra held together by strong springs:
(flextegrity by Sam Lanahan, sculpture by Glenn Stockton)
Of course no classroom has such a thing, as we're looking at a prototype fresh from the factory, woven to suit. With 4-frequency, you first get a nucleus.
The edge and face centers are easy to apprehend. 4-frequency means five members along an edge, plastic icosahedra in this case. So there's a middle one that's easy for students to grasp.
Having something heavy and robust adds measurably to the quality of the kinesthetics upon passing this around -- makes a lasting impression.
This is not salesmanship as I can't offer any for sale. More like advertising my Radical Math agenda, a product I believe in (obviously) and consider open source in large degree i.e. I'm not the gatekeeper (nor bottleneck). The truth is out there.
What might be a real world application for getting into polyhedral numbers ala sphere packing? The tetrahedral numbers? The triangular numbers?
Well, here's a write-up of a recent nanotechnology lecture we got in Portland, Oregon recently, where I think the connections are pretty clear (note Coxeter paper for further reading, if you want to go deeper into this stuff):