On Sun, Sep 2, 2012 at 6:27 PM, Paul Tanner <email@example.com> wrote:
<< snip >>
> Let's generalize this all the way from a field to a ringoid, so we can > really see what's going on. A ringoid is simply a set under two binary > operations such that they are "connected" with a distributive > property, one of these operations distributing over the other. *By > convention*, we call the one being distributed over "addition" and the > other one "multiplication". >
Note that when we linger around GCD, play more with primes, bigger primes, do more with totatives, stray into Group Theory, I'm not suggesting a full semester on those topics necessarily, as if "in for a penny, in for a pound" is always operative. No, you can be in for a penny, a quarter, five dollars, ten... i.e. your level of time/energy investment in group theory / abstract algebra is controllable. You don't have to get into "ringoids" just because you watched 'Lord of the Rings' in your planetarium / movie theater (what every school has by fictional 1999+).
>> >> (a/b)(c/d) = a (bc/d) so you can always isolate an integer and then >> say you're adding (bc/d) to itself a times. >> > > But this is a serious redefinition of viewing "multiplication" as > "repeated addition". >
It may seem like radical surgery to you but I assure you the avid teachers of the "repeated addition" meme took this step long ago. As I was saying above, even your "scaling" which you present in cartoon form *in contrast* to repeated addition, is just more repeated addition (of some unit of vector / length / distance).
> Jump to >> Reals. >> > > I don't think so. > > Take e(pi). > > What does it mean to have pi *instances* of e or e *instances* of pi? >
You have an ungodly huge Avogadro Number of atoms and you divvy them so that 3.14159... of them (some trillions) get to be a "unit" in some way, with respect to some multiplier. Numbers like pi and e are just fractions that "never end" in this vague hand-wavy extension of Q to R.
> Talk about having to bend over backwards to redefine things! >
I think you're just culturally isolated and to you it's big news that some people think this way.
> > I'd say that we have to really redefine what it means to do something > n "times". >
Well, maybe we should really do that then.
In any case, I'm more interested in keeping it short, whatever we do. The movie directors have trended towards short cutting and subject change, with TV also pioneering in that direction. I don't want to be a mile wide inch deep but nor do I want to be a mine shaft in the wrong area, or any number of other poor designs i.e. there are many more ways to get it wrong than right and "mile wide inch deep" is but one of them.
So do some permutations, as an excuse to get good with hash tables (Python's dict) more than anything, and touch on their closed nature and the existence of group properties, and then move on. Come back to group properties another time, when looking at totatives modulo N ala 'Vegetable Group Soup' segments **, and move on again, but take a few moments to allude back and forth i.e. point out these bridges.
It's like Ikea (furniture store): there's the "long route" where you see everything, and then you have these "short cuts" that connect you in a kind of "hyperspace".
So in STEM we have our linear tracks, which don't go on and on for too long, and their hypertext topologies (other ways of "stringing it together").
Remember we have RSA (topic in crypto) to hit, still within the K12 sphere. Grades 13-16 provide time to go deeper, before hopping into trucking (a business school gig) and then into some mix of management and finance. You still draw on those crypto skills and some of the products you work with are about routing trucks (an important area in software development, especially with real time GPS getting involved).