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Re: Square root of six
Posted:
Aug 28, 2012 11:52 AM
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Yes, I'm all in favor of roots and root finding. Raising to powers is
an important concept. I'm also a big believer in "infinite series"
i.e. the sums and sequences portion of precalc, where you're both
converging and/or your partial sums are converging... I'm a big fan of
that stuff.
However, I think it's very important that we get away from numbers,
meaning anything in the N, Z, Q, R, C pool, and focus on other objects
that many also be multiplied, such as permutations.
A permutation, in this namespace, is like a mapping of the letters A
to Z unto themselves, such that each letter is paired with another (or
itself). The identity permutation points every letter to itself.
When you multiply two permutations, you get the one that does the work
of both, e.g. if A -> R in the first and R -> K in the next, then
their product goes from A to K in one step.
These kinds of object are somewhat tedious to work with by hand and
students rapidly get carpel tunnel or throw fits in frustration,
twirling and foaming on the floor, as if possessed by demons.
We don't get that behavior with bright talking screens are enlisted,
and when we let computers do the guts of the operations. We program
though, which means we take control, have insight. Programming your
permutations, teaching them how to "multiply" is not just "for
programmers" (the way they say in Florida), it's for anyone learning
math, as this is group theory / abstract algebra, right at the core of
the disciplines.
So the nth root of a permutation: that makes sense. We can do "power
tables" which show how often permutations cycle, when multiplied by
themselves.
There's this thing about using the GCD (greatest common divisor
algorithm) to get the totatives of a number, those positive numbers <
N with no factors in common with N. If you multiply totatives modulo
N, you get a group. You are also laying some stepping stones for
understanding cryptography, RSA in particular, within reach by senior
year. This is the digital math track I talk about, used in the better
schools with state of the art STEM curricula.
There's a gap between schools with digital savvy and real STEM, and
lower quality laggard schools that don't have such quality curricula.
The USA is for the most part far behind, but here and there has some
pockets of excellence. We can't really break it down by race though,
as the better STEM schools are not furnishing us with exactly those
kinds of statistics. They're not even really "schools" in many cases,
just learning environments in cyberspace with no zip code, so more
like Facebook (a socially engineered meeting space).
Kirby
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