On Fri, Jun 21, 2013 at 4:04 PM, Robert Hansen <firstname.lastname@example.org> wrote:
> > On Jun 21, 2013, at 3:48 PM, kirby urner <email@example.com> wrote: > > This one is somewhat visual, a couple mistakes on the > sound track: > > https://www.youtube.com/watch?v=jb1VMGrxPWg > > This is more algebraic and well presented. Better than > a chalkboard. > > https://www.youtube.com/watch?v=fwuj4yzoX1o > > > > While the intent is sincere neither of these help to explain how they > work. Mathematics isn't a set of magic tricks. > > Bob Hansen >
I agree, neither is sufficiently transparent.
There's an added glitch, in English at least, that bites in the first video: the difference between "dividing evenly" and "dividing an even number of times". To a teacher it may be obvious there are two meanings here: "without remainder" and "a multiple of two times" respectively, but a student watching the demo may be confused as to how three of something "evenly divides".
I'd build up to it slowly with more "brick layer" stories, having longer and shorter bricks and wanting to come to a common edge. Dividing without remainder is satisfying anyway. In modulo arithmetic it's your zero, nothing left.
GCD(p, q) == 1 means p and q have no factors in common which means p/q and q/p are in lowest terms if perhaps "improper" (a top-heavy fraction is supposed to simplify to N + r/s if q/p > 1).
GCD(q, p) == 1 also means p and q are "strangers" and/or "relatively prime". That's useful terminology.
Then I may define all the positive numbers less than N as the totatives of N. This becomes so expressive in the Python notation where you can just say:
>>> def totatives(N): return [T for T in range(1, N) if GCD(T, N) == 1 ]
>>> def totient(N): return len(totatives(N))
in other words, I can write a one-line function that immediately gets me a list of the totatives of a number. This presumes GCD(a, b) is already implemented, which is where' Euclid's Method comes in.
The games you can play with Totatives to demonstrate group properties of Closure, Associativity, Inverse and Neutral (identity element) are quite fun on computer and yet the 1900s curriculum said "you must prove your metal by traversing Calculus Mountain before we let you rest by these clear pools of Group Theory."
Asset (b) below is a breath of fresh air because it takes us to RSA (a cryptography algorithm) as a goal, as something to understand, versus a lot of calculus. Students have a mountain to climb that's more digital, less analog.
(a) Divided Spheres (STEM primer, geometric, Popko) (b) Digital Mathematics (STEM textbook, Litvin & Litvin) (c) King of Infinite Space (a bio, Siobhan)
Does this mean I'm against teaching calculus? No.
Does it mean I think there's room for more tracks in the high school years besides just "vocational" (not college) and "college prep" (precalc / calc) -- yes of course I do.
But the states are not taking this agenda seriously and instead are going with Common Core State Standards as their PR to the voters.
Do I think all the states should stop what they're doing and defer to me? No.
Do I think we have excellent ways of competing with mediocre state designs, using shared (sometimes copyleft) curricula that are better and more relevant by orders of magnitude? That's obvious.