On Thu, Dec 24, 2009 at 1:15 PM, Anna Roys <email@example.com> wrote:
> RE: On Thu, Dec 24, 2009 at 10:29 AM, Hecman Gun <firstname.lastname@example.org>wrote: > >> For students of all ages, definitions of basic >> mathematical concepts have to be framed with care: not too formal, not too >> informal... >> >> >> ...What should be argued for a class (learning) the real numbers, >> fractions, etc., is to learn by intuition. >> > >
Hi Anna --
Let me try with the topic of real numbers again. How might we spark the intuition in some curriculum?
REALS AS EDGE LENGTHS:
First, I agree with Bill and Dr. Wu that the number line is important, but why exactly? Because it's graphical, not lexical, is part of the answer, and because it relates number to length, a distance. The relation of numbers to length is primal and should drive many of our early lesson plans.
However, instead of starting with just any old lengths, it'd make sense to look at some specific distances in a geometric context. We'll want to build some specific objects, will need specific lengths to do so.
FRACTIONS AS VOLUMES:
What's a common context for fractions, one with practical applications and associations? Cooking skills, following recipes, in terms of whatever units. Half cups, three quarter teaspoons etc. Nothing new here (so far).
But what might our measuring bowls actually look like? Here's one version:
There's much to recommend this bevy of inter-connecting concepts, in that we've got spatial geometry going in tandem with both fractional and real number relationships. In addition to lengths, we have angles. We could add rotation matrices, spherical coordinates.... looking down the road, yet keeping something already familiar.
We have some specific fractions relating to volumes, measuring discrete amounts, motivating computations and detection of equivalancies.
We have irrationals relating to lengths, motivating a discussion of Real Number, their appearance in the history of ideas. Bill's use of intervals might feature here.
MEETING A STANDARD
With polyhedra in view, we've got a state standard to hit: understand how area and volume change as a 2nd and 3rd power of changes to linear scale. Double edges, four fold area, eight fold volume. Have edges, reduce area 4x, volume 8x.
Published sources for using polyhedra in this way include a 1965 paper in Math Teacher by MIT crystallographer Arthur Loeb. Robert Williams in his The Geometrical Foundation of Natural Structure (Dover), and of course Bucky Fuller himself (the dome guy). Lotsa street cred here, websites, books, teaching supplies... everything one might need (including free computer source code if you're taking a more digital approach -- on tap at my site, with expository video).
I go into more biblographic detail here. Bill, this post mentions our discussion and links back to this thread FYI: