On Sat, Sep 8, 2012 at 3:56 PM, Paul Tanner <email@example.com> wrote:
> No one is saying don't use repeated addition *as a model* in the > naturals, and no is saying don't use the repeated idea later with the > redefinitions. But just don't exclude the one and only model that > needs no redefinition all the way up through multiplying two > irrationals, especially the one and only model that has the added and > probably vastly more important benefit of teaching proportional > reasoning from day one via reasoning on equal or equivalent ratios or > fractions (which is a great foundation for working with percentages - > and I mean using the equivalent ratios or fractions x/100 = y/z.) >
Scaling can also be about volume as volumes scale as surely as lengths do, and so multiplication, presented as scaling, could show a sphere getting bigger and smaller. You will agree that the radius, not just the circumference could be pi, but so could the volume in the continuum of algebra, i.e. we can set the volume to pi and compute the radius accordingly (some irrational number). In this way, a growing sphere might represent two irrationals being multiplied. No need to get hung up on length.
Then, if we zoom in and see the sphere contains many discrete particles, this doesn't have to be seen as subverting the fact of incommensurability. The XYZ coordinate system offers the same view: a matrix of closest packed cubes in which floats said sphere. But instead of always that rectilinear matrix, the STEM experts I hear loudly and clearly are asking for a sphere packing matrix as well. No problemo. It shall be so (is already so).
> The lack of fluent proportional reasoning in a large percentage of > students all the way up through adulthood is one of big problems that > must be addressed, and a least including this scaling way of modeling > multiplication using equivalent or equal ratios or fractions from day > one is a ready-made way of attacking this problem. >
My pet peeve is to always cast multiplication in terms of numbers, always using a number set. That's terribly unimaginative and not worthy of a second look. You need to see the addition operator used like this: "ABC" + "RFP" == "ABCRFP" (concatenation), and the multiplication operator used like this: "TA" * 3 == "TATATA". If you see nothing like that in the wood pulp math text they're planning to use on your children, consider going on-line and looking for more tech savvy schools with more intelligent curricula. If multiplication is always presented in terms of "numbers", be sure that's an inferior school in that way.