On Mon, Sep 10, 2012 at 10:19 AM, kirby urner <email@example.com> wrote: > On Sat, Sep 8, 2012 at 3:56 PM, Paul Tanner <firstname.lastname@example.org> 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. >
The set of volumes that are bigger or smaller is still a set under a total or linear order.
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. >
No, but I was using "length" because it's not easy to get away from a set under a total or linear order when we are talking about scaling. If it's only under a partial order, then how does scaling work? Sure one could perhaps come up with a whole lot of definitions for "scaling" when the set is only partially ordered, but at least for kids I would think that that would be out.
>> 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".
But this is multiplication in terms of numbers, using a number set as the domain of one of the factors.