On Mon, Sep 10, 2012 at 6:45 PM, kirby urner <email@example.com> wrote: > On Mon, Sep 10, 2012 at 2:03 PM, Paul Tanner <firstname.lastname@example.org> wrote: >> >> >> > 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. >> > > The intermediate value theorem or something like it might be used. >
I know, but what I'm trying to get across is that "length" as in distance from 0 is a metaphor for the fact that with volume or anything else you care to try to use to get away from "length", you're still talking about absolute value or magnitude, which is distance from 0, which is a "length".
>> >> 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. > > > It's very easy to use volume or area in place of length. >
Not to get away from from what I'm talking about above, which is "length" in terms of absolute value or magnitude, which is distance from 0, which is a "length".
> Since you agree scaling covers the Reals (R) as well as the Rationals > (Q), this kind of scaling (using volume) should be no problem.
Yes, I agree.
But it does not escape what I'm talking about.
>> >> 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 multiplication in terms of numbers, using a number set as the >> domain of the one of the factors. > > Yes, an integer appears. But so does a character string. It's an > actual use of the multiplication operator that is common in many > notations and should be mixed in as one of the many examples of > multiplication in the wild. > > What I was saying is that text books which are exclusively numeric in > their presentations of maths, and not alphanumeric, are deficient, not > STEM-worthy, not STEM-compliant. If we're to align with STEM (as all > but the Americans might be doing), then we need to expand our horizons > beyond use numbers exclusively, even with respect to "the four > operations". Another reason why scientific calculators are not > suitable. >
I agree that it's a good idea to show a number of good examples of these operations using non-numbers as at least one of the factors.