On Mon, Sep 10, 2012 at 2:03 PM, Paul Tanner <email@example.com> 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. For the volume to go from 3 to 4, it must go through pi along the way, i.e. all real numbers between 3 and 4 are hit, as the volume increases from 3 to 4.
> > 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. Just show it growing and shrinking with no change in surface and central angles i.e. the *shape* is held fixed while the "size" (sometimes related to "frequency" in STEM) is a variable.
We are keen to keep the 1, 2, 3 powering of line, surface, volume in view, i.e. take a shape as complicated as a sewing machine and scale it up, linearly, by 3.445.
The volume will increase by a factor of 3.445**3 (i.e. "to the third power") exactly. We don't even need to know the initial volume to know what scale factor applies.
We might want to (definitely want to) use a growing / shrinking tetrahedron in some segments.
Since you agree scaling covers the Reals (R) as well as the Rationals (Q), this kind of scaling (using volume) should be no problem.
But then we also have our discrete units of volume, and our factional quantities.
A lot of the "repeated addition" model will propagate over, i.e. when we show a shape growing and shrinking, we can talk about how much volume is being added or taken away, with repetition involved.
> > > >> 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.