On Mon, Sep 10, 2012 at 4:03 PM, Paul Tanner <firstname.lastname@example.org> wrote:
<< snip >>
>> >> 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". >
I'm fine with that idea, that all the cases we're talking about involve relative size in space and that's what we call "distance" or "length".
You're saying I'm in the same "realm of blobs" as you are, or "res extensa" and I'm agreeing with you.
I have a propensity to focus on volume because in some of the discrete non-continuous sand-castles (metaphor for "math systems") we allow even lines to have volume i.e. they're very thin in aspect ratio, as thin as you like, but not "infinitely thin" as "no volume in principle" is less conceptually clear than "some volume in principle".
Likewise we don't need infinitely thin planes that go on and on forever -- in every math.
In "Euclid's math" we want those. In "Democritus's math" we maybe don't.
So in this view, volume is more primitive in the sense that all shapes have some, even points. We call them all "lumps", following mathematician Karl Menger and his essay postulating such material (we've come pretty far since that pioneering essay).
> 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". >
That's fine. The volume of a sphere can be shown on a line, where the line is understood to represent fixed "units of volume" (which might be tetrahedrons).
>> 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. >
I'm not trying to escape. I'm happy with "scaling". We never agreed that repeated addition and scaling fade into one another leaving no seam, but I don't think it's critical that we reach agreement on that matter. It's mostly a matter of definition and people have different definitions.
>> 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.
Also, and I'm not sure how many turns of the spiral into it we are when we get there, varies with the student, we need to show multiplication modulo N.
We can even define sets of "modulo N numbers" that have an internalized sense of what it means to multiply i.e. we don't make it be a different operator applied to familiar numbers, but a different "type of number" (the kind that multiplies modulo N).
Around the time we introduce primes and composite numbers, we need to pause and go more deeply into a few things, including modulo arithmetic, the concept of totient and totative, and Fermat's Little Theorem.
The idea of a totative is not at all hard to communicate and reinforces the GCD concept. The totatives of N are all those positive integers < N that have no factors in common with N (other than 1). What's true is the totatives of N, multiplied together, form a finite group. How cool. And how easily accessible. We're not talking high IQ genius. We're talking about an excuse to write short few-line programs. These problems may be aimed to the middle of the bell curve.
Also: I already introduced meanings of multiplication where *neither* element was a number. Permutations for example. A permutation takes all 26 letters plus the space to a scrambled copy, i.e. A->S, B->R, C->D.... a huge number of possibilities. These "mappings" may be treated like objects and "multiplied" such that the product is another permutation. Another finite group. Another meaning of multiplication. Another excuse to write short few-line computer programs.