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Re: Non-Euclidean Arithmetic
Posted:
Sep 10, 2012 10:19 AM
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On Sat, Sep 8, 2012 at 3:56 PM, Paul Tanner <upprho@gmail.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.
Kirby
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