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Re: Non-Euclidean Arithmetic
Posted:
Sep 10, 2012 5:03 PM
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On Mon, Sep 10, 2012 at 10:19 AM, kirby urner <kirby.urner@gmail.com> wrote:
> 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.
>
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.
Message was edited by: Paul A. Tanner III
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