On Sat, Sep 8, 2012 at 4:35 PM, kirby urner <firstname.lastname@example.org> wrote: > On Fri, Sep 7, 2012 at 4:14 PM, Jonathan Crabtree > <email@example.com> wrote: >> Thank you Joe! >> >> That is the 'what and why' of ASAPANS (as simple as possible and no simpler). >> >> Multiplication should have been clarified by Einstein. >> >> Multiplication is NOT scaling. >> > > I think a fairly common mistake among mathematicians is to think their > job is to develop "global meanings" in the sense of their being a top > level namespace that anchors all the others, a fixed firmament. >
I think you are making a mistake of confusing the drive to generalize things with this attempt to impose as you say.
First, I don't think mathematicians do this imposing. It's this imposing that mathematicians are actually against. Sure, the drive to generalize may result in trying to see whether there is any "common denominator", but that is not this imposing.
This type of imposing is what Devlin I think is arguing against, that the operation of binary multiplication is - where "is" means the sense of "is identical to" - the operation of repeated addition. Sure, these different operations sometimes give identical results in some contexts, but that is different than saying that they are identical operations, that one operation is the other operation.
It seems to me that mathematicians say that modeling an operation is an OK thing, but if there are going to be models then please don't exclude the ones that have more generality, as the scaling model has more generality than the repeated addition model, as general as any set on which we can define magnitude or absolute value or if we wish, the geometric language of distance or length from 0.
Side note: No one is saying that scaling is what multiplication is - only that scaling is a model for multiplication.
With repeated addition as a model we have to keep changing the definition of repeated addition as we go to more and more general number systems and to the breaking point when we hit multiplication of two irrationals. But with scaling we don't have to change the definition of scaling at all. Even for multiplying two irrationals e and pi, where 1, e, pi, and e(pi) are viewed as distances from 0, we use the same exact model of scaling we use for multiplying two naturals: We have these ratios as a model: pi is to 1 as e(pi) is to e, and via commutativity we have e is to 1 as e(pi) is to pi. Or if we wish to use equality on fractions, pi/1 = e(pi)/e, and via commutativity we have e/1 = e(pi)/pi. And we can model this by construction on a graph: Connect 1 on the x-axis to pi on the y-axis, and, parallel to that drawn line segment, connect e on the x-axis to the y-axis, and this point on the y-axis is e(pi). And via commutativity we have the other way: Connect 1 on the x-axis to e on the y-axis, and, parallel to that drawn line segment, connect pi on the x-axis to the y-axis, and this point on the y-axis is e(pi).
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.)
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.