
Re: NonEuclidean Arithmetic
Posted:
Sep 9, 2012 1:25 AM


On Sat, Sep 8, 2012 at 11:34 PM, Joe Niederberger <niederberger@comcast.net> wrote: >>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, > > SO you are going to address it by assuming kids just learning to multiply two integers are ready to be told "you just scale one number by the other"? > > Let me know how it works out. > > Joe N
When would you teach this model? Never?
Regardless:
Kids are smarter than you think.
In my post
http://mathforum.org/kb/message.jspa?messageID=7886496
I did not say what you said I say. I said that teaching the model of repeated addition is fine, but that scaling also should be taught, but I did not say that we should say "you just scale one number by the other" when teaching it.
Kids are smart enough to see that if we start at point 1 and make that length 3 times longer we end up at point 3. Nothing wrong with using repeated addition here to count up to the target point, and I think Devlin is fine with that. They are smart enough to then see that if we start at point 5 and make that length of 3 times longer then we end up at point 15. Again, nothing wrong with using repeated addition here to count up to the target point, and I think Devlin is fine with that.count up to the target point. Finally kids are smart enough to see the same pattern being done: "Stretching a length of 5 out to make it 3 times longer is the same pattern as "stretching" a length of 1 out to make it 3 times longer. 15 is to 5 as 3 is to 1.
If you think 7 year old kids are too stupid for this, then OK, go ahead and believe it. I don't believe it.
Side note: This teaching of patterns, or mathematics as patterns, is in line with the theme of Devlin's book "Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe". (Scientific American Library)
I reiterate that Devlin did not actually say don't teach or use repeated addition. His position is only that we should teach that although these two operations of multiplication and repeated addition get the same results, that does not mean therefore that they are the same operation.
His main beef I think is teaching that multiplication *is* repeated addition is the same as teaching that they are not different operations but are the same operation simply because they get the same results. A great example to illustrate this idea that different operations can get the same result is 2 plus 2 equals 2 times 2.
His other beef I think is that repeated addition even as a model does not generalize to the later real numbers well  and not at all when we get to the point of multiplying two irrationals. And so we need a another model to add to the repeated addition model, where this other model holds up without having to change at all through the multiplying of two irrationals.
I reiterate what I said in my post above, especially this:
"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 xaxis to pi on the yaxis, and, parallel to that drawn line segment, connect e on the xaxis to the yaxis, and this point on the yaxis is e(pi). And via commutativity we have the other way: Connect 1 on the xaxis to e on the yaxis, and, parallel to that drawn line segment, connect pi on the xaxis to the yaxis, and this point on the yaxis 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 readymade way of attacking this problem."

