On Sun, Sep 9, 2012 at 10:03 AM, Robert Hansen <firstname.lastname@example.org> wrote: > > On Sep 9, 2012, at 1:25 AM, Paul Tanner <email@example.com> wrote: > > 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. > > > That is not what Devlin said.
I said that it's his position based on what he said.
I said that I think his beef is that 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.
I also said that another beef I think he has 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 another model to at least add to the repeated addition model, where this additional model holds up without having to change at all through the multiplying of two irrationals.
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.
To reiterate part of what I said in that above post:
"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 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. 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."
This has them think proportionally, a good thing, not a bad thing.
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.
But here is another problem with the idea that we should not include scaling as a model for multiplication:
The crux of all this is that those who think that "multiplication is repeated addition" is just fine for R seem to treat the irrationals as not all that important, just a side curiosity, just "those numbers over there" that "don't really count" for what is really important (which seems to be finite computation for those who think this way about multiplication), and all that. But again, it's the irrationals and especially the non-algebraic irrationals that make up almost all of R, and is the basis of calculus and much of everything else higher up in analysis and algebra and topology.
This disrespect for the irrationals and especially the transcendental (or non-algebraic) irrationals is something I think bothered Devlin enough to write what he wrote. There is no disrespect for the irrationals here by the "multiplication *is* repeated addition" crowd that "multiplication is one and the same as repeated addition" and never agrees with the idea that we should at least include an extra view of multiplication that accommodates all of the reals not just almost none of them? There is. What this crowd doesn't get is this:
To ban or at least shove off to the side as unimportant those pesky irrationals and especially those pesky transcendental (or non-algebraic) irrationals is to ban or at least shove off to the side as unimportant all of mathematics based on continuous functions, which is almost all of algebra, calculus, and differential equations and much, much more that they could study on through college in math, science, and engineering.
Great idea, huh?
Sorry, but it's not a great idea. Devlin is right. We need to teach students another model for multiplication, at least in addition to a model based on repeated addition.
> Devlin said that he doesn't know how to teach > mathematics to elementary aged students and then proceeded to tell us how we > should teach mathematics to elementary aged students. >
Just because someone doesn't know how to teach kids doesn't mean that that someone cannot say nothing of value about the content, of what should be taught.
Or are you now of the position that all those mathematicians who don't know how to each kids but who like you have had a problem with reform mathematics education at that level of teaching kids and therefore spoke up about it - and not only have you have agreed with them you are glad they spoke up - should have shut up and should forever stay shut up on this issue and other such issues?