On Sun, Sep 9, 2012 at 7:46 PM, Wayne Bishop <firstname.lastname@example.org> wrote: > At 12:10 PM 9/9/2012, Paul Tanner wrote: > > Is the multiplication of two irrationals, especially two non-algebraic > irrationals like the product e(pi), a computable procedure? > > Computation is fine when you actually do it, but when you can't, you > will need a definition of multiplication that holds up on all of the > reals, not just on almost none of them. > > > Forget the "Non-" part of the thread title and the Greeks told us how to > multiply any two (positive) numbers some time ago, no approximations - or > even convenient notation for approximations - needed. That said, it's no > way to begin multiplication with little children. Repeated addition is the > way to go even with fractions > m/n = m(1/n) so that: (k/l)(m/n) = km(1/lm) with the nice picture to go with > it, and is analogous with a nice picture that goes with mn. > > The problem with Devlin's strawman "is" was explained by that great > communicator former Pres. Clinton. > > Wayne > > >
Neither Devlin nor I say what you say we say. I believe that he said that teaching repeated addition is fine, just not as one and the same thing as multiplication, since this repeated addition model fails utterly when confronted with multiplying two irrationals later on. And it has to be totally defined for two rationals - as I said repeatedly, with two rationals we can no longer use the two original factors, where with before rationals, the two original factors are directly used: ab means a instances of b added together, but with (a/b)(c/d), we have to abandon both a/b and c/d and derive two new factors to do this repeated addition thing, one factor being a and the other being c/(bd). Yes, it;s still so-called "repeated addition" but so what? There has to be a major redefinition.
But with scaling, there has to be *no* redefinition even for two irrationals.
Kids are smarter than you think.
As an extra model to go with the repeated addition model, this extra model being able to hold up totally unchanged even with multiplying two irrationals:
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 as a model 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 as a model 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.
If you think 7 year old kids are too stupid for this, then OK, go ahead and believe it. I don't believe it. You shortchange them, I think, in terms of what they are capable.