On 2/20/10 at 10:07 am, Alain Schremmer wrote in reply to me:
> On Feb 19, 2010, at 10:04 PM, Jonathan Groves wrote: > > > I agree that multiplication is not repeated > addition and that > > exponentiation > > is not repeated multiplication--not for real > numbers anyway. > > 1) This "is not" makes no sense for any number of > reasons, Devlin > notwithstanding. > > One is: Should one define a quarter as a coin whose > head is > Washington's or as a coin whose tail shows a spread > eagle? Should a > Boolean Algebra be defined in terms of two binary > operations or in > terms of two binary relations? > > Another one is that multiplication is not a single > one operation; > multiplication on natural numbers is not the same as > multiplication > on positive integers. > > Yet another one is that multiplication on natural > numbers is not > thought of in the same manner by Newton, Euler, > Leibnitz, etc as by > Bourbaki. > > Yet another one is that, even today, multiplication > for a six years > old is not the same as multiplication for a > Bourbakist (Disclosure: > such as I once was.) > > Etc. > > 2) Any such "is" is a totalitarian means to enforce > and "justify" a > particular view as just about everything "is" at the > intersection of > many different things---as well as people as Amartya > Sen forcefully > argued in "Identity and Violence". > > Not that I am thinking that Groves is anywhere near > remotely a > totalitarian of course and indeed, another > disclosure, as late as ten > years ago I would have strenuously agreed with > Devlin. But, if > perhaps somewhat indirectly, David Tall got me > thinking . . . again. > For which I have been grateful to him ever since. > > Regards > --schremmer
Alain, thanks for your comments. You had mentioned several good pointers that no one in the math-teach discussion has mentioned yet, and I will address these below.
The argument that had begun with my post there several days ago continues to rage, and it is not yet clear if the arguments are beginning to slow down. I do have to admit that several really heated discussions on math-teach last year had begun with my posts. One of them about this same time last year was about the website Mathematically Sane. The arguments raged on for a while then quit and were later reignited. That cycle has repeated a few times already. Another one this past summer was about teaching pure mathematics in K-12, and soon arguments over that idea and what a proof means and how to teach proof to kids in K-12 raged on. And these arguments led to additional arguments about visual proofs and their roles and their level of rigor. And arguments about standardized tests continue to rage, die down, and get reignited again.
I think that these ideas you had mentioned didn't arise in the math-teach discussions because the question is primilarly about how to teach what multiplication "is" to kids who are learning multiplication for the first time. The formal approach certainly wouldn't work with them.
Strictly speaking, I must agree: Multiplication of natural numbers is different from multiplication of positive integers and is different from multiplication of positive rational numbers and so on. And different mathematicians have given different definitions over the years.
Speaking of definitions, at least one mathematician had raised a fair question when he had asked what a definition really is and what separates a definition of X from an equivalence theorem about X and had asked who is to say that a particular definition about X is the "official" definition of X. We see different definitions because they had arisen from different problems, and one problem suggests one feasible definition, and another problem suggests a different feasible definition (sometimes this alternate definition is equivalent to the first one and sometimes not). For example, some books, especially older ones, do not require that a ring have a unit, and other books do. But in teaching or presenting mathematics, we have to start somewhere, and the choice of which definition to use depends on what is being discussed, how the ideas are developed, and even to whom you are teaching. It appears that the teaching of multiplication of natural numbers as repeated addition has trapped many students into thinking that any form of multiplication is repeated addition. So fraction multiplication, for example, is hard for them to understand because the repeated addition idea no longer works.
Is it possible that multiplication of integers or rational numbers is taught too soon (before they have developed their mathematical thinking skills enough to learn that context in mathematics affects which definitions and theorems apply)? That is, is the main problem not really about teaching multiplication of whole numbers as repeated addition before teaching multiplication of integers and rationals and reals but about the timing and how it is taught? Perhaps.
Related to this question about definitions is the second point you had made here, and it is a valid one. One problem with the teaching of mathematics is that often students are rarely told that different definitions of the same concept exist and that sometimes these definitions are not equivalent. Furthermore, the students aren't told that one definition isn't necessarily any better or more correct than another one. And they certainly are not told why these different definitions exist and certainly aren't given even the slightest idea of why one definition might be chosen over another. It is possible that some teachers in K-12 or at the beginning college level teach this, but I know such teachers are rare. I hate to admit that I am one of the many guilty ones, but I'm thankful I'm aware of that now because I wasn't previously.
Mathematicians generally agree (at least according to what I have learned in abstract algebra) that the operations on a subset of a ring R are the same operations in R restricted to this subset. Thus, if we view the integers Z as a subring of Z[x]*, then the addition and multiplication in Z can be defined as addition and multiplication in Z[x] restricted to Z. Likewise, the whole numbers form a subset of the ring of real numbers (which is also a field, of course), so whole number arithmetic is, by this idea of definition, real number arithmetic restricted to the whole numbers. Actually, we could go further with this and give definitions of complex number arithmetic and then define arithmetic on any subset of C as complex number arithmetic restricted to that subset.* I wonder if developing definitions of these different multiplications along these lines (with the appropriate level of formality so that students get it) would work better than the multiplication-as- repeated-addition idea for whole numbers.
I would be interested to know more about what David Tall had taught you. Did you read one of his books or attend one of his lectures?
---------- *Note: Strictly speaking, we know that R is not a subfield of C but instead is isomorphic to a subfield of C. Likewise, Z is not really a subring of Z[x] but instead is isomorphic to a subring of Z[x]. But it is common practice in abstract algebra to say that "a ring R is a subring of S" if R is isomorphic to a subring of S. This shorthand is not always used but is generally used for many familiar rings. To make the above more rigorous, we would have to say that the product ab in Z (where a,b are in Z) is the product phi(a)*phi(b) in Z[x] where phi is an isomorphism of Z with the subring of Z[x] containing the polynomials c+0x+0x^2+0x^3+... (where c is in Z). And we can say something similar about the second example I had given. Finally, we can always find an even larger ring or field that contains the numbers in question; for example, we could find an even larger ring that contains C as an isomorphic copy. But we can't define operations on all rings using this idea here because we can always extend a ring R to another ring S so that S contains an isomorphic copy of R as a subring.