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Topic: Multiplication Is Not Repeated Addition
Replies: 45   Last Post: Sep 30, 2010 7:11 AM

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 Jonathan Groves Posts: 2,068 From: Kaplan University, Argosy University, Florida Institute of Technology Registered: 8/18/05
Re: Multiplication Is Not Repeated Addition
Posted: Feb 20, 2010 1:40 PM

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
> > 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
> 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

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 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
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
teaching multiplication of integers and rationals and reals but about the
timing and how it is taught? Perhaps.

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-

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?

Jonathan Groves

----------
*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.