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 Jonathan Groves Posts: 2,068 From: Kaplan University, Argosy University, Florida Institute of Technology Registered: 8/18/05
Posted: Mar 24, 2010 7:09 PM

Dear Fellow Mathematicians and Educators,

Here is a copy of my first post to math-teach that I had posted on
February 16 of this year in which I had started a hot argument about
multiplication as repeated addition. The argument on math-teach
continues to rage.

The text of the message follows, but you can access the message and its
replies at http://mathforum.org/kb/message.jspa?messageID=6982992&tstart=0.
issue, so this particular thread does not contain the entire debate.
I think this issue is relevant to adults who are trying to learn basic
numeracy, and I think this issue explains why students struggle to
learn fractions and proportional reasoning.

the issue and see how many here on numeracy are interested.

The text of that message follows:

I had seen recently several articles on "Devlin's Angle" arguing that
multiplication is not repeated addition and should not be taught that way
(at least not as the definition of multiplication). And he makes similiar

June 2008: http://www.maa.org/devlin/devlin_06_08.html

July-August 2008: http://www.maa.org/devlin/devlin_0708_08.html

September 2008: http://www.maa.org/devlin/devlin_09_08.html

January 2010: http://www.maa.org/devlin/devlin_01_10.html

I agree that multiplication is not repeated addition and that exponentiation
is not repeated multiplication--not for real numbers anyway. These are
properties of whole number multiplication and whole number exponentiation,
but, as Devlin says, these properties are not definitions.

He essentially argues that these properties should not be taught as definitions
because then the students will be confused later because they will have to undo
all they've learned about multiplication and exponents of whole numbers
(strictly speaking, we have to disregard an exponent of 0) to
learn multiplication and exponentiation of integers, rational, and real
numbers. In fact, he mentions that he has corresponded with multiple readers
who think multiplication is repeated addition and that exponentiation is
repeated multiplication. So I think his point is valid.

He is correct in that the field properties of the real and complex numbers do
not tell us what the operations are but simply how they behave. Definitions
of the real and complex number operations exist, but the formal, precise
definitions are far too abstract for elementary school and certainly are for
most high schoolers and even adults.

However, teaching students these operations without giving definitions of some
sort doesn't make sense because students want to know what these operations are
and not just how they behave. And it would be difficult for them to
understand these operations conceptually if they don't see any definitions.
I didn't see any ideas in these articles about how we should teach these
operations to students, especially operations with whole numbers.
The usual way is to teach these operations for whole numbers and then extend
those ideas to operations on the integers, rational numbers, and real and

I'm sure Keith Devlin is not saying that it's a bad idea for students to
learn that multiplication of whole numbers is repeated addition and that
whole number exponentation (ignoring 0 as an exponent) is repeated
multiplication but is saying that it's a bad idea to teach these properties as
definitions. Does anyone have any ideas on how we can teach these properties
to students and help them to understand that these are not definitions?

Jonathan Groves

Date Subject Author
3/24/10 Jonathan Groves
4/7/10 Jonathan Groves