Date: Mar 24, 2010 7:09 PM Author: Jonathan Groves Subject: Is Multiplication Repeated Addition? 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.

However, there are other discussion threads on math-teach about this

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.

I will add more comments later, but I had at least wanted to bring up

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

comments about exponentiation and repeated multiplication.

Here are the links to his columns addressing this topic:

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

complex numbers. Does anyone have any ideas about this?

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