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:
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?