What the heck is multiplication anyways?

Speaking of misusing words, Keith Devlin issued a call to teachers to stop teaching multiplication as repeated addition, and instead, to teach multiplication simply as one of the basic operations we can apply to numbers. The crux of Devlin's argument is the following:

Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not. Multiplication of natural numbers certainly gives the same result as repeated addition, but that does not make it the same. Riding my bicycle gets me to my office in about the same time as taking my car, but the two processes are very different. Telling students falsehoods on the assumption that they can be corrected later is rarely a good idea. And telling them that multiplication is repeated addition definitely requires undoing later.

Of course, Devlin is right in that what multiplication really is is one of two operations defined for a algebraic structure called a ring. He is also correct in asserting the need to undo the perception of multiplication as repeated addition, especially when it comes times to multiply things like complex numbers, matricies, or functions.

However, Mark Chu-Carroll and Jason Rosenhouse both highlight a key problem: Devlin really doesn't really provide any real answers as to what to teach elementary school students when they first encounter multiplication. They correctly point out that taking an approach other than "multiplication is repeated addition" would likely fail to build students' intuitions about what multiplication is and how to carry out the mechanics of the process.

To a certain extent, I disagree with Devlin that continually "redefining" the rules of multiplication leads to frustration and "thinking mathematics is just a bunch of arbitrary, illogical rules that cannot be figured out but simply have to be learned"; like Jason Rosenhouse, I think most of the frustration comes from the fact that mathematics forces us to think using specific abstractions we aren't entirely comfortable with. Moreover, I (and I suspect many others) do eventually stop considering multiplication to be repeated addition without really thinking about it. For me, this unconscious transition happened when I started working with formulae that involved units, such as those from physics. By that point, it was no longer clear how repeatedly adding two kinds of units (e.g., mass and acceleration) gave a third kind of unit (e.g., force), and multiplication became a basic operation performed on two numbers to get a third number.

On the other hand, like Devlin, I'm not a fan of the lie to the children approach to teaching, primarily because doing so leaves "brain bugs" that are notoriously hard to get rid of. Going back to Devlin's argument, defining multiplication as repeated addition implies that the inverse operations are related as well, i.e., that division is repeated subtraction; as we discovered one day at work, this leads to enormous complications when trying to explain certain things, like why dividing by zero is undefined. Another good example Devlin points to is the teaching of exponentiation as repeated multiplication; because this "definition" has been beaten into my head, I have tremendous difficulty understanding what matrix exponentiation (i.e., e^A where A is some matrix) really means.

I'm not entirely sure as to what to make of all this. Devlin wrote a follow up article, saying essentially "however you teach it…, don't do anything that is counter to the way the mathematicians do it." While I agree in principle, I'm still not sure how to translate this to the classroom. So far, the only good solution I can think of is simply to attach and repeat a caveat like this when teaching multiplication:

Aside from addition, the other thing we can do with numbers is to multiply them. When working with the counting (natural) numbers, multiplying works like adding over and over again. But multiplying and adding over and over again are not always the same thing, as you will see in higher grade math.