I've been getting peppered with requests to comment on a recent argument that's
been going on about math education, particularly with respect to multiplication.
We've got a fairly prominent guy named Keith Devlin ranting that
"multiplication is not repeated addition". I've been getting mail from both
sides of this - from people who basically say "This guy's an idiot - of
course it's repeated addition", and from people who say "Look how stupid
these people are that they don't understand that multiplication isn't repeated
In general, I'm mostly inclined to agree with him, with some major caveats. But since he sidesteps the real fundamental issue here, I'm rather annoyed with him.
You see, the argument isn't really about multiplication, but about math education. The argument isn't really about whether multiplication is repeated addition - it's about whether or not we should teach kids to understand multiplication as repeated addition. And that's a tricky question, because the answer is both yes and noe.
Is multiplication repeated addition? Sometimes, it is. But multiplication isn't just repeated addition. It includes cases where it makes sense to talk about it as repeated addition, and also cases where it doesn't.
What's exponentiation? Is it repeated multiplication? Sometimes. And sometimes it isn't. Try to give me a simple definition of exponentiation, which is understandable by a fifth or sixth grader, which doesn't at least start
by talking about repeated multiplication. Find me a beginners textbook or
teachers class plans that explains exponentiation to kids without at least starting with something like "52=5×5, 53=5×5×5."
With respect to multiplication, it's the same question, only with even younger kids: how do you explain multiplication to a third grader? How can you start to tell a kid about 2×2=4 and 2×3=6 without showing them that 2×2 = 2+2, and 2×3 = 2+2+2?
Multiplication isn't really a simple thing. What mathematicians mean by multiplication is, roughly, one of the two fundamental operations over the field of real numbers. Outside of the realm of abstract math, multiplication actually has
multiple meanings, which each work in different contexts. But they're all concrete
applications derived from the fact that multiplication is the second field operation
in the field of real numbers.
But how are you going to explain that two a third grader?
Just think of one of the classic word problems that every kid sees in second or third grade when they start doing multiplication. Every kid in class has three apples; so how many apples does the class have?
When you're using that problem, repeated addition makes excellent sense. It also
matches the mechanics of what the kids are doing. So it's a good intuitive way to
get them started on understanding multiplication. It's not the whole picture - but it's an initial intuition that provides some concrete handle to grab on to.
Of course, pretty soon, you have to break that intuition at least a little bit - because there are plenty of places where repeated addition just doesn't really make sense. Look at the figure over to the side. There's a triangle with a base five
inches long, and it's two inches high, with the highest point being three inches in. What's the area of that triangle? 1/2 base×height, in square inches. How can you describe that by repeated addition?
For the triangle, you can do a geometric explanation of multiplication. The two numbers being multiplied are the sides of a rectangle, and multiplying is creating the area inside the rectangle. You can use that intuition to explain the area of a triangle, by showing how to create a rectangle by cutting the triangle into pieces, and re-arranging them. That gives you a geometric intuition about multiplication.
But neither of those is particularly good for explaining how multiplication can tell you what 3/5ths of $25 is.
So the real question isn't "Is multiplication repeated addition?". The answer to that is "sometimes". The real question is "How do we introduce multiplication to children?"
Professor Devlin doesn't have a good answer for that - and in fact, he weasels out of answering it entirely, which really bugs me. After a long argument about how it's all wrong to teach kids to understand multiplication as repeated addition, and lecturing teachers on how the way that they're teaching is all wrong, he wimps out and says, in essense, "But I don't know anything about teaching, so I can't tell you the right way to do it. All I can do is tell you that you're doing it wrong."
So what are teachers supposed to do? Professor Devlin is very forceful in telling teachers what to do: the last line of his article is: "In the meantime, teachers, please stop telling your pupils that multiplication is repeated addition." But he won't tell those teachers what they should teach their pupils.
You can't tell a teacher to change the way that they're teaching math without
giving them any clue of what the right way to teach it is. What happens in a classroom if the teacher stops using repeated addition to explain multiplication? One of two things will happen. Either the teacher will switch to a different, and
equally incorrect intuition about what multiplication means; or they'll do away with trying to provide any intuition at all.
The right answer is to say that simple multiplication can be understood intuitively in terms of repeated addition. Teachers should do their best to be
clear that it's just an intuition, not the full meaning. Ideally, they should show
multiple ways of understanding it, so that students understand that no one intuition about multiplication is the whole truth. But given a choice between teaching children no intuition, and teaching them a pretty good beginners intuition, I'll take the latter.