Recently I have been spending some time browsing ScienceBlogs, and there’s one post on which I particularly feel I have something to say. This is the July 25 post on Good Math Bad Math about whether multiplication should be explained as repeated addition, even though it’s more than that when you start dealing with fractions.
I have a bias. You see, when I was seven or eight years old, having been taught that multiplication is essentially repeated addition, I came up with the idea of exponentiation all by myself. So I say, of course multiplication should be taught as repeated addition, because if it hadn’t been then I wouldn’t have had that creative insight. Later on, when I was taught that other people had already thought of exponentiation, I came up with the idea of “operational algebra”, which was my term for a scheme in which numbers are assigned to operations so that addition is operation #1 and operation #(p+1) with integer operators is repeated operation #p. I had fun playing with this idea. I didn’t make any earth-shattering mathematical discoveries, but who cares? A teenager had fun! Surely that’s significant enough. (In the future, I may tell you more about what I did with this idea, but it was a long time ago and the difficulty is remembering.)
Of course, when you start dealing with fractions, there’s more to multiplication than repeated addition and there’s more to exponentiation than repeated multiplication. But I say, that’s a wonderful opportunity to teach students about the beauty of mathematics, how it’s full of patterns that just seem to fall into place. One and a half to the power of one and a half doesn’t make much sense if you rephrase it as one and a half multiplied by itself one and a half times, and yet, with a bit of very simple bit of algebraic trickery, you can show that it does have a meaning after all, and that meaning is perfectly consistent with the original definition, so that in a manner of speaking you can multiply something by itself one and a half times. It’s all very beautiful – to find that you can define something in terms of integers and have it turn out, unexpectedly, that it can be re-defined in a consistent and simple way to work with all numbers. But appreciating that beauty begins with the definition in terms of integers, so absolutely I say, multiplication should be taught as repeated addition, and exponentiation as repeated multiplication, because those definitions are the foundations of the lookout tower from which the deeper patterns can be seen.
A related post on this blog: “Calculator“.