June 2008
It Ain't No Repeated Addition
In my column for September 2007, which was titled What is conceptual understanding? I remarked that I wished schoolteachers would stop telling pupils that multiplication is repeated addition. It was little more than a throwaway line, albeit one that I feel strongly about. I put it in to provide a further illustration for the overall theme of the column, to indicate that there are examples beyond the ones I had focused on. In the intervening months, however, I've received a number of emails from teachers asking for elaboration. Their puzzlement, they make clear, stems from their understanding that multiplication actually is repeated addition.If ever there were needed a strong argument that professional mathematicians need to interest themselves in K-12 mathematics education and get involved, this example alone should provide it. The teachers who contact me do so because they genuinely want to know what I mean, having been themselves taught, presumably either in schools of education or else from school textbooks, that multiplication is repeated addition.
Let's start with the underlying fact. 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.
How much later? As soon as the child progresses from whole-number multiplication to multiplication by fractions (or arbitrary real numbers). At that point, you have to tell a different story.
"Oh, so multiplication of fractions is a DIFFERENT kind of multiplication, is it?" a bright kid will say, wondering how many more times you are going to switch the rules. No wonder so many people end up thinking mathematics is just a bunch of arbitrary, illogical rules that cannot be figured out but simply have to be learned - only for them to have the rug pulled from under them when the rule they just learned is replaced by some other (seemingly) arbitrary, illogical rule.
Pretending there is just one basic operation on numbers (be they whole numbers fractions, or whatever) will surely lead to pupils assuming that numbers are simply an additive system and nothing more. Why not do it right from the start?
Why not say that there are (at least) two basic things you can do to numbers: you can add them and you can multiply them. (I am discounting subtraction and division here, since they are simply the inverses to addition and multiplication, and thus not "basic" operations. This does not mean that teaching them is not difficult; it is.) Adding and multiplying are just things you do to numbers - they come with the package. We include them because there are lots of useful things we can do when we can add and multiply numbers. For example, adding numbers tells you how many things (or parts of things) you have when you combine collections. Multiplication is useful if you want to know the result of scaling some quantity.
You don't have to use these applications, but both are simple and familiar, and to my mind they are about as good as it gets in terms of appropriateness. (I do think that you need to present simple everyday examples of applications. Teaching a class of elementary school students about axiomatic integral domains is probably not a good idea! This column is not a rant in favor of the "New Math", a term that I use here to denote the popular conception of the log-ago aborted education reform that bears that name.)
Once you have established that there are two distinct (I don't say unconnected) useful operations on numbers, then it is surely self-evident that repeated addition is not multiplication, it is just addition - repeated!
But now, you have set the stage for that wonderful moment when you can tell kids, or even better maybe they can discover for themselves, this wonderful trick that multiplication gives you a super quick way to calculate a repeated addition sum. Why deprive the kids of that wonderful piece of magic?
[Of course, any magic trick loses a lot once you see behind the scenes. In the very early days of the development of the number concept, around 10,000 years ago, there were only whole numbers, and it may be that the earliest precursor of what is now multiplication was indeed repeated addition. But that was all 10,000 years ago, and things have changed a lot since then. We don't try to understand how the iPod works in terms of the abacus, and we should not base our education system on what people knew and did in 8,000 B.C.]
Mathematics is chock full of examples where something that is about A turns out to be useful to do B.
Exponentiation turns out to provide a quick way to do repeated multiplication - wow, it's happened again! Is this math thing cool or what!
Anti-differentiation turns out to be a quick way to calculate an integral. Boy, is that deep!
I can just hear some pupils wondering, "Hey, how many more examples are there like this? This is really, really intriguing. It all seems to fit together. Something deep must be going on here. I've gotta find out more."
I assume the reason for the present state of affairs is that teachers (which really means their instructors or the writers of the textbooks those teachers have to use) feel that children will be unable to cope with the fact that there are two basic operations you can perform on numbers. And so they tell them that there is really only one, and the other is just a variant of it. But do we really believe that two operations is harder to come to terms with than one? The huge leap to abstraction comes in the idea of abstract numbers that you can do things with. Once you have crossed that truly awesome cognitive chasm, it makes little difference whether you can do one abstract thing with numbers or a dozen or more.
Of course, there are not just two basic operations you can do on numbers. I mentioned a third basic operation a moment ago: exponentiation. University professors of mathematics struggle valiantly to rid students of the false belief that exponentiation is "repeated multiplication." Hey, if you can confuse pupils once with a falsehood, why not pull the same stunt again? I'm teasing here. But with the best intentions of drawing attention to something that I think needs to be fixed.
And the way to fix it is to make sure that when we train future teachers, and when authors write, or states adopt, textbooks, we all do it right. We mathematicians bear the ultimate responsibility here. We are the world's credentialed experts in mathematical structures, including the various numbers systems. ("Systems" here includes the operations that can be performed on them.) Our professional predecessors constructed those structures. They are part of our world view, things we mastered so long ago in our educational journey that they are second nature. For too long we have tacitly assumed that our knowledge and understanding of those systems is shared by others. But that isn't the case. I have a file of puzzled emails from qualified teachers that testifies to the gap.
I should end by noting that I have not tried to prescribe how teachers should teach arithmetic. I am not a trained K-12 teacher, nor do I have any first-hand experience to draw on. But the term "mathematics teaching" comprises two words, and I do have expertise in the first. That is my focus here, and I defer to others who have the expertise in teaching. The best way forward, surely, is for the two groups of specialists, the mathematicians and the teachers, to dialog - regularly and often.
In the meantime, teachers, please stop telling your pupils that multiplication is repeated addition.
