[Photo by SuperFantastic.]
Keith Devlin’s latest article, It Ain’t No Repeated Addition, brought me up short. I have used the “multiplication is repeated addition” formula many times in the past — for instance, in explaining order of operations. But according to Devlin:
Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not.
I found myself arguing with the article as I read it. (Does anybody else do that?) If multiplication is not repeated addition, then what in the world is it?
Mathematician vs. Teacher
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.
What sort of undoing? I mean, the “repeated addition” formula works with algebra, too: . Why can’t we use that as a definition?
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, yeah. “Repeated addition” works fine as a model for , and for , and even for . But we have to use a different model to make sense of or . And if the model doesn’t work universally, then it certainly cannot be used to define the operation.
The teacher concedes: Multiplication ain’t repeated addition.
So How Shall We Teach It?
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… 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.
Huh? I am supposed to tell my students these are “just things you do to numbers”? Do it because I’m the teacher, and I say you have to, and someday you will see that it makes sense?!
No, that can’t be what Devlin means. I think. Although I could be wrong. He wrote in an earlier article:
I think many mathematical concepts can be understood only after the learner has acquired procedural skill in using the concept. In such cases, learning can take place only by first learning to follow symbolic rules, with understanding emerging later, sometimes considerably later.
But for now, back to the multiplication article. Devlin gives us an example of each operation…
Addition Is “And”
For example, adding numbers tells you how many things (or parts of things) you have when you combine collections.
The basic process of addition is combination:
Or, to put it graphically:
Multiplication Is “Of”
Multiplication is useful if you want to know the result of scaling some quantity.
The basic process of multiplication is scaling, as in definition 2:
v. scaled, scal·ing, scales (v.tr.)
1. To climb up or over; ascend: scaled the peak.
2. To make in accord with a particular proportion or scale: Scale the model to be one tenth of actual size.
3. To alter according to a standard or by degrees; adjust in calculated amounts: scaled down their demands…
Or, to put it graphically:
It makes sense to scale a model up or down. We can easily have a fraction of a unit. If we can put an irrational number on the number line, then we can use it as a scale factor. No problem there.
The “Aha!” Factor
And the diagrams show something else. We can see how repeated addition of the same number “morphs” into multiplication of a unit. Or as Devlin wrote:
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?
But I Still Have a Problem
Devlin goes on:
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?
So now, I need a simple way of explaining exponents to my middle school students, without falling back on “repeated multiplication.”
Edit: A Comment from Devlin
Keith Devlin was kind enough to read this article and offer a suggestion. He pointed out that the diagram I drew for multiplication still strongly implies the repeated addition of an integral number of units.
I suspect you’d need something dynamic to show scaling by an arbitrary amount, eg. a continuous volume knob or slider for a radio. With a fixed diagram, you are going to have to use some basic unit of measurement, and then your diagram will turn out to be repeated addition of that unit.
He is right, but I am not quite sure how to draw that. The basic idea of scaling is that there are two measurements: the real world and the scale model.
- Imagine a fixed number line, and mark off the distance from 0 to whatever we are multiplying. That is our unit.
- Then imagine a stretchable number line, expanded or contracted to make the interval from 0 to 1 match the size of our unit. That sets the scale.
- Now, find on the scale line the factor we want to multiply by. Whatever number on the real line matches that will be our answer.
In place of a dynamic drawing, I offer this modification of my original diagram:
What do you think: Does this communicate the idea of multiplication better than the first drawing? Can anyone suggest another way to show it?
Update: Still Wondering About Addition and Multiplication
I continue to revise my thinking on this subject. Check out my new post: