Via Mark Chu-Carroll, I just finished reading this article by mathematician Keith Devlin. He writes:

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.

Reminds me of a joke.

A man facing a mid-life crisis sells everything he has to finance a trip to Tibet. He seeks the counsel of the High Lama, certain that the Holy One will tell him the meaning of life. His friends and family implore him not to do anything so foolish, but he ignores them, certain that he is doing the right thing. His travels to Tibet are a litany of trials. Bad weather, flight delays, passage through war zones, attacks by wild animals, but he perseveres. He *knows* the wisdom of the Lama is worth any sacrifice. Finally he enters the chamber of the great Lama and says, “Holy One! What is the meaning of life?” The Lama strikes a dramatic pose, and after a moment’s thought says, “Life, is like a tree limb!” Then he falls silent. The man is horrified. Greatly annoyed he says, “After all the trials, the ridicule, the personal sacrifice, you tell me that life is like a tree limb?” The Lama gives him a puzzled look. He says, “You mean it isn’t like a tree limb?”

That was my initial reaction to Devlin’s statement. It’s news to me that multiplication isn’t repeated addition. It certainly is that, *sometimes*. Picking up where the last quote leaves off:

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.

I still don’t see the problem. When teachers tell elementary school students that multiplication is repeated addition, they are not making a philosophical statement about the nature of multiplication generally. They are giving instruction on how to evaluate problems involving the multiplication of natural numbers. The statement “Multiplication is repeated addition,” captures something very important about multiplication, and it is something that young children, not yet ready for abstract reasoning, can understand.

If there are any elementary school mathematics teachers reading this, I’d be curious to know if Devlin’s fear has any merit. Do students feel they have been lied to upon learning that multiplication of fractions is different in fundamental ways from multiplication of natural numbers? Do students come away from their early lessons thinking that numbers are just an additive system? I’d be very surprised if that were the case.

If your whole world is positive integers, then you will never go wrong thinking that multiplication is repeated addition. When you enlarge your world, why should it be confusing that the old concepts have to expand as well? Telling students that multiplication of natural numbers can be understood in terms of addition in no way suggests that it is not also a full-fledged operation in its own right, or that it serves different purposes from addition.

So how do we answer the question in the title of this post? Abstractly speaking, multiplication is one of the basic operations that define the algebraic structure known as a “ring.” Rings are required to have two binary operations. One of these operations is called addition, and it is required to satisfy certain axioms. Specifically, we require that addition be commutative, associative, possess an identity (think 0 in the integers), and possess inverses (think negative numbers, again in the integers). Multiplication is a second operation that is required only to be associative, and to interact with addition via the distributive property.

The mechanics for carrying out these operations will depend on the specific ring you are studying. Multiplication of integers is one thing, matrices something different, and continuous functions something else still.

Regardless, if we are to give a definition of what multiplication really is we would have to say that it is an abstract binary operation on a set that satisfies certain axioms.

There was a time when people seriously thought it was appropriate to explain this sort of thing to young students. Devlin makes it clear he has no sympathy for that approach:

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.

Quite right.

Having read Devlin’s column several times, I still have no idea what teachers are supposed to be telling their students. Devlin seems to have had some teaching experiences very different form my own:

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.

Really? Exponentiation *is* repeated multiplication, so long as we are talking about positive integer exponents. Personally I have never found it difficult to explain to students that if you wish to discuss non-positive-integer exponents, you have to alter your notion of exponentiation.

I think what Devlin is getting at here is that there is a distinction between understanding conceptually what mathematics is all about on the one hand, and being able to carry out the mechanics of solving actual problems on the other. When teachers say, “Multiplication is repeated addition,” or “Exponentiation is repeated multiplication,” they are addressing the mechanical aspects of the subject. Devlin’s concern, I think, is that these mechanical considerations sometimes transgress their boundaries, and get improperly applied to the conceptual side of things. This makes it difficult to give students a broader understanding of what these operations are really all about.

I do not believe this is correct. I think the reason students have so much trouble with abstract mathematics has nothing to do with conceptual barriers we put in their way by poor arithmetical pedagogy. Rather, they find these abstractions difficult because they *are* difficult. They require a style of thinking that simply does not come naturally to people.

Ideally we want to balance mechanical and conceptual understandings of mathematics. Both are important, and both have a big roll to play in a proper mathematical education. But after many frustrating years of trying to teach calculus to freshmen, I’ve come to the conclusion that conceptual understanding is vastly overrated. Give me robots who can carry-out basic arithmetical and algebraic operations with quickness and accuracy and I’ll be happy. If they have strong mechanics there is a platform for teaching the concepts. Without strong mechanics there is little hope.

I don’t know a whole lot about how mathematics gets taught in elementary and middle school. The bits and pieces I’ve picked up on the street suggest to me that things have gone too far in the direction of conceptual understanding, without enough emphasis on mechanics. The irony is that in calculus classes things have probably gone too far the other way. We spend a lot of time with the mechanics of cranking out derivatives, learning techniques of integration, and grinding away at Taylor series, which makes it difficult to make students take a big picture approach to the subject.

For elementary and middle schol students I want proficiency in arithmetic and algebra, even if that proficiency is entirely at the level of mindless repetition. (I would prefer deep conceptual understanding too, of course, but I will take what i can get.) For college students not specifically majoring in mathematics or science, I want them to be able to explain what calculus is all about, what sorts of problems it addresses, and what it’s big ideas are, but I don’t care so much whether they remember the formula for integration by parts. Likewise for other branches of higher mathematics.