Devlin's Right Angle, Finale

It seems ironic that Devlin, who belongs to a group that "finally learned to sidestep that unanswerable 'What is it?' question," is now telling us what something is not--that multiplication is not repeated addition.

Not only is he confidently answering an apparently unanswerable question here, he's also dispensing advice based on that answer: "please stop telling your pupils that multiplication is repeated addition." This would seem to break—if not fundamentally, then at least technically—what Devlin refers to as a kind of 100+-year-old rule requiring mathematicians to never again think about or speak about the "what is it" that they're doing.

But it only seems that way.

However, even though I am not a professional mathematician, my saying that Devlin is right would only make me an accomplice to this unlawful behavior. Therefore, while I won't specifically say he's right, I will say—and have said—that he has the "right angle" on this issue.

Walking to Paris

From what I've seen, when people argue that it is okay to teach kids that multiplication is repeated addition, they generally try to do so in two ways (at least mathematically). They either argue that multiplication is defined as repeated addition, using the Peano axioms--or something similar—or they argue that multiplication and repeated addition both get the same results.

The second argument is undoubtedly true, but being so is not a justification for telling kids "multiplication is repeated addition"—even when you're talking about just the counting numbers (1, 2, 3 …). Yes, the multiplication function y = 3p (or 3 × p) and the repeated addition function y = p + p + p both have the same "functional value" when you plug in a number for p. (For example, 3 × 4 = 12, and 4 + 4 + 4 = 12.) But we can rewrite the repeated addition function above as y = p &minus (^-p) &minus (^-p). Shall we take this as a justification to tell kids that "multiplication basically reduces to the repeated subtraction of negatives"?

Even the simple multiplication function y = ax can be expanded to


If we plug in any counting number (1, 2, 3, …) for a and x into both equations, both will have the same "functional value." Does this mean that now we are justified in calling upon factorials, square roots, and squares to explain or define multiplication?

Of course not. Just because multiplication and repeated addition get the same results does not make them the same operation.

The first argument I mentioned—that multiplication is defined as repeated addition—is trickier, in part because it seems to depend somewhat on what text you're reading. I quoted from the text I have in front of me at the moment here. Ultimately, whether or not you agree or disagree with the idea that Peano—or anyone else—essentially defines multiplication as repeated addition (I would disagree), this definition is all but useless for solving problems.

For example, most sixth or seventh graders (and certainly some students in lower grades) would know that an equation like p + p + p + p + p = 85 can simply be rewritten as 5p = 85. Then, to solve the equation, they would know to divide (the inverse of multiplication) both sides by 5 to find p = 17. But it is impossible to solve that equation using only repeated addition (and its inverse, subtraction)—unless you guess or use a model or something like that. If all you know is repeated addition and subtraction, then you could not even get started by rewriting the expression p + p + p + p + p as 5p, because you simply wouldn't know to do that.

There are a lot of things in mathematics that you can "basically reduce down to" or "think about as" repeated addition. Theoretically, a plane trip somewhere is the exact same thing as a bunch of repeated steps to get you to the same destination. But if you're in the U.S., and that destination is Paris, they most certainly aren't the same. You can argue all day long that multiplication as repeated addition is "rigorously defined" or that the historical development of multiplication ran through repeated addition and then was extended to other number systems. Even if all that is true, understanding multiplication as repeated addition will leave you mathematically impotent in the modern world.

Multiplication is fundamentally different from addition (and of course repeated addition) because it helps us understand fundamentally different things. Things change. Get over it.

Mathematics Is Not a Matter of Opinion

From various comments on this issue I can see that a big stumbling block for many people is their assuming that (a) mathematics never ever changes, (b) that mathematics is nothing more than what is or has ever been written down, and (c) that mathematicians thinking about the subject today only have credibility by being entirely consistent with everything that has happened in mathematics for the past thousand years--no matter how unuseful it might be.

As far as (a) is concerned, it seems popular to argue that every bit of mathematics that was ever thought of is simply an "extension" of some earlier mathematics. Obviously, this is complete rubbish. One would be hard-pressed, I think, to call Greek mathematics an "extension" of Babylonian mathematics unless one were using the word extension very, very loosely.

The assumptions in (b) and (c) are interrelated. For (b), I'm reminded of a great quote from Uri Leron:

According to the algebraic image of functions, an operation is acting on an object. The agent performing the operation takes an object and does something to it. For example, a child playing with a toy may move it, squeeze it, or color it. The object before the action is the input and the object after the action is the output. The operation is thus transforming the input into the output. The proposed origin of the algebraic image of functions is the child's experience of acting on objects in the physical world. . . . Inherent to this image is the experience that an operation changes its input—after all, that's why we engage it in the first place: you move something to change its place, squeeze it to change its shape, color it to change its look.

But this is not what happens in modern mathematics or in functional programming. In the modern formalism of functions, nothing really changes! The function is a "mapping between two fixed sets" or even, in its most extreme form, a set of ordered pairs. As is the universal trend in modern mathematics, an algebraic formalism has been adopted that completely suppresses the images of process, time, and change.

Now, certainly, the idea of a function didn't start out like this. It was almost certainly more informal, incorporating concepts like "process, time, and change." But eventually, as Leron notes, those concepts were stripped away from the pure mathematical notion of a function. So where are the axioms and theorems written down throughout the history of mathematics that TELL US as much? Where in the theorems and axioms of Dedekind or Peano or Landau can I directly find the notion that functions (or anything) really don't include the concept of "process, time, and change"?

Well, I'll tell you where. Nowhere.

But this idea that, for example, 5 &minus 2 isn't really considered to be some kind of process where 2 things get taken away from 5 is FUNDAMENTAL to our understanding of mathematics. Yet, some people would argue that because it isn't written down, it must not be true. And therefore it's okay to fill our kids' heads with wrong ideas.

Which brings us to (c). This assumption--that the ideas promoted by mathematicians today MUST be completely consistent with what you learned in Kindergarten or high school or college or else they're wrong--sounds a lot like the creationist argument that everything that can be explained about modern evolutionary theory we can find in The Origin of Species.

It's just not true. Both biology and mathematics--and a lot of other fields for that matter--are growing, learning, dynamic fields. I hate to use the ethotic argument here, but I find it interesting that people who aren't professional mathematicians--say, for example, a K-12 Web site developer or an actuarial analyst--would not first suppose that they didn't know everything about mathematics and start asking questions.

Unfortunately, it's not what happened.

Even something as simple as this (from the Park, Nunes paper I referenced earlier in the series):

"Amy's Mum is making 2 pots of tomato soup. She wants to put 3 tomatoes in each pot of soup. How many tomatoes does she need?

is a multiplication problem, whereas this (again, from the same paper):

Tom has three toy cars. Ann has three dolls. How many toys do they have together?

is a repeated addition problem. They are different ideas, fundamentally. The "processes" of finding a product and finding a repeated addition sum are the same for both problems, but the ideas involved--INCLUDING THE MATHEMATICAL IDEAS--are very, very different.

Mathematics is not the notation. Mathematics is the meaning behind the notation. And much of that meaning cannot be captured by the notation itself. No doubt, 3 + 3 = 2 × 3. But multiplication and repeated addition are STILL fundamentally different.

What we are doing now in elementary mathematics education is blurring (or even erasing) this distinction when we first introduce multiplication. And that's bad math, not good math.

Connections, Connections, Connections

Devlin supposes (I think correctly) that at least one reason why people have such a problem with this idea—that multiplication is not repeated addition—is because of their desire to make mathematics tangible:

Many people feel a need to make things concrete. But mathematics is abstract. That is where it gets its strength. Multiplication simply IS NOT a generalized addition, and exponentiation IS NOT a generalized multiplication. Just as you can't really say what the number 7 IS in concrete terms - it's a pure abstraction - so too you can't say what addition and multiplication and exponentiation ARE. They are BASIC, not derived. A significant part of mastering mathematics is coming to terms with that.

I blame it on the obsession with connections:

It is obviously extremely important and useful for teachers and for curricula to try to "connect with where the students are" and to turn expertise "into building blocks for teaching." A sad sort of hilarity ensues, however, when "connection" and "building blocks" consume all thinking about mathematics education—as they have—to the extent that we completely disregard both the need for students to have a deep understanding of mathematics and the need for students to gain formal knowledge in the discipline of mathematics.

Just Accept the Challenge

But nowhere in Devlin's articles on this issue (the way I'm reading them) does he suggest that we teach multiplication without connections to repeated addition and without concrete representations. On the contrary,

(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! . . .

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.

What he's simply suggesting is that we introduce multiplication and addition as distinct operations and then come back to the connections between multiplication and repeated addition later. Perhaps he wasn't as clear as he could have been, but it seemed pretty straightforward to me.

And, given that, and the fact that he breathlessly disclaimed knowing much if anything about K-12 education, it would be silly to read his articles as though they were lesson plans and then turn round and declare either that he didn't give us enough help or that the whole thing is impossible.

My advice would be to just accept his challenge. I promise I'll do the same.

Correspondence

For those of you still holding on to the idea that while the math might be correct, exposing children to the idea that addition and multiplication are distinct operations might make their little heads explode, let me tease you with a neat little study that I referenced in an earlier post in this series. The authors are Jee-Hyun Park and Terezinha Nunes, and the title is "The development of the concept of multiplication." Here's the (chopped-up) abstract:

Two alternative hypotheses have been offered to explain the origin of the concept of multiplication in children's reasoning. The first suggests that the concept of multiplication is grounded on the understanding of repeated addition, and the second proposes that repeated addition is only a calculation procedure and that the understanding of multiplication has its roots in the schema of correspondence. . . . Pupils (mean age 6 years 7 months) from two primary schools in England, who had not been taught about multiplication in school, were pretested in additive and multiplicative reasoning problems. They were then randomly assigned to one of two treatment conditions: teaching of multiplication through repeated addition or teaching through correspondence. . . . At posttest, the correspondence group performed significantly better than the repeated addition group in multiplicative reasoning problems even after controlling for level of performance at pretest.

I should note here that, although it may read that way above, the ultimate aim of this study was not to compare the effectiveness of the correspondence and repeated addition treatments; it was to test two hypotheses about the "origin of the concept of multiplication in children's reasoning." Obviously, one of the hypotheses says that the origin is in repeated addition, and another says that it is in correspondence.

In other words, the "significantly better" performance of the correspondence group over the repeated addition group was taken by the researchers not as evidence of the superiority of the correspondence treatment, but as evidence of the fact that children begin to think about multiplication NOT as repeated addition but as a "one-to-many correspondence."

That's about it. It's been real.

Part I | Part II | Part III | Part IV | Part V | Part VI | Finale