Is multiplication repeated addition? The answer may seem obvious to you, but I assure you there’s a raging debate inside of math education about whether it’s true and how this should be taught.

This article describes some aspects of the current debate and presents some high-level explanations of the concepts involved.

A Brief History

Keith Devlin at Devlin’s Angle has written many times about how multiplication is not repeated addition and how he thinks it is a detriment to teach children this falsehood.

I strongly suggest going and reading all of his articles before reading this to see a full description of his viewpoint. I’m going to summarize it here, but some might find it disingenuous. And if you don’t see it with your own two eyes, you might think I’m just making up this viewpoint.

This is not supposed to be a direct response to him. All mentions to him and his articles are due to him having the most thorough and prominent articles on the topic.

I assure you that many mathematicians and educators are on his side. This is not an attack on him despite using his name as a convenient stand-in for a certain form of the argument.

A Preliminary Overview

In the linked article, Devlin makes the analogy that he can take a car or a bike to work. The end result happens to be the same, but we shouldn’t pretend like the process is the same.

I agree.

Just because two processes happen to produce the same results does not mean we can conclude that they are the same process. The main claim is that multiplication and addition are fundamentally different, yet related, operations on numbers.

• Addition is an operation that corresponds to combining in the real world.
• Multiplication is an operation that corresponds to scaling.

Proponents of this viewpoint want to claim multiplication happens to give the right answer to repeated addition as a useful tool, but it’s wrong to define multiplication as repeated addition.

My point of view is that this is false. Repeated addition and multiplication don’t just happen to come out with the same answer. They come out with the same answer because they are actually the same!

We’ll do a deep dive into why this is the case in the next main section.

An aside on teaching

Before we begin I want to give a disclaimer on the research about how to teach this topic in K-12 education.

I don’t teach at that level, so I can’t comment on whether or not I think the results are valid. But it is amazing how much bias — yes, even among professional research scientists — happens in experiments involving observing and quantifying how well children understand things.

If teaching children in a certain way increases understanding, then, by all means, go ahead. I’m no expert on early childhood education.

The idea of how to teach the concept has little to do with whether or not multiplication is repeated addition and that is what will be addressed for the rest of this article.

What is Multiplication?

First, I agree with Devlin on one point. We can abstractly define multiplication to be a basic operation that is separate from addition. He never really makes this precise, so let’s do that first so we see exactly why they are different operations and why he is saying we confuse the two.

Let’s say we have a set:

The sole purpose of this is because our notation is causing the problem. We should just think that aⱼ is the number j.

We can define an addition and a multiplication as abstract arbitrary rules that satisfy certain axioms. Note that the multiplication operation a priori has absolutely nothing to do with the addition (and even a posteriori in most cases).

• We’ll use the standard plus sign for addition: aᵢ+aⱼ.
• We’ll use the × multiplication sign for multiplication: aᵢ × aⱼ.

Yes, we’re going to use that large clumpy symbol to try to avoid future confusion. It’s worth noting here that S could be any set. It doesn’t have to be “numbers” like ℕ, ℤ, ℚ, ℝ, or ℂ.

When a set is given addition and multiplication like this that satisfies a bunch of the usual properties of addition and multiplication, it’s called a ring in math.

In arbitrary rings, the multiplication and addition actually are separate things. There is not an interpretation of one as repeating the other. When people say multiplication is not repeated addition, this is what they mean!

Despite the multiplication being completely arbitrary, notice that we can still define something that we’ll call additive multiplication.

We’ll denote this with a dot. In other words, 3⋅aⱼ=aⱼ+aⱼ+aⱼ. No matter what integer we pick we can define this additive multiplication. Since our set S might not even have the number 3 in it, we see very clearly that this additive multiplication is not the same as multiplication.

But there is no reason to stop there.

We can define (1/3)⋅ aⱼ to be some element of the set, call it aₖ, with the property that aₖ+aₖ+aₖ=aⱼ.

We can even do one at a time to extend that: (5/2)⋅ aⱼ means (1/2)⋅ (5⋅ aⱼ) or maybe even better it is adding aⱼ “two and a half times”: aⱼ+aⱼ +(1/2)⋅ aⱼ.

We can extend this to the negatives.

I won’t worry about the interpretation of this because Devlin admits negatives pose problems in any interpretation. In algebra, we’d call this giving our set a ℚ-module structure since we’ve defined a multiplication ℚ× S→ S.

It doesn’t ever even mention the multiplication, so indeed it is completely separate.

Is Multiplication Repeated Addition?

Let’s cautiously reintroduce normal notation.

We have these two arbitrary operations on the integers: addition 3+2=5 and multiplication 3×2=6.

As was pointed out already, we also have this other thing 3⋅2=6 and it is very good to point out that this other thing need not agree with the multiplication rule we’ve defined.

The reason we don’t usually notice that they are two different things is that when we do the dot and think of repeated addition we don’t usually specify with different symbols. In fact, the number 3 itself is being used in two different ways!

The times operator goes ×: S×S → S. It takes two elements of the set and produces a new element of the set.

The dot operator (repeated addition), goes ⋅: ℕ×S → S (and as we saw, ℕ can be extended once we have that much). It takes a natural number and an element of the set and produces an element of the set.

When the natural numbers are a subset ℕ ⊂ S, as in the case of S=ℤ or ℚ or ℝ, we are accidentally reusing symbols in different ways.

Unfortunately, when we teach addition and multiplication, we are implicitly using S=ℤ or ℕ. This means 3 is an element of the set on which we defined × multiplication.

Now you see why I originally went through the trouble of using the notation aⱼ to keep things separate at first.

I applaud the people who claim multiplication is not repeated addition at this point for pointing out the conflation of these two separate things (although most of these people never tell us this was the real reason for the confusion).

The Problem

Let’s start pointing out some issues with defining multiplication in terms of “scaling” instead of repeated addition.

First off, most articles that claim multiplication is not repeated addition start by reminding us that the concept of multiplying by repeated addition only works for positive whole numbers.

Hold on just a second.

Now that we’ve figured out what’s really going on we see this is not true at all. Our set could be anything. Suppose S is the real numbers, ℝ.

We still have this ℚ-module structure on it. So it makes perfect sense in the repeated addition paradigm to do (3/2)⋅(-π).

This is a far cry from only positive integers. We just used rational numbers, irrational numbers, and negatives and didn’t run into any problems.

In fact, even the alternatives usually proposed will have issues with real numbers and the way it is overcome is exactly the same way it can be overcome in this situation.

Multiplication as Scaling

The second major problem comes from using “multiplication as scaling” as the definition to get around the repeated addition concept.

Let’s dig into the concept of scaling. We’ll use the example given in Devlin’s article instead of making up my own. I’ll just quote verbatim the relevant section:

Take a piece of elastic, and tie two knots in it, one near each end. Ask the child to measure the distance between the knots. Suppose it turns out to be 5 inches. Now, as the child watches, slowly stretch the elastic until the distance between the two knots is 10 inches. Get the child to measure it again. Now ask the child to write down a mathematical description of how the new length depends on (is related to) the original length. I would hope the child writes down

10 = 2 x 5

In more general terms, what you did was double the length, or, as an equation:

new length = 2 x old length

I would be very surprised if the child wrote down

10 = 5 + 5

corresponding to

new length = old length + old length

and if he or she did, you would have your work cut out trying to put them right before they have serious trouble in the math class. Sure, these addition equations are numerically correct. But so what? What you have just shown the child when you stretched the elastic has absolutely nothing to do with addition and everything to do with multiplication.

What horrifies me here is that this little example actually shows exactly the opposite of what he claims.

He thinks if the child says the new length is the old length plus the old length they are in serious trouble, but I’ll rephrase the example to show that if they think in any other terms they will be in big trouble.

What exactly does it mean to “double”?

Really. Seriously. I want to know. Take some time to think about this yourself.

Here is what I mean when I use it. The number 5 is causing problems again because it gives us something tangible in which to apply our arbitrary multiplication rule 2×5.

Take a piece of elastic. Take a stick of unknown length. Tie knots at where the ends of the stick are. Stretch the elastic.

When using the stick to see how far the elastic is stretched you find that you use the stick exactly twice. In other words, the new length is the old length plus the old length.

This isn’t semantics or a rhetorical trick. Now that we can’t accidentally apply our multiplication rule, we see that when we use the word double, the only possible meaning it can have is 2⋅(length).

The dot operation is what we mean when we double!

Now we’ve come to the amazing part of all of this. The people who advocate for multiplication as not repeated addition want us to define the × multiplication operator to mean “scaling” which is a priori completely separate from the “additive multiplication” of the dot.

But when we go to define what we even mean by scaling we find that the only conceivable definition is repeated addition.

This is why the two agree! They are actually the same by definition even for people who want to use a scaling notion.

If the child tries to tell you that the elastic is 2×(length), in which they mean the length is an arbitrary multiplication rule unrelated to (length) + (length), then obviously the child is going to be in way more trouble because they have completely missed the point.

Devlin seems to have worked out that in abstract math land the “additive multiplication” and the × multiplication are very often different.

From this, he seems to be concluding that because of this they are different for the number systems we teach children.

This is not correct.

In principle, they could be different, but in reality, the dot defines a perfectly valid multiplication and we take that to be the definition (even abstractly) of the times multiplication.

They don’t “just happen to give the same answer.” They are the same. Even if we take multiplication to mean “scaling,” when we go to figure out what we mean by that word, we come back to repeated addition.

The Peano Definition

If you thought the debate ended with the above, you’re mistaken. A new tactic for claiming multiplication is not repeated addition surfaced on how multiplication is defined.

The debate shifted to more abstract topics about how to rigorously define multiplication within the axioms of Peano arithmetic using recursion:

f: ℕ× ℕ→ ℕ

The claim is that since the “definition” of such a function as repeated addition does not yield a function on the whole natural numbers it cannot possibly be the definition.

This means that, in the strictest sense, the definition is not repeated addition because it is this other recursively defined function. Of course, that is true.

Repeated addition only works for a finite stage and you must rely on something else to get the full function. The problem comes from what we mean by “definition.”

I’ll forgive you if this semantic game is intolerable and you decide to leave now.

Devlin wants to say that because it is necessarily the case that to define an honest multiplication function axiomatically in Peano arithmetic we must make some part of the definition that is not repeated addition, then we must say that the definition is not repeated addition.

This should be a completely absurd idea to a mathematician, and since the debate has veered into this territory, it’s important to disentangle a few concepts.

The Definition vs Proof of Existence

This argument conflates the two mathematical notions of definition and existence.

You can make a definition of anything you want in math, but that doesn’t mean it exists. You then need to prove that such a thing exists.

I claim that the above “not repeated addition” definition of a function is actually the proof of the existence of a multiplication function and not a definition.

Recall that the context of this conversation is all about whether or not we should teach children that multiplication is repeated addition.

The whole point of teaching children multiplication is precisely so that they can apply it to real-world situations. The only sense in which multiplication can be applied to real-world situations is if we know that it produces the same answer as repeated addition.

Even the analogies about stretching and scaling being a continuous rather than a discrete process only make sense after interpreting it as repeated addition.

Now that we’ve established the difference between definition and existence, I’ll decompose what is going on to show that multiplication is repeated addition (by definition).

Recall that definitions in mathematics are technically biconditional statements.

Definition: A function m:ℕ×ℕ→ ℕ is multiplication if and only if it satisfies m(n,k)=k+k+… +k repeated n times for any choice of n,k ∈ ℕ.

Note that this is not only the definition that will tell us the multiplication function is useful but a priori we don’t actually know that such a function exists.

As Devlin points out, we can use recursion to prove that such a function exists. The definition is the one I give above. If we try to conflate the existence and definition, then we have no idea if the recursively defined function can be used to compute something in reality.

For all we know, that nice recursive definition could yield f(n,k)=2 for all inputs. Are we going to call that multiplication? Of course not!

This might look like a flippant thing to say, but it illustrates the necessity of separating definition from existence and is precisely the point of disagreement in this whole debate.

Now, to try to summarize briefly, in order to have a function defined on ℕ×ℕ it must be defined recursively without reference to repeated addition. I agree completely.

I just disagree that the definition of such a function should be considered the definition of multiplication. The actual definition should be the one I gave above. The proof that such a function exists is the recursive construction of such a function.

Thus, multiplication is repeated addition, unless we are going to use a non-standard idea of what it means to “be” something.