ôô Learn what it really means to multiply two numbers and whether or not you should think of multiplication as repeated addition.

By
Jason Marshall, PhD
Episode #33

In past articles we’ve talked at length about three of the big four traditional processes used in basic arithmetic: addition (see also “How to Add Quickly”), subtraction, and division. However, for one reason or another, we haven’t yet said much about the fourth and final basic arithmetic process: multiplication. So, without further ado, today we’re going to begin taking a closer look at multiplication.

## What is Multiplication?

Along with addition and subtraction, most of us learn how to multiply two numbers together at a fairly early age. As such, I wouldn’t blame you for thinking that the question “What is multiplication?” sounds a little overly simplistic. But the truth is, it isn’t. Although we learned to multiply early on, most of us never stopped to think about what it really means. As it turns out, this meaning has been a rather hot-button issue in parts of the math education community for the past few years. And as you’ll see, the answer still isn’t exactly clear. So then—what is multiplication?

Let’s start by thinking about a simple problem like 3 x 2 (“three times two”). What does it mean? The way many of us learned to multiply in school was to think of 3 x 2 as meaning the same thing as “three of the quantity two.” By that I mean if you have a box with two rocks in it, then 3 x 2 is the total number of rocks contained in three boxes that each contain two rocks. In other words, 3 x 2 is the same as 2 + 2 + 2, which of course is 6. Aha! So we can just think of multiplication as adding some number together some other number of times, right? Multiplication is just repeated addition. That seems to make perfect sense. Or does it? Well, we’ll get back to that question in a minute.

## Multiplication as Repeated Addition on the Number Line

First, I’d like to talk about what this picture of multiplication as repeated addition looks like on the number line. So, go ahead and get that image of the number line back in your head—zero in the middle, positive integers extending indefinitely to your right, and negative integers to your left. Now, what does a problem like 3 x 2 look like on this number line? Well, imagine a stick of length 2 laying along the line (one end at zero, and the other at positive two). Since 3 x 2 is the same as 2 + 2 + 2, we need to set two more sticks of length 2 end-to-end next to the first—giving us a total length of 6. Alternatively, you can think of taking a certain sized step some number of times. For 3 x 2, if you take three length 2 steps along the number line, you’ll end up at 6—exactly as before. Perfect! It all makes sense, right?

## A Problem with Multiplication as Repeated Addition

Well, not exactly. Everything about our interpretation of multiplication as repeated addition seems to work fine, but we’ve only been working with integers. What happens with fractions? How about a problem like 3 x 1/2? Well, actually, that still works. We can think of 3 x 1/2 as 1/2 + 1/2 + 1/2, which is equal to 3/2 or 1 1/2. So where’s the problem? Well, what if we multiply two fractions? Say, 1/3 x 1/2? Uh oh. This is now a problem since is doesn’t make sense to think of adding 1/2 to itself 1/3 of a time! The interpretation of multiplication as repeated addition has broken down—it doesn’t work for all numbers.

## Multiplication as the Scaling of Numbers

Are we out of luck then? Is there an alternative meaning for multiplication that does work for all numbers? To keep things simple, let’s again start with integers—in fact, let’s again use 3 x 2 as our example. And let’s start right away by thinking of how multiplication works on the number line. So, instead of thinking of 3 x 2 as the total length of 3 sticks that are each 2 units long lined up end-to-end, let’s think of 3 x 2 as the new length that the single 2-unit long stick will have after it is stretched to be 3 times its original size. In other words, let’s think of multiplication not as repeated addition, but as a process that scales the size of a number. So, that 2-unit long stick that has been stretched to be 3 times its original length will have a new length of 6-units—2 is scaled by a factor of 3, so 2 x 3 = 6.

And here’s the great news: this type of scaling works for fractions too! Remember the problem 1/3 x 1/2 that didn’t make any sense in terms of repeated addition? Well, let’s now think of this multiplication problem as asking you to scale 1/2 to be 1/3 of its original size. Yes, that is a perfectly reasonable interpretation—it makes sense! And if you think about it in terms of the lengths of sticks on the number line, you’ll see that the answer is 1/6 (we’ll have more on how to multiply fractions in the next article).

## Is Multiplication the Same as Repeated Addition?

So, back to our original question: What is multiplication? Is it the same as repeated addition? Is it the same as scaling one number by another? Can it be both? Herein lies the controversy I spoke about earlier. Many teachers have used the idea of repeated addition to help explain the meaning of multiplication. But in June 2008, Stanford mathematician and NPR’s “Math Guy” Keith Devlin wrote a column called “It Ain’t No Repeated Addition” in which he argues against this practice. That led to a great deal of healthy debate, including a great blog post (and ensuing stream of comments) entitled “If It Ain’t Repeated Addition, What Is It?” If you’re an educator, or a curious bystander, it’s an interesting read that I highly recommend.

So what was the outcome of this debate? Well, to be honest, it’s unsettled—each side is holding to their convictions. And, from my perspective, that is fine because while the ultimate “meaning” is interesting (and no doubt mathematically important), it doesn’t change how you use the tool of multiplication in practice. It is healthy, however, to be aware that the debate exists since once you understand and can explain why it exists, you won’t be confused in the future when you find that one meaning breaks down and another is required.

## Wrap Up

Okay, that’s all the math we have time for today.

Please email your math questions and comments to mathdude@quickanddirtytips.com. You can get updates about the Math Dude podcast, the “Video Extra!” episodes on YouTube, and all my other musings about math, science, and life in general by following me on Twitter. And don’t forget to join our great community of social networking math fans by becoming a fan of the Math Dude on Facebook.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans! 