In past articles weve talked at length about three of the big four traditional processes used in basic arithmetic: addition , subtraction, and division. However, for one reason or another, we havent yet said much about the fourth and final basic arithmetic process: multiplication. So, without further ado, today were going to begin taking a closer look at multiplication.
But before we get to that, the podcast edition of this article was sponsored by Go to Meeting. With this meeting service, you can hold your meetings over the Internet and give presentations, product demos and training sessions right from your PC. For a free, 45 day trial, visit GoToMeeting.com/podcast.
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 wouldnt blame you for thinking that the question What is multiplication? sounds a little overly simplistic. But the truth is, it isnt. 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 youll see, the answer still isnt exactly clear. So thenwhat is multiplication?
Is Multiplication Repeated Addition?
Lets start by thinking about a simple problem like 3 x 2 . 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, well get back to that question in a minute.
Multiplication as Repeated Addition on the Number Line
First, Id 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 headzero 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 . 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 firstgiving 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, youll end up at 6exactly 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 weve 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 wheres the problem? Well, what if we multiply two fractions? Say, 1/3 x 1/2? Uh oh. This is now a problem since is doesnt make sense to think of adding 1/2 to itself 1/3 of a time! The interpretation of multiplication as repeated addition has broken downit doesnt 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, lets again start with integersin fact, lets again use 3 x 2 as our example. And lets 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, lets 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, lets 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|units2 is scaled by a factor of 3, so 2 x 3 = 6.
And heres the great news: this type of scaling works for fractions too! Remember the problem 1/3 x 1/2 that didnt make any sense in terms of repeated addition? Well, lets 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 interpretationit makes sense! And if you think about it in terms of the lengths of sticks on the number line, youll see that the answer is 1/6 .
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 NPRs Math Guy Keith Devlin wrote a column called It Aint No Repeated Addition in which he argues against this practice. That led to a great deal of healthy debate, including a great blog post entitled If It Aint Repeated Addition, What Is It? If youre an educator, or a curious bystander, its an interesting read that I highly recommend.
So what was the outcome of this debate? Well, to be honest, its unsettledeach side is holding to their convictions. And, from my perspective, that is fine because while the ultimate meaning is interesting , it doesnt 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 wont be confused in the future when you find that one meaning breaks down and another is required.
Wrap Up
Okay, thats all the math we have time for today. Thanks again to our sponsor this week, Go to Meeting. Visit GoToMeeting.com/podcast and sign up for a free 45 day trial of their online conferencing service.
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Until next time, this is Jason Marshall with The Math Dudes Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
