### Isn't multiplication repeated addition?

I just found out an interesting column by Keith Devlin... he tells elementary teachers to stop telling the students that multiplication is repeated addition.

Why? His point is, this idea does not carry through. As soon as the student encounters multiplication of fractions (or of decimals), it won't work. You can't think of 3/4 x 6/11 as repeated addition.

He feels it's better to portray multiplication as a scaling process: say 5 x 9 means 9 is scaled by a factor of 5. Then, students can have a true "aha" moment as they discover for themselves that you CAN use addition to find the answer to 5 x 9. But, Devlin says, they should be taught and shown the multiplication idea as a scaling process.

Now, I feel that Devlin has a point here... so since I'm constantly in the process of writing math materials for my Math Mammoth series of books, and right now I'm writing lessons on multiplying decimals for 5th grade, I took this idea just yesterday and tried to go with it.

Unfortunately, I immediately ran into problems.

Let me illustrate.

I thought, we need to illustrate the idea of scaling. So I thought, kids might know scaling from computer programs such as scaling images, or scaling maps. I was going to use a picture of a toy car: and scale it by a factor of 2, to let it be TWICE as big.

I scaled its width and length by a factor of 1.414 so as to make the AREA to be twice the original: .

But I realized, kids would feel that that's NOT twice as big... they might feel it needs to be scaled like this, doubling the width and height: . But then, of course, the area is quadrupled (which 2nd or 3rd grades wouldn't automatically know).

Right there I gave up. I SURE don't want to get them confused by doubling the width, height, or area... Later on they need to learn that IF you multiply the two dimensions by some factor

I did write for my lesson an illustration of scaling a "stick", or a line. In that one-dimensional situation we won't run into this problem. But even so, I would be extra careful of using it a lot, because surely some student will ask about scaling two-dimensional images, and then we have confusion.

I would find it more natural to present the idea of multiplication to 2nd and 3rd graders as "multiple copies", such as 2 × = .

Even our word "multiply" refers to multiple copies of the same... people and animals "multiply", we talk about multiples, etc. We use the word "times" referring to doing the

Then, when it comes to multiplication of fractions and of decimals, one has to bring in the idea of taking a part:

1/2 x 7 means 1/2 OF 7. I do not see a problem there.

Later one can tie these "two meanings" of multiplication together with the scaling idea... maybe... somehow... I just do not know myself how to do that without confusing the idea of scaling the width/height by some factor and the area being scaled by the square of that factor.

Or maybe I'm all wrong and it IS possible to use the idea of scaling images?

Maybe someone should TRY it on a few classrooms of kids and see what happens over the years.

Essentially, we can define multiplication of whole numbers (and integers) as repeated addition. We have to define multiplication of fractions in a different way - but that is not a problem. It is

Sideline... in other words, multiplication of fractions can be defined as

a/b * c/d = (ac)/(bd) which is the familiar rule. (BTW, definitions vary. That's why I can't say that multiplication of fractions would always necessarily be defined this way.)

If you have integers y and z, they can be written as fractions as x/1 and y/1, and multiplying them using the definition of

y/1 * z/1 = (yz)/(1*1) which equals yz.

Later on, multiplication of real numbers and that of complex numbers are defined still differently, as "extensions" of the idea of basic multiplication.

Other bloggers have their take, too:

If it ain't repeated addition... by Let's Play Math, Devlin on Multiplication by Rational Math Education, and Devlin's Right Angle at Text Savvy.

Why? His point is, this idea does not carry through. As soon as the student encounters multiplication of fractions (or of decimals), it won't work. You can't think of 3/4 x 6/11 as repeated addition.

He feels it's better to portray multiplication as a scaling process: say 5 x 9 means 9 is scaled by a factor of 5. Then, students can have a true "aha" moment as they discover for themselves that you CAN use addition to find the answer to 5 x 9. But, Devlin says, they should be taught and shown the multiplication idea as a scaling process.

Now, I feel that Devlin has a point here... so since I'm constantly in the process of writing math materials for my Math Mammoth series of books, and right now I'm writing lessons on multiplying decimals for 5th grade, I took this idea just yesterday and tried to go with it.

Unfortunately, I immediately ran into problems.

Let me illustrate.

I thought, we need to illustrate the idea of scaling. So I thought, kids might know scaling from computer programs such as scaling images, or scaling maps. I was going to use a picture of a toy car: and scale it by a factor of 2, to let it be TWICE as big.

I scaled its width and length by a factor of 1.414 so as to make the AREA to be twice the original: .

But I realized, kids would feel that that's NOT twice as big... they might feel it needs to be scaled like this, doubling the width and height: . But then, of course, the area is quadrupled (which 2nd or 3rd grades wouldn't automatically know).

Right there I gave up. I SURE don't want to get them confused by doubling the width, height, or area... Later on they need to learn that IF you multiply the two dimensions by some factor

*r*, the area will be multiplied by*r*.^{2}I did write for my lesson an illustration of scaling a "stick", or a line. In that one-dimensional situation we won't run into this problem. But even so, I would be extra careful of using it a lot, because surely some student will ask about scaling two-dimensional images, and then we have confusion.

I would find it more natural to present the idea of multiplication to 2nd and 3rd graders as "multiple copies", such as 2 × = .

Even our word "multiply" refers to multiple copies of the same... people and animals "multiply", we talk about multiples, etc. We use the word "times" referring to doing the

*same thing*over and over, such as "I opened the door three times".Then, when it comes to multiplication of fractions and of decimals, one has to bring in the idea of taking a part:

1/2 x 7 means 1/2 OF 7. I do not see a problem there.

Later one can tie these "two meanings" of multiplication together with the scaling idea... maybe... somehow... I just do not know myself how to do that without confusing the idea of scaling the width/height by some factor and the area being scaled by the square of that factor.

Or maybe I'm all wrong and it IS possible to use the idea of scaling images?

Maybe someone should TRY it on a few classrooms of kids and see what happens over the years.

**Update:**Joe Niederberger has left an excellent comment on the issue at Let's Play Math blog. I feel I need to quote him... hope he doesn't mind:Devlin unfortunately makes the mistake of thinking of multiplication as one "thing". It’s true multiplication of any two real numbers cannot be simply reduced to repeated addition, however, the multiplication of any two integers *can* always be reduced (or thought of, or defined by) repeated addition. Even though we call them both "multiplication" technically they are different functions.

In fact, we learn somewhere along the mathematical way that functions (like multiplication) are only properly defined by specifying their domain (among other things). Two functions that have different domains cannot be the *same* function. One function can be the extension or restriction of another, but they are not the same.

This is the basis of the confusion. Multiplication of integers *is* repeated addition, in some form or other (Peano uses a recursive definition - recursive, repeated; I say to-mae-toe, you say to-mah-toe.) Multiplication of rationals is a different animal (related, but different.) Same for multiplication of reals, complex numbers, etc. All different functions even though they build on one another.

Again, multiplication of rationals is technically a different function, in fact, an extension of multiplication on integers. Defining it requires that multiplication of integers has already been accomplished — and that, yes, means that repeated or recursive addition has already been put in the soup.

Essentially, we can define multiplication of whole numbers (and integers) as repeated addition. We have to define multiplication of fractions in a different way - but that is not a problem. It is

*extending*the idea of multiplication in a way that it will "match" or "work" for integers as well.Sideline... in other words, multiplication of fractions can be defined as

a/b * c/d = (ac)/(bd) which is the familiar rule. (BTW, definitions vary. That's why I can't say that multiplication of fractions would always necessarily be defined this way.)

If you have integers y and z, they can be written as fractions as x/1 and y/1, and multiplying them using the definition of

*fraction multiplication*we get:y/1 * z/1 = (yz)/(1*1) which equals yz.

Later on, multiplication of real numbers and that of complex numbers are defined still differently, as "extensions" of the idea of basic multiplication.

Other bloggers have their take, too:

If it ain't repeated addition... by Let's Play Math, Devlin on Multiplication by Rational Math Education, and Devlin's Right Angle at Text Savvy.

## Comments

on the number line. (I added a section on the end of my article about this.)I think the most useful thing for homeschoolers to take from the idea of scaling is that multiplication can be represented by the word "of." This is something many people have never heard, and it is hugely useful through all the different tough topics of arithmetic: fractions, decimals, percents, negative numbers...

But the point made by Niederberger (and Todd) is important, too --- that the rules have to change, as our understanding grows. The new rules aren't arbitrary; they grow out of the old ones, as we adapt to new situations.

Multiplication of Squares

Of course, it can work having the imaginary component equal to zero. It's really a matter of drawing a new figure where all of the unit squares are replaced with the actual square you are multiplying by. With the squares, you can see that the sides have been multiplied, and that the area has been multiplied. Try it!

If you wanted to show 1/2 X 1/2 = 1/4, you can draw a square, cut it in half vertically and in half horizontally. Put thin vertical lines in one vertical half and horizontal lines in one horizontal half. You'll note that only 1/4 of the square is crosshatched (intersected) and 1/4 is the answer.

1/4*2 = 1*2/4*1 = 2/4 = 1/2...so multiplying by 1/4 is the same as dividing by the the reciprocal 4/1.

Further, multiplication is scaling because scaling is unit conversion, and that includes advanced situations such as multiplication of vectors that converts them to scalars.

Conversion of units always requires an higher dimension, so by multiplying you are creating a grid in the next higher dimension: length to area, area to volume, volume over time.

Multiplication is conversion of larger units to smaller ones. Division is conversion of smaller units to larger ones. Think of a measuring cup, and the idea becomes clear. You multiply to convert CUPS to OUNCES (large to small), and divide to convert OUNCES TO CUPS (small to large).

-Keith

Scaling: You just have to know what you are adding. If you are adding cars, then 2 * 1 car = 2 cars. If you are add length, then 2 * 12ft = 24ft. So if you still want one car, then it will be 24 feet long.

Devlin makes a ridiculous example:

===

If I take an elastic band of length 7 inches and stretch it by a factor of 3, its final length is

[3] x [7 INCHES] = 21 INCHES

But I did not take 3 copies of a 7 inch band and join them together (addition), rather I scaled (stretched) the 7 inch band by a factor of 3.

===

Of course you didn't make three bands, you even wrote down the units and nowhere in the units was a band, that's because you didn't add bands you added length. If you did 3 * 1 band then you get 3 bands -- maybe they're not even the same length! Decide what you're adding.

Fractions: 2 1/2 * 4 = 10. This is "two and a half times 4", which is one times 4 (4) and another times 4 (8) and another half of four (2). Intuitively, it is adding of fours.

The algorithms are shortcuts to repeated addition. The connection between them and the addition are somewhat disconnected because they rely on realizing some properties of numbers.

Multiplication gets a little hairy against non-numbers like matrices and sets, but Devlin doesn't talk about those. Against numbers multiplication is always repeated addition.