[Photo by Alejandra Mavroski.]

Myrtle called it The article that launched a thousand posts…, and counting comments on this and several other blogs, that may not be too much of an exaggeration. Yet the discussion feels incomplete — I have not been able to put into words all that I want to say. Thus, at the risk of once again revealing my mathematical ignorance, I am going to try another response to Keith Devlin’s multiplication articles.

Let me state up front that I speak as a teacher, not as a mathematician. I am not qualified, nor do I intend, to argue about the implications of Peano’s Axioms. My experience lies primarily in teaching K-10, from elementary arithmetic through basic algebra and geometry. I remember only snippets of my college math classes, back in the days when we worried more about nuclear winter than global warming.

I will start with a few things we can all agree on…

We All Agree on Some Things

Elementary students will use addition to solve beginning multiplication problems. What else could they do? They have not yet learned multiplication.

The multiplication table is initially built by repeated addition, as our students add up (or count by) the numbers in each row and column. Even adults, when they get stumped by a math fact they’ve forgotten, will use repeated addition to figure out the answer.

Repeated addition does give the correct answer for any multiplication of whole numbers. Sometimes it is more trouble than it is worth — who wants to add up 957×842? But by using the Distributive Property, any multiplication of whole numbers can be reduced to a repeated addition calculation:

4 × 3 = (1 + 1 + 1 + 1) × 3 = 3 + 3 + 3 + 3

Rational number multiplications can be calculated as addition of parts. This is how the Egyptian scribes handled multiplication of fractions. But be careful! To calculate the parts, one needs multiplication — or rather, its inverse, division. So to use this as a definition, one must resort to circular reasoning.

Finally, I think we can all agree that repeated addition is an important problem-solving tool. Repeated addition can help students think their way through simple multiplication word problems. It doesn’t always work, but sometimes it is the quickest way to understand a situation.

So What’s the Problem?

So what is wrong with the definition, “Multiplication is repeated addition”? Perhaps wrong is too strong a word — perhaps in some deep, theoretical sense, the statement is true, at least for the whole numbers — I’ll leave that argument to the mathematicians. (See, for instance, How multiplication is really defined in Peano arithmetic.) But speaking as a teacher, the phrase can definitely be misleading.

To define multiplication as repeated addition is to make multiplication a sub-species of addition.

It is as if there were two types of addition: regular, random, “wild” addition and the specially-bred variety of addition to which we give the name multiplication. Is that really how we want our students to think? Multiplication is not a mere sub-species of addition. Multiplication is its own animal, an independent operation.

The operation of addition has its identity element.

The operation of multiplication has its identity element.

And they are not the same.

Every number has its own additive inverse.

Each number has a multiplicative inverse, too.

And they are not the same.

Addition has its inverse operation, subtraction.

Multiplication has its inverse operation, division.

And they are not the same, because the operations are not the same.

[Sidetrack: Oops! I forgot that zero does not have a multiplicative inverse. Jonathan pointed out my mistake.]

[Another side note: I find it interesting that repeated subtraction can be a useful tool in solving some division problems, just as repeated addition can be a useful tool in understanding multiplication. Subtraction plays an important role in the algorithm for long division. I suppose someone will argue that this is evidence multiplication is repeated addition after all.]

Dimensional Reasoning

Dimensional analysis means looking at the dimensions (units of measurement) of a quantity to help you solve a science or engineering problem.

Addition requires identical units. The sum must always have the same units as the addends:

2 apples + 3 apples = 5 apples

2 apples + 3 oranges = ??

What does that second equation give you? Fruit salad? In order to add quantities with unlike units, we need to find a common denominator. Apples and oranges are both pieces of fruit, so…

2 apples + 3 oranges =

2 pieces of fruit + 3 pieces of fruit = 5 pieces of fruit

Multiplication requires different units. The product does not have the same units as either the multiplier or the multiplicand.

2 baskets × 3 apples per basket = 6 apples

How can we make multiplication come out the same as repeated addition? The only way to do it is to change the units.

3 cm + 3 cm = 6 cm

But…

2 cm × 3 cm = 6 cm2

We need…

2 lengths × 3 cm per length = 6 cm

We do not normally think about dimensional analysis when we work with plain numbers in math class. But the fact remains that multiplication changes things in a way that addition does not.

Addition is one-dimensional, but multiplication is multi-dimensional.

This is why the rules for fraction addition and fraction multiplication are so different. When you add positive rational numbers, you always get a sum that is bigger than either addend. But when you multiply rational numbers, all bets are off — the product may be bigger, smaller, or somewhere in between the numbers.

Language Does Matter

Addition: addend + addend = sum. The addends are interchangeable. This is represented by the fact that they have the same name.

Multiplication: multiplier × multiplicand = product. The multiplier and multiplicand have different names, even though many of us have trouble remembering which is which.

multiplier = “how many or how much”

= “how many or how much” multiplicand = the size of the “unit” or “group”

Different names indicate a difference in function. The multiplier and the multiplicand are not conceptually interchangeable. It is true that multiplication is commutative, but (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.

A New Type of Number

In multiplication, we introduce a totally new type of number: the multiplicand. A strange, new concept sits at the heart of multiplication, something students have never seen before.

The multiplicand is a this-per-that ratio.

A ratio is a not a counting number, but something new, much more abstract than anything the students have seen up to this point.

A ratio is a relationship number.

In addition and subtraction, numbers count how much stuff you have. If you get more stuff, the numbers get bigger. If you lose some of the stuff, the numbers get smaller. Numbers measure the amount of cookies, horses, dollars, gasoline, or whatever.

The multiplicand doesn’t count the number of dollars or measure the volume of gasoline. It tells the relationship between them, the dollars per gallon, which stays the same whether you buy a lot or a little.

By telling our students that “multiplication is repeated addition,” we dismiss the importance of the multiplicand. But until our students wrestle with and come to understand the concept of ratio, they can never understand multiplication.

How Then Shall We Teach?

If we accept this argument, if we agree to no longer define basic multiplication as repeated addition, then what? How does that affect the way we teach?

Mainly, we need to change our focus from how to why.

We can teach multiplication in much the same way that we do now, using manipulatives arranged in groups or rows, pictures of multiplication situations, and rectangular arrays of dots or blocks. But instead of drawing our student’s attention to the process of adding up the answer, we want to focus on the fact that the items are arranged in equal sized groups.

In other words, we teach our students to recognize the multiplicand:

Teach children the useful word “per” and how to recognize a “this per that” unit.

Have them label the quantities in their workbook: 3 cookies per student, 5 flowers per vase, 1 eye per alien, or whatever.

If we need a simple, elementary-level catchphrase to replace “multiplication is repeated addition,” how about this?

Multiplication is counting by this-per-that groups.

As with any such phrase, this statement fails to capture all that multiplication entails. The definition will have to be expanded as students learn about rational numbers. “Oh, look! We can count just part of a group, and we can measure with a unit that is not a whole number.” Our students will someday have to learn about real numbers, complex numbers, and matrix multiplication. Even so, the phrase captures an important aspect of many multiplication situations our students will meet in K-12: that there is a multiplicand, some “this per that” quantity.

This approach should be especially helpful to those frustrating students — you know, the ones with the blank stare — who read a word problem and then ask, “Do I add or multiply?”

A Useful Tool

Completing the circle, I come back to the point of my first “repeated addition” post. I would like you to consider the teaching power of bar model diagrams to represent arithmetic operations. These diagrams are used in the Singapore Primary Math books, and according to one commenter they are popular in Russia and Australia. They are even beginning to show up in newer American textbooks, where they are sometimes called “tape diagrams.”

Here are some advantages of the bar diagram model:

Bar diagrams chunkify the number line and make number relationships less abstract.

They provide elementary students with a pictorial algebra that can help them think through complicated word problems.

It is easy for students to see the inverse relationship between addition and subtraction, or between multiplication and division.

Because they are based on the number line, the diagrams extend naturally to rational and real numbers, growing in application with your students’ growing understanding.

Addition is “this AND that”: putting two (or more) amounts together. This is the basic addition/subtraction diagram:

Multiplication is “how many or how much OF the unit”: measuring or counting parts of a given size. Here is the diagram for multiplication/division:

To learn more about modeling arithmetic problems with bar diagrams, check out the Mad Scientist’s Ray Gun model of multiplication:

And here is an example of the multiplication bar diagram in action:

OK, Now It’s Your Turn

I’ve talked long enough. What do you think:

Is there really a difference between multiplication and repeated addition, or am I tilting at windmills here?

Is it even necessary for teachers to define multiplication? Or is the teacher’s job to provide plenty of examples of multiplication in action? Should we let the students intuit their own definition(s)?

Will it help students if we change our focus from “how to get the answer” and teach them to identify the multiplicand, the “this per that” unit? Or will that introduce new difficulties I haven’t considered?

Or do we already teach this way, only in different words?

If you are an elementary teacher, how do you teach multiplication to your students?

Are some students clueless because, no matter how we explain it, they just don’t pay attention?

Have you tried using bar diagrams to model elementary arithmetic situations? And if so, how did your students respond?

If you’re interested in digging deeper into how children learn addition and multiplication, I highly recommend Terezina Nunes and Peter Bryant’s book Children Doing Mathematics.

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