I wish Maths teachers would stop telling students that “** multiplication is repeated addition**“.

Multiplication of natural numbers certainly *gives the same result* as repeated addition, but that does not make it the same. Its like riding a bike and a car to office. Both may take the same time, but the two processes are very different.

Why can we not say, that there are at least two basic operations you can do to numbers: ** you can add them and you can multiply them**. (not counting subtraction and division here, since they are simply the inverses to addition and multiplication, and thus not “basic” operations.)

**Adding **and **multiplying **are simply things you do to numbers.

- Adding numbers tells us
*how many things (or parts of things) you have when you combine collections.* - Multiplication is useful if you want to know the result of
*scaling some quantity*.

Telling students falsehoods on the assumption that they can be corrected later is not a good idea. And telling them that multiplication is repeated addition definitely requires undoing later.

As soon as the child progresses from whole-number multiplication to multiplication by fractions (or arbitrary real numbers). At that point, you have to tell a different story.

“Oh, so multiplication of fractions is a DIFFERENT kind of multiplication, is it?” a bright kid will say, wondering how many more times you are going to switch the rules. No wonder so many people end up thinking mathematics is just a bunch of arbitrary, illogical rules that cannot be figured out but simply have to be learned – only for them to have the rug pulled from under them when the rule they just learned is replaced by some other (seemingly) arbitrary, illogical rule.

SO MULTIPLICATION IS NOT REPEATED ADDITION– PERIOD

**Elementary students will use addition**initially to solve multiplication problems since they have not yet learned multiplication.**The multiplication table is initially built by repeated addition**, as 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 is the issue?** The issue is that:

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.

- 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.

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## Dimensional Reasoning

Dimensional analysis means looking at the 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 cm^{2}

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”**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 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 fully 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

Consider the teaching power of bar model diagrams to represent arithmetic operations. These diagrams are used in the Singapore Primary Math books, and they are popular in Russia and Australia. They are sometimes also 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:

## By the Way

It is interesting to observe 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. But don’t say that Division is Repeated Subtraction.