## Question 5

### “Multiplication is just repeated addition.” Explain why this statement is false, giving examples.

Now this is when things get sticky.

It is a strong and presumptuous claim to say what an idea *is.*

In recent years, I have come to an understanding of why *repeated addition* is not the strongest foundation on which to build the idea of multiplication. But that is a far cry from making claims about what it *is*.

You see, any rich enough mathematical idea has multiple meanings. What *is *subtraction? Is it the inverse of addition? Is it the distance between two points on a number line? Is it *takeaway*? Subtraction is all of these, sort of.

And what is a fraction? The only definition of a fraction that is good enough to withstand the test of time and mathematical scrutiny is that a fraction is an *equivalence class* resulting from the equivalence relation,

if and only if

Is that what a fraction *is*?

No. But I am off task.

I suspect that my answer may vary from some others out there. (Although perhaps it will not.)

Repeated addition is shaky ground for establishing multiplication because it doesn’t capture the unique structure that multiplication represents.

There is additive structure, and there is multiplicative structure. Additive structure is about comparisons and changes involving the same units. Apples plus apples gives apples. Miles plus miles give miles.

Multiplicative structure is about comparisons and changes involving different units. *Hours* times *miles per hour* gives *miles. *Three different units; one of them a unit rate. Always.

These are related structures but they are different.

Multiplicative structure is captured better by this idea: means *A* groups of *B *(I am pretty sure I first ran across this particular characterization in Sybilla Beckmann’s textbook for math courses for elementary teachers). *A*, in this interpretation, is expressed in one unit. *B* is the unit rate (*things per group*). The product is expressed in a third unit.

This difference shows up in the following conversation between a mom and her daughter as they count the number of things in this array of meatballs.

Image from The New York Times.

Maya counted the top and bottom, 4 + 4 = 8. Then she counted L and R. 3 + 3 = 6. 8 + 6 = 14. 14 + 2 in the middle = 16. When I asked her why, she said, “Because you double count the corners when you count an array.”

She asked me to count so she could show me how. I counted 4 across the top and 3 down the side. “See, Mommy! You’re counting the corner one twice.”

Why *do* we count the corner one twice in this scenario? This seems to violate a fundamental principle of counting—*one-to-one correspondence*. One number word for each object, and one object for each number word.

The answer is that mom really did *not* count the corner meatball twice. The first time, she counted the *meatball* to establish that each row has 4 meatballs. The second time, she counted the *rows*. There are 3 rows, so there are 3 groups of 4 meatballs.

Much, much more on arrays in many places in my writing. Especially these:

Beyond the textbook wrap up (or *What does this have to do with mathematics?*)

Twister (on sister site, *Talking Math with Your Kids*)