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# Ways of thinking and ways of doing

By William McCallum

Somewhere back in days of Facebook fury about the Common Core there was a post from an outraged parent whose child had been marked wrong for something like this:
$$6 \times 3 = 6 + 6 + 6 = 18.$$
Apparently the child was supposed to do
$$6 \times 3 = 3 + 3 + 3 + 3 + 3 +3 = 18$$
because of this standard:

3.OA.A.1. Interpret products of whole numbers, e.g., interpret $5 \times 7$ as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as $5 \times 7$.

The parent had every right to be upset: a correct answer is a correct answer. Comments on the post correctly pointed out that, since multiplication is commutative, it shouldn’t matter in what order the calculation interpreted the product. But hang on, I hear you ask, doesn’t that contradict 3.OA.A.1, which clearly states that $6 \times 3$ should be interpreted as 6 groups of 3?

The fundamental problem here is a confusion between ways of thinking and ways of doing. 3.OA.A.1 proposes a way of thinking about $a \times b$, as $a$ groups of $b$. In other words, it proposes a definition of multiplication. It could have proposed the other definition: $a \times b$ is $b$ groups of $a$. The choice is arbitrary, so why make it? Well, there’s an interesting discovery to me made here: the two definitions are equivalent. That’s how you prove that multiplication is indeed commutative. It’s not obvious that $a$ groups of $b$ things each amounts to the same number of things as $b$ groups of $a$ things each. At least, not until you prove it, for example by arranging the things into an array:

You can see this as 3 groups of 6 by looking at the rows,

and as 6 groups of 3 things each by looking at the columns,

Since it’s the same number things no matter how you look at it, and using our definition of multiplication, we see that $3 \times 6 = 6 \times 3$. (We leave it as an exercise to the reader to generalize this proof.)

None of this dictates the way of doing $6 \times 3$, that is, the method of computing it. In fact, it expands the possibilities, including deciding to work with the more efficient $3 \times 6$, as this child did. The way of thinking does not constrain the way of doing. If you want to test whether a child understands 3.OA.A.1, you will have to come up with a different task than computation of a product. There are some good ideas from Student Achievement Partners here.