The meanings of division

I was talking with Griffin one day when he was in third grade.

Me: Do you know what $12\div 2$ is?

Griffin (8 years old): 6

Me: How do you know that’s right?

G: 2 times 6 is 12.

Me: What about $26 \div 2$?

G: 13

Me: How do you know that?

G: There were 26 kids in Ms. Starr’s class [in first grade], so it was her magic number. We had 13 pairs of kids.

Me: What about $34 \div 2$?

G: Well, 15 plus 15 is 30…so…19

My notes on the conversation at this point only have (back and forth), which indicates that there was probably some follow-up discussion in which we located and fixed his error. The details are lost to history.

Our conversation continued.

Me: So $12 \div 2$ is 6 because $2 \times 6$ is 12. What is $12 \div 1$?

G: [long pause; much longer than for any of the first three tasks] 12.

Me: How do you know this?

G: Because if you gave 1 person 12 things, they would have all 12.

Me: What is $12 \div \frac{1}{2}$?

G: [pause, but not as long as for 12÷1] Two.

Me: How do you know that?

G: Half of 12 is 6, and $12 \div 6$ is 2, so it’s 2.

Me: OK. You know what a half dollar is, right?

G: Yeah. 50 cents.

Me: How many half dollars are in a dollar?

G: Two.

Me: How many half dollars are in 12 dollars?

G: [long thoughtful pause] Twenty-four.

Me: How do you know that?

G: I can’t say.

Me: One more. How many quarters are in 12 dollars?

G: Oh no! [pause] Forty-eight. Because a quarter is half of a half and so there are twice as many of them as half dollars. 2 times 24=48.

So what do we learn?

Mathematical ideas have multiple interpretations which people encounter as they live their lives. As we learn more mathematics, we become better at connecting these different ways of thinking about ideas.

In this conversation, Griffin relies on three ways of thinking about division:

1. A division fact is a different way of saying a multiplication fact. ($12 \div\ 2$ is 6 because $6 \times 2$ is 12).
2. Division tells how many groups of a particular size we can make (Ms. Starr’s class has 13 pairs of students).
3. Division tells us how many will be in each group if we make groups that are the same size. (When he was working on $34 \div 2$, Griffin put 15 in each group to start off with.)

We were just talking for fun, not homework or the state test. So I wasn’t worried about his connecting those ways of thinking. I was just curious how he would apply them to some more challenging tasks, such as dividing by 1 or by a fraction.

I was surprised by how difficult $12 \div \frac{1}{2}$ was for Griffin. Not because it is an easy problem, but because he could have applied his how many of this are in that? idea, or his multiplication facts idea. But he did neither and reinterpreted the task as twelve divided by half-of-twelve.

I was also surprised at the length of the pause he took for $12 \div 1$. It makes sense in retrospect. After all, are you really making groups if it’s just one group? I imagine he had to think that through, rather than the number relationships involved.

Starting the conversation

When the opportunity presents itself—when you and your child are not under homework stress, not rushing to get out the door or find the dog’s leash; when you happen to be talking about number anyway—ask follow up questions. Even a simple set of division problems got a lot of good thinking out of Griffin. Problems involving 1, 0 and $\frac{1}{2}$ are especially challenging.

Vary the size of the numbers.

Don’t worry about whether the answers are right or wrong.

Keep asking How do you know? and listening to your child’s answer.

Offer a few ideas of your own.

Quit before anybody gets frustrated or bored.