I found some notes on a conversation I had with Griffin last fall. I do not remember the context for it.

Me:Do you know what 12÷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÷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÷2?

G: Well, 15 plus 15 is 30…so…19

Here we see the role of cognitive load on mental computation. Griffin is splitting up 34 as 30 and 4 and finding pairs to add to each. Formally, he’s using the distributive property: .

He wants to choose and so that .

But by the time he figures out that , he loses track of the fact that and just adds 4 to 15.

At least, I consider this to be the most likely explanation of his words.

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

Our conversation continued.

Me:So 12÷2 is 6 because 2×6 is 12. What is 12÷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.

Let’s pause for a moment.

This is what it means to learn mathematics. Mathematical ideas *
*have multiple interpretations which people encounter as they live their lives. It is (or should be) a major goal of mathematics instruction to help people reconcile these multiple interpretations.

Griffin has so far relied upon three interpretations of division: (1) A division statement is equivalent to a multiplication statement (the *fact family* interpretation, which is closely related to thinking of division as the *inverse* of multiplication), (2) Division tells how many groups of a particular size we can make (Ms. Starr’s class has 13 pairs of students—this is the *quotative *interpretation of division) and (3) Division tells us how many will be in each of a particular number of same-sized groups (Put 12 things into 1 group, and each group has 12 things).

This wasn’t a lesson on multiplication, so I wasn’t too worried about getting Griffin to reconcile these interpretations. Instead, I was curious which (if any) would survive being pushed further.

Me:What is ?

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÷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.