Division and fractions with a third grader

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.

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Partitive fraction division

As promised, more notebook pages on fraction division. This is based on the work I did a while back on trying to write authentic partitive division problems with fractional divisors. (As I wrote that last sentence, I reminded myself what a bizarre niche market I am trying to occupy on this here blog.)

I settled on situations involving fractional values of unit rates, such as the following.

If of a lawn takes of an hour, how much can I mow in one hour?

Before we begin, remember that if the problem were about 2 lawns in 3 hours, we would easily and naturally divide by 3. Only the numbers have changed, so the mathematical structure remains the same and we need to find .

Click each image to see it full size. If you’re into this sort of thing.

page 1

page 2

Christopher cries “Uncle!” (but doesn’t give up the fight)

Last week I called out this problem (whose existence was implied by a web search that brought a reader to my blog):

In how many ways can 7 peanuts be shared among 3 people?

In particular, I argued,

Any problem that uses everyday language [such as “share”] or imagery that will mislead if taken seriously is a bad one in my view.

Readers took me to task for too narrow a view of the verb share.

I concede.

Not all sharing is equal sharing. I probably use the word sharing to mean equal sharing far too often. This makes my point while simultaneously implicating me. Sweet.

But no way am I going to let that crummy problem off the hook.

Chris Hunter argues in the comments that the problem has gotten the implicit stamp of approval from the National Council of Teachers of Mathematics (about which more in the next couple of weeks), by way of being in an article published in Teaching Children Mathematics:

Danny, Connie, and Jane have eight cookies to share among themselves. They decide that they each do not need to get the same number of cookies, but each person should get at least one cookie. If the children do not break any of the cookies, in how many different ways can they share the cookies?

But that’s not the peanut problem.

Danny, Connie and Jane are likely to be satisfied with their share of eight cookies. Indeed (equal sharing aside), it is likely possible to find some way to share these cookies so that everyone’s appetite is sated.

Were they sharing peanuts, it would be tougher going. When was the last time you stopped at the seventh peanut?

And by the way, what’s the unit here? Is this one peanut or two?

The sharing will proceed differently in each case, I would imagine.

Here’s what I’m saying. Context matters. Dan Meyer will argue that context matters for motivation and for intellectual honesty. Karim Ani will argue that context matters for motivation and so that kids understand that math is power.

All true.

But I want to argue that context matters because people bring intuitive mathematical ideas to class. More often than not in K-12 schooling (and beyond), those intuitive ideas are based on their experiences in the real world. If we don’t build on those ideas, then we alienate students from mathematics.

If we do build on those ideas, then we’re helping students to make their ideas better. There is efficiency in this, but also an opportunity to avoid the well-documented effects of instruction that doesn’t connect to everyday experience. Namely, that said instruction has absolutely no effect on people’s views of the world, nor on their ways of operating in it.

It’s not so much that students end up choosing not to use their mathematics education in their lives, it’s that it never occurs to them to do so.

Because no one shares seven peanuts among three people.