The tale of Tabitha and the two division problems

Consider these two division problems:

Problem A: 22 cookies. Each kid gets 10 cookies. How many kids can get a full share? How many are left over?

Problem B: 22 cookies. There are 10 kids. How many cookies does each kid get? How many are left over?

These are not copied verbatim from Tabitha’s third-grade homework this week, but the numbers and context are the same. (Forgive me; I didn’t think about the potential for large-group discussion until the homework went back to school.)

The point is this: One of these problems was very easy for Tabitha, and the other was very challenging.

Do you know which is which?

We talked about this on Twitter today. (Click through for some really outstanding discussion….seriously.)

I have written about the two major types of division problems before, and they are relevant here.

Problem A was a snap for Tabitha. She skip counts well, and she is a whiz with place value. How many 10s in 268? Why 26 of course! This is the sort of thing I’m talking about.

So Problem A above is a piece of cake for her. This problem—for Tabitha—is very clearly asking How many tens are in 22? For her, this isn’t really even a question worth asking. Each kid gets one ten. There are two tens. QED.

Problem B doesn’t submit to this strategy in an obvious way. It requires her to keep track of 22 things as they get shared among 10 kids. One for you, one for you, one for you, etc. That’s taxing work, and so it’s a much harder problem for her.

When we discussed this problem together the other night, I made the argument that you use up 10 cookies each time you give everybody one cookie. I wanted to help her see how her strategy from Problem A would be useful in Problem B, while respecting that—for her—the sameness of these two problems is not at all obvious.

What’s the moral of the story? Let me know your thoughts in the comments.

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Several people have observed that they would love to have audio of the conversations I report under the heading Talking Math with Your Kids. I agree that this would be helpful. But here’s the rub. These conversations are a natural part of our day, and they have to be natural.

My kids have no interest in being show ponies. Oh, they’ll show off for the recording device, but it won’t be natural. Observe Griffin as a young lad in this video, for instance.

He’s totally making faces for the camera and watching himself in the monitor. (You may also note that the spinach is washed; and please forgive the praise style—I know better now!) This mugging behavior has only gotten worse with time.

I do have some ideas for getting good audio, but these will require funds (Do you have a couple thousand dollars for a good cause? Tweet me! We’ll talk!) So in the meantime, we’ll continue the transcribed conversations.

Today’s conversation is a brief one, but I want to make a comparison. The discussion in the following video is not a natural one. (Tip o’ the hat to David Wees for the find.)

Here’s what I mean. The woman discusses cookies because she thinks they will interest the child in question, not because cookies are already under consideration. The question of multiplication (or repeated addition—I have no interest in that distinction here) doesn’t arise naturally either. It arises in the context of putting four chocolate chips on each of three cookies.

To be clear, I have no problem with any of this. But it’s different from the kinds of conversations I am hoping to encourage. The ones I hope to encourage go more like this…

Tabitha (5 years old) and her mother made cookies from one of those frozen-cookie-dough-school-fundraiser things over Thanksgiving weekend. These cookies were stored (unwisely) in a transparent Pyrex container on an open shelf. This led to a desire on Tabitha’s part for a cookie before dinner.

Tabitha: How many cookies can I have? One or two?

Me: Zero. You can have zero cookies.

T: A half?

Me: No. I said zero.

T: Zero whole ones and a half cookie?

Me: Zero halves.

T: And a quarter?

This is a natural conversation about cookies. The opportunity to turn it into a mathematical one was Tabitha asking, How many can I have? I could have played the role of rule enforcer and replied, We don’t eat cookies before dinner; you may have one for dessert. Responding with zero in answer to her question gave her some mathematical wiggle room to play with. And we are far enough along in this talking math adventure that she’s going to play with it nearly every time.

For the record, Tabitha and I have spoken about zero before. And several times, we have had conversations about fractions.