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January 19, 1990

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Deborah Ball's class

Friday, January 19, 1990

Third grade, Spartan Village School, East Lansing, MI

January 19, 1990

(1:05:31)

Ball:

Shea:

More comments about the meeting? I'd really like to hear from as many people as possible what comments you had or reactions you had to being in that meeting yesterday. Shea? Um, I don't have anything about the meeting yesterday, but I was just thinking about six, that it's a. . . I'm just thinking. I'm just thinking it can be an odd number, too, 'cause there could be two, four, six, and two, three twos, that'd make six ... Uh-huh . . .

And two threes, that it could be an odd and an even number. Both! Three things to make it and there could be two things to make it.

And the two things that you put together to make it were odd, right? Three and three are each odd? Uh huh, and the other, the twos were even.

So you're kind of--I think Benny said then that he wasn't talking about every even number, right, Benny? Were you saying that? Some of the even numbers, like six, are made up of two odds, like you just suggested. (pause) Other people's comments? (pause) Tina?

I disagree with Shea when he says that six can be an odd number. I think six can't be an odd number because . . . look--(she gets up and comes up to the board) (interrupting) Jillian, Bernadette?

Six can be an odd number because this is (she points to the number line, starting with zero) even, odd, even, odd, even, odd, even, how can it be an odd number because (starting with zero again) that's odd, even, odd, even, odd, even, odd. Because zero's not a odd number.

Because six, because there can be three of something to make six, and three of something is like odd, like see, um, you can make two, four, six . Three twos to make that and two threes make it . But that doesn't-- Kevin?

That doesn't necessarily mean that six is odd. Yeah.

Why not, Kevin?

Just because two odd numbers add up to an even number doesn't mean it has to be odd.

What's the definition--Shea?-- what's our working-- Two odd numbers make--

Shea? What's our working definition of an even number? Do you remember from the other day the working definition we're using? What is it?

It's, um, that (pause) . . . I forgot.

Could somebody help us out with this? Because we need in the group to have an idea that we're working with. What's the working definition we're using? (pause) Do other people know it besides Liz and Shekira? (pause) I think other people do. Marta, do you know what the definition is that we've been using for an even number?

Shea:

Ball:

Shea:

Kevin: Ball:

Kevin:

Students: Ball:

Kevin: Ball:

Shea: Ball:

Ball:

Shea:

Ball:

Tina:

Shea: Ball:

Ball:

Tina:

(1:08:16)

At this point I thought that Shea was just confused about the definition for even numbers. I thought that if we just

reviewed that, he would see that six fit the definition and was therefore even. I assumed that after this we would be able to get on with our discussion.

Ball:

Shea:

Ball:

Shea:

Ball:

Within a couple of minutes, we had settled on a definition of even numbers. Jillian said:

If you have a number that you can split up evenly without having to split one in half, then it's an even number. So I turned to Shea in order to make the connection and clarify things:

Can you do that with six, Shea? Can you split six in half without having to use halves? Yeah.

So then it would fit our working definition, then it would be even. Okay? There was a pause.

And it could be odd. Three twos could make it. Okay. One of the points here is that if it fits the definition then we would call it even. If it fits our working definition, then we would call it even. It fits the definition for odd, too.

I began to see that the issue was more complicated than I had thought.

What is the definition for odd? Maybe we need to talk about that?

We discussed a definition for odd numbers--before this we had had an explicit definition for even numbers only. I had assumed that this was sufficient. I think now that I was wrong. So we agreed that odd numbers were numbers that you could not split up fairly into two groups. But this still did not satisfy Shea yet. He persisted with the observation he had made about what made six special

(1:11:24)

Shea:

You could split six fairly, and you can split six not fairly. You can like cut six in half, um .... there's like, say there's two of you and you had, and you had, um, six cookies and you didn't want to split it in half and so that each person would get three and you wanted to split it by twos. Each person would get um, two and there would be two left. For which number now? For six? Uh huh.

So, are you saying all numbers are odd then? No, I'm not saying all numbers are odd, but . . . Which numbers are not odd then?

Um. . . Two, four, six . . . um, six can be odd or even . . . eight No. . . !

I don't know how. Show us.

Because there's three twos. One, two. Three, four. Five, six. Prove it to us that it can be odd. Prove it to us. Okay. (He rises and comes up to the board.)

Does everybody understand what Shea's trying to argue? He's saying six could be even or it could be odd. I disagree . . .I don't think so . . .

Well, watch what he's going to prove and then you can ask him a question about it.

Shea:

Ball:

Shea: Ball:

Shea:

Students: Kip:

Shea: Kip:

Shea: Ball:

Students: Ball:

Shea:

Well, see, there's two, (he draws) number two over here, put that there. Put this here. There's two, two, and two. And that would make six.

I know, which is even!

I think I know what he's saying. Which is even, Shea.

Lin? (to Shea) Could you stay there? People have some questions for you.

I think what he is saying is that it's almost, see, I think what he's saying is that you have three groups of two. And three is a odd number so six can be an odd number and a even number.

Do other people agree with that? Is that what you're saying, Shea?

Yeah.

Okay, do other people agree with him? (pause) Lin, you disagree with that?

Yeah, I disagree with that because it's not according to like . . . here, can I show it on the board? Um hm.

(She comes up to the board.) It's not according to like. . . Rania, can you watch what Lin's doing?

. . . how many groups it is. Let's say that I have (pauses) Let's see. If you call six an odd number, why don't (pause) let's see (pause) let's see--ten. One, two . . . (draws circles on board) and here are ten circles. And then you would split them, let's

Shea: Lin:

Shea: Lin: Liz: Lin:

say I wanted to split, spit them, split them by twos. . . One, two, three, four, five . . . (she draws)

then why do you not call ten a, like- I disagree with myself.

. . . a, an odd number and an even number, or why don't you call other numbers an odd number and an even number? I didn't think of it that way. Thank you for bringing it up, so --I say it's--ten can be an odd and an even. Yeah, but what about ... Ohh!!!

What about other numbers?! Like, if you keep on going on like that and you say that other numbers are odd and even, maybe we'll end it up with all numbers are odd and even. Then it won't make sense that all numbers should be odd and even, because if all numbers were odd and even, we wouldn't be even having this discussion!

Kip: Lin: Kip:

Ball: Lin:

Ball:

Shea:

Ball: Lin:

(1:15:40)

Deborah Ball's class

Friday, January 19, 1990

Third grade, Spartan Village School, East Lansing, MI

Class list as of January, 1990 NAME GENDER RACE COUNTRY ENGLISH PROFICIENCY fluent fluent native speaker developing native speaker native speaker fluent fluent native speaker beginning native speaker fair beginning good developing native speaker native speaker native speaker native speaker Ansumana Benny Bernadette David Jillian Kevin Kip Lin Liz Marta Mick Ogechi Pravin Rania Safriman Shea Shekira Tiffany Tina

M M F M F M M F F F M F M F M M F F F African-American White White Asian White African-American African Black Asian White Latina White African Black White White Asian White African-American White African-American U.S.A./South Africa Ethiopia Canada Indonesia U.S.A. U.S.A. Kenya Taiwan U.S.A. Nicaragua U.S.A. Nigeria Nepal Egypt Indonesia U.S.A. U.S.A. U.S.A. U.S.A. HOW LONG AT THIS SCHOOL1 2 weeks 3 years 4 months 3 years 3 years 2 weeks 3 years 2 years 3 years 4 months 2 years 3 years 9 months 3 years 16 months 2 years 4 months 2 weeks 16 months

NOTE: This column reflects the length of time the child had been in this school as of 1/19/90. No one had been in this class longer than 4 months (since September).

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