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Knowing and Learning Mathematics for Teaching Appendix E: Transcript of Ball Videos Figure 1. Class Seating Arrangement on 9/19/89 1:03:38 Ten minutes into class. 86. Ball I wonder if someone can think of a number sentence that uses more then two numbers here. Just so we have a bunch of ideas of how we could do this. Who can make a number sentence that equals 10, but has more than two numbers adding up to 10? 87. Kip One plus one plus one plus one— 88. Ball Okay, wait-wait, slow down, I've got to h—to write it. One plus one plus one—

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Knowing and Learning Mathematics for Teaching 89. Rania —one plus one— 90. Ball I want Kip to tell me. 91. Kip Plus one. Plus one. Plus one. Plus one. Plus three. 92. Ball Okay, why does that equal 10? Why does that equal 10? Safriman? 93. Safriman Plus one plus one plus one— 94. Ball Sorry? 95. Safriman That's just plus one plus one plus one. 1:04:43   96. Ball Okay, can someone add to what Safriman is saying? How do we know that that equals 10? Rania? How do we know that that equals 10? 97. Rania Because one plus one plus one plus one plus one plus one plus one and plus three equals 10. 98. Ball Well then you're just sort of reading it, but how could you prove it to somebody who wasn't sure? 99. Rania Because I counted it. 100. Ball What did you count? What did you find out? 101. Rania There's one and the next one is two and the next one is three, next one is four, next one is five, next one is six, next one is seven, next one is—seven, and then three more, eight, nine, ten. 102. Ball Do you see the difference in Rania's second explanation? Did you see how she really showed us how it equals 10? The first time you just read it. And the second time you explained it. That was really nice. Okay? That's enough ideas for right now. I want you to write down as many different ones as you can and you—try to see if you can come up with some that you think nobody else in the class will think of. 103. Student Can we work with each other? 104. Ball You can work with other people at your table, but you should write it in your own notebook. 1:05:47 Students work individually and in small groups for the next 30 minutes. 1:36:30 43 minutes into class. 199. Ball Let's see if we can share some of the things you can come up with. Look over your list, and pick a couple that you feel especially—are especially interesting. That you think other people might not have thought of or that you were pleased to get or something. And I would like you to listen to each other's. If you liked one that somebody else brings up you can add it to your list. Liz, do you have one that you like? 200. Liz Yeah. 201. Ball What is it? 202. Liz 100 divided by 10 equals 10. 203. Ball Bernadette, could you hear Liz? 204. Bernadette Yeah. 205. Ball Okay, this is the time I want you to listening to other people. She said 100 divided by 10 equals 10. Liz can you explain that?

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Knowing and Learning Mathematics for Teaching Liz What do you mean by explain? Ball How do we know that that's right? 206. Liz Um, cause I have, like when I did 20 divided by, um—well I don't know really— Lin I think I can explain it. 1:37:26   207. Ball Lin thinks she can explain this. Is there anybody else who thinks they can explain this? How do we know that what Liz told us is right? Sarah, you think you could explain it. Sarah Um-hmm. I—I agr— Ball Pravin and Kip can you be looking and listening right now? 208. Sarah I agree with Liz about what she said about the math sentence. 209. Ball Why do you think that's right? 210. Sarah I don't know. I just think it's right, for some reason. 211. Ball Lin? 212. Lin I had it—I agree with Liz too because I had it on my paper. 1:38:00   213. Ball But how do you know that it's right? Just because a couple people have it on their paper, doesn 't mean it's right? Liz 214. Liz Lin said her mom taught her what—about dividing by, and so I believe Lin. 215. Ball Okay, so Lin told you and her mom taught her and so you believe her. Lin—. Rundquist Liz has another reason too. Ball You have another reason too Liz? Liz Uh-uh. 216. Rundquist What did you think about that ___ the ones you told me. 217. Liz Um, well um, I have a lot of other ones, and like, if I had 20 divided by 2, Lin said it was right, so then—when I said that, she said it was right, so if I—if I have 10 divided—10 divided by 1 equals 10, and then I have it—and then I have like 50 divided by 5 equals 10, and Lin said that it was right, and so, like if you have 100 divided by 10 it would be right because—because if you went from like 5 then it would be 5 more, and then—cause 50 is 5 less than 100 so—so it would be 10 cause if it's 5 divided by—50 divided by 5 it would equal 10. 1:39:21   218. Ball Comments from other people? Bernadette? 219. Bernadette I have ano—I think I know what she's saying is, like the 50 she just put—she just plussed 50 plus 50 and made it into 100, 5 plus 5, made it into a 10, and then she divided it and put it into 10, made it 10. 220. Ball Is that what you're saying Liz? 221. Liz Yeah, sort of.

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Knowing and Learning Mathematics for Teaching 1:39:47   222. Ball Does anybody, um, have anything they could say that would help us know if we should believe this or not? Pravin and Kip? Lin, you think you have something that would help us believe it? 223. Lin Yeah. 224. Ball What? 225. Lin Cause see, really divided, those kind of problems are really the opposite of times. 226. Ball Uh huh. 227. Lin And like if—10 times 5, it would be 50. So, 50 divided in 5 would be 10. 1:40:20   228. Ball What does she mean 10 times 5 is 50. What does she mean by that? Anybody? Anybody have any thoughts about that? Jillian? What do you think? What does she mean 10 times 5 is 50? 229. Jillian When she writes it up there it looks like—like she writes, you know, if the number has three numbers she writes the first two numbers. And if it has two numbers, she writes the first number. 231. Ball It seems like maybe we should go on right now. I'm not sure that people—Rania and Bernadette—other people seem to be working on other things and I'm not sure that everyone is thinking about whether or not we should believe this. I'm not sure should—we should have included this on our list until we have some way of showing that we know that it's right. Like, remember before when Rania explained one plus one plus one plus one plus one plus one plus one plus three. She proved to us that it made sense. But right now we don't have any way of really knowing if these are right or not. And I'm not sure we should have them on our list unless we have a way to show that they make sense. Lin? 232. Lin I have one. 233. Ball Okay, I'm going to just put a bracket around this for right now. That doesn 't mean that it's wrong, but until we have some way of deciding if it's right we're not sure. Okay? Who has something different? 234. Student Um, 1-200 take away 190 is 10.

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Knowing and Learning Mathematics for Teaching Table 1. Class list as of September 19, 1990 NAME GENDER RACE COUNTRY ENGLISH PROFICIENCY HOW LONG AT THIS SCHOOL1 Benny M White Ethiopia fluent 3 years Bernadette F White Canada native speaker just arrived Charles M Asian Taiwan developing 2 years Christina F African-American U.S.A. native speaker 1 year David M Asian Indonesia developing 3 years Jillian F White U.S.A. native speaker 3 years Kip M African Black Kenya fluent 3 years Lin F Asian Taiwan fluent 2 years Liz F White U.S.A. native speaker 3 years Marta F Latina Nicaragua beginning just arrived Mick M White U.S.A. native speaker 2 years Ogechi F African Black Nigeria fair 3 years Pravin M White Nepal beginning 1 month Rania F White Egypt good 3 years Safriman M Asian Indonesia developing 12 months Sarah S White U.S.A. native speaker 2 years Shea M White U.S.A. native speaker 2 years Shekira F African-American U.S.A. native speaker just arrived 1NOTE: This column reflects the length of time the child had been in this school as of 9/19/89. 9/21/89   1:22:38 30 minutes into class.   Ball has just asked Bernadette, Rania, Mick, and Benny to tell the class their idea from 9/19/89. 92. Bernadette See, what we did is we would take any number, it wouldn't matter what number, say 200. And then we would minus, 200, then we would plus, 10, and it would always equal 10. So you could go on for, oh, a long, long time, just keep on doing that. You can get up to this. Shea How about 2000 __. Bernadette And then you minus it, and then you plus it, and then it equals 10. So, since numbers they never stop, you could go on and on and on and on and on and on and on, and I got this one right here. It's—right here. Pravin On and on and on and on—

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Knowing and Learning Mathematics for Teaching 93. Students Whoa! Student Let me see. Let me see. Student Did you put ___? Student Bernadette, we can't see it over here. Student Sarah can't see. Ogechi I can't see it anymore. 95. Ball Could you say what your idea was again? I want to write it down for us to think about. 1:23:45   With students' help, Ball writes Bernadette's idea on a poster. 1:26:50   119. Ball Okay. Now, I put a title on this piece of paper, it's important. I wrote, “Bernadette, Rania, Mick and Benny's Conjecture.” Shea What does that mean? Ball Conjecture is an important word that we're going to use this year. A conjecture is when you come up with an idea, something you think is true, that you're trying to prove to other people. How did they prove to us that they think this is true? Who understands what they were doing, that tried to prove to us that the answers would go on forever? What did they show us? What did Bernadette do on the board to try to make you believe that this is true? Jillian, what did she do? 120. Jillian She wrote it up on the board until we thought that they were right. 121. Ball Well, what kind of example did she use? Do you remember what she showed? 122. Jillian She took away 200, and then that's zero, and then she added 10. 123. Ball Right. Table, um, 4, Ogechi and Chris—I know—and Shekira— Shea That's 3. Students 3. Ball 3. The numbers are not matching the tables today. That's the problem. I'd like you listening to Jillian. She said she had a number—Bernadette had a number, then she took it away and that was zero, and then she added 10. Why does that—why does that help us to think that this is true? Why does that show that the answers would go on forever? I thought maybe people weren't sure about that. Can you—this group say anything more about why you think—oh, Sarah you think you know why that proves it? 124. Sarah Because numbers go on forever, and you can go like infinity take away infinity plus 10 equals 10. Because numbers go on and on. 1:28:42  

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Knowing and Learning Mathematics for Teaching 9/25/89   1:00:00 Three minutes into class. 12. Ball Bernadette, you want to read what you said? 13. Bernadette We'd pick any number, it doesn't matter what number, say 200. Then we'd write, 200 minus 200 plus 10 equals 10. And it always equals 10. 14. Ball Go on, there's a little more there. 15. Bernadette And numbers go on forever and—and so you could go on and on. 16. Ball Actually, she said, “So you could go on and on and on and on, and on” is what she said. Do you remember that? Do the rest of you remember Bernadette explaining that? Students Yeah Ball She said you could take any number and you could minus that number, and then you could add 10, and it would always equal 10. I want to show you a way of writing what Bernadette said, and I'd like you to copy it down too. A way of writing—she had to use a lot of words to say what she said, didn't she? But it made sense to everybody. But there's a way she could write that that mathematicians would use to write her idea, her proof. And I want to show you how you could write that. She wanted to say “any number,” didn't she? She said you could do this with any number, she just used 200 as an example, right? She could have used 50 as an example. She could have said—how would that one work? Could somebody tell me how it would work with 50? Kip? 19. Kip 50 minus 50 plus 10 equals 10. 20. Ball Right. She could have picked 74. How would it work with 74? Christina? 21. Shekira Shekira. 22. Ball Shekira. Sorry! 23. Shekira 74 take away 74, plus 10, equals 10. Ball Shea, what other number could she have picked? Shea Um, you—she could have picked like—like over a—over 10 million? Uh— 24. Ball Pick a number so we can write down one more example. 25. Shea 10 million. 26. Ball 10 million. Shea Uh— Ball Then what? Shea Take away 10 million. Equal—plu—plus 10, equals 10. 1:02:14   27. Ball Okay. But she was really saying to us any number—it doesn't matter what number. Do you see where she said that? Any number, it doesn't matter what number. And when mathematicians want to say

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Knowing and Learning Mathematics for Teaching   that instead of saying any number, it doesn't matter what number, they just sometimes use another symbol to stand for that, and we could use x, for example. And you could say x minus x—and that means you're taking any number—any time you pick a number it would fill in the place for x. Any number minus that same number, plus what? Student 10. Ball Plus 10, would equal what? Shea 10! Ball That's what she was saying. She was saying, any number minus the same number, plus 10 would equal 10. And she also said something else. She said, you—there are—the numbers that you could use to fill in for x go on and on forever. That's the other part of what she was saying. 1:03:04     Pause in the tape, but the tape continues from where it stopped. 1:03:04   Ball She assumed you knew something. What did she assume you knew? What does everybody in this class understand that allows you to make all these examples from her proof? There's something you guys all know but nobody said because you all assumed it. Does anybody—can anybody figure out what you all know? There's something you all know that we haven't even had to say, when we looked at this. Rania? 29. Rania Numbers go on and on. 30. Ball That's one thing, but we did say that. I guess a lot of people know that numbers go on and on forever, but she did say that. There's something she didn't say that you all know, that makes this easy for you, that a kindergartner might not know. Something in this number sentence that you knew that we didn't even have to talk about, and in this one, and in this one. What was it? And in this one, and in x minus x plus 10— Student Oh, I know. 1:04:00   Ball There's something you guys all know in this class that you don't even have to talk about because you all know it. Shekira, do you know what it is? 31. Shekira Anytime you take away a number, then you add a number, it's just gonna be a zero in there so— 32. Ball That's right. What you all know, and I'm going to write it here because it's important. Everybody knows—everybody in this class knows that x minus x equals zero. What do I mean by x minus x

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Knowing and Learning Mathematics for Teaching   equals zero? You all know that, and it made this easy for you. Benny, do you think you know what I mean when I wrote that? 33. Benny Well like, any number—any—any number, um, plus, uh— 34. Ball Not plus, minus. 35. Benny Not—minus, uh, another number— 36. Ball Not another number— Benny x, x— Ball The same number. Any number minus the same number— 37. Benny Would be zero. 1:05:00   38. Ball Would be zero. Is that true? Does everybody in the class know that? Students Yes Ball You all knew that any number minus itself would be zero. Right? And that was part of her proof and you—she didn't even have to tell you that because you all assumed it. That was one assumption she could make. Okay? So this is a general way that she could write what she was saying. It's easier than writing all those words. And sometimes in mathematics we can find really easy ways to say very big ideas. Bernadette had a very big idea, and there's an easy way to write her big idea. See how short this idea—this is? Lin Um hmm. 1:05:32   Ball Can somebody try explaining in their own words what that says? Looks kind of complicated but I bet everybody in here could explain it. Who'd like to try explaining what x minus x plus 10 equals 10 means? Could you try, Lin? 39. Lin Yeah. 40. Ball What? 41. Lin ‘Cause, x is like—you could pick any number, like, even in the thousands or millions — 42. Ball Right, and then what? 43. Lin Take away—you take away—the same number that—that you picked, and then—and then it would be zero ‘cause all—cause any number take away the same number would be—be zero. 44. Ball Um-hmm. And then? There's some large part that you didn't finish explaining. Here you are at zero and then what? 45. Lin And then you plus 10, so there's 10. 46. Ball Okay. I'd like everybody to write down this idea that we talked about in your notebook. And write down the part about that we assumed, that everybody knew, too. Write down that everybody in this class knows that any number minus itself is zero. You can write it in that mathematical way instead of writing all those words. 1:06:35