# Description: PUP Math Brandon interview Location: Conover Road School Colts Neck, NJ Researcher: Professor Carolyn Maher

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1 Page: 1 of 8 Line Time Speaker Transcript 1. Narrator When the researchers gave them the pizzas with four toppings problem, most of the students made lists of toppings and counted their combinations. But researcher Amy Martino noticed that one student, Brandon, used a highly unusual and insightful method of keeping track of his combinations. Brandon first made a chart with the toppings arranged vertically in columns. Moving down the page, he worked methodically row by row to create his pizzas. He wrote a one in each column to represent the inclusion of a topping and a zero to indicate when a topping was not present. 2. Brandon...I'm making a graph. 3. Amy Martino What does that mean, one-zero, one-zero? 4. Brandon Well, instead of using, like, you have pepper down, or sausage down, I'm just going to put, like, a one, for like, "Yes, it's going on," and zero for "No I'm not." 5. Narrator One month later, in an with Amy Martino, Brandon was asked to recreate his chart and account for all possibilities. 6. Carolyn The was to validate what we already found in the classroom, and Amy wanted to push it further. We did not expect Brandon to do what he did. It was spontaneous. 7. Amy Okay. You want to tell me about what you're doing here, and how these turn out to be pizzas, these zeroes and ones? 8. Brandon Well, since there are three, four toppings, that is. Nothing on the pizza. And you could have one pepper on the pizza with nothing else, one mushroom on the pizza with nothing else. Then you could have a

2 Page: 2 of 8 couple sausages on the pizza with nothing else, maybe a couple pepperonis. And if you don't want to have that, you could start getting fancy and go into twos. So have a pepperoni and mushroom, nothing else, then a pepperoni-sausage, nothing else. 9. Amy Mm-hmmm. 10. Brandon Pepper and pepperoni, nothing else, and so on. Then, since we're all done with pepperoni, you could have a mushroom and sausage with nothing else. 11. Amy What do these zeroes and ones mean? Like what does the zero represent here? 12. Brandon You have nothing on that - that's nothing. I don't know why I chose to use zeroes and ones. 13. Amy Mm-hmm. I was going to ask you about that, where you got this idea from? 14. Brandon I don't know how I got it. It just popped into my head. 15. Amy Oh. 16. Carolyn Some of my colleagues were saying to me, at the time, "Maybe his father is a computer scientist, and he is exposed to binary numbers, and that's how he knows his ones and zeroes." Well, his father is a businessman. His mother was a homemaker. And as we pushed that, nope, we eliminated that possibility. Brandon didn't have a computer at home. He wasn't a person who worked on the computer all of the time. Literally, the idea of zero and one popped into his head, just as he said it. 17. Amy Can you show me what - you have them in groups here - can you show me what those groups are on here? 18. Narrator Brandon divided his chart into groups,

3 Page: 3 of 8 organized by the number of toppings. 19. Amy Okay. And what group is that? 20. Brandon Okay. Here's the "ones" group. 21. Amy Okay, and what does that mean, the "ones" group? 22. Brandon You only have one topping in the group. 23. Amy Okay. 24. Brandon Then you could have the "twos" group, which will go about - The "twos" group is like the most. 25. Amy What do you mean, "the most"? 26. Brandon You get the most out of two, because you get more choices than one, and you get more choices: pepperoni and mushroom, pepperoni-sausage, pepper-pepperoni, and that so on... So the "two" group is, like, the biggest. 27. Amy Can you convince me that there aren't any more in the "twos" group, that there aren't seven or eight? 28. Brandon You go, pepper-mushroom, that's one. Pepper-sausage, that's two. Pepperpepperoni, three. Then you can't do any more, because you already used sausage once and mushrooms once. And to tell that you already - And to see that you made duplicate, look over there, and "one." Because if you just look there, you'll see another one. But if you see a zero there, that means it's not a duplicate, because you've got nothing there. 29. Amy Okay. 30. Brandon So if there's a "one/one", then that would be the same as there. Then you get into mushrooms Carolyn He decided to keep track of his pizzas by saying it either had a particular topping, or it did not. And he did it in a very systematic

4 Page: 4 of 8 way. And as his chart reveals, he accounted for all possible pizzas, and he had 16. It was the notation he used that helped him. 32. Brandon So then your only choice left is having an "all" pizza, with everything. 33. Amy Interesting. And what are we calling this group? 34. Brandon The "all"...i don't know what I call that. The "total." 35. Amy Okay, the total. You call these the "zeros," the "one toppings," right? 36. Brandon Yeah. "Two toppings," "three toppings," "four toppings." 37. Amy You call it four toppings, right? Sure. Does this problem with pizzas remind you of any other problems we've done this year? 38. Brandon It kind of a little reminds me of the blocks, because you Narrator When Amy asked Brandon if this problem reminded him of any other problem. He asked for manipulatives, and started making towers. He showed how each topping column in his chart corresponded to one position on the tower, with a "one" on his chart representing yellow, and orange represented by a "zero." Brandon organized his answer by categories, based on the number of blocks of each color. 40. Brandon It's kind of like the pizza problem. You start off with the group. Like this would be the "ones" group. Oh yeah, I see this now. This is like the "ones" group. You only have one of the opposite color in there. This isn't how I did it, but I just noticed this. 41. Amy This is fascinating to me. 42. Brandon I just noticed it. Then you would have - that would be the "ones" group - you only

5 Page: 5 of 8 have one Carolyn He did exactly the same rebuilding of towers at that session as he did in the classroom. He found the tower and an opposite, the tower and an opposite. And he found all 16. But something happened; something happened in his head. Because he said, "Wait, I just thought of something. Just a minute." And he had these tower models right in front of him, and he reorganized them in a way that they mapped into his chart for pizzas. 44. Brandon... you have one pepperoni. That would be like - one pepperoni is like. Since we were looking at yellows, a yellow would be "one", the reds would be "zeroes." You could have one pepper, like I chose here, and right there. Then it's like stairs. If I draw a line down Amy You need a pen? 46. Brandon If I draw a line down here like this, it would go like - sort of look like stairs. 47. Amy I see. 48. Brandon Then you'd go across, draw a line down there, go across, draw a line down there, across, draw a line down there - across - So you would have, like, "one," "one," "one," "one." It's sort of like here. You have one pepperoni, one mushroom, one sausage, one pepper. 49. Amy Oh! Is what you're saying to me then that, like, the yellow cube here is like a number one on your chart? 50. Brandon Yes. If we were focusing on red, a red would be a number one. 51. Amy Okay. Well let's continue with yellow. This is interesting. I think this is really neat. Now, what would come next, with what we have here, if we want to reorganize.

6 Page: 6 of 8 You said these would be like the one - yellows. 52. Brandon Yeah. These are the "ones" group. 53. Amy Okay. What about Brandon Now you would start with the "two" yellow group. 55. Amy Okay. 56. Narrator Brandon referred to his notations, and demonstrated an exact correspondence between each tower he had built and each pizza on his chart. 57. Brandon Yellow-yellow, red-red. Same here. Because if you wanted to stand them up, it would be harder to have to stand up the paper. So it's yellow-yellow, one-one Amy I understand. 59. Brandon That would be a "two." Then you could have 'em 60. Amy Yeah, what would the tower be that would like this pizza? 61. Brandon Right here you would have yellow stand for "one." So you would have a yellow "one," red "zero", yellow "one," red "zero." 62. Amy I see. 63. Brandon That would be another one. 64. Narrator When two problems that might look different on the surface, like towers four high and pizzas with four toppings, have the same underlying mathematical structure, this is called isomorphism. 65. Carolyn Brandon recognized the isomorphism after working on pizzas. What students sometimes do is they think of one problem one way, they think of the other problem the other way, and don't see the equivalence in structure. So to recognize the isomorphism is to disclose that

7 Page: 7 of 8 equivalence by looking at both problems in very deep ways. 66. Brandon If we're just focusing on yellows, then the pizza with everything. 67. Amy Oh, I see. Okay. And are we missing any? 68. Brandon No. 69. Amy You know what I'm wondering? We have this guy left, right? 70. Brandon Yeah, because we're not focusing Amy Because he's the opposite of this guy? 72. Brandon Yeah, we're not focusing on red. 73. Amy If we had to call him a name, though Brandon Oh, this will be the "zero." Oh yeah. Since the reds would stand for "zero," this would be a "zero" guy. 75. Amy This is neat. This is really neat, Brandon. 76. Brandon I finally found out what the red would be. Red: "zero" guy. 77. Amy I wanted to ask you. Could we have done it the other way around? Could we have focused on red and gotten it to work the same way? 78. Brandon Same way. It would just look like this. Here's the "ones" group, "twos" group Amy One red. Okay. 80. Brandon The "twos" group would be the same. And then all you'd do is Amy What would these be? What would these things be? 82. Brandon That would be the "threes" group. And just switch those around. Same thing. 83. Amy Neat! Now, would we be changing the number names for red and yellow? In other words, when we did this Brandon Yeah. Now the reds would be "one" and the yellow would be "zero." 85. Amy This is really nice. Are you convinced that

8 Page: 8 of 8 you found all the towers and all the pizzas? 86. Brandon Yeah. All the towers, all the pizzas. Yeah. 87. Amy They both come out to how many? 88. Brandon It's 16. Two, four, six, eight, ten, twelve, fourteen, sixteen. 89. Amy Are you convinced of this now? 90. Brandon Yeah. 91. Amy Yeah? This is really very nice. 92. Carolyn Brandon had an opportunity to think deeply about a problem. And he had an opportunity to talk to someone about his ideas. I think we have to remember - We see Brandon and we all so impressed with what he did. And what he did was very impressive. But at that time, the schools grouped students according to math ability. They don't do that anymore. This was many years ago. And Brandon was in the lowest group. And when later we went to the teachers with what we found, with our of Brandon, and we said, "Look. Look at this! This is just absolutely brilliant. This is wonderful; this is amazing!" And they hadn't seen anything like that, they told us. Well, I think we don't see these things because we don't give students an opportunity to show us their thinking. I think the world is full of Brandons. We just don't take the time to find them and to listen to them. We don't have mechanisms to pull them out. I think they're all over.

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