Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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Private Universe Project in Mathematics

Workshop 1
Workshop 2
Workshop 3
Workshop 4
Workshop 5
Workshop 6


Workshop Sessions


Workshop 3: Inventing Notations

Watch the video:


Part 1. "Putting it on Paper: Elementary Students Invent Notations"

[Video: Students presenting at board.]

NARRATOR: In mathematics, how do we make visible an idea, or keep track of a line of thought? From kindergarten to calculus, mathematics involves notations, symbols as surrogates for abstract ideas.[Student: ..I saw it like that. So that's A plus 2B.] [Student voices, computer noises.] Often, the goal of math education is to give students the standard notation: the written language of symbols and equations that is commonly accepted in the math profession. But, how often have you seen students come up with unique and surprising ways to express a mathematical idea? It happens more often than many teachers realize. In this program, we'll examine how giving students a chance to create their own notations can help advance learning.

STUDENT:.. starting with... peppers.


CAROLYN MAHER: What we've been finding from our research is that students find a way to represent their ideas. [STUDENT: ..and pepperoni I got again.] And these ways of representing, I'm going to call notations. Notations are a natural part of people's lives. It's a convenience. It's a handy way of keeping track, of remembering, and for sharing with other people.

ENGLEWOOD TEACHER: And then at one point we realized that we hadn't ...

NARRATOR: Last time, we observed a two-week summer workshop for teachers in Englewood, New Jersey. These teachers presented solutions to problems and tried to justify them to the group, using their own notations.

ENGLEWOOD TEACHER: ...and then, we just pretty much play with them, and so that there are no other possible combinations...

NARRATOR: In the fall following the workshop, some of these teachers began to try out activities from the workshop with their own students.

STUDENT: ...and these are the same two...


NARRATOR: How might this new approach to teaching and learning help these teachers discover more about the notations their students invent?

[Student voices in classrooms.]

CAROLYN: Children surprise us. They have wonderful ideas. They can represent their ideas in very interesting ways, in ways that would not even have occurred to us. So the teacher, in a sense, has to become a learner. The teacher is learning about other ideas, the ideas of all of those students. And they may differ, and they will differ, from the way the teacher thinks about those ideas. And they are often very brilliant, if we take the time to listen to what children do, and what they say, and what they write, and what notations they use.

NARRATOR: At the Lincoln Elementary School, fourth grader teacher, Blanche Young, introduced a challenge she brought from the workshop.

BLANCHE: Imagine yourselves as a pizza owner. You've just opened a pizza shop and you have four toppings...

NARRATOR: The problem asks, starting with the basic cheese pizza, how many different pizzas can be made by selecting from four additional toppings - sausage, pepperoni, mushrooms, and peppers?

BLANCHE: ..Okay. What were our four toppings, again?

STUDENT: Sausage...

NARRATOR: Before the class, Blanche had expected each group of students to make a written list of the different combinations they had found. However, she gave the group free rein to come up with their own notations, asking only that they share their results with the rest of the class.

BLANCHE: ... toppings, and this work together to try and get the toppings.

CHRISTINA: Oh. That looks like a "J."

JASMINE: That looks like a "T."

STUDENT : Want me to write it? ...Peppers and mushrooms...

CAROLYN: There are limitations on what a teacher can do, given classroom time. What teachers can do is encourage students to write. When their ideas get put on paper and they're recorded in a particular way, in the particular notations they choose, now there's an opportunity to share. So the notations are very, very important; it becomes the text, if you like, that is negotiated.

JASMINE: ...how plain? You see plain on the board?


JASMINE: So you have to do that whole...

ALVIN: You can put plain pizza.

JASMINE: We can?

ALVIN: Yeah of course. Don't you, when you order pizza from the store you can say, "I want a plain pizza."

JASMINE: Yeah, but it was not on the board.

ALVIN: So? When the professor was here, it was on the board.

JASMINE: No it wasn't.

ALVIN: Yes it was.

JASMINE: No it wasn't!

CHRISTINA: OK...stop arguing.

ALVIN: Forget it. Anyway, I'll just put the thing on it.

JASMINE: Make that whole half over everything on it.

BLANCHE [handing out transparency sheet]: Just in case...

JASMINE: Put all the mushroom and pepper and stuff like that over each half

ALVIN: Now we can make the ones with the two toppings....


ALVIN: Two toppings.

CHRISTIAN: That's good.

NARRATOR: Blanche is not working to make these changes alone. After the summer workshop, the facilitator, Arthur Powell, continued to support the Englewood teachers by visiting classrooms and holding after school seminars.

ARTHUR POWELL: Can you guys explain to me the list that you have?

BLANCHE: I think they need something other than the words, to keep them categorizing...

ARTHUR POWELL: I was wondering whether you thought some might want to do a report back now?

BLANCHE: Okay. All right.

BLANCHE: Is there a group that would like to get started with sharing their work with us as to what they did today?

NARRATOR: Each group that came to the overhead had a slightly different way of presenting their findings.

LISVER: ...the fifth one? We put sausages with pepperoni. The next one we put pepperoni and mushroom.

NARRATOR: Some students simply made a list. .

BLANCHE: Is anyone looking at her toppings? Do you notice anything that she has there?

CHRISTINA: She said something over.

BLANCHE: Which one is repeated?

CHRISTINA Sausage and pepperoni.

BLANCHE: Which pizza got repeated?

CHRISTINA: That one right there, and then -

LISVER: And now you're saying this one?


BLANCHE: Do you see it, Lisver?


BLANCHE: You have one of them repeated, okay? Now let Christian explain his work.

CHRISTIAN: Right over here I put pepperoni and pepper.

NARRATOR: This group made a drawing that showed all of their possible combinations on one circle.

CHRISTIAN, CON'T: ... Over here I put pepperoni and mushroom. Here I put sausage and pepper, pepperoni and sausage -

NARRATOR: Categorizing by the number of toppings, this group drew a different circle for each category that included all the toppings that would be available.

STUDENT: OK, in this pizza, in each slice, I put the different toppings. First, in this one I put pepper and mushrooms, sausage and pepperoni...

BLANCHE: So am I correct in asking you that this is the one topping choice?


BLANCHE: That's my one topping choice pizza? Okay. That's my one topping choice pizza.

DAVID: I put pepperoni... Then I put tomatoes, mushrooms...

STUDENT: This is the three choice...

BLANCHE: This should have been the three choice pizza...I think all of you have made very, very good observations with this pizza problem, and you did a tremendous job representing it in your drawings, putting it on the overhead, and explaining it to the group. But again, it's a problem that needs to be revisited again. We may see this problem again. I'd like to save your overheads. And those students that did not have an opportunity to put their work on an overhead, I will make sure you get that opportunity the next time we revisit this problem.

NARRATOR: After the presentations, Arthur and Blanche met briefly to discuss the activity. Arthur started by bringing up one of Blanche's previous concerns.

ARTHUR POWELL: ...there was some disappointment, you had some disappointment about those students who attacked the problem through drawing.

BLANCHE: Yes. When the problem was first posed to them, we had asked them to come up with a list of the different combinations of choices they could offer. And when I walked around and children were making drawings, I panicked, because that's not a list. In my mind, it wasn't a list. Yet, in their minds, it was as clear to them as the list I thought they would have, you know, come up with. And the picture list was just as valid as the word list, which has led a lot of them to come up with combinations.

CAROLYN: Children are natural thinkers. If you give them something to think about, if you give them an investigation, a problem to pursue, they have ideas. All children have ideas. And unless you know what those ideas are, you're not going to know what the appropriate intervention is, what the next step is, what the question is that you should be asking. Where to take that idea, to help the understanding grow for that child.

NARRATOR: Back in the spring of 1993, Carolyn Maher and her research team began work at a new site, the Redshaw Elementary School, in the urban district of New Brunswick, New Jersey.

ALICE ALSTON: What we're going to do today, I had some ideas and the ideas were completely changed because Dr. Davis and Dr. Maher were particularly interested in seeing how you all would go about solving some problems that some other students worked on....

CAROLYN: The Redshaw students had just finished working on the towers problem: Building towers four tall, selecting from two colors. The children did not want to pursue building towers five tall and six tall, as the children had done in other sites. They were more interested in how can you build towers four tall when you can select from three colors or four colors. So this is what they were concentrating on before they worked on the pizza problem.

ALICE: This is a problem about pizza and what I'm going to do first is hand it out, let everybody read it -

CAROLYN: You still have to be in tune for what the students are ready to do, or what they're patient with doing at that time, particularly if they have something else in mind that's also valuable and important. For them, it was the exploration with several colors.

PATRICK: "PM" means pepperoni and mushrooms.

NARRATOR: Understanding the students' interest in selecting from multiple choices to make combinations, the researchers presented the pizza problem - How many different pizzas can you make when selecting from four toppings? One of the questions they wanted to address was: In what ways would these 5th graders use notations to represent their ideas?

ARTESHIA: We have this one.

NARRATOR: Arteshia and Desiree used blocks, with yellow to represent cheese and different colors to represent each of the additional toppings.

DESIREE: ..see if my solution matches with your solution.

NARRATOR: Eboni and Kersa simply wrote out the names of the toppings.

AMY MARTINO: You have boxes around these. What's in this first box here?

KERSA: One cheese pizza, one pepper and cheese pizza, one sausage and cheese pizza, and one mushroom and cheese pizza, and one pepperoni and cheese pizza. And the total is five pizzas.

NARRATOR: Frederick wrote out the names while his partner, Marcel, put the symbols inside circles. Patrick and Benny used a series of abbreviations to represent the toppings.

PATRICK: Look. "PM" means pepperoni and mushrooms. Peppers and mushrooms. Pepperoni and sausage, mushroom alone, and just sausage alone.

CAROLYN: We deliberately wrote the problem so that two of the toppings would begin with the letter "P": peppers and pepperoni. That was deliberate, because that, in essence, forces students to think even beyond "Let's just use the first letters. Now we have to make a modification here." And they do. They do. There will be, of course, some students who write out and spell out the toppings. But students do not like to do more writing than they have to. Just as we adults look for abbreviations and shorthand, so do they, in very natural ways.

NARRATOR: Like most of the groups, Patrick and Benny organized the combinations they were finding into categories, based on the number of toppings present.

ALICE: What are these?

BENNY: A pizza with 2 toppings.

ALICE: And how many of them are there?

CAROLYN: You see so many different kinds of notations coming from individual students. And it shows the power and the potential of the students. They like very much being asked to be creative, and they responded. They were creative.

LATIMA: This is cheese - regular cheese pizza, and this is cheese with peppers. This is cheese and mushrooms-

EBONY: Just say we had cheese with all of them.

ALICE: Sure. Yeah, I understand that. And so these are the four single toppings? And what about those six?

LATIMA: These two have the two toppings; the rest of them have three. And this one has four.

NARRATOR: After about an hour of working in small groups, the students presented their findings to the class. The researchers found that each of the 9 pairs of children had at least one representation that differed from all the others, including this unique drawing, produced by Marcel and Frederick.

MARCEL: ... And this is a cheese with the pepperoni and sausage,[ALICE: That's 2 toppings.] and cheese, pepper and pepperoni, cheese, pepper and mushroom, cheese pepperoni and mushroom, cheese, sausage and pepperoni...


CAROLYN: It's so fascinating to see what these children can do. And they naturally do invent their own notations, and they naturally do invent their own ways of communicating to each other. What we learned is that their mathematical thinking was very parallel to the mathematical thinking in the other sites.

BHARPUR: ..we tried to make another in the "3's", we got a duplicate.

CAROLYN: Maybe with little twists to interpretations of the problems or with interest in pursuing it or the problems in different directions, but the findings were very parallel in all of the sites.


NARRATOR: In March of 1993, the researchers brought their investigations to a 4th grade class at the Conover Roads School in Colts Neck, a small town in rural New Jersey.


[Student voices]


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.

BRANDON: ...I'm making a graph.

AMY MARTINO: What does that mean, one-zero, one-zero?

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."]

NARRATOR: One month later, in an interview with Amy Martino, Brandon was asked to recreate his chart and account for all possibilities.

CAROLYN: The interview 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.

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?

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 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.

AMY: Mm-hmmm.

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.

AMY: What do these zeroes and ones mean? Like what does the zero represent here?

BRANDON: You have nothing on that - that's nothing. I don't know why I chose to use zeroes and ones.

AMY: Mm-hmm. I was going to ask you about that, where you got this idea from?

BRANDON: I don't know how I got it. It just popped into my head.

AMY: Oh.

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.

AMY: Can you show me what - you have them in groups here - can you show me what those groups are on here?

NARRATOR: Brandon divided his chart into groups, organized by the number of toppings.

AMY: Okay. And what group is that?

BRANDON: Okay. Here's the "ones" group.

AMY: Okay, and what does that mean, the "ones" group?

BRANDON: You only have one topping in the group.

AMY: Okay.

BRANDON: Then you could have the "twos" group, which will go about - The "twos" group is like the most.

AMY: What do you mean, "the most"?

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.

AMY: Can you convince me that there aren't any more in the "twos" group, that there aren't seven or eight?

BRANDON: You go, pepper-mushroom, that's one. Pepper-sausage, that's two. Pepper-pepperoni, 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.

AMY: Okay.

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 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.

BRANDON: So then your only choice left is having an "all" pizza, with everything.

AMY: Interesting. And what are we calling this group?

BRANDON: The "all"...I don't know what I call that. The "total."

AMY: Okay, the total. You call these the "zeros," the "one toppings," right?

BRANDON: Yeah. "Two toppings," "three toppings," "four toppings."

AMY: You call it four toppings, right? Sure. Does this problem with pizzas remind you of any other problems we've done this year?

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.

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.

AMY: This is fascinating to me.

BRANDON: I just noticed it. Then you would have - that would be the "ones" group - you only have one...

CAROLYN: He did exactly the same rebuilding of towers at that interview 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.

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?

BRANDON: If I draw a line down here like this, it would go like - sort of look like stairs.

AMY: I see.

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.

AMY: Oh! Is what you're saying to me then that, like, the yellow cube here is like a number one on your chart?

BRANDON: Yes. If we were focusing on red, a red would be a number one.

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. You said these would be like the one - yellows.

BRANDON: Yeah. These are the "ones" group.

AMY: Okay. What about -

BRANDON: Now you would start with the "two" yellow group.

AMY: Okay.

NARRATOR: Brandon referred to his notations, and demonstrated an exact correspondence between each tower he had built and each pizza on his chart.

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.

BRANDON: That would be a "two." Then you could have 'em

AMY: Yeah, what would the tower be that would like this pizza?

BRANDON: Right here you would have yellow stand for "one." So you would have a yellow "one," red "zero", yellow "one," red "zero."

AMY: I see.

BRANDON: That would be another one.

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.

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 equivalence by looking at both problems in very deep ways.

BRANDON: If we're just focusing on yellows, then the pizza with everything.

AMY: Oh, I see. Okay. And are we missing any?


AMY: You know what I'm wondering? We have this guy left, right?

BRANDON: Yeah, because we're not focusing...

AMY: Because he's the opposite of this guy?

BRANDON: Yeah, we're not focusing on red.

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.

AMY: This is neat. This is really neat, Brandon.

BRANDON: I finally found out what the red would be. Red: "zero" guy.

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?

BRANDON: Same way. It would just look like this. Here's the "ones" group, "twos" group -

AMY: One red. Okay.

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?

BRANDON: That would be the "threes" group. And just switch those around. Same thing.

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."

AMY: This is really nice. Are you convinced that you found all the towers and all the pizzas?

BRANDON: Yeah. All the towers, all the pizzas. Yeah.

AMY: They both come out to how many?

BRANDON: It's 16. Two, four, six, eight, ten, twelve, fourteen, sixteen.

AMY: Are you convinced of this now?


AMY: Yeah? This is really very nice.

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 interview 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.


NARRATOR: What do these three examples --- Englewood, New Brunswick, and Colts Neck --- have in common in terms of how notations help students justify their solutions?




STUDENT: In this pizza, in each slice I put...

NARRATOR: We've seen some of the thoughtful and creative approaches the students used as they uncovered the mathematical similarity between the towers and the pizza problem. [Brandon: ...red- zero; yellow-one; red-zero.]

In mathematics, just because everyone agrees on an answer, it doesn't mean they're right. How can you teach students the difference between feeling you're right and proving you're right?

CAROLYN: Okay. Hi, everybody. There's a problem on your table. And if you'll all take a copy - and you night want to read it yourself; then we could be sure you understand it. You might talk among yourselves.

CAROLYN: It seems, on the surface, like a very simple problem: how many different pizzas can you make when you select from two toppings? However, as in many restaurants, you're allowed to order a different topping on half of the pizza, if you choose. So how many choices do you have? So, this was a very real problem. It was something that they would encounter in their normal, everyday life. And they never thought about all possibilities before. So it was not difficult to engage them in this problem. And they quickly saw that it got complicated very fast.

NARRATOR: The students were divided into two groups, and worked on the problem for about 45 minutes.


ANKUR: Wait. O.K. Look, the plain pizza that's one. [BOBBY: Half a plain.] Then half sausage, and half pepperoni.

BRIAN: One whole plain. [ANKUR: No, wait.] One whole sausage, one whole pepperoni. There's three.

ANKUR: Now, half plain and half sausage.

BRIAN: One plain, one sausage.

ANKUR: OK, Mike you draw the pizza, and then Amy will write underneath ... Make a pie, and make it whole plain. Just put, like cheese - a cheese pizza.

BRIAN: Here, Ankur. Half pepperoni and sausage, half pepperoni. Half sausage and pepperoni and -

ANKUR: No, half plain and sausage, half pepperoni.

BOBBY: What are you doing?

ANKUR: Forget the flames, Mike. Okay? There. Now make - now put one sausage, like a sausage one.

NARRATOR: While one group of students tried to write or draw all possible combinations, the students at the other table argued over the best way to organize their answer.

STEPHANIE: Matt, that kind of graph isn't right, because it's a cheese, pepperoni, and sausage. All you're going to get there is cheese, pepperoni and sausage. You cannot put - Because it's not organized. You can't put cheese and sausage in a group. You'd have to put the cheese over here and the sausage over here. So why don't you just make - OK a little graph like this.

JEFF: Because you're going to put all in one column, and then you're going to put the same amount in the next column, and then you're going to put the same amount in the next time, and then you're going to be crossing out two column's worth. It's a waste of time.

JEFF (VO): We didn't know if we were right, most of the time, you know? I would have an idea of how to get to a certain point, and you might have the same idea how to get to it, but we'd have to - Getting there was the hardest part, and that's what we were arguing about - the right way to get there or the right way to make sure that you'd covered all the bases. You know, anyone could pick up a pen and get the right answer. But knowing how to get there, that was what we were arguing about: the right way to get there and how to make sure, how to prove. That was a big question at the time, how to prove what we needed to accomplish.

MATT: Why don't you just draw it, like -cheese and.. But that isn't organized. Keep it organized, it'll be easier.

STEPHANIE: Well, that's not - Well, how can you organize it? How do I know whether to put this under cheese or sausage? How do I know whether to put this under cheese or pepperoni

JEFF: Your graph was great. Like, you said, we should make a graph with the one toppings and the two toppings and the threes.


MATT: But it's not organized.

JEFF: It's more organized than going like this!

STEPHANIE: Yeah, because Matt [JEFF: Nobody knows what that means.] - how do you know? How do I know? You know, how do I know whether I put this under cheese or sausage? Or how do I know whether I put this under cheese or pepperoni?

MATT: Put it under the column.

STEPHANIE: But, yeah, but there's not going to be a cheese and pepperoni column, I mean, or a cheese and sausage column. That's a pizza. You don't have to make a column for that one little pizza. Do you know how many graphs that is? You know, you'd have to make, like, tons of little, separate, eeny-weeny [JEFF: Eeny-weeny.] [Laughter] graphs.

MATT: Steph! I'm just talking about this.

STEPHANIE: Yeah, but you can't put that under a column, because you don't know which column to put them under. If you tell me how to...

MATT (VO): Maybe you took your idea, and put this on it. Okay. So then you go around - another person. "What do you think about this?" "Do that and that." And he'd say, "Well, what if you put this on it?" And it kind of comes into one big, whole thing that you use to solve your problem.

BRIAN: I'm saying one plain -

ANKUR: Do what Brian says!

BRIAN: One sausage and pepperoni pizza.

BOBBY: We already have that.

ANKUR: We have that.

BRIAN: Mixed!

BOBBY: That is mixed, almost.

BRIAN: It's half and half! I mean mixed.

ANKUR: I know what he means.

[Students agree]

NARRATOR: Even though they had some disagreement over their methods, by the end of this session, both groups had come up with a preliminary answer: ten combinations.

ANKUR: Ten. Now that's seven, eight, nine, ten, ... .

CAROLYN: Okay. I think that -

JEFF: Don't tell me we're out of time!

CAROLYN: I know. Isn't that awful, Jeff?

JEFF: Ooooaaahhhh!

CAROLYN: It's really kind of disappointing to me that we do get out of time so fast.

JEFF: Why don't we eat lunch here and come back after lunch?

CAROLYN Can we come back tomorrow morning?

CAROLYN (VO): They're so committed to working these problems out that they don't want to be disturbed, and that they say "Let us have the time." Isn't it lovely? I mean, schools aren't structured to do that. But isn't it so nice when we can do that?

CAROLYN: This is a real problem, by the way. In fact, we have here Mrs. Weir, who's given the same problems for a college class. So we're not really giving you things that aren't important and the kinds of things we want you to do in the future. So think hard about this. You know, it's one thing to find them - "I think I have them all." Remember the towers, "I think I have them all?" But then there's the next question. How could you convince us that you have all possible ones?

JEFF: Why do you always have to ask that question?

MILIN: Yeah.

CAROLYN: Because that's the mathematics of it; that's when you become mathematicians. That's when you become real problem solvers.

MILIN: IF everybody agrees, then - if everybody agrees in this whole class, then can you guys -?

JEFF: Yeah, but this is just a class of 12 kids. If you go to ask another class, they might not all agree.

STEPHANIE: Besides that, you know, the person that doesn't agree could be right.

CAROLYN Let me say it another way. I have you on film in certain grades where you've all agreed, and you've been wrong. So that's the challenge to you now. That's what it is to do mathematics. That's what mathematicians do. You've taken it to the level of trying to convince, and that's what we're asking you to do. So kind of put your names on your papers, and leave them there, and we'll see you tomorrow.[BELL]

SHELLY : Like, with the Rutgers, a lot of times, we found an answer. And that usually wasn't good enough. They wanted to know, well, how did you know it was the right answer? And because there was no teacher there to tell us, "Yeah, that was right" or "That was wrong," and they didn't just tell us how to do it, you had to look at it and look at it over and over again, and compare it to everybody else's answers, and see how they came about their answers and how it compared to how you got your answer. And you went through the whole process over and over again, and then you started to branch out to different answers to see if they were right. And a lot of times, in the end, you ended up with your original answer, but you were more secure, knowing that was the right answer.

NARRATOR: The next day, the students returned to the same problem for another 45 minute session.

ALICE ALSTON: Would you all mind if we sort of worked together, if some how we worked out a way of checking your lists and your pictures and each other's list and making sure that we all agree that everything we got is right?

BRIAN: Here, a person can read out one of them, and we could say if [ANKUR: check them] we wrote them or not.

ROMINA: One plain. [Wait.] [Check.] One sausage. [Check.] One pepperoni. [BRIAN: Check.] Half pepperoni- half sausage.

MATT: What are we doing?

STEPHANIE: Figuring out our charts.

MATT: Here's what we'll do.

NARRATOR: The students spent a few minutes negotiating their justifications, and preparing charts to help them present. By now, both groups had confirmed that there were ten possible combinations.

CAROLYN: Can you sort of, in a very general way, tell me why you think ... ? You know, you really were -

STEPHANIE: We can't get any more. We've been working, we've been -

CAROLYN You should be able to have a picture in your head ... of why ... -

STEPHANIE: We've proved everything to everybody in this group. All right. What we did is we put them into columns of one - which is a whole pie, [JEFF: I just wrote mine out.] two - which is two toppings on a pie, [JEFF: Put that in you key.] and three - which is three toppings on a pie. Okay?

NARRATOR: Stephanie's group made notations to account for all of their combinations. Notice that they treated the plain, or cheese pizza, as a topping. They listed three categories of pizzas, based on the total number of toppings that were used.

STEPHANIE: Now for a whole pie, you can have cheese, you can have pepperoni, and you can have sausage. You can't have it any other way. There's no other way you can get a one topping whole pie. [MICHELLE: Why!?] Because there's only three toppings.

JEFF: Explain why.

STEPHANIE: Because there's only three toppings.

JEFF: How are you going to convince me?

JEFF: I'm not convincing you. I'm convincing her. Are you convinced?


STEPHANIE: See, she's convinced. Okay. [CAROLYN: Jeff, you're convinced, too, aren't you? JEFF: Yeah.] Two, we have halves and two toppings. Two toppings, okay? Plain old two toppings. And we have pepperoni, and then on the other side, sausage.

TEACHER: Now is cheese on there also?

STEPHANIE: Yeah, cheese is automatically on there.


STEPHANIE: Then we just put cheese on there to show you that there's, like, cheese on it, you know?

JEFF: Yeah, she was sitting there crying before, "that's not cheese, why do we call it cheese!?"

STEPHANIE: Leave me alone. You can put cheese, and then on the other half, pepperoni; cheese, and then on the other half, sausage. Or all together, mixed, no one-half the other, sausage and pepperoni.

JEFF: You sure that's it?

TEACHER: All right.

MILIN: Jeff if you ask another-

STEPHANIE: Are you convinced? Okay. Then for three, we have sausage and pepperoni on one side, and sausage on the other.

TEACHER: Oh, so you're allowed to mix the sausage and pepperoni on one side?

STEPHANIE: Yeah. Okay. And then we have sausage and pepperoni on one side and pepperoni on the other. Then we have sausage and pepperoni on one side and cheese on the other.

MICHELLE: Or half of the side is plain.

TEACHER: All right. I think I got it.

CAROLYN ... OK you're convinced? You all convinced? Okay, that's great.

NARRATOR: Brian's group also divided the pizzas into categories: whole pizzas with single toppings, halves with different single toppings, and mixed. Pizzas with two toppings, both sausage and pepperoni.

BRIAN: We know that there's no more wholes, there can't be any more.

ANKUR: There can't be wholes. We know there's no more halves. And no half and mixed.

ALICE: How do you know there's no more halves?

ANKUR: In halves, because we used all the, like, ingredients in the pizza.

ANKUR [VO]: When the Rutgers program comes over here, they always ask us to convince them or they always ask us to convince the other people in our group. While we convince, we realize that we're actually learning more, we understand the concept better, and we help others understand the concept, and everyone in the group learns together.

BRIAN: Because, uhhmm, plain, that's like considered like a topping.

ALICE: Sure.

BRIAN: Yeah, plain, you can only use two other toppings, because that's all they give you.

ALICE: Yeah.

BRIAN: So you use pepperoni as half and half, or half pepperoni and half plain. And then you use the other topping, which is sausage to put on half a pizza. Not mixed on whole ones, like half pepperoni and half sausage.

ALICE: Okay. And you had one more category?

ANKUR: Yes. We had...

BRIAN: Half pepperoni and sausage, mixed.

ANKUR: Half one side and the other side mixed. One side is half, the other side is mixed.

ALICE: OK, now say that again.

ANKUR: One side is like, mixed and the other side is, like, a whole -no, wait.

ROMINA: Like with colors.

BRIAN: Like with colors. Like one side could be all different colors.. [ANKUR: and another side the same color.]

ALICE: So one side is the mixed sausage and pepperoni?

BRIAN: Yeah.

ANKUR: And the other side is [BRIAN: just, like, one thing.] Just one thing. And so how do we write that?

BRIAN: ..it could be sausage or pepperoni.

ALICE: And that's all it could be?

BRIAN: Right. [ALICE: Why?] At the end, the one's that are non-mixed. That's all the toppings.

ALICE: Because one side is either sausage or pepperoni?

[Bell rings]

CAROLYN: Always, we try to push students a little beyond where they were. It was never about solving a particular problem. It was about looking at other problems, maybe, in this class, and seeing if they could come up with a generalization. So very early on, they were doing this. They might not have had the - quote - "standard notation" to do this. They sometimes did it in words. And when we thought they had the idea, we thought that would be the opportune time to now bring in the standard notation and see if they now re-represent their idea with the standard notation.

NARRATOR: One month after working on the problem of pizzas with two toppings and halves, the same group of 12 students met for an extended session, lasting approximately 2-1/2 hours. This time, the researchers began with the simpler problem: How many different combinations could be made when selecting from four toppings, with no half pizzas?

ALICE ALSTON: We have to make a decision. Did they say anything about halves or is this just pizzas?

JEFF: Oh, wait there's no halves. Yes, hallelujah!

ALICE: Read it, what do you think it says?

ANKUR: Wait, but it says how many different choice does...

JEFF: I don't think they make halves there. [Wholes! Wholes! Wholes!]

ALICE: I think it's just whole pizzas.


JEFF: Thank the Lord!

MATT: Cross it out!

ALICE: OK, do you all want to work for a couple minutes, and see if you can come up with something?

[Student conversation and writing]

ROMINA: ... and the plain, too.

ANKUR: What about the mixed?

JEFF: The plain!

[Student conversation]

ALICE: And then what was this pattern?

ANKUR: I started with the first one, and mixed it with the second one. That's "P" slash "S." Start with the first one mix it with the third one: "P" slash "M." And then "P" slash "PE." And then start with "S": "S" slash "M", "S" slash "PE," then "M" slash "PE."

NARRATOR: Approximately 15 minutes later, the students were confident that they had found all possible combinations.

ALICE: Did everyone come up with a solution to this one?


CLASS: Sixteen.

ALICE: Okay. If you're going to do 16, who's going to convince me of it? [ANKUR: I will. I already did.] Stephanie and Matt?

STEPHANIE: All right, uhhmm. Well, we have whole and then we have a mixed column.

MATT: Well, we have - They're thinking we have [STUDENT: Sub-titles.] the whole column and the mixed column. The sub-title. [STUDENT: That's what we got, too.]

ALICE: Okay. Whole and then mixed, and then sub-titles? Is that what you're saying?

STEPHANIE: And when we started out, we did, like, ... And then cheese, we did pepperoni, we did sausage, we did peppers and we did mushrooms. And each one of them was all by themselves. You know, nothing was ... .

ALICE: Okay. This was in your singleton category?


ALICE: How many were in that category?


ALICE: Five?


ALICE: So that was an easy one, wasn't it?


ALICE: Okay. Now, Stephanie and Matt, you're saying that your second category was sub-divided? Tell me what your first sub-division was.

MATT: Our subtitle was "the mixed ones." And what we did for the mixed ones was we started with the topping, and we added a topping. So we had -

ALICE: Ankur, this is sounding a little bit like the way you described it to me, too. How did you do it?

ANKUR: I had a pattern.

ALICE: What was your pattern?

ANKUR: I started with the first one and mixed it with the second. Like, so my first one was peppers and sausage. So I took peppers slash sausage. So I skipped the second - I started with the first one again, skipped the second one, and took the third one, "P" slash "M". And then I put peppers and skipped the second and third, and I went with the fourth one, "P" slash "PE." And then I started with the "S" and -

ALICE: And then you're sure you were finished then. And what did you do?

ANKUR: And then I started with the next, the second one. I started with S, sausage, and mixed it with mushrooms. And then sausage and pepperoni. Then I went down to the next one, mushrooms - mushrooms and pepperoni.

NARRATOR: Ankur's idea of holding one topping constant and changing the others is a strategy that Matt noticed and will use again in the next problem.

MATT: We started with peppers and pepperoni, and added.

ALICE: Okay. You say peppers and pepperoni?

MATT: And then we added.

ALICE: And you added -

MATT: Sausage. Peppers and pepperoni, with mushrooms. Then we had - then we couldn't do any more with peppers and pepperoni. So then we figured out a peppers, sausage and mushrooms.

ALICE: Peppers, sausage, and mushrooms. Yeah. Is that all?

MATT: No. And there was no more for peppers. We were convinced there was no more for peppers.

ALICE: That was all you could do with peppers? Yeah.

MATT: There was only one thing you should do with pepperoni -

ALICE: Which was?

MATT: Pepperoni, sausage, and mushrooms.

ALICE: And then you were done?

MATT: And then you have the big one, the four topping pizza, which was the pepperoni, the peppers, the sausage and the mushrooms.

CAROLYN: In most of our other sessions, and even with adults, and even with college students and high school students, if you give students the pizza problem and you ask them to account for all possibilities, that takes at least a session to do. These students did it in a matter of 10, 15 minutes, if that long. And I suspect that the complexity of the pizza with halves made this a very trivial problem for them. Just had to write it up and tell us what it was. They had to think about the idea of whole pizzas in solving the pizza with halves. And generalizing it to four toppings was very easy.

ALICE: Pizza Hut feels like they didn't get their money's worth from their consultants, and so [Student: Another pizza problem.] they're saying, OK, [Groans from Students] now I want to see if...

NARRATOR: About half an hour into this session, the researchers introduced a final problem, one that included half pizzas.

ALICE: Sure, Robert, would you read it for us ?

ROBERT: At customer request, Pizza Hut has agreed to fill orders with different choices for each half of a pizza. Remember that they offer a cheese pizza with tomato sauce. A customers can then select from the following toppings: pepper, sausage, mushrooms, and pepperoni. There's a choice of crusts: regular, thin or Sicilian, thick. How many different choices with pizzas does the customer have? List all the possible choices. Find a way to convince each other that you have accounted for all possible choices.

ALICE: Is this going to be more? Or is this going to be less?

STUDENT: It's going to be more.

STUDENT: What you do is you times it by two.

NARRATOR: The researchers deliberately chose this problem to stretch the students' thinking. The number of combinations is much larger than in the previous problems, too large to accurately count out, using trial and error. The students built on their past work, and Matt immediately came up with a system that could find the answer.

ALICE: Before you start working on it, Matt, you have an answer?

MATT: Well, I'm going start with - What you could do is you start with the cheese, and then you put a half, then you add all the rest of the toppings, the peppers, all the rest of the toppings, the pepperoni, all the rest of the toppings, the mushrooms, all the rest of the toppings.

ALICE: OK, you all want to work on it for a little while? Remember...

NARRATOR: The other students ignored Matt's solution at first, and attempted to find their own answer.

NARRATOR: A few minutes later, the researchers asked Matt to explain his strategy in more detail.

MATT: We got 120 pizzas. I figured it out. I figured it out. Some way I thought I might have been right. What I did was I got the half cheese, the half cheese- divided it in half; then I took each topping and I put it in the half. Then I went to the peppers, each topping, put it on that, put it on the side. Then to pepperoni, same thing.

ALICE: Okay, Matt, explain to me what you're saying. You're saying that you started with your cheese, and it could be with all of the others? Okay, that was how many?

MATT: That was 15. It's like Ankur, it's like Ankur did ..with the last problem. He moved down the line, and added all the other toppings as he went. So it was like this.

CAROLYN: If you think about, you know, Matt's solution, and if you think about Matt's reference to the idea that he gives credit to Ankur for presenting in the two topping choice of the earlier problem, think of what he does. You know, he makes use of all of the ideas, from the more complex problem to the simpler problem, to, again, a more complex problem, and he introduces a strategy of controlling for variables. Now he says "Well I have all the sixteen, you know?" But he talks about holding one topping constant. And then you can, on the half, you have all your choices.

NARRATOR: Matt knew, from the previous problem, that there are 16 possible combinations of toppings for whole, undivided pizzas. Matt next considered all the possible pizzas that are made up of two different halves.

MATT: So it's half cheese, and half -

ALICE: And half each of those other things.

NARRATOR: He started with a pizza that is half cheese and half other toppings. Since he had already counted a whole cheese pizza, he couldn't use cheese on the other half, and so he had to count only 15 possible combinations.

ALICE: And on this page, what do I have here?

MATT: Half pepperoni.

NARRATOR: Then he moved on to his second topping, pepperoni, holding that constant on one side of his pizza. Since he couldn't repeat either cheese or pepperoni, he counted the remaining 14 toppings.

MATT: ...pepperoni and sausage - like that...

NARRATOR: Going through his list, he eliminated the toppings that would have made duplicates, eventually accounting for each of the possible remaining combinations. Finally, he added up the numbers in each column: 16 plus 15 plus 14 plus 13, and so on, all the way down to one.

ANKUR: Is it possible to write out all different combinations?

MATT: Well, if you wrote out all the different combinations that I had -

MILIN: You'd die!

MATT: - your hand would be pretty sore.

BRIAN: All right Matt.

ALICE: Are there any duplicates in Matt's approach?


ALICE: Is everybody convinced that you've got a solution?


CAROLYN: Matt's notation was particular to Matt. You know, he had his elaborate lines to show the detail of the possibilities. He said, "Well look, you know, if you keep this constant you could have it with this topping, with this, with this - Notice the care. Now an adult might say "You could have it with any of those 15 toppings," or "Now you have 14 left." Now Matt eventually said that, but Matt, remember, was part of a group, and he had to express his idea to others. And in order to do that, he had to provide detail. And the detail was provided in the notation he used.


NARRATOR: We've seen students spontaneously creating ways of keeping track of their solutions to a problem. What notations are students using to represent their ideas and organize the pizzas?


[End of program]


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