Workshop Sessions

PROBLEMS AND POSSIBILITIES

Workshop 2: Are You Convinced?

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TRANSCRIPT

PART 1. "TEACHERS BUILDING PROOFS"

NARRATOR: Englewood, New Jersey is a small city located just across the George Washington Bridge from Manhattan. Englewood students have recently been scoring poorly on the statewide standardized assessments. Superintendent Joyce Baynes, a former math teacher, was hired with a mandate to improve student achievement in the district's public schools.

JOYCE BAYNES: I really felt very confident that I'd be able to make a difference in the teaching and learning here in Englewood. And I really feel that we're making some impact in that way, really concentrating on student learning and concentrating on teachers understanding students so that they can learn much better than they had been learning in the past.

The city of Englewood has a very diverse picture to it. In fact, it is a very interesting city. We have families who are homeless; we have families who are "middle class;" we have families who are extremely affluent. We have youngsters who may go to private schools or other non-public schools such as parochial schools outside of the city.

Approximately 55% of the eligible school-age youngsters actually attend the public schools, so that gives us cause to be concerned about the other 45% who are not in our schools.

When I came on board, I didn't see a solid staff development plan for the teachers. The teachers, I felt, were open to new ideas. I wanted to bring on board a philosophy about teaching and learning so that I could trust when a teacher went, you know, into his or her classroom and closed the door, that there was a certain feeling about teaching and learning that would take place throughout the district.

As a former math teacher, I actually felt encouraged about my ability to move the mathematics achievement to a higher level. So, seeing mathematics as a challenging subject for many teachers and students here, I began to feel I could put myself into this; I can really share what I've done with others, and bring in people and resources that I've used to move the district ahead.

NARRATOR: To raise achievement in mathematics, Joyce is implementing a professional development initiative that encourages teachers and students to think deeply about problems and to justify and convince others about their solutions.

ARTHUR POWELL: "Englewood signal" [raises his hand, laughter]. I think we're probably ready to return to some work that we left off with yesterday...

NARRATOR: To begin the initiative, Joyce Baynes has invited Arthur Powell, Associate Professor at Rutgers University - Newark, to work with Englewood teachers, grades K through eight.

ARTHUR POWELL: I'd like to ask that you spend seven minutes reflecting and writing on the following two questions...

NARRATOR: During the summer, Arthur begins the project by engaging 30 teachers in a two-week professional development workshop. Arthur has defined clear goals for this initial work:

ARTHUR POWELL: My goals for the workshop are many-fold. One has to do with introducing to the teachers a new way of working in the classroom whereby they pay very careful and close attention to what students actually do and say. The other goal of the workshop is to get teachers to rethink how they view themselves in front of mathematics - that is, if they use their powers of perception and action while they're working with materials and then make statements about what they're doing and what they see, that eventually that will lead them to making all sorts of discoveries about mathematical relationships and patterns.

ARTHUR POWELL: ...so, spend some time in your groups talking about where you are. Finish up preparing for your presentations, and then we'll have presentations probably in about 10 or 15 minutes. OK?

NARRATOR: During the workshop, Arthur asked the teachers to work on combinations problems such as - How many different towers four blocks high can you make by selecting from blocks of two colors? For each solution, teachers explained and attempted to convince the others that they had found all possible towers, four high.

ARTHUR POWELL: In terms of teaching mathematics and in terms of mathematics itself, it's very important to get involved in looking for justifications because at the heart of mathematics is the idea that we can look at patterns and relationships and try to understand the underlying reasons why those patterns and relationships exist, given the particular mathematical objects. And in reasoning - in understanding why they exist - one is developing ideas of proof.

NARRATOR: This group is developing a justification that they have found all possible towers four-high by building up from towers one-cube high.

ANITA: ...so, the only reason we did this was in order to demonstrate how we knew there were no more, and also the reason why we set it up so organized like this, was to show how exactly you can double things, where the doubling comes from. And the way to convince others is this - is the one tower, two tower, three tower, four tower.

NARRATOR: After discussing the problem in the small groups, the groups presented their findings to the other teachers.

ARTHUR POWELL: I'd like to invite this group here to present what you found. Now remember - what we're going to be after - we're after trying to provide for each other some way of convincing each other that what you found is, in fact, all that there are.

GROUP 1 TEACHER 1: How do you know if you have all possible duplications? Well, pretty much we went through observations. This is pretty much what we based everything by. I made a little chart of how many possible blues and yellows we could have-

NARRATOR: The first group began by making a table of all possible numbers of blocks of each color. They made a table of towers four cubes high, showing five different cases for the combinations of blue and yellow cubes.

GROUP 1 TEACHER 1: I arranged the towers this way according to this chart, and then we just pretty much played with them so that there are no other possible combinations. We end up with 16 possible combinations.

ARTHUR POWELL: Yeah, you say all possible combinations. I don't quite understand how you found, how you are justifying that you found all possible combinations.

GROUP 1 TEACHER 2: It's all based on this chart, and then we just did it through trial and error. If we got a combination that was a duplicate, we knew we already had it, so it didn't work, and we just basically went on to the next one, and that was it.

GROUP 1 TEACHER 1: In this one, two, three, we kind of use a checkerboard to kind of just check and balance ourselves. And then we did the same thing here and then here and then here and then the same thing here. And these two are opposites, and that was it.

ARTHUR POWELL: Any questions for them? Okay. Robin, Tobey, and Helen, yes?

NARRATOR: Organizing by cases, this group approached the problem differently. They arranged towers by ìopposites,î pairing the cubes in each tower with cubes of the opposite color in a second tower.

TOBEY: We chose to do this in pairs, so we took four of one color and then took the opposite of it. And after we did that, we knew we'd done every possible combination of four of one color together. Then we went to using three of color together. So we took three blues and we matched it with the one yellow, and then we immediately took the opposite of it and took the three yellows and matched it with the blue, and these are the only possible combinations that you can - well, no, they're not. Sorry.

Then we took - we put the three blues on the bottom and put the opposite color on top and the same in pairs; everything was done in pairs. So, we have the single color on the bottom and the single color on top. And these were the only possible combinations that we have found where you can use three single colors together.

We then went to try and use two single colors together and this row shows all the possible combinations that you can make. And after we've exhausted all the combinations of two single colors together, we were left with using alternating colors together. We made one; we turned it opposite and matched it and felt that when we finished it, we were pretty well convinced that we have made all combinations possible.

ARTHUR POWELL: Any questions for Tobey? Comments?

ARTHUR POWELL: They're beginning to understand that in order to develop a convincing argument that they have to say more than, "Well, I've considered all the possibilities, so there can't be any others." Their level of justification, the way in which they're justifying themselves, is far more sophisticated than what they were doing before.

ARTHUR POWELL: Caroline and Patricia.

CAROLINE: Okay, what we did is we started off with four yellow cubes to make the tower. And when we realized that there was no other possibilities, we decided to go to three yellow cubes and one blue, and then what we did-

NARRATOR: Reporting for their group, Caroline and Patricia justified their solution by combining strategies used by the first and second groups. They looked carefully at the number of cubes in each category to be sure that there were no more combinations before they moved on.

CAROLINE: We started off with the yellow at the top. Is that what we did?

PATRICIA: We had blue on the top.

CAROLINE: We started with blue at the top and then we had our three yellows and then we moved down...

TEACHER: There are only four towers, cubes, so there are only four possible positions for that blue to be in, so you can only come up with four different patterns.

ARTHUR POWELL: So, you've now talked about the case of towers that contain four yellows...

CAROLINE: Three yellows and one blue.

ARTHUR POWELL: And now you're talking about the case of towers containing three yellows and one blue.

CAROLINE: Mm-hmm.

ARTHUR POWELL: Okay, so you're taking a kind of case analysis here. And then where did you go from there?

CAROLINE: After three yellows? We went with two yellows and two blue. We started off with the yellow/yellow/blue/blue and that was..

ARTHUR POWELL: As you have there.

CAROLINE: Right. And then we just, you know, we switched this around to make the next one. And we had them all lined up so we knew that there were no other combinations but the ones that we listed.

ARTHUR POWELL: You're saying because you had lined up you knew that there weren't others?

CAROLINE: Yes, just looking at them, once we had all six of them in front of us, we looked and we saw what - in ascending order going one way and you know, there were no other combination.

ARTHUR POWELL: Yeah, I don't feel as convinced as I do in your other two cases.

ARLENE: If you had yellow/yellow/blue/blue and then blue/blue/yellow/yellow, if you had to try to find another combination, to move the two again, it would give you the same thing; it would give you the same possibility. Because there are only two colors, so you have, you do the rotation another time and you know that you would get the same color again, same combination - yellow/yellow/blue/blue.

ARTHUR POWELL: So, Maria, you've helped me see that if I'm looking at towers four tall in which two adjacent cubes are to be yellow and the other two to be blue that those are the only two. But you're claiming that there are six towers four tall with two of each color. The third one that you have there?

ARLENE: ... trying one blue and still keeping the same, you'd have this.

ARTHUR POWELL: So, you're keeping two yellows together?

ARLENE: Still keeping two yellows together.

ARTHUR POWELL: Okay. So, is it the case that you thought about this - the first three that you have there - as keeping two yellows together, looking at the different ways of keeping two yellows together? Is that how you thought about it?

CAROLINE, OTHERS: Yes.

ARTHUR POWELL: Teachers often times, at least in this workshop and in others, feel uncomfortable with the way in which I'm conducting the workshop. And it raises anxiety in them, it raises frustration, and in some case, anger, because they come to the workshop with an expectation of being told: "This is how you should teach. Here's a mathematical task; you've worked on it; now let me confirm for you that you're right. Or, If you're not quite right, let me tell you how to get on the right track, or, Here's the answer." So, that's their expectation.

But instead, they're getting something very much to the contrary. I'm asking them questions; I'm asking them: are they sure when they make a response? I'm asking that they listen to each other, that the conversations that we have are not just different atoms put out into the air, but that there is some connection, there is some link in what's being said. And so they feel uncomfortable.

ARTHUR POWELL: ...Yeah, I was confused. So, that helps me through the first three of those.

CAROLINE: When we did the blue/yellow/yellow/blue, the fourth one became a reversal of that one. And then when we did the alternating, the sixth one became a reversal of the fifth one.

ARTHUR POWELL: Ah...so that's a different scheme, a different way of thinking about how you organize this. See, what we're trying to do is I'm trying to understand how did you think about the construction of these four tall towers? All right? And I'm sometimes hearing different things. Maybe you thought about them in both of these ways, I'm not sure. And then: for the case where there's one yellow and three blues underneath?

TEACHER: Like I said before with the one blue and three yellows, there are only four possible places for the one yellow cube to be in. It can be at the top, the second, third, or fourth position. Once you've done all those positions, it cannot be in any other, so you know you've done all of the possible combinations.

ARTHUR POWELL: And then?

CAROLINE: And then when we had no more yellows, we went with the four blues.

ARTHUR POWELL: And there could only be one -

CAROLINE: - way to do the four blues. And then if you look, you know, going up, you'll see that the same -

ARLENE: We get the same pattern as we did with the yellow; we used all four blues, then three blues, then two blues, then one blue, then going to no blues.

ARTHUR POWELL: Oh, interesting. So, you're saying that you could read your chart either from top down or bottom up.

CAROLINE: Yes.

ARTHUR POWELL: Uh-huh. When you worked, which way did you actually go?

ARLENE: We started with the yellows.

ARTHUR POWELL: You started top down with the yellows. Uh-huh. Does anyone else see any other patterns or have any comments to make about the chart that they have up? Does anyone see a different way of talking about that chart than what has been described? Martha?

MARTHA: They had up at the top zero blue, then they had one blue in each case, but in a different position. And that pattern occurred again. They kept, say, the blue constant, the two blue together, change the yellow. Whatever way they did it, the way they were explaining it, that's what came to mind, the concept of keeping something constant and changing another.

ARTHUR POWELL: Does any other group care to present what they have which is different from any of the presentations that we've heard so far? Yes, Anita?

ANITA: We went through a whole series of, like a chapter book... So I'll start at the beginning. When we first made all the combinations, we arranged them in eight pairs, like a lot of people did, each with its opposite. And then, when we wanted to answer the questions, we wanted to demonstrate how we knew there were only 16. And we knew that from prior experience, we wanted to connect that to the doubling effect, and that if we went to the lower tower number possibilities, we could show that it was doubling in a pattern. So we did that. We showed all the combinations of three towers and all the combinations of two towers and the combination of one tower -

ARTHUR POWELL: I'm sorry. When you say you showed all the combinations of two, you mean -

ANITA: We took yellow and blue and made two tower combinations.

ARTHUR POWELL: So, choosing from two colors, you made -

ANITA: The same two colors.

ARTHUR POWELL: - towers two tall.

ANITA: Right, and then we made towers three tall.

ARTHUR POWELL: All the different ones.

ANITA: And then we began trying to arrange the four tower combinations, the three tower combinations, and the two tower combinations in some sort of a way that would make it clear, visually, that we were doubling each time, and why we were doubling each time.

Basically, you can make all the combinations by reproducing this and then putting either a yellow or a blue on top. So, you have yellow/yellow or yellow/blue. This one, you can reproduce this, put down two blue cubes, and then on top of each one, just put either a yellow or a blue, because those are your only two other choices. And then you can do the same thing going up.

NARRATOR: Anitaís group first arranged towers in pairs, each tower with its opposite. However, to justify that they had found all possible towers, they investigated what they called a ìdoubling effect.î They started with towers one-cube high and then systematically built towers two-high and so forth.

Because each time a new cube is added to a tower, it can be either of two colors, the number of different towers doubles with each new tower height.

The resulting set of towers can be placed in an arrangement that resembles a tree.

ARTHUR POWELL: So, you're going from towers two tall to towers three tall.

ANITA: Towers two tall to towers three tall. So, each tower that you have in the two cube row becomes two more - because you have two colors. So that's how it doubles.

ARTHUR POWELL: Is anyone confused by this? [simultaneous conversation]

DORIS: I'm thinking, wow, you seem to be going through a lot of work making these shorter towers to get to the bigger ones, when the ultimate goal was a four tower, four cube tower.

ANITA: In reality, we started with the four. We made the 16 four tower combinations. But then when the question was, "Well, how do you know? how can you prove this? how can you demonstrate that there's only 16, no more than 16?," We knew that was related to the fact that it was doubling, that this was a pattern, and it wouldn't suddenly become 17. But the question then is, "Well, how do you know? Where does this pattern come from? How do you show that?" And we rearranged our little cubes so many times, and this was an easy way to see how each one in the lower level turns into two at the next level up if you have one new cube allowed to be added.

BLANCHE: I want to thank Anita's group because your explanation clarified it for me. It became a lot clearer. And we did take the doubling concept into mind and dealt with the 16 from the onset.

ARTHUR POWELL: Are there any comments or questions that you have? ... What you can do is would you create towers 10 tall with one color? [Laughter] So that I can then put them back in their boxes...

PAULA HAJAR: It's not about fun activities. It's about a whole way of looking at mathematics and mathematics teaching. And I know some of the tensions we have, even in seminars, is between coverage and going deeply. One teacher, she was so dear, she said, "This is lovely, but we don't have time for it." And he said, "Well, what's the it?" And she said, "Well, you know. Deep understanding." And then she caught herself. And by the end of the seminar, she was very much an advocate for what we were doing, and tried very hard to incorporate some of the activities. But also, she knew it was a whole way of looking at things. It wasn't just about activities.

ARTHUR POWELL: I have a question: Do you think that this problem could effectively be worked on by young students? What do you think?

TEACHERS: Yes. Yes.

ARTHUR POWELL: What do you think they might find? Do you think they might come up with the kinds of justifications and organizing schemes that you came up with?

ARTHUR POWELL: I think that, in fact, the kindergarten teachers in the workshop are the most comfortable ones with this particular approach. In some ways, kindergarten teachers see the importance of allowing students to talk, getting students to express themselves, asking students to work with each other. It's as you get into first and second and third grade that teachers begin to think of the classroom and students as working more individually and focusing their attention more on the teacher, less on students working in groups.

What we're talking about is a different pedagogical outlook, a different way of approaching the teaching of mathematics. The content is different. But the way in which you want students to engage their minds, the way in which you want students to verbalize and to externalize what's in their minds, are the same.

I would expect to see, teachers using one or two activities that we've worked on, and look into the curriculum and try to find one or two kinds of lessons that they might redirect in this new way. They will sort of incrementally begin to see that they can add on to their repertoire more and more open-ended activities, more and more tasks that have students working in groups, more and more opportunities to be asked to develop convincing arguments, more and more opportunities that they'll give to students to reflect, using writing or drawing

Hopefully each month and each semester as we go along, that they will slowly, incrementally, increase the repertoire of lessons that they give, which has a change to it.

NARRATOR: We've seen teachers presenting a number of arguments for finding all the combinations of towers four high when selecting from two colors. Which arguments are convincing, and why?

ANITA: ...how can you demonstrate that there are 16, no more than 16...

PART 2. "STUDENTS BUILDING PROOFS"

NARRATOR: The "Towers" problem that the Englewood teachers were working on came out of the Rutgers long-term study. The researchers originally presented this problem to the Kenilworth students in October, 1990.

AMY MARTINO: All right. We're going to do something a little different today. We're going to build towers today that have four stories to them. You're going to get two colors of Unifix® cubes. Your job is-

NARRATOR: The question was - How many different towers four blocks tall can you build when selecting from blocks of two colors?

TEACHER: ..and again, it's like the shirts and pants. You have to be convinced that you've found them all...

NARRATOR: On this particular day, the students spent about an hour working on the problem. The researchers wanted to find out how the focus group of students would build mathematical ideas, not just today, but over a long period of time. And this was the first in a series of carefully linked activities.

CAROLYN MAHER: We start with at least four tall. The students keep trying to do the problem until they can't find any more, even if they haven't come to organize their findings in a way that would account for all possibilities. We do not believe that you start them building towers one tall, two tall, three tall, four tall, and then they see this pattern. That was not what we were trying to do. That, to me, is a programmed way of proceeding.

NARRATOR: The researchers always tried to challenge the students with problems that would force them to invent new strategies.

CAROLYN MAHER: For instance, in the four tall tower problem, when you have two of a color inside that tower, and you produce all possible towers with two of a color, making the argument that you have them all is demanding of some interesting reasoning, like controlling for variables, keeping one row constant and changing the other. So, it pushes them to invent other approaches, heuristics - methods of solving problems - like "guess and check" is a heuristic; a random method is a heuristic; working backwards is a heuristic.

AMY MARTINO: You guys working together? Do you have any of the same towers as each other?

STUDENT: Yeah.

AMY MARTINO: Yeah? Which ones are the same?

STUDENT: This one...

NARRATOR: After spending less than five minutes making random combinations, the students started to compare their towers and eliminate duplicates.

STEPHANIE: Everything we make, we have to check. Everything we make... Let's make a deal. Everything we make we have to check.

DANA: All right. I'll always make it and you'll always check it.

STEPHANIE: Okay, you make it and I'll check it.

AMY MARTINO: How's it going guys?

JEFF AND BRIAN: We're done.

AMY MARTINO: Okay, you- What'd you get?

BRIAN: We found 17 towers.

AMY MARTINO: 17? Is there a way that you can check to be sure?

BRIAN: No.

AMY MARTINO: Is the way that you could, you know-

BRIAN: We like laid them down and we saw if they're the same or not, and they weren't. They weren't the same.

AMY MARTINO (OFF CAMERA): Stephanie, what makes you sure that you got everything?

STEPHANIE: I don't know.

DANA: Well, we just test it, like we used all of our blocks and then we had matches and the ones that matched - because one of them that matched, and we eliminate them.

AMY MARTINO: Could you have missed one?

DANA: No.

AMY MARTINO: How come? How do you know?

DANA: Because we double-checked about four times.

STEPHANIE: Okay, Dana, I'm going to try and make one more.

NARRATOR: The students recorded their findings, and most agreed that there were 16 combinations. The following day, the researchers returned to the problem. In a whole group discussion, they asked the students whether there would be more, fewer, or the same amount of combination for towers three tall.

ALICE ALSTON: So, do you think there'd be more than 16 or fewer than 16?

STUDENT: More.

ALICE ALSTON: You think there'd be more?

CAROLYN MAHER: It is very interesting to say to students - "Now you've built all towers four tall; you've convinced us, or you haven't, that you've found them all. What about three tall?" The three tall is interesting because very young children predict that there will be more towers three tall, and that surprises many teachers, many researchers, who expect them to think there are fewer.

NARRATOR: Matt suggested taking off one cube from every tower.

MATT: Take one block off each pattern. And then count up how many of these things...

ALICE ALSTON: Could everybody with your partners see if you can figure out how many there would be if there were only three. Remember that each one is to be different from all the others at your place...

CAROLYN MAHER: If you really listen carefully to what the students are saying and ask them why do they think there are more, well, what they're imagining is removing one of the rows of cubes from their four-tall towers and having more cubes to create new towers. And until they try to do that, they are really unaware that they end up duplicating towers of three that are already built. And that's very important for them to recognize why there are fewer rather than more or the same. It's important for them to understand that reversibility in their thinking.

DANA: Amy, we figured out that it's less.

AMY MARTINO: You think so? Why?

STEPHANIE: Because if we took one away, we had these. If we took one away from these, so-

AMY MARTINO: And you can't have those?

STEPHANIE: Well, we can't have them the same because-

AMY MARTINO: That's right. They have to be different.

BRIAN: ...'cause, if you take like one off the bottom or one off the top, you might have another one that's the same as that. And then you have to make like - then you can't use that because it's a match, and they have to be different...

NARRATOR: Jeff shared an interesting way to look at the problem.

JEFF: You could link - first of all, you could do this as a math problem, because you could do 16 minus 8 is 8.

ALICE ALSTON: You mean, there's something about math to it?

JEFF: Yeah, because 16 minus 8 or 8 plus 8 equals 16. And when you took the one away from each, it would be minus 1, so instead of saying minus 1, minus 1, minus 1, you could have just said 16 minus 8 or 8 plus 8.

ALICE ALSTON: I see. So it has something to do with 16 minus 8.

CAROLYN MAHER: ...and so let's talk about what they should be. First of all, how tall should they be...

NARRATOR: 16 months later, in the fourth grade, the Kenilworth students investigated towers five cubes tall.

CAROLYN MAHER: ...and you have to be able to convince us that you have found all possibilities - that there are no more or no less. Got the problem? Have fun!

STEPHANIE: Okay, we'll start out with the easiest one. One, two, three, four, five reds and five yellows.

DANA: One, two, three, four, five.

STEPHANIE: I only have four. Okay, well, stand them up straight so we know what we have.

CAROLYN MAHER: In towers five tall, to make a convincing argument that you found them all is harder except for when you have all of a color or one of a color.

SHELLY: Now we take one of these, one of these.

NARRATOR: Building towers five tall offered a richer, more complex challenge for the students to investigate. Students spontaneously invented strategies, such as making a tower and then building its "opposite".

BRIAN: ...this one matches with this.

ROMINA: Put the pairs.

BRIAN: Like the opposites.

DANA: And then I got another idea.

STEPHANIE: Well, tell me it so I can do the opposite.

DANA: I'm going to do the red - this, that-

STEPHANIE: Show me. Oh, okay, and I'll do the red - and I'll do it with the red at the top.

CAROLYN MAHER: They were holding one variable fixed, constant, and then varying the other. It was exciting that these children at a very early age were showing evidence of controlling for variables. It's lovely. And they were being exhaustive.

BRIAN: I have to do the opposite. I'll do this-

STEPHANIE: We made a pair!

DANA: No, look. Look, that's fine. That goes with this one.

STEPHANIE: No it doesn't because if you turn it around, it's the same, so that doesn't go with that one.

DANA: That one goes with that one.

STEPHANIE: Wait, let me check. Let me make sure...No that doesn't because...

ROMINA: I think we have them all.

CAROLYN MAHER: Do you think it's possible to have an odd number?

STUDENT: No.

CAROLYN MAHER: They have an odd number - 35.

MIKE: You're not supp- You can't because when you have a number, you could have the opposite. And if you would have one of this, you have another one because it's the opposite. And if you have 10 of these, you have another one that's opposite, so that makes 20.

SHELLY: We found 32.

CAROLYN MAHER: You found 32? How did you do that?

JEFF: Easy. You just go this way and then-

CAROLYN MAHER: You're tired, Jeff. Jeff, how do I know that you don't have duplicates?

JEFF: You can check: all you want.

CAROLYN MAHER: Because you checked it. How? ... That's how you checked it, you compared? How do you know you there're not 34?

JEFF: I can't make any more. My brain is tired!

CAROLYN MAHER: Your brain is tired?

CAROLYN MAHER: So you might ask us - Why did we ask them to convince us? Why do we ask them to justify? Well, we do that because beginning when they start, they solve their problem randomly. It's sort of guess and they try something. When you don't know what to do, you try something, so you'll build something. And maybe you'll notice certain kinds of patterns in your building; maybe you won't. You might just do trial and error, trial and error, trial and error. We want students to get past trial and error.

CAROLYN MAHER: Okay, let's take another set and try to convince me the same way.

JEFF: We'll show you the other set.

CAROLYN MAHER: Okay. I believe this one, too - you can have one red, right? And you have the other possibilities. I buy that.

JEFF: There's only two kinds of these because there are alternates.

CAROLYN MAHER: Okay, I buy that. All right. You're convincing me. That's great.

JEFF: This, we just ... How are we going to convince her about this one?

CAROLYN MAHER: You've got to convince me about this one. Why don't you think about this? I'm convinced about these that there are no other possibilities when you have one of a color - either one yellow or one red. Okay? I'm not so sure I'm convinced if there's two reds or if there's two yellows, so why don't you work on convincing me of that? You think about it and I'll be back, and you can call me.

CAROLYN MAHER: But they're thinking was still very, very exhaustive and it was very organized - when they had to justify their solutions. What it does, then, is it enables them to look at what they have, that they did just by hard work and drive, which we skip in school; we skip that piece of it. How awful - because we don't have time. You know, we skip that drudgery of that going through this hard, hard work we might not see the point of. We don't look enough. Because as they're going through this real intense, hard work, they're noticing things about the structure of the problem - maybe not seeing it overall, but they begin to notice relationships, they begin to notice sub-patterns, and they invent names for these. They really get to know the task well. This is what we expect mathematicians to do in their work.

JEFF: ...we tried to make like patterns...

NARRATOR: The following day, the researchers interviewed the students about their thinking on the towers problem.

JEFF: First we did this - we started out by moving the block up in each one.

CAROLYN MAHER: What did you come up with as a solution to the problem? Did you finally decide on how many?

JEFF: We decided on 32. And we kept on going up -

CAROLYN MAHER: Right.

JEFF: - And then we did the opposite of it.

CAROLYN MAHER: So, you decided on this pattern, that there were how many like that?

JEFF: It would go up to five.

CAROLYN MAHER: The interviews served multi purposes for us. It validated some of what, to us, were just theories. We had certain theories about what they were doing and what they were thinking based upon what we observed, what the cameras caught. But we weren't sure how aware they were of that thinking.

STEPHANIE: Well, 'cause this one, we had the pattern, the two, and then the two blocks up, and then the two blocks up.

CAROLYN MAHER: Yes.

STEPHANIE: And then we had the two in the middle, and we had the two here, and the two here.

CAROLYN MAHER: Okay. So, that was the other pattern you had. I'm confused though. How do you know that some of these aren't these?

STEPHANIE: Oh, thatís right. This one is this one. So, this one's -

CAROLYN MAHER: How did you deal with that yesterday? Did you end up counting things?

STEPHANIE: We ended up counting a lot over. We had 34 and we had it so we subtracted, I think, three groups, because we were down to 28 and then we added two groups.

CAROLYN MAHER: So, so, you think that that's what was happening yesterday?

STEPHANIE: Yeah, we kept, we kept finding different patterns, but we didn't check it with the other patterns.

CAROLYN MAHER: Uh-huh, okay.

STEPHANIE: It's really the same pattern in different places.

CAROLYN MAHER: Right.

STEPHANIE: It's taking one - building on one pattern. It's, okay, say I started with the pattern at the top. You're taking that pattern and you're moving it down one and then you're moving it down another and another.

CAROLYN MAHER: I'm just wondering if we can come up with a way that would make it easier to remember, because it's a nice way of trying to find them. I like your patterns, but I worry about if we're missing some or counting some twice. That's tricky. You might want to think about that: a way of trying to come up with a way to do that.

STEPHANIE: Yeah. How could you be absolutely, positively... A guess, a very lucky guess.

CAROLYN MAHER: Yeah, but math isn't a guess. Math you should be able to figure it out and be convinced. And you promise you're going to work on this?

STEPHANIE: Yeah.

CAROLYN MAHER: Okay, that's going to be great. I can't wait to talk to you about it some more. So, imagine the four towers and imagine the five towers and if you have time, imagine six.

STEPHANIE: All right, and I'll work on a way to be definite about your answer.

CAROLYN MAHER: Yeah, that would be exciting. That would be really fun. Terrific. Well, thank you so much.

STEPHANIE: You're welcome.

CAROLYN MAHER: You have a good weekend. I enjoyed this very much.

STEPHANIE: Thank you.

CAROLYN MAHER: It was good thinking, very good.

NARRATOR: About one month later, the researchers held a group interview with fourth graders: Jeff, Michelle, Milin, and Stephanie. In this session, they were especially interested in what made the students' reasoning convincing?

CAROLYN MAHER: The last one you did in class - remember what that was about?

JEFF: Robin Hood? That was the last one we did.

STEPHANIE: Towers of five.

CAROLYN MAHER: Towers of five. Do you remember what you did with those towers of five?

JEFF: Yeah.

STEPHANIE: Mm-hmm.

CAROLYN MAHER: And tell me about it. What was the problem?

MICHELLE: You had to figure out how many towers you could make, different towers you could make from five blocks up.

CAROLYN MAHER: Any five blocks?

STUDENTS (ALL): They had to be two colors.

CAROLYN MAHER: Two colors, okay. And did you figure out?

STEPHANIE: Yes.

CAROLYN MAHER: And what is it? Do you remember?

STEPHANIE: 32.

CAROLYN MAHER: You're sure of that?

STEPHANIE: Yes.

CAROLYN MAHER: How can you be so sure?

MILIN: 'Cause we checked!

CAROLYN MAHER: How can you be so sure?

JEFF: Remember we did all those - the chart, the thingies, for like all the different patterns? Remember I convinced you up in the whatch-a-ma-callit?

CAROLYN MAHER: Yes, in the room.

JEFF: So, I don't feel like convincing you again.

CAROLYN MAHER: You don't feel like convincing me again. [LAUGHTER]

NARRATOR: This session was a performance assessment to learn more about the students' reasoning.

CAROLYN MAHER: And Stephanie did try to work on towers of six and I asked all of you if you -

MILIN: So did I.

CAROLYN MAHER: You did too? If you were building towers of six, how many would there be?

MILIN: Probably 64.

CAROLYN MAHER: Why do you think 64?

MILIN: Well, because there was a pattern.

CAROLYN MAHER: What's that?

MILIN: You just times them by two.

CAROLYN MAHER: Times what by two?

MILIN: The towers by two, because one is two, and then we figured out two is two, I mean four, and then -

JEFF: You're not making much sense!

MICHELLE: See, if you had only one block up and two colors, then you would have two towers, and we figured out that the other day -

JEFF: Everything is opposite!

MICHELLE: - that you keep on doing, like two times two would be four and then...

CAROLYN MAHER: So four would be for what?

STEPHANIE: All you have to do -

MICHELLE: - four for, there would be four towers for two high.

CAROLYN MAHER: Okay.

JEFF: They're always opposite, though.

CAROLYN MAHER: Wait, let me hear what Michelle is saying.

MICHELLE: And then for this three high, you would eight towers.

CAROLYN MAHER: Do you agree with that?

JEFF: I don't know what you're talking about.

CAROLYN MAHER: Okay, let's get a piece of a paper and write down what you're saying and see if you all agree. I think Jeff hasn't been with us for a while and he doesn't know what we're talking about. But let's take one at a time.

CAROLYN MAHER: There are a couple of ways of approaching the problem. There's the notion of you can start with towers one tall, each color. And once you have the tower - let's say red and blue - you start with a red tower and now you're making a two tall tower. Well, you can either put a red on the top or a blue on the top. So, from that one, you get two. Likewise for the tower with the blue on the bottom. You could either put a red on the top or a blue on the top. From that one, two.

So, from the two towers - one tall - you generate four towers two tall. Each of those towers you can choose to either put a red on the top or a blue on the top, red on the top or blue on the top. From the eight, two more, two more, two more, two more. You could begin to generalize this idea and, later on, you can come up with a nice way of expressing your justification, which leads you to a kind of way of reasoning that we call inductive reasoning.

NARRATOR: For about half an hour, the students shared their different approaches. When it was Stephanie's turn, she presented a version of Proof by Cases to justify her solution for the number of towers three high.

CAROLYN MAHER: How about you, Stephanie?

STEPHANIE: I found it like this. I drew my lines and then I went red/red/red, blue/blue/blue, blue/red/blue, red/blue/blue, blue/blue/red, red/red/blue, red/blue/red, blue/red/red.

CAROLYN MAHER: So, what I'm hearing you say is that you're just -

MILIN: Guessing!

CAROLYN MAHER: - you believe they are eight, but you say guessing. Now, why does that sound like guessing?

MILIN: Because what if you can make more?

STEPHANIE: Okay, this is the three high, right? And you're convinced you can make eight?

MILIN: Yep!

STEPHANIE: I'm convinced I can make eight.

MILIN: But could you convince her?

STEPHANIE: Convince who? Michelle? Him?

MILIN: No. Her.

STEPHANIE: Yeah. All right. I've done this before. OK.

CAROLYN MAHER: Take another piece of paper if you want to. You've got to convince me there are eight and only eight, and no more or fewer.

STEPHANIE: All right, first you have without any blues, with just red/red/red.

CAROLYN MAHER: Okay, no blues.

STEPHANIE: Then you have with one blue - blue/red/red or red/blue/red or red/red/blue.

CAROLYN MAHER: Anything else?

MICHELLE: And you would do the same pattern for -

STEPHANIE: No, not with the blue, not with one blue -

MICHELLE: You would do it, you would do it with one red and two blues?

JEFF: You would alternate-

MICHELLE: You would do it the other way around.

CAROLYN MAHER: Let her finish. That's what you would do. You would alternate. Let's see what Stephanie does. Maybe she's-

STEPHANIE: Well, there's no more of these because if you had to go down another one you'd have to have another block on the bottom. Then you have with three blues - well, not with three blues. I'll go like this.

CAROLYN MAHER: You have no blues and now you have exactly one blue.

STEPHANIE: Now you have exactly two blue. You could put blue/blue/red; you could put red, blue and blue.

MILIN: You could put blue, red and blue.

STEPHANIE: Yeah, but that's not what I'm doing. I'm doing it so that they're stuck together.

CAROLYN MAHER: Okay.

JEFF: There should be one - there could be one with one red and then you could break it up and there's one with two reds and there's one with three reds.

MILIN: Ah, but see - you did the same thing, but there's the blue.

JEFF: There's all reds and then there's three reds, two reds. There should be one with one red. And then you change it to blue.

STEPHANIE: Well, that's not how I do it.

CAROLYN MAHER: Let's hear how Steph - We'll hear that other way; that's interesting. Okay, now, so what you've done so far is-

STEPHANIE: One blue, two blue.

CAROLYN MAHER: Okay, no blues-

STEPHANIE: One blue, two blue.

CAROLYN MAHER: One blue, and two blues, but Milan just said you don't have all two blues, and you said - Why is that?

STEPHANIE: All right, so show me another two blue. With them stuck together, because that's what I'm doing.

MILIN: In that case, here.

CAROLYN MAHER: Okay, so now what are you doing, Stephanie?

MICHELLE: What if you just had two blues and they weren't stuck together:

STEPHANIE: But that's what I'm doing. I'm doing the blues stuck together. Then we have three blues, which you can only make one of. Then you want two blues stuck apart - not stuck apart - took apart.

CAROLYN MAHER: Separated?

STEPHANIE: Yeah, separated. And you can go blue/red/blue -

CAROLYN MAHER: The pattern that Stephanie chooses in justifying that she found all possible of three tall towers selecting from two colors is an organization by cases. Now, her organization by cases does not map into the organization that some of her classmates want. She is adamant about her organization, and it works. It's not "the most elegant" organization by cases, but it's an organization by cases.

JEFF: I have a question. Do you have to make a pattern?

MICHELLE: No.

JEFF: So, then why is everybody going by a pattern?

MILIN: Because we liked it.

STEPHANIE: It's easier.

STEPHANIE: It's easier to find than just going, Ooh, there's a pattern!

MICHELLE: 'Cause if you, 'cause if you just keep on guessing like that, you're not sure if there's going to be more.

STEPHANIE: It's easier, maybe, like Shelly and Milan's pattern was to go put this in a different category.

JEFF: We know their pattern.

STEPHANIE: Okay, but what I'm saying is it's easier, it's just easier to work with a pattern since it's like -

MILIN: Oh, here's another one. Let's see -

MICHELLE: Because you might have a duplicate, and then you may not know.

STEPHANIE: It's harder to check. It's harder to check just having them like come up from out of the blue.

MILIN: Then just going like this and getting two -

JEFF: How do you know there's different things in the pattern?

CAROLYN MAHER: We have naturally, in the solution of these problems, thinking of these ways of organizing that account for all possible towers that can be built.

I'm building towers of a particular height, let's say four tall. In building these four tall towers, I could say, "How many towers can I build that have no red? Only one. How many can I build that have exactly one red?" Well, students will argue, eventually, that there are four of those, if they're four tall, one in each position. "How many can I build that have two red?" That's a little tougher. That kind of reasoning pushes students to begin to have to control for variables.

Is anything else possible? Can I build five? The students will say, "No, I can't do that because you told me the towers had to be four tall." In that kind of reasoning, they're beginning to do proof by contradiction. We see that kind of reasoning also, which is an indirect method of proof.

MILIN: And you can't make any more in this one, so you go on to the next one.

JEFF: How do you know you can't make any more?

MILIN: Because there's not any more colors.

STEPHANIE: Look, okay, start here. Start here - okay, you have the three together. The one, one blue, right? You have the one blue. How could I build another one blue?

JEFF: You can't.

STEPHANIE: All right. So, I've convinced you that there's no more one blue. All right -

MICHELLE: But if you didn't have that pattern, it would be harder to convince you.

STEPHANIE: Okay, so I've convinced you that there's no more one blue?

MICHELLE: Then you have to go to two blue.

STEPHANIE: Two blue. Here's one - all right? Two blue - you have one, blue/blue/red, then we have red/blue/blue. How am I supposed to make another one?

JEFF: Blue/red/blue.

STEPHANIE: No, this is the other. Milin gave me that same argument.

JEFF: But the thing is does it matter that they're together?

MICHELLE: She means stuck together.

STEPHANIE: Stuck together, that means like - Okay, so can I make any more of that kind?

JEFF: No.

MICHELLE: Then you have to move the three, which you're trying to make one.

STEPHANIE: Yeah, you can only make one and then you can make without blue with the three red.

MICHELLE: And then you can make two split apart.

STEPHANIE: Two split apart, which you can only make one of, and then you can find that you can - you can find the opposites right in this. All right, so I've convinced you that there's only eight?

JEFF: Yes.

CAROLYN MAHER: When Jeff says, "Why do you have to have a pattern?" that's really a very important question. My theory is that in Jeff's own exploration with the towers and his finding of many patterns and the opposites of those patterns, they got him in trouble when he was eliminating duplicates. And so all of his pattern finding became really an enormous task of trying to keep track. And I have always wondered if when he says, "Why do you have to have a pattern?" if he doesn't mean that in frustration. But there are other interpretations of what Jeff means. I wonder what other people think. Why do you have to have a pattern?

CAROLYN MAHER: How many if you're making towers of four?

MICHELLE, MILIN: 16.

CAROLYN MAHER: Do you agree, Jeff?

JEFF: Yeah. [laughter]

CAROLYN MAHER: Jeff, why do you agree? Don't let them get by so easily. This could be pressure here.

MICHELLE: Just because say you add a red or a blue, you can add a red or a blue here -

CAROLYN MAHER: Make a "Y" or something to show me...

JEFF: I understand because you can only -

MICHELLE: You can put two colors here, two colors there, two colors - and keep on going.

JEFF: Yeah, you can keep on doing two colors for each one. And that's two, four, six, eight, ten, twelve, fourteen, sixteen.

CAROLYN MAHER: And so that's the towers of-?

JEFF: Four.

CAROLYN MAHER: You made towers of five in class and what did you get?

STEPHANIE: 32.

CAROLYN MAHER: Does that work the same way?

STEPHANIE: Yeah.

MICHELLE: If you get towers of four -

STEPHANIE: The hard part is to make the pattern. Like, from now, we know how to just - Oh, you could give us a problem like how many in 10 and we could just go -

CAROLYN MAHER: How many in 10, and you'd know the answer?

STEPHANIE: Yeah, I know the answer. I figured it out. It's 1,024!

CAROLYN MAHER: 1,024.

ALICE ALSTON: Are you sure?

STEPHANIE: Uh-huh.

JEFF: Don't try to convince her.

CAROLYN MAHER: Try to convince me. [laughter]

JEFF: No!

STEPHANIE: Just give me a couple thousand Unifix® cubes!

CAROLYN MAHER: You can do that later.

NARRATOR: We have seen fourth graders naturally develop two mathematically sound methods of proof-making:

One, Inductive --- building taller towers from shorter towers, one at a time, and using known results about the shorter towers to derive results about the taller ones;

and Two, organizing towers of a given size into different cases which could be studied separately.

What are some of the similarities and/or differences in the mathematical reasoning by the teachers and students that you have observed in this program?

[End of program]