Workshop Sessions

PROBLEMS AND POSSIBILITIES

Workshop 1: Following Children's Ideas in Mathematics

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TRANSCRIPT

PART 1. "THE YOUNGEST MATHEMATICIANS"

[Student voices]

Narrator: As teachers, we rarely get to see what happens to the mathematical ideas our students have built after all the hard work we we've put in together in the course of a year.

Student voices:...3, 4, 5, 6!

Narrator: Are these ideas extended in subsequent grades? Will the mathematics my students have learned help them in their careers?

Teacher: OK, 6 times 6..

Narrator: In this workshop series, we'll have an opportunity to follow a group of children - beginning in first grade and continuing through graduation from high school - to see how students build mathematical ideas over time.

Student: ..you're going to get one more, because when we did the pattern with six, you got one more..

Student: ..and there's 2 with a b...

Narrator: Over the course of the six workshops, we'll follow step-by-step as these students connect ideas that they have built in earlier years to new thinking of ever-increasing complexity.

Narrator: And we'll visit teachers who are changing classroom practice to help their own students make the same kind of connections.

[Bell ringing]

Student: .. until the end of class.

Teacher: Until the end of class? I'll see what I can do.

[Music]

[Preschool]

Narrator: In this program, we'll set the stage for this long-term progression. Let's start by looking at very young children - before they begin their formal education. The child psychologist Lawrence Frank said, "Play is a child's work." Is there mathematical thinking in this play?

[Music]

[Voices of pre-school children]

Narrator: Prof. Herbert Ginsburg, a psychologist at Columbia University Teachers College, spends a lot of time in pre-schools.

Herb: I don't know how to do these, so you have to tell me how to do them, O.K. ?

Narrator: He investigates what children, at the age of 3 or 4, do during free play.

Herb: So I went to visit a pre-school in Manhattan. If I watch the kids when they are in their free play, they seem to be doing some really interesting things. They got involved in who has more. They got involved in building beautiful block designs.

Herb: They engaged in counting. What are they really thinking about? What are they really learning? So I started to do a detailed study of this.

Herb: We made video tapes of these kids as they're engaged in free play. We break down the 15 minute segments into one minute segments. We carefully code each of those segments.

Herb: When we do that we find that half the time they're doing something that we think we can legitimately call mathematical. By mathematical I mean everyday activities that involve numbers, that involve size comparisons, that involve dealing with patterns, with shapes. When kids are playing with blocks, they will line up a number of blocks in parallel, and they'll be all blocks the same size. They will be very careful to make a square. They'll be very careful that the right angles are correct. So we think when they're doing activities like this, they are learning some very fundamental geometric ideas.

Herb: With play-dough, they will often start out with something like a sphere. They will start systematically to cut this thing up. Okay? They cut it into pieces, which may be more or less even. Now what are they doing here? Well, there are ideas about division, there are ideas about taking away. Well, what happens when you take this thing off? It gets shortened. Cutting apart allows you to reverse the transformation, too. Say, "All right. I can cut this apart, but I put it together."

Herb: So, in effect, they're dealing with ideas about equivalence, about conservation of volumes and quantities. It's like our basic mathematical idea of changing things - when you add, when you take away, when you divide it up, when you put it back together. These are really fundamental mathematical ideas. Later on, when we deal with calculus, for example, we're dealing with ideas about change.

Herb: So they're doing little scientific experiments, they're learning how to observe, they're interested in physical phenomena, or how fast something turns, how much water do you need to put in. There's the measurement ideas. So in that kind of play there's an awful lot going on that kids don't encounter in school until much later. When these kids in pre-school are doing it, it's real; it's literally hands-on. It's something you do. It's not just reading about some of this in textbooks.

[Music]

Narrator: We've seen that students enter school with the ability to do some surprising mathematical thinking. Is it possible for students to keep up this early enthusiasm for math?

[Kenilworth, NJ]

[Band playing, voices cheering]

Narrator: Part of the answer might be found in Kenilworth, a small New Jersey town. Some of these students, now in their senior year in high school, are part of a focus group of children whose mathematical thinking has been followed in depth, year after year, for 12 years. What kind of a town is Kenilworth?

[Voices cheering]

Joanne Gocel: It's the quintessential small town. Everybody knows everybody. Lots of school spirit, lots of town spirit, lots of pride. We have a lot of different nationalities which compliment each other. If there's something at school, if there's a show at school, everyone comes, the senior citizens come, the grandparents come. We get huge crowds for football games. It's a great place to raise kids.

Mike Aiello: Kenilworth is the same as it was probably 20 years ago. It's basically the same thing. Like, everybody knows each other in this town. It's a little boring, there's not much to do. All there is is restaurants and banks. That's all we really have in this town.

Narrator: Kenilworth, with a population of under 10,000 and a median household income under $50,000, has a mixture of housing and light manufacturing.

Narrator: In 1985, the principal of the Harding Elementary School, in Kenilworth, invited Carolyn Maher from Rutgers University to work with his teachers in mathematics. At that time, no one could have guessed that what started as a project with teachers would open the door to a 12-year study about the development of mathematical ideas.

Carolyn Maher: I began my career as a mathematics teacher. I taught high school for several years, and then taught mathematics at the 2-year college level for another six or seven years. Then I moved to mathematics education at Rutgers University and discovered that I really didn't know about little children. My own work was with older students, high school students, and college-age students, and I had to learn more. So I began to go to classrooms and work in classrooms with younger students.

Carolyn: When I visited Harding elementary school in Kenilworth in 1985, I really had not fully realized how rigid the math instruction was, it was very rote; and it was sort of sad because the principal at that time was so aware that the students who moved on to the high school - their very best students - were barely making C's in mathematics. And he was very concerned. This is really before standards, this is before the reform movement in mathematics. The curriculum was very rigid, very atomistic, and very much computation-driven. And there was really no attempt to develop understanding on the part of those students.

Carolyn: We worked in the elementary school with the teachers in K-8 for 3 years. That was really a wonderful time. Now we worked with the teachers to think more deeply about the mathematics they were teaching. And we were aware that there was really no research about how students developed mathematically, when they had an opportunity from a very young age to do thoughtful mathematics.

[Voices of students in classroom]

Narrator: In 1989, just as small, reasonably priced video cameras were becoming available to researchers, Carolyn's group began to videotape the Kenilworth students working on mathematical activities. These activities, which were separate from the standard school curriculum, were designed to reveal the students' mathematical thinking.

Carolyn: Video cameras became an important tool for us - became not only a tool for us so that we could share what was happening in certain instances with the kind of exciting discoveries children were making, and the inventions they were doing, and how much fun they were having, and how deeply their thinking was. But it was also important for me to share that with my own students and my colleagues so we could talk about this and learn from it ourselves.

[Voice of student]

Carolyn (on camera): Oh did anyone else here get 34?

Narrator: The Rutgers team has collected over 2000 videotapes following the same group of students, from first grade through high school. This is the longest study about the long-term development of mathematical ideas ever conducted.

Amy Martino: This is the question. How many of the little tiny blocks make up the big block?

Narrator: One of the researchers was Carolyn's former student, Amy Martino:

Amy: My affiliation with Rutgers started back in 1988. I started out as a graduate student of Carolyn Maher's. And I was currently a middle school teacher; currently a middle school math teacher. And I came back to school full-time. I just felt, after teaching - going out in the field and teaching for a year - that there's got to be more. I've got to know more in order to be able to do this better.

Amy [to students]: The next one was the one with the,.. where Stephen had 3 different colored shirts...

Amy: I was thrown into classrooms my first year, doing research.

Amy [to students]: ... And I was curious as to how you did this, because you drew a nice picture here. Can you explain the picture to me?

Amy: I went from being more of an assistant or a helper to actually going on and doing my own research in the classrooms . Things were pretty primitive back then, 12 years ago. You had to beg, borrow, steal to get people to come that day to film. You know, they had to make time in their schedule. And, you know, we basically had these very primitive microphones. I didn't even have microphone holders.

Dana: [tapping on microphone] Testing. Testing.

Stephanie: Hi!

Amy [to students]: ...What was the order? You put down the number of pencils and the number of erasers...

Narrator: To study how mathematical thinking develops, the research team began by asking: How might children solve problems that use mathematical ideas, before the procedures to solve them were formally introduced in school? In the second grade, the researchers introduced the problem: Shirts and Pants.

[Shirts and Pants. May 1990 - 2nd grade]

Narrator: Stephen has a white shirt, a blue shirt, and a yellow shirt. He has a pair of blue jeans and a pair of white jeans. How many different outfits can he make?

Let's see how the students in the Rutgers' study approached this problem.

Stephanie: I'm going to make a shirt and I'm going to put a "W" for white.

Michael: Yeah, white shirt, white pants.

NARRATOR: The students spontaneously began to make drawings.

Dana: Blue and... a yellow shirt

Stephanie: He has a pair of blue jeans and a pair of white jeans. How many different outfits can he make?

Michael: Well, he can only make two outfits.

Stephanie: No, how many different outfits? He can make a lot of different outfits. Look, he can make white and white...

Dana: He can make all three of these shirts with that outfit.

Stephanie: Shh! You can do it in a lot of different ways. You can do white and white and that's one. By doing "W" and "W".

Stephanie: Two. Blue. Blue jeans and a white shirt.

Shh! Yellow shirt...number three could be a yellow shirt-

Dana: It can't be a yellow. A yellow shirt can't go with the white.

Stephanie: Yeah but how many outfits can he make? It doesn't matter if it doesn't match, as long as it can make outfits.

Stephanie: It doesn't have to go with each other , Dana.

Dana: What outfits can it be? It can make more if you put them mixed up. Look, I'm on my fourth one. Number four, it could be a blue shirt and blue pants.

Stephanie: Number five. It can be a white shirt, and...It can be a blue shirt...wait, did I do blue and white?

Dana: What's two?

Stephanie: It can be a yellow?

Dana: What's two?

Stephanie: Two's a blue shirt and white pants. Wait a yellow shirt... Wait did I do yellow and white? A yellow shirt and blue pants. Yellow shirt and blue pants...

Michael: Well I'm gonna do it the way you want.

Stephanie: 'Cause look. There's five combinations. There's only five combinations.

Dana: Lemme see your paper...

Stephanie: You can do this. Listen, Michael. Michael, will you listen for once? Five combinations: Number 1 - white and white. Number 2 - Blue and white. Number 3 - Yellow and white. Number 4 - Blue and blue. Number 5 - Yellow and blue.

Michael: (inaudible)

Stephanie: You can do four combinations, I'm sure of it.

Dana: Five!

Stephanie: I mean five.

Dana: Amy. Amy, we're done. We made five combinations.

Amy: What did you do? what are your combinations?

Stephanie: [to Amy] I've got white and white...

Narrator: Let's look at this segment again and see what the researchers found out about each of the students' thinking:

Narrator: Very early in the tape, Dana drew lines to show combinations.

Narrator: At that time, the school's curriculum did not include teaching multiplication in the second grade.

Narrator: But Dana's approach to the problem shows that she is capable of the kind of mathematical thinking that underlies multiplication.

Narrator: Dana's initial graph shows three shirts, with all but one, the yellow shirt, connected to two pairs of jeans. At this point Dana was influenced by her sense of fashion!

Dana: It can't be a yellow. A yellow shirt can't go with the white.

Stephanie: Yeah but how many outfits can it make? It doesn't matter if it doesn't match, as long as it can make outfits....

Narrator: She believed that yellow and white don't "go together."

Stephanie: They don't have to go with each other, Dana.

Narrator: As teachers, how often have we seen cases like this - where students come up with a logic that makes sense to them but is completely different from what we expected?

Narrator: Stephanie made drawings and wrote letters and numbers to keep track of her random attempts to find outfits. Using these representations, she found five outfits. Notice that Stephanie, on her fifth combination, first wrote "W" over the "B", then wrote a "Y". Does this notation stand for one or two outfits?

Narrator: For Michael, perhaps his sense of style required that outfits have matching colors. He created a new color of pants - yellow - that wasn't included in the problem.

Narrator: The researchers were wondering: How were these students influenced by each others' ideas?

Amy: What was fascinating was that nobody seemed to be truly bothered by the fact that they'd come up with different numerical answers to the question. To me that was very important, because it really did say you do need to give children time to build, just to build and to think. And that's what that session was for. That was really the purpose. So we really kind of left it open.

Amy (to students): Is that all the ways you can make it?

Dana: ... Yes, that's all the ways you can make it. I have the same thing.

Amy: I see that. That's really good! Okay, Stephanie, put your name on this, okay? And write to me that you found five ways, okay?

[Shirts and Pants Revisited: October 1990 - 3rd Grade]

Narrator: The first opportunity to revisit the problem came four and a half months later, when the students were in third grade.

Amy: Are we ready to start?

Students: Yes!

Narrator: In the meantime, the student's classroom teachers, who cooperated with the study, were careful not to tell the students how to solve the problem.

The wording of the problem was identical. Let's see how their thinking has grown.

Stephanie: Want me to read it out loud?

Dana: No, I'll do it... he has a pair of white jeans and a pair of blue jeans? How many outfits can he make?

Stephanie: Why don't we draw a picture?

Dana: He has a white shirt, a blue shirt and a yellow shirt.

Stephanie: He has a pair of blue jeans and a pair of white jeans.

Narrator: First the students made drawings and used letters to show the colors.

Stephanie: All right, let's find out how many different outfits you can make. Well, you can make white and white, that would be one...

Narrator: When they started counting the outfits, it was Stephanie, not Dana, who started drawing lines to connect the different shirts and pants, and both of them solved the problem this way.

Dana: ...the blue and the white...

Stephanie: You mean the blue pants and the white shirt?

Dana: Now we could have three with the blue and the blue pants. And the yellow could go with the....

Stephanie: Hold on Dana, you're going too fast. And we could have the blue and the white. That would be three. The blue and the blue, that could be four. We could have the yellow and the white, that would be five...

Dana: One, two, three, four, five, six. I have six so far.

Stephanie: I've got one, two, three, four, five. What are your other combinations? I have white and blue. I've got white and white. I've got blue and white. I've got yellow and white. What were your two other combinations?

Dana: I mean I have six. Six.

Stephanie: What were your two other combinations?

Dana: You mean one other combination. The yellow and the blue.

Stephanie: The yellow and the blue.

Dana: And the yellow and the white.

Stephanie: I put the yellow and the white.

Dana: All right.. Then the blue and the white?

Stephanie: Four, five, six. I have six too. I have six.

Dana: Amy! We're done. It's six.

Amy (off camera): Are you both convinced of that?

Dana and Stephanie: Yes

Amy: Can you explain it to me?

Stephanie: Well OK. If we have three shirts and two pants-

Dana: -and we just drew lines they all made combinations. Six combinations.

Stephanie: We made white and blue. We made blue and white.

Dana: We made yellow - blue. and Yellow - white. Six.

Amy: Is that six

Dana: Yes

Amy: What are these lines that you drew? You drew lines between the shirts and the pants.

Stephanie: So that we could make sure; so instead of we didn't do that again and say, "Oh, that would be seven, eight, nine, 10." We just drew lines so that we can count our lines and say, "Oh we can't do that again, we can't do that again."

Amy: Oh. That's very nice... And you're sure that there are six?

Dana and Stephanie: Yeah.

Amy: Positive that there aren't seven or five?

Dana: No six!

Amy: OK, now what I want you to do then, as a pair, is to open this up and take a fat marker and take either the blue or the red and draw this for me.

Dana: How about we each do one thing?

Stephanie: Dana, you can write and I'll draw, 'cause you're a better writer...

Narrator: Michael solved the problem with lines too. But he didn't make drawings. Rather, he drew lines between the words. This is interesting because last time, when he came up with a completely different answer, it wasn't clear whether Michael was hearing or noticing the solutions of his partners.

Amy: The second time around is interesting. Say, a strategy that one child had come up with in second grade, in third grade, another child had modified that strategy and was making use of it. And it's fascinating, because most of the class, that time, many children did not even recall having done that exact problem. The ones who did recall it all remembered having come up with the exact, correct number of combinations, which goes to show that, really, that first session was just so important for getting them to build strategies and to modify. But those two tapes had a heavy impact on me.

Cups, Bowls, and Plates

April, 1991 - 3rd Grade

Narrator: Since the students were able to come up with a way to solve this problem on their own, would they be able to use this strategy to solve a more complex problem?

Six months later, when the students were in April of the third grade, the researchers presented an extension of the Shirts and Pants problem.

Amy: ...first, we're going to pretend that there's a certain situation. We're going to pretend that today is somebody's birthday in your class.

Narrator: Called "Cups, Bowls, and Plates," the problem adds another choice to be considered.

Narrator: "Let's pretend that there's a birthday party in your class today. It's your job to set the places with cups, bowls, and plates. The cups and bowls are blue or yellow. The plates are either blue, yellow, or orange. Is it possible for 10 children at the party each to have a different combination of cup, bowl, and plate?"

Amy: ... it's supposed to be finger food, so we're not going to have forks and knives today. And what your job is going to be to do is that you're to make different combinations...

Dana: B, yellow, yellow.

Narrator: Stephanie and Dana worked together on the problem .

Stephanie: B, B, Y.

Dana: No, it isn't "B, yellow, yellow..."

Stephanie: B, Y, Y. B, yellow, yellow.

Dana: B, yellow, blue?

Narrator: Using their own system of notation based on letters, they came up with the answer that there are at least 10 different combinations.

Stephanie: ...and we found 10 by recording our answers. We found 12 by looking at the recording from the first problem...

Narrator: The next day, researcher Alice Alston interviewed Stephanie to explore her thinking in more depth. After reviewing her previous work with Shirts and Pants, Alice went on to ask how Stephanie could justify her total number of combinations - 12. Stephanie used the same strategy that she had used with Shirts and Pants - linking choices by drawing lines, then counting the lines.

Stephanie: ..so then, I guess what you'd do is that you'd go like this...

Narrator: Then she had an idea for a different way to solve the problem: multiplication!

Stephanie: ...and then I just thought of it at the end when I was done figuring out that that was 12 I thought, "Hey, there's three fours and four times three equals 12."

Alice: OK, what does the three represent?

Stephanie: OK. The three represents the three fours that we got. Or the three plates

Alice: OK, the three represents the one, two, three plates? What does the four represent?

Stephanie: The four represents, see each one could have could have four different ways, so you then multiply three times four and then you got 12.

Alice: ah...

Carolyn: Students, we have found, in a very natural way , like to represent their ideas symbolically. Algebraic thinking must begin early. In our first task, shirts and pants, the origins of algebraic thinking are there. The notion of the generalized solution is there. The notion of controlling for variables is there. The notion of extending that particular solution structure to more complicated problems like cups, bowls, and plates is there. So that's very, very important.

Narrator: We've just seen the teacher/researchers repeating the same, or similar problems, three times over the course of almost a year.

What can we say about the changes in the students' methods over time?

Amy (to student): ...why, why couldn't you find some more, couldn't you just look for more?

Dana: Well...

Stephanie: Before we had recorded what we had found already...

End of Part 1.

PART 2. "FROM TOWERS TO HIGH SCHOOL"

Narrator: Right after the shirts and pants activity, the Rutgers team introduced a new challenge: Towers. Students were asked to select from stacking cubes of two colors and assemble ìtowersî of a given height. This problem is from a branch of discrete mathematics called combinatorics- which is usually taught in high school or college as part of probability.

Narrator: Towers would become one of the central activities of the study.

Prof. Carolyn Maher: The towers problem is a nice introductory counting problem because you don't really need to know anything to start it.

Towers of Five - February, 1992 - 4th grade

Brian (off camera): Look at these cameras. They look they're TV cameras.

Romina (off camera): Look at that one.

Brain (off camera): They're huge.

Romina (off camera). We're going to be on TV!

Narrator: When the students were in the 4th grade, Rutgers received funding from the National Science Foundation and began an ambitious program of regular taping sessions, beginning with towers.

Carolyn: Okay, I wonder why you have these? Any idea? What do you think?

Jeff: We're gonna build with them.

Carolyn: You're going to build with them. Okay we're going to try to build towers that have certain characteristics...together. Now let's talk about what they should be. First of all, how tall are they going to be? The rules here say they are going to be five cubes high. And they also said what else about them? They're going to be-

Student (of camera): As many different towers.

Carolyn: As many different towers. And what colors are you allowed to use to build your towers? What do you think, Sebastian?

Sebastian: Red and yellow?

Carolyn: Red and yellow. Okay. Can somebody make a tower that satisfies those conditions and hold it up?

Narrator: The researchers referred to this task as "Towers of Five": How many different five-tall "towers" can the students make by selecting from blocks of two colors?

Carolyn: We choose four tall to start with because we thought it was not obvious. Then we moved to three tall. We move up to five tall because it forces them to need other strategies that they didn't need for four tall. They have more to manage and they have to organize their thinking more carefully in order to keep track.

Carolyn: Many of the problems in the strand of combinatorics came from my earlier teaching of high school and college finite math and probability. In those days, teaching those students, I found it very frustrating that they didn't find these problems easy. That they looked for formulas and they were not very self reliant in terms of building solutions. And yet I believe they should have been able to. So my earliest work with those problems began in my own teaching at the secondary and post secondary levels. So, as you can see, it was really very risky of me to pose these same tasks in a variety of different forms where children would have materials available where they could actually build and explore.

Carolyn: So it's not that we chose topics that were not school math we just chose them earlier looking to see how students did.

Carolyn: See these two? Are we going to call these the same or different?

Jeff: Same

Carolyn: We are going to call these the same - even though I am holding different blocks. I really am holding different blocks. For our purposes we are going to call these the same. And we didn't say that. Jeff, what do you think is going to be another condition about our towers?

Jeff: That you can't turn it around.

Carolyn: You can't turn it around. It has like a chimney on top. You see that? OK. How many do you think you can do under those conditions? Do we have any good estimates of how many we can build? Dana, what do you think?

DANA: Ummm.... 13.

Carolyn: Dana thinks 13. Okay. Do we have any other guesses?

Jeff: 25.

Carolyn: Jeff thinks 25. Anybody think something else? Steve?

Steve: Ahhh.. 10.

Carolyn: Steve thinks 10. What do you think Jamie?

Jamie: 10.

Carolyn: Jamie thinks 10.

Student: 15.

Student: 12.

Student: 26.

Carolyn: Sebastian?

Sebastian: 12.

Carolyn: But were not agreeing on this. Jennifer?

Student: 15.

Jennifer: Matthew?

Jennifer: 8.

Carolyn: Well I think what your going to have to do, is work on it and see how many, but remember: You have to be sure that you have no duplicates. Can't have two of the same 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 ones. 1-2-3-4-5 Reds and five yellows.

Dana: 1-2-3-4-5. Okay well stand them up straight so we know how many we have.

Narrator: The students were asked to attack the problem in their own way, and the researchers simply recorded what they did. At the end- whatever strategy they used- the students would have to justify their answers and convince the others that their thinking was correct.

Jeff: First we'll do the four big ones then we'll do the four easy ones.

Shelly: Two of these and three of them.

Jeff: Then we'll take three of these and two of these.

Shelly: And we can switch these all different ways. Then we'll take one of these and four of these.

Narrator: At the end, whatever strategy they used, the students would have to justify their answers and convince the others that their thinking was correct.

Carolyn: When the decision was made that students ought to be challenged in their thinking about math, that it shouldn't just be rote computation, memorization, the interest on our part was to see what was possible, because we didn't know. And the driving question for us was to see how particular mathematical ideas developed. We chose key mathematical ideas that made sense because it wasn't too far away from what ultimately we would expect kids to learn in school. The vision was for the students to be successful. And success - and I think that's still true today, but certainly it was at that time - was to be successful in algebra. Algebra is the first great separator. Students who don't succeed in algebra, they usually don't continue their study of mathematics. And then that's a path for them, they don't go on an academic track, and so they are eliminated from other programs as well. So, we were interested in the beginning of algebraic thinking.

Brian: Blue. White. Blue. Blue. These things stick.

Romina: Do we have them like this we only have white on the top?

Brian: Four blues and a white. Oh I got it I got the four blues.

Romina: You got the blue.

Brian: Oh God, this is a lot!

Narrator: The researchers found that after the students tried creating random towers, they quickly switched to using more sophisticated approaches - like pairing towers with opposites.

Allston: Can you tell me what your thinking about?

Brian: Well once when we find one we do just do the opposite of it.

Allston: what do you mean the opposite?

Brian: Like when we found this one out we just put two blues on the top and two whites in the middle

Teacher: Do they always have an opposite?

Brian: Yes, well not - yeah. Let's see if we have any one without an opposite.

Romina: Yeah, that's a good idea.

Brian: I found one already.

Romina: Which one?

Brian: Three blues one white and one blue.

Brian: ìTowers" is my first memory. It seems like we've been doing "Towers" since I was about eight. Back then, I thought it was pretty much fun. I enjoyed building little towers. It was very child-like, but I really pretty much had more fun than thought of it as math.

Brian: We don't have it.

Romina: Are you crazy? We have to have that!

Brian: We don't have it, we don't have it... That ain't it. Oh.

Romina (older): We called ourselves your guinea pigs. (laugh)

Brian: This one better fit in there.

Romina: Well give me some of these whites and blues. In case that we find another.

Romina: We thought that you thought that we were smart. And when you came up I thought you just wanted to experiment with what our capabilities are. And that was very scary when you're young, because none of us had any confidence.

Brian: Let's see if we have some of the same. Let's put the sames in order.

Romina: Ummm.... no. Well some of them could be the same.

Brian: Oh good idea.

Romina: This matches this.

Brian: Put the pairs like the opposites.

Narrator: Stephanie and Dana used a strategy of making pairs they called "duplicates".

Dana: We just did this one

Stephanie: And here's the duplicate ,see?

Carolyn: What do you mean by duplicate?

Stephanie: Well it's not the duplicate, it's the upside down version.

Dana: It's like if we turn this upside down it matches this one.

Stephanie: See, it goes with this.

Carolyn: I don't quite understand, would you go through that one more time?

Stephanie: Instead of going, okay look. Okay we take the design and instead of just like making a new design we take one and make the same design upside down.

Carolyn: Yeah, but how did you get this one?

Dana: This one goes to this one.

Stephanie: We were just fooling around when we came... and it does that. See?

Stephanie: So it's not actually the same but when it is turned upside down it is the same.

Carolyn: A-ha. Is that how you did all of these?

Dana & Stephanie: Most of them, yeah.

Carolyn: Most of them? Show me another.

Carolyn: We had certain ground rules in our research, because we were trying to study the development of mathematical ideas. But there were consequences of what we were doing. We said we would not show them-- we didn't intend to show them how to think about a problem. That was not in our set of rules for being researchers. We did not believe that if they worked on a problem for one or two hours, now that when they were all finished we had to tell them if they were right or wrong. We didn't have to follow that rule. We could say, "Well, maybe you need to think about that some more."

Student (off camera): 22, 24, 26, 28, 32

Narrator: By building the towers and checking for duplicates, most of the students agreed that there were 32 different combinations.

Carolyn: Now how many do you have?

Child: 33

Child: 32. Oh yeah, 33.

Carolyn: We've been interested in creating conditions whereby we can give children an opportunity to show us how they think about mathematics. So the only thing we're doing is making conditions possible. The children are doing the rest.

Carolyn: And you're all convinced you could take a stranger and tell them there are no more. Right? Now remember when I walked around and you were convinced cause you worked so hard doing this problem. Remember you really worked hard. I don't think I have ever seen a 4th grade class work so long on a math problem, ever - as you worked without stopping. Your incredible. I think your great. Let's give you a round of applause?

[4th - 8th grade]

Narrator: The study continued from 4th to 8th grade. The research team was able to videotape the same group of students year after year, solving a variety of math problems in combinatorics, probability, and algebra.

Jeff: ...man, you're wasting your time doing that. Because you're going to put all in one column, and then your gonna put the same amount in the next column, then the same in the next column, and then, of course, you have two columns worth...

Jeff: We got so in-depth on topics that that leaves an impression you know. We can talk about, you know doing the blocks in first grade. We could almost go through the problems we did: Shirts and Pants in second grade. How many people can tell you the math they were doing in second grade?

Amy: I need you to write your reasons for why you picked that. Okay? Why you picked seven yellow and three blacks?

Jeff: I have no idea.

Amy: She left you space, Jeff, yes you do.

Jeff: I don't know.

Amy: You told me why.

Jeff: Oh yeah.

Mike: It was basically like, what we thought. Like, you know, they were giving us a probability, like, questions. And those-- I liked those kind of questions. Those really get you to think.

Mike: Well since there are 16, - umm to make with those toppings you put a Sicilian crust on it and that's 16 plus...

Carolyn: These students were always up to the task. We'd give them a problem, they would work hard on it and would often take it to higher levels themselves.

Romina: Then you're going to go two three oh two three...

Romina: We thought we were real weird. What fourth grader's interested in math and arguing with their own friends about it?

Ankur: That's how she told me I still don't agree with her.

Student (off camera): It is, it's half and half.

Romina: Look okay, you have a whole pizza and you split it in half...

Amy: And we'd come in four or five times a year for a couple days each time and just film what went on. Children working in small groups, encouraging them to discuss and to argue and to justify.

Romina: First when you had the pepper you got four and then when you had the sausage you got three and the mushrooms you got two and the pepperoni you had one.

Carolyn: Here is this group of students, just, you know, kids going to school, doing what was expected. And now we're challenging them with problems, and they were taking them, and solving them, in ways that were very impressive to us. We pulled back and gave them more, and they responded to more. And we pulled back and gave them more, and they responded to more. ... I hope that we can show that under certain conditions, and these conditions were not in place all of the time, remember we went in six times a year in the first few grades and then maybe four times a year in the middle grades and so on and so forth. That that kind of an intervention seemed to be very very important for this group of students And it wasn't - We did not change all of the curriculum, we did not change, we didn't throw out textbooks. We didn't say teachers don't do what you do. What we tried to do is bring into the curriculum thoughtfulness and the notion it's so very important to understand fundamental ideas. The building blocks really have to begin very early.

[Fall, 1996]

Narrator: When the students began 9th grade the Kenilworth high school was closed.

Nancy Baton: The regional district served six different communities and had four high schools. Because the cost per pupil was about $15,000, the taxpayers were very concerned about it, and the superintendent had a very difficult decision to make, but he decided to close one of the four regional schools. And they picked on Kenilworth, which was the newest building in the regional district. And so I closed the building. But the residents of Kenilworth were very upset and concerned. After all, this school is really the hub of the community.

Narrator: The study continued during this period in living rooms and kitchens. The students attended 9th grade at the regional school. The subject was geometry, the approach: strictly textbook based.

Mike: I'd sit in class. My name starts with an A, so I'm in the front. The teacher would just tell us, "These are the laws of geometry." I don't remember half the stuff she'd say.

Brian: In the first year of high school, I had geometry. And the way we learned was sick, I guess you can call it, because it was not taught, it was read from a book. And we never got involved. It was "Read this and see if you can do it yourself, and then I'll tell you the answers the next day." Which I don't think anybody learned too much from it.

Jeff: She had, my math teacher, had a plan to get through the entire book in one year. That was her goal. That's what we were going to accomplish. She gave us a sheet the first day of school, saying, "All right, on day 154, the homework from this chapter is due." And, you know, you totally - you could do all your homework in one night if you had the time to do it.

Carolyn: Particularly in high schools, deviations from the curricula are not encouraged for teachers. So I think we have to be careful - but I wonder what we can learn by listening to what those students are telling us. I wonder what we can learn about their own assessment of their own learning over that year.

Romina: Ask me one question about geometry, because I won't know it.

Narrator: When the students were in 10th grade, protests in the town were successful in re-opening the local school. A core group of students - Romina, Mike, Ankur, Brian, and Jeff - continued to meet in the evenings and after school.

Loretta Malina: As the time went by a lot of this was done after school and it was done on their own time. And they weren't always always just dying to get there. But they went and they had the option of dropping out whenever they wanted to, but they didn't. They kept going and kept going and going.

[Romina and Parents - October 1999 - 12th Grade]

Romina: Daddy, I'm sending this one out. This week. I need a check actually. (Laughter)

Romina: I want to go to college. As far as I can go in schooling I want to do that and everything necessary. I want to probably major in economics in something math related. Hopefully in four years graduate, get a good job, and live well. I guess that's everyone's dream though.

Romina [to parent]: Rutgers I'm done, I'm almost done with. They're going to tell me. This is the first school I'll find out. They are going to tell me by January first, but if I don't make it into that one...

Romina: My parents never were able to actually be like, "Oh this is how you or let me correct your paper or let me read this." But they're always so encouraging. They don't think I have any, I can't be held down in any way but they give me the moral support I need to do whatever I can.

Mario D'Andrea: She likes to study, you know she goes hard on the books and you know I want the best for her.

Yolanda D'Andrea: It's important that kids today learn mathematics. It's the future of this country. It is essential for everyone.

Romina: And I have to call 1-800-SATSCORE. They took them last week, and my guidance counselor told me I needed to break at least 600 in math, because I'm going into economics.

Romina: The SAT's... Everyone knows the SAT's are the most important test you'll ever take. And I've been in classes with people that have teared during the test. Because they understand how important it is. And they've come out, like - hey, I've never seen anyone look sicker. And it's just like they were put through the worst thing of their life. And it was only a test. It was a three hour test.

Romina: Daddy get out a credit card.

Mario: What, you need a credit card?

Romina: Yeah.

Narrator: Romina's score on the math section of the SAT, a test where 30% of the problems cover geometry, was a little under 600.

Mario: How much do you need? 1170?

Romina: 1170.

Mario: 1170?

Romina: That's not good Dad. I just can't take those tests. I just can't do them.

Mario: 1170.

Yolanda: Yeah.

Romina: Yeah. They told me I should be getting at least 1250 - 1300. I'm a little bit off.

Romina: It's upsetting, because I work so hard in every other aspect of my life. And then, taking a test, one test, that takes up four hours - not three hours - of my time, on a Saturday, and that's going to decide whether I get into college or not? That's really upsetting. Because those three hours, like, reflect over, like, 100,000's of hours that I put into everything else.

Mario: So what are you going to do now?

Romina: I'm going to hope they don't look at my SAT scores? (laughter)

Mario: You better hope.

[The next day at school]

Narrator: There are many different skills involved in being successful in mathematics. Experienced teachers, like Romina's AP Calculus teacher, Ralph Pantozzi, know that not all math skills get measured on standardized assessments.

Ralph Pantozzi: Doing well involves being able to actually do mathematical inquiry. To approach a problem and then say "all right, I'm approaching it this way and I am going to try different ways of going about a problem; or I'm going to look for patterns - or I'm going to look for whether the change in this number creates some change in some another number." Looking for things that might be related to the general question that their asking. And when you see kids looking for that and generating those questions themselves, before you ask them those questions, that is really the key. And that is what I've seen this class do.

Romina: ...the area at zero is zero, so you don't actually subtract anything, so your answer is "D"..

Brian (off camera): Oh man!

Narrator: Nationally, only a small percentage of students take four years of mathematics, culminating in AP Calculus. Yet, all of the students who participated in the Kenilworth study, that remained in the district, chose to take advanced placement calculus in 12th grade.

Ralph: ...OK, the area from "A" to "B", between the graph of f of x and the x axis. In between those two points. OK...

Narrator: Today, Ralph is reviewing problems involving integrals, including one that centers on finding the difference of the two areas under two similar curves.

Ralph: Draw what that says...

Romina: This year I am doing calculus and I'm actually liking it only because I feel that I understand it and we just talk about it in class like it's just, we don't talk about like it's math. We talk about it like it's an issue.

Ralph: That area in the bottom that's already shaded in is A + 2B. If we add on that top area which is always 5 above the bottom one, what's the new total going to be? The integral from A to B of F of X plus 5?

Romina: B minus A times 5, all plus A plus 2B. Only because...okay, if we move it by five up, there is going to be another rectangle there. The length of the rectangle is five and the width is B minus A.

Ralph: Romina, I didn't see, I don't see any rectangle there. All I see is that curvy thing on top. How do you know that that is b minus A times 5?

Romina: I didn't take it, I didn't do it like that, I kinda just assumed, like your just moving it...

Ralph: Go ahead.

Romina: This part, like, like this part moved up right? so this all moved up, so this part right here takes up this much. Like I saw it like that so that's A plus 2B. Because this part, the top of the graph just did that

Ralph: Oh oh oh all right, I didn't see that at all.

Romina: And underneath here is where you added the plus five and you added just a regular because like you go straight down so there's no curves and there's just a straight block.

Ralph: Fascinating.

Romina: No?

Ralph: I've never seen anyone do that before. Yeah that's right

Romina: Yeah, okay

Ralph: Do you understand what she just did?

Students: Yep. Yes. I got it.

Jeff: I was on top of everything.

Ralph: I was thinking that you would say the area of this shape up here was B minus A times 5. But you saw it in a completely different way. And just as valid.

Ralph: They end up suggesting the questions that I have in my lesson plan. And that tells me that they're doing well, because they're actually doing mathematical thinking. And that's the best evidence you can get.

Romina: So I'm thinking I'm going to have to use a lot of my background to figure out a lot of the problems that I'm going to have to deal with. I'm not sure what they're going to be. I actually have no clue. Pretty sure my career is going to revolve around mathematics. But I think it's going to revolve around other things too, but mainly mathematics. And maybe not the math everyone thinks, but the math that I've learned, in the last 10 years.

Romina: I don't even know what I can go into without math. So anywhere, any path I choose is probably going to have to deal with math..but we'll see.

[On screen text: In April, Romina found out that she was accepted by all of the colleges to which she applied, including top choices Boston College and University of Pennsylvania.]

Carolyn: It's so fascinating to see what these children can do when you give them the opportunity. When you don't tell them how to think. They naturally want to think. They naturally want to solve problems. They naturally want to make sense of their mathematics.

Narrator: We've had a chance to look at an overview of the Rutgers' long-term study. In what ways are the students in the study similar to your students? How are they different?

[End of program]