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Private Universe Project in Mathematics

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

 

Surprises in Mind Transcript

December 12, 2000
Copyright © 2000 Smithsonian Institution
Astrophysical Observatory

 

Kenilworth street scenes.

Narr: It looks like the start of a pleasant spring day in an ordinary American town.

Scenes of children walking to school.

Narr: Children here, like children all over the country, head to elementary school, ready and eager to learn.

Scenes of kids entering Harding Elementary school.

Narr: But in this school, as in most American schools, there is a problem: Many of these young students are well on the way to trouble with mathematics. By the time they graduate from high school, many will join a familiar chorus.

Graduating woman in yellow gown.

Young woman: I just cannot do math. (Laughs.) I can do anything else, but not math.

Two graduating seniors in blue gowns.

Two men: (Together) I hate math.

Man One: I failed it.

Man Two: Yeah, I hate it.

Family member at graduation.

Woman: Math...ok, now, math was my difficult subject, and it still is today. (Laughs.)

Scene of bored teens in math class.

Narr: If this is the way you remember math class … it doesn’t have to be this way.

Preschoolers

Kids: 6, 8, 10!

Rutgers footage: Kids with

blocks

Narr: We’ll see that some children are learning math — and enjoying it.

 

Stephanie: Alright, so I’ve convinced you there are only 8?

Baby picks out car from box

Narr: In fact, new research shows we all have a natural ability for mathematics.

Boy writes on blackboard

Chuck Walter in front of overhead projector

Boy leaps down line on floor

Narr: If teaching gets away from drill and practice,

Chuck Walter: What if we mark this out on the floor?

Narr: and lets students search for the meaning of mathematical ideas -

Jeff writes on blackboard

Narr: children will do better in school …

Patagonia plans

Orchestra

Narr: …and become more creative problem solvers … on the job …

 

Narr: …and in the realms of imagination.

Bilbao Guggenheim

Narr: So if you think learning math is pointless, or even hopeless … it’s time to think again.

SURPRISES in MIND

 
   

Babies at NYU

Narr: The first surprise is that math starts very young.

Lisa begins experiment placing buckets, biting cracker and placing them in the bucket while getting the attention of the infant.

Lisa Feigenson: See my buckets, they’re empty, there’s nothing in there…

 

Narr: At New York University, grad student Lisa Feigenson is finding that even infants know the difference between one and two.

 

Lisa : Ok, Maya, look at this. Maya, look at this. Look.

Shot over Lisa’s shoulder, she is sitting on the floor and placing the second cracker in the bucket, then mom sets infant free.

Lisa: You can set her free...

...good girl. What did you get?

 

Narr: This investigation into the origins of our number sense was devised by psychologist Susan Carey.

Lisa: Are you going to give some to mommy?

Susan Carey interview.

Lisa putting crackers in buckets again.

Susan Carey: You want not only that babies can discriminate two different numbers, but that it has numerical meaning to them, so this is clearly a very simple numerical meaning. 3 is more than 2. No one’s ever shown that in babies this young before.

 

Narr: This time Lisa put the two crackers in first, and on the left side.

Baby walks to bucket on left.

Narr: But Jenna still makes a beeline for the bucket with two.

Lisa: What did you get? Did you get some crackers?

CU baby eating crackers

Narr: Though Carey has systematically varied all the factors that might influence infants, 80% of them choose two over one. But there is a limit to their expertise

Susan Carey interview.

Lisa puts box on table

Susan: The fact that they fail at three versus six tells us quite a lot. It tells us that they’re not just paying attention to how long you’re putting things in the box, because obviously it takes twice as long to put six in as three. They are not summing up the total amount of cracker stuff. And their number capacities are really sharply limited.

Lisa with box on table in front of baby in high chair.

Lisa: Look at that box. Look at that box…

 

Narr: Even more unambiguous evidence for infants’ concept of number comes from another experiment in Carey’s lab.

Lisa and baby are in ‘box’ experiment room, Lisa places keys in box to get baby ready for experiment, baby finds keys in box.

CU of baby sitting in high chair.

Lisa: Where’d those keys go? Where’d they go? Can you find them? Can you find those keys?

Lisa explaining experiment in the room.

CU of baby pulling car out of box.

Lisa: So we’ve set up this room to measure what babies know about number in a very natural task.

Lisa: Look. (makes clicking sound) Look.

CU of Lisa’s hands putting cars in back of box, pull out and up to Lisa explaining experiment.

Lisa pulls car out from back of box, places it in infant’s view then places it in box, where baby reaches for and gets it.

Lisa: We give the baby a chance to see the object see what color it is, see how large it is.

Lisa: There it goes.

Lisa: And then we allow the baby to reach.

Susan Carey interview.

Shot over infant’s shoulder, Lisa places two cars, one at a time, on top of box, then in box.

Susan: The babies will show by their patterns of searching, that if they’ve only seen one object go into the box, then they only expect one object in the box.

Lisa: Yeah, look at that.

Lisa: We can provide evidence for the baby that there are in fact two toys in the box.

Lisa : Yeah, you ready for your turn?

CU of baby’s hand pulling out car that is larger than other car.

Lisa explaining experiment in the room.

Two shots of infant determinedly searching for second car, Lisa moves car to front of box, infant gets car.

Lisa : So we allow the baby to reach in, retrieve one of the toys.

Lisa : And now the question is, does the baby recognize that there is in fact another toy hidden in the box and decide to search back in there.

Narr: To see if Jenna is really sure there’s another toy car in the box, Lisa is holding it out of reach.

Lisa: Here it is sweet girl.

Susan Carey interview.

Baby pulls car out of box and places it on top of box and then smiles.

Susan: And if they have seen two different objects go into the box, then they search persistently for two. When they have gotten two, they're satisfied. So they're representing exactly two. We know for sure that it is number that they are paying attention to. Because if they see a small one go in, plus another small one go in, and they bring out a big one, they're not satisfied. This work is part of a real revolution, of realizing that children bring much, much more to the table. Children are sense makers. Children have the basis for conceptual understanding.

Lisa: Yeah, you found it.

Susan: Those capacities should be engaged in education.

Baby babbling

Sign:Quiet, experiments in progress

Babies in waiting area.

Cu of baby on the floor, with a rattler in his mouth.

Mom picks up baby and follows grad student into experiment room.

Narr: So a kind of natural number sense emerges in the first year of life. And in another NYU lab, cognitive scientist Gary Marcus is discovering that babies also have a surprising ability to perceive patterns.

Two grad students are setting up an experiment, placing a video camera and adjusting lights.

 

Set up of experiment continues.

Gary Marcus in interview

Gary Marcus: When we designed this experiment, we already knew that infants could do one particular kind of learning. Infants could do something like counting or some kind of statistical analysis. And what we wanted to see was whether infants could do something else, which I call algebraic rule learning.

Set up of experiment continues.

CU of baby in experiment listening and looking at the lights.

Narr: To find out if infants can detect these abstract rules, Marcus uses sound patterns. Speakers play computer-voiced strings of nonsense syllables. When a baby’s paying attention, he looks at the lights that flash along with the sounds..

Computer generated sounds.

GRAPHICS: LAY LEE LEE

Shot from behind of Marcus and Grad student at controls of experiment.

GRAPHICS:

A B B

Gee Dee Dee

CU of baby in experiment listening and looking at the lights.

Shot from behind of baby crying.

Computer generated sounds.

Narr: Notice that each syllable string has the same pattern: a,b,b.

Computer generated sounds.

Narr: Soon enough, young Oliver gets bored.

Baby cries.

Restless baby on mom’s lap.

GRAPHICS:

A B A

Jay Day Jay

Pan to Marcus and grad student at controls of experiment

Narr: Now Marcus switches to a new set of nonsense syllables. If the pattern is still a, b, b, Oliver gets restless quickly.

Narr: But, when the pattern changes to a, b, a...

Computer generated sounds.

Narr: his interest perks up.

Over shoulder shot of mother and baby in experiment.

CU of flashing light.

Pan from tight screen shot of mom and baby in experiment to shot of Gary looking on.

Narr: 46 out of 48 infants tested showed this same surprising ability to detect abstract patterns and grasp underlying rules.

Sound of light clicking on and off.

Narr: That’s what makes the implications of this experiment so intriguing.

Computer generated sounds.

Slow shot of CU of baby’s face.

Marcus interview.

Street scene of adults and kids walking.

Gary: They just sit there, and they're interested in these sounds and they're trying to get the information that's there. They're trying to analyze the information that's there. And if they do that in this sort of sterile lab situation they must be doing it all the time in the real world. They must always be analyzing the world, looking for deeper, more abstract patterns.

Exterior of Harlem school. Mother and child enter school.

Girl reaching for wooden blocks with boy

Herb kneeling in block area with kids.

Narr: Do kids continue to analyze the world? Do they keep developing mathematical concepts on their own?

Narr: Questions like these prompted Herbert Ginsburg, professor at Columbia Teacher’s College, to take a close look at pre-schoolers.

Anna Housley reading a counting story to children.

They count along with her.

Anna Housley: She could count dogs. And she carefully pointed her finger at each one and said: One, two, three, four, five, six, seven!

Girl: Seven, yeah!

Herb Ginsburg observing children, slow zoom out to kids.

Shots of kids playing.

Herb Ginsburg interview.

Herb Ginsburg: Several years ago I started to do observations in a day care center in Manhattan. I started looking at their free play very closely to find out what they are doing on their own.

Herb: It became evident to me that they were spending much more time than I ever would have expected, doing interesting mathematical work.

Kids playing

Herb: It is the kind of thinking they do when they are faced with certain kinds of challenges.

Anna interview.

CU faces of kindergartners.

Anna: Kids are thinking mathematically when they are counting how many peas there are on their plate. And they are thinking mathematically when they notice that a box fits inside another box, but it doesn't fit in this box over here. That's all mathematical thinking.

Anna leading circle.

Anna: It's good to see everyone today. We’re gonna do some stuff like we’ve done before…

Narr: Graduate student Anna Housley is helping Herb Ginsburg explore how four year- olds think about math.

CU faces of kids counting

Anna doing "counting by twos" with kids.

Narr: All over New York City, from private day care centers to this public school kindergarten…

Anna: Good job!

… they find the same surprising results

Anna interview.

Anna: We saw a lot of children just naturally like want to count things. They like to count high.

Kids counting with Anna.

CU of children's faces.

Anna and kids: 68,69, ..Who knows what number comes after 69? Daryl?

Daryl: Sixty-ten.

Anna: Sixty-ten is exactly right. It is sixty plus another ten. But we don't say sixty ten...

Boy: 70!

Anna: Thank you, 70 comes next.

Kids with wooden blocks

 

Monitor shot of kids on video.

Over shoulder shot of Herb watching video, pan over to grad students watching.

Narr: And beyond counting, Ginsburg and his students find that children spend up to 50% of their play time on activities that reflect their natural math sense. That includes things that are relatively easy to spot, like comparing sizes …

Girl demanding blocks from others.

Shot of grad student’s on-looking face.

Girl leaning over structure and adjusting the triangles on the roof.

Herb: She knows the difference clearly between little ones and big ones.

Narr: … and things that most parents might not recognize as mathematical, like identifying shapes…

Girl: The triangle… oooohhhh.

Shot of Herb watching video, holding remote control.

Girl placing block to balance roof of structure.

Narr: … using spatial relations…

Herb: See, she knows she needs to support that, that was planned ahead.

Girl and boy working on structure.

Narr: … and building with balance and symmetry.

Tilt down block structure.

Narr: So from infancy to pre-school, children naturally develop more and more sophisticated concepts. These are firm foundations they can build on in the future.

French school playground area, teachers

 

Sign "Gymnase de Mondetour"

Narr: But in most places, for most kids, school doesn’t build on their natural affinity for mathematics.

One lone kid runs into school.

 
 

French kids chanting in French.

Shots of hallway, filled with shoes and coats.

Narr: In this French school, as in most American schools, the accent is on rote learning, on number facts and formulas

Several shots of kids in class, apparently working on a math problem.

French kids in class speaking French.

Stanislas Dehaene: Now why do some children succeed very well in mathematics and why do others fail? My personal opinion is that the critical factor is whether you learn to love or you learn to hate mathematics.

Teacher writing on chalkboard

Ecu writing sums on a pad

Narr: How do you learn to love math? For neuroscientist Stanislas Dehaene, it’s breaking through to the meaning behind the numbers. .

Little girl writing sums

Pan to little boy

Stanislas Dehaene interview.

Stan: I remember very vividly, one day, when the teacher asked us to bring the bikes to school. And we measured the diameter of the wheels. We also measured how far they would go when the wheel would do one turn. And there was this miracle, that regardless of the size of the kid and regardless of the size of the bike, the division always got the same results, 3.14, which of course is pi. That was wonderful, I remember this day as a wonderful day where the magic of mathematics was really shown to me.

Stan and assistant are measuring Emile’s head with a tape measure and marker.

Narr: Now Dehaene is trying to uncover just how our brains perform their mathematical miracles.

Shots of Stan and assistant place electrodes on Emile’s head.

Stan and assistant speaking French.

 

Stan: Basically we were trying to find evidence that there really are two different ways of doing calculation. One way is to rely on your rote memory. You know by rote that 3 times 9 is 27, you’ve learned that by rote, it’s just words.

 

Stan and assistants and Emile speak French.

Stanislas Dehaene interview.

Pan from CU of foam head to Stan and assistant making final adjustments to Emile’s electrodes.

Stan: The other circuit is totally non-language specific. It is really a circuit that gives you the meaning independently of the words.

 

Stan and assistants and Emile speak French.

Placing electrode hairnet on Emile’s head

Narr: Dehaene's experiment is designed to show that our number sense combines the activity of these two different parts of the brain.

Stanislas Dehaene interview.

Stan: We are trying to understand exactly what we have in the mind when we think about a number. And the fact that we can open up the brain case in a living brain and see exactly what areas are being active, is simply, I find, fantastic.

Pan from CU of hands attaching electrode ends to device, to wide shot of Emile moving into place for experiment.

Stan: The experiment was about doing exact calculation on the one hand, versus doing approximation on the other hand.

Stanislas Dehaene interview. He is referring to information on computer screen.

CU of screen shot, with pen pointing to numbers.

Stanislas Dehaene interview. He is referring to information on computer screen.

Stan: In this experiment what we are doing is present, exactly the same additions, here, the example is 4 + 5, then we have a little blank, and then we flash two proposed results. Both close to the correct result. One is correct, the other is false, you have to select the correct one.

CU ‘Ready’ on a screen, pull out to Emile in experiment room.

CU of Stan’s eye.

Shot of Stan speaking French, to begin experiment.

Two numbers appear on screen.

CU hands pressing button.

CU of Emile’s face.

Shot over Emile’s shoulder.

Screen shot of information (waves) on computer screen.

Stan, in French, tells Emile to begin the "exact calculation" part of the experiment.

Dehaene referring to information on computer screen.

CU of screen shot, with pen pointing to numbers.

CU of screen shot, with pen pointing to numbers.

Stan: Whereas in the approximate case, the two results are false and what you have to do is select the one which is closest. And because the other one is very false, you see it’s very far off, you can decide very quickly, using your intuition or your approximation without doing the exact calculation.

Shot over Emile’s shoulder, of him participating in the experiment, zoom into Emile’s reflection on screen.

 
 

Stan: You don’t have to do the exact calculation to know that 17 + 23 cannot be 95. You know immediately that it is false, you are using some kind of intuition.

Stanislas Dehaene interview.

CU of waves of data on screen, pull out to Stan talking to assistant as they watch the screen.

Stan: If you think of it, you’re seeing exactly the same additions. You’ve noticed that, it’s the same additions in the exact and the approximation case, but you have to do different strategies with them. This was sufficient to create a strong difference in the brain activation patterns. Within the first 1/5 of a second that you are seeing these additions, your brain is already activating very different circuits. That was a big surprise to us.

Pan over electrodes on Emile’s head, to connection.

Screen shot of brain.

Narr: It turns out that exact calculations activate the brain area that processes words and rote rules. But approximation — where the meaning is expressed — takes place in a totally different area — the parietal lobe.

Computer image of Stan’s brain

Stan: So what we see here is the left hemisphere of a brain, it is actually my brain, as it turns out, but its not important. The eyes should go here and this is the back of the brain, and here is this parietal region that we are talking about. Let’s turn it around to see a little bit better.

 

Narr: This is where visualization and intuition make sense of the numbers.

Shots of French kids in class, speaking French.

French kids in class, speaking French.

Narr: Teaching math as rules and procedures ignores this part of the brain — and makes it harder for students to find the meaning in what they’re being taught.

Boy in yellow writing on chalkboard

Stan: Some schools actually discourage children from using their intuition. They will discourage children from using their fingers to count. They will begin with the notion that children come at school with absolutely no abilities at all, which is clearly very false. And that will discourage children very much.

Kid raising hand in class

Narr: But maybe school doesn’t have to stamp out children’s developing interest in math. Maybe there’s a way to build on their natural ideas and intuitions.

Romina at ice-cream store

Romina (on phone): Jeff? Oh, lucky me, I got a hold of you. Uh, I'm at work buddy.

Kids coming into ice-cream store.

Narr: Investigating this possibility has changed the lives of these high school seniors.

Kids at counter.

Romina: What do you want?

Kids together: Coke float, 2 coke floats, 2 vanilla ice creams. Can I have another float? Can I have a rootbeer?

Romina at soda fountain.

Narr: Here, in the small working-class town of Kenilworth, New Jersey, they’re part of an extraordinary experiment.

Romina at soda fountain.

Girl: Ro, I want root beer, not Coke.

Romina: All right Lauren, I can handle it.

Girl: Just reminding ya.

Romina and kids at cash register

Romina: Five-thirty.

Narr: When these kids entered first grade, they were randomly selected to become part of the study. They weren’t chosen for any special talents.

Romina: My tip.

Kids at table in ice-cream store.

Narr: But, 12 years later, the experiment has profoundly affected the way both Jeff and Romina think about math.

CU Romina in store.

Sync Romina interview.

Romina: No one has ever made it, like, hard or difficult for me. No one has ever made me dislike it. It has always been, like, positive encouragement, always in math, so how could I not like it?

Jeff in CU in store.

Sync Jeff interview.

Jeff: It brought out different qualities in all of us, you see it...it brought...it made us different people than we would have been if we never did it.

Kids in ice-cream store.

Narr: Their story may change the way we think about every child's potential to learn mathematics.

Rutgers footage of Stephanie, Dana, and Michael at table.

Stephanie: How many different outfits can he make?

Michael: He can only make 2 outfits.

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

CU Carolyn.

Carolyn, Linda and Elena watching television and discussing.

Cu of tv —tower building

Narr: At Rutgers University, professor of math education Carolyn Maher has devoted most of the past 12 years of her life to finding out how Jeff and Romina and a dozen other Kenilworth kids learn math.

Screen shot of Stephanie in Rutgers study, holding towers

Carolyn Maher interview.

Rutgers footage of Stephanie, Romina and ? at table

Carolyn Maher: We were intererested in following how particular mathematical ideas developed...to see what was possible, what students were capable of doing.

 

Carolyn: Our intent was to find a representative group of students from this community, randomly selected, and look to see over time the development of particular mathematical ideas.

Camera follows Carolyn into office, Linda comes out of office and heads toward the tape room.

CU of videotapes on shelf, pull out to Linda grabbing two video tapes off a shelf and handing them to Carolyn.

Carolyn: Linda, can you get a tape for me?

Linda: Sure.

Carolyn: Do you know, the '92 Romina and Jeff working on the towers?

Narr: Maher has compiled a unique archive — 2,000 video tapes, tracking the mathematical development of the same small group of kids, from first grade all the way through high school.

CU of deck, as tape is put in.

Rutgers footage of Stephanie and girl at desk, with blocks on their table.

Girl: Are we doing a problem?

Carolyn: When Amy comes here, what do you do?

Stephanie: Math!

Carolyn: That’s right.

Rutgers footage of CU of Romina working with blocks.

Narr: When the Kenilworth kids were in fourth grade, Carolyn Maher challenged them with a problem she knew would stretch the limits of their thinking: How many towers can you build, 5 blocks high, using blocks of just two colors?

Stephanie with partner

Stephanie: Ok, we’ll stand them up straight so we know what we have.

Rutgers footage of Stephanie and girl at desk, with blocks on their table.

Rutgers footage of Carolyn at front of classroom.

Rutgers footage of Jeff and girl at desk, with blocks on their table.

Rutgers footage of Carolyn at front of classroom.

Rutgers footage of Jeff and girl at desk, with blocks on their table.

Rutgers footage of Carolyn at front of classroom.

Carolyn: We have many good estimates of how many we can build. Dana, what do you think?

Dana: Uhhhhh….13

Carolyn: Dana thinks 13. Jeff?

Jeff: 25.

Carolyn: Jeff thinks 25. What do you think, Jamie?

Jamie: 10.

Carolyn: Jamie thinks 10. But we’re not agreeing on this. Jennifer?

Jennifer: 15.

Carolyn: Well I think what you’re going to have to do is to work on it and see how many…But remember, you have to be sure that you have no duplicates, you can’t have two 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!

Rutgers footage of Stephanie and girl at desk, with blocks on their table.

Stephanie: Ok, we’ll start out with the easiest one. One two three four five reds, and five yellows.

Girl: One two three four five.

Stephanie: I only have four.

Girl: OK.

 

Narr: They dive in — even though they’ve never been taught how to solve a problem like this.

 

Stephanie: I have too many. I can’t get one off.

Girl: Then let’s do one of these.

Stephanie: No, what we can do is just put one on the top, see? Tada!

Girl: Tada!

Stephanie: Now we’ll put one in the

middle.

 

Narr: They’re already savvy enough to look for patterns, and they quickly come up with some clever strategies.

Cu Stephanie’s hands with blocks

Rutgers footage of Stephanie and girl at desk, with blocks on their table.

Girl: And then I got another idea.

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

Girl: I’m going to this, that…

Stephanie: Huh? Show me. Ok, and I’ll do the red kind.

 

Narr: These kids have held on to their natural pleasure in thinking hard about interesting puzzles.

Rutgers footage of Romina and Brian at table with blocks.

Brian: Wait, take the kinds that are like this...

 

Narr: They get excited about new theories — and exchange their ideas with pride.

 

Brian: That’s what I was thinking of!

Romina: Well, I thought of it first!

Brian: You don’t have it.

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

Brian: We don’t! We don’t have it!

Rutgers footage of Romina and Brian at table with blocks.

Narr: Maher creates the conditions that make these surprises possible, by pushing the kids to define their own ideas, and to challenge each other’s.

Romina D’Andrea interview.

Romina: We're not scared of being wrong. They liked us being wrong, like when we were wrong they asked us why we thought of doing that. And you like always had like logical reasons behind why you did it, so it like turns into something.

Rutgers footage of Jeff explaining his towers to Carolyn.

Narr: Most of all, she demands that the students justify their answers. As she constantly says, "Convince me!"

 

Jeff: There’s only two kinds of these because they’re alternates.

Carolyn: OK, I buy that, alright, you’re convincing me, that’s great.

Rutgers footage of Jeff and girl explaining their towers to Carolyn.

Jeff: We’re making our own theorems. You know what you did the whole way through. You really have a good memory of what you did and where it came from. No one really knew what was going on and we just tried to make sense of it, and I think we made, we made good sense of it. We kind of came to a good answer just by ourselves.

 

Jeff: We know this can’t be any of these, so we skip that, because they’re like that pattern.

Carolyn Maher interview.

Rutgers footage of Jeff and girl explaining their towers to Carolyn.

Carolyn: This is what it is to do mathematics. It’s the reasonsing, it’s the making senses of the ideas. It’s the way they fit together. Our job was to observe what they did, to hear their arguments and to leave it to them to support and justify their thinking.

 

Carolyn: How do you know there’re not 34? How do you know that?

Jeff: Because I can’t make any more. My brain is tired.

Carolyn: Because your brain is tired.

Carolyn Maher interview.

Carolyn: Some of the most dramatic data came when the students were coming up with justifications for their solutions and inventing the idea of mathematical proof.

Rutgers footage of ‘Gang of four’.

Narr: By the time they’ve all solved the towers problem … the kids are so excited about their proofs they can hardly sit still.

 

Arguing.

Carolyn: Ok, one at a time.

Stephanie drawing

Narr: Stephanie drew a diagram to show all the possible towers three blocks high.

.

 

Stephanie: First you have without any blues, which is red.

Rutgers footage of CU of hand drawing proof.

Narr: She’s invented an effective argument mathematicians call "proof by cases."

 

Boy: And you can’t make any more with this one, so you go onto the next one.

Rutgers footage of CU of hand drawing proof.

Narr: Millin and Michelle take a different approach: It's easy to see that with two blocks you can only build four towers.

Rutgers footage of ‘Gang of four’.

Girl: You can add a red or a blue here.

Carolyn: Make a ‘y’ or something.

Rutgers footage of CU of hand drawing proof.

Narr: But here’s their powerful insight: To make a tower one block taller, you can add a blue block or a red one.… Do that to all four, and you get all the possible towers three blocks high.

Rutgers footage of ‘Gang of four’.

Stephanie: All right, so I’ve convinced you that there’s only eight?

Jeff: Yeah.

Stephanie: YES!

Carolyn: How many if you’re making towers of four?

Rutgers footage of CU of hand drawing proof.

Narr: The same procedure works for towers of four. And the students realize they can keep going, multiplying by two each time.

Rutgers footage of ‘Gang of four’.

Narr: They’ve discovered what mathematicians call "proof by induction."

Zoom in on Stephanie

Stephanie: Oh…you could give us problem, like how many in ten and we could just go…

Carolyn: And you know the answer?

Stephanie: I know the answer, I figured it out, it’s 1,024.

Carolyn: Are you sure?

Stephanie: Uh-huh.

 

Carolyn: You didn’t wait for the authority, for the teacher to say if your thinking was right or wrong, or there is a way to think about this problem. And if you thought about it this way, you’re a better thinker, a better student, you were a more successful mathematics student if your thinking followed a particular…expectation. I think that freed students. They were now able to trust themselves, that everyone’s ideas were important. And they were able to pull from themselves and surprise themselves in what they could do.

5th grade pizza, from 2

Narr: Throughout elementary school, the Kenilworth kids met about six times a year with Carolyn Maher and her colleagues.

Math class

6th grade towers of Hanoi, from 3

Putting discs on stick

Teacher: Somebody come down and do it with 3.

Jeff: I’ll do it with three.

Teacher: 3,4,5,6,7

Kids eating pizza,

starting to work

Narr: And a smaller group continued two or three times a year in high school. Here, as sophomores, they’re working on a problem close to adolescent hearts: how many different pizzas can you make from five toppings?

Kids at work on pizza problem

Michael: Thirty-one plus cheese.

 

Jeff: Thirty-two plus cheese.

 

Michael: Thirty-one plus one is thirty-two. That's with cheese. It's thirty-two, that's the answer.

Romina, Jeff and class

Pointing to tower diagrams

Carolyn: They knew when their ideas were being respected. It was very powerful for these students. Powerful enough for them to come back on their free time. They weren’t getting paid for it. They weren’t getting extra credit for grades. There must have been something that was very intrinsic about their reward. They knew when they were doing something they felt good about.

Kids doing towers, working at table

 
 

Jeff and Romina: So it’s 36.

 

Narr: When the Kenilworth kids were juniors, Maher pitched them a real brain-twister. It’s the World Series problem: how many different ways can a baseball team win a best-of-seven contest?

Jeff and partner

Student: We have 8 ways of winning, but it’d be over…

Jeff: Then 8 over 2 to the 5th…

Hands writing combinations

Narr: This is a classic combinations problem that might be taught — in college. Even Carolyn Maher was often surprised by just how far the kids could take their own mathematical insights.

   

Rutgers footage, Jeff and Romina in class

Carolyn: Children could do much much more than we have imagined they could do. And they always went much, much further than children had in regular classes in school and that was very encouraging. And as we were encouraged by their performance, what they revealed to us by their mathematical thinking and by their reasoning, we challenged them more and more and more.

Jeff Gocel interview.

Jeff tutoring Gerardo.

Jeff: Whenever anyone had to give an answer, it was assumed that someone who never heard of math walked in here and you had to explain this problem to them.

Jeff and Gerardo sit down at a desk.

CU of math in text book.

Jeff: The Rutgers study taught me how to do that. It makes life easier, ‘cause even when you’re arguing a point that has nothing to do with math, you break it down to where someone understands it, and you can build your argument from there.

Pan up from math text book to Jeff and Gerardo.

 

Jeff Gocel interview.

Jeff: This year and last year was one of the first times that I really experienced teaching somebody something and them really understanding it and doing it. And it’s just one of the most unbelievable things that I’ve ever experienced. It’s pretty decent.

CU of text book.

Narr: For most students, math isn’t taught in a way that breeds this kind of confidence in their ability to understand and solve problems.

Various campus shots of BYU campus.

Tilt down from WS of Bob in class computer/math class to shot of Chuck with students at computer.

Narr: So after struggling through high school math, most college students are terrified of calculus. Here at Brigham Young University, Bob Speiser and Chuck Walter are researching ways to change that.

Girl looking puzzled in front of computer

Girl: So that doesn’t make any sense.

Narr: They know how useful calculus can be to describe the physical world, and they want to make it accessible and meaningful to students.

Bob Speiser and Chuck Walter interview.

Bob Speiser: If we were going to do that, that calculus course would have to have a very different feel and a very different way of working.

Meeting scene.

Sara Lee and John Marshall walk into meeting.

Narr: So they designed an experiment much like the Rutgers study.

Sara Lee Gibb: Are we late?

Narr: Only instead of kids, the test subjects are professors. From philosophy to administration, industrial arts, and dance, none of these participants has studied higher math.

Bob Speiser and Chuck Walter interview.

Shot of people getting settled at the meeting.

Chuck Walter: The people we talked to were somewhat apprehensive about this. But on the other hand, the chance to really have a voice here was something that they really had to take advantage of.

CU of Sara Lee in meeting.

Various shots of dancers in class.

Narr: For example, Sara Lee Gibb, chair of the dance department, wants to help her students overcome their fear of math.

Dancing students

Sara Lee: They think, I can’t do this, and somewhere along the line they’ve heard that or thought that, that they just didn’t have the mind for it.

 

Bob: Why couldn’t her students, with their richer experience of movement, be able to understand the mathematics that describes motion and change?

Hum-V drawing in industrial design workshop, pan up to model with John and student talking.

Narr: Professor of Industrial Design John Marshall wants to broaden the experience of his students.

Shots of students in industrial design class.

John Marshall: They work all the time with engineers who qualify and quantify things like mathematics, and they sit in amazement, and I thought it would really be great if they understood what they were doing when they went to calculate.

Shots of people in meeting looking at paper with cat photos on them.

Cat Animation.

Narr: The group is shown a famous series of photographs taken by Edweard Muybridge. 24 cameras snap frames of a cat running past a grid background.

The problem: how fast is the cat running in frame 10? Finding the average speed would be easy: that’s just distance divided by time. But determining the speed at a particular instant promises to be a real puzzle.

Bob at white board explaining graph.

Narr: Speiser starts with this view of the cat’s progress …

Bob: This is frame 1, this is frame 2, this is frame 3 …

Narr: — conventional, for a mathematician. But what do his new students make of it?

Shot of Sara Lee at meeting.

Sara Lee: That's a very flat chart...compared to what's really happening.

Shot of Bob across table.

Bob: So for you, this graph goes out the window. How could you represent the cat’s motion so that it does make sense?

Circular saw in shop, pan up to see John working.

Narr: For John Marshall, making sense of the graph means going to the familiar territory of his design shop, to acquire a hands-on feeling for the information.

Shots of cutting and arranging blocks.

John: I had to arrange blocks to give me some idea of what the concept was, what we were dealing with visually.

Creating graph with blocks.

John: The width of one was representing time, the other one was representing distance. Then, where they were placed in relationship to each other was representing the difference.

Bob taping floor in dance studio.

Narr: The seminar moves on to Sara Lee’s dance studio. To translate the graph here, they lay out a line representing the total distance the cat traveled.

Sara Lee walking in dance studio.

Bob walking over tape in dance studio.

Sara Lee: OK, now if you were a cat you could make that in 7/10ths of a second, from beginning to end, right?

Laughs.

Shots of Bob, Chuck and Sara Lee marking frames and distance.

Bob: That looks reasonable.

Narr: Then they mark off the distance between each frame.

 

Bob: 13……14.

Sara Lee explaining to Bob and Chuck.

Sara Lee: See, when you realize that each one is a frame, that’s here, and they look equal, then you really see how it is, now that is very revealing.

Sara Lee running on tape line.

Beating of drum.

Sara Lee counts: 1,2,3,4,…

Sara Lee Gibb interview.

Sara Lee: Modern dance, we deal all the time with abstraction, and how to make meaning of it Well, the mathematicians are doing the same way. And I understand it, because it connects somehow with the experiences that I’ve had.

Shot of Bob explaining in dance studio.

Narr: For Bob Speiser and Chuck Walter, traveling in the cat’s footsteps gives the mathematics a whole new dimension.

Bob Speiser and Chuck Walter interview.

Chuck: There is such a difference between imagining, intellectually, a change and doing something where you in fact feel that change, where you embody that change.

Bob running along the tape.

Kenilworth kids put down masking tape.

Narr: For the mathematicians and their students, experiencing the cat’s motion in a personally meaningful way was a breakthrough. But … they didn’t solve it ...

Kenilworth kids looking at Muybridge cat photos

Narr: … so Speiser and Walter brought the cat problem to Kenilworth.

Kenilworth kids at tables, surrounded by graphs and pictures

Narr: The kids, attending a summer workshop with other area students, go right to work.

 

Boy: Like right there, he’s meaning 4 centimeters a second. It doesn’t make sense…

Jeff interview

Jeff: This was a class Rutgers kind of situation. Within the first 10 minutes, everyone has an answer. And in the first 20 minutes, everyone scrapped their answer and is totally lost.

Romina looking at graphs

Narr: But soon they’re back on the track, generating graphs and calculating cat speeds.

Romina standing up, explaining, with overhead projection

Romina: And then starts accelerating faster and faster till it reaches a climax…

Kids putting tape on floor

Narr: When the chance comes to act out the cat’s motion, it’s the same in New Jersey as at BYU — the graphs really come alive.

Kid leaping across floor

Romina: We didn’t understand how something could change speeds so fast, what was going on, why it would change speeds so fast. But when you do it like a real-life version of it, you can see what the cat’s doing so you can understand it, like it just makes sense of all the math.

 

Narr: And when Romina translates the cat’s movement into the even more familiar terms of the Garden State Parkway, she suddenly hits on the real answer.

Kids in class discussing solution

Romina: At exit 9, you're going 30 miles per hour. And exit 11, you're going 60. How fast were you going at exit 10? You don't know.

 

Jeff: You could have sped up to 120 miles per hour for exit, for that exit 10, and slowed down to get 60. I mean, all you know is the beginning and the end. You have no clue what you did in between.

Kids at table with calculators and Muybridge photos

Kid leaping down hall along tape

Narr: What made Jeff and Romina’s insight possible? It’s the essential math lesson of the cat problem: Don’t start with rules or symbols — first get a feeling for what the problem really means.

"Red Violin" clip.

Narr: It’s an insight that many artists understand intuitively — like composer John Corigliano, who won an Oscar for scoring "The Red Violin."

Corigliano talks to conductor

John Corigliano: John Carlo? The bassoons and oboe, mark it "piano."

Orchestra sections rehearsing

Narr: Now Corigliano is rehearsing with the Minnesota Orchestra to stage the world premiere of his newest work, "The Ghosts of Versailles."

Corigliano with conductor

John: The thing that’s important really at this point, is not the big sections and not the ghost sections, so much I’m worried about…

 

Narr: We often hear of music and math as related talents. But the link may be even deeper than we’ve realized.

John to orchestra

John: Yes, legatissimo, slurred.

John Corigliano interview

CU of a hand adjusting a microphone.

Musicians.

John: A lot of my composing takes place with me lying on a bed with a pillor over my head. To isolate inside my brain and hear the symphony playing inside my head. Then I have to translate to music paper so a real live symphony can play it for other people.

Truck down orchestra

Harpist plucking instrument

Cellist playing and marking score.

Narr: In other words, some kind of inner process — what Stan Dehaene might call developing intuitions about a problem — has to come first. Then, as in math, comes the long struggle to get it right.

Cellist writing on score.

Players readying themselves

John: You get the idea but you don’t play it. You look at it and say, is that the best it can be, what can I do to make that more what I really need for this piece. So you cross out this and you cross out that and you rewrite this and you rewrite that. And then you look at it again and then you look at it again, and 20 sketches and 30 sketches later, you say, OK, now it’s ready, now I can put it into the piece.

 

Narr: Like a math student, the composer first has to get hold of a vision, the meaning he wants to convey.

 

Orchestra plays.

 

Narr: Only then can he transform that intuitive sense into musical notes he can give to the orchestra.

 

Orchestra plays.

MS of xylophone players.

Corigliano interview.

MS flutist.

John: When I wrote "The Ghosts of Versailles," the first job I had to do was to create a world of ghosts. I said, I want a world of smoke. Now what I meant was, you see it and then it disappears, sometimes into the air, and you don’t see it and a little higher you’ll see more of it, and then it will be curling and curving and drifting and you can see through it but you can also see it.

How do you do that in an orchestra?

MCU Giancarlo conducting.

John: I can draw this ghost music. I can’t hear it yet, but I can draw it.

Corigliano pointing at color line.

Violinist, with score

John: If I start with a little tiny wisp of orange, and it just goes like that, in the middle of the orange pencil I shifted to a red pencil, and I drew a little more and then the orange had disappeared again, and then the orange and red became purple and violet and green and blue. As this line moves, it changes color.

John’s finger following lines across notes in the sheet music of his ghost music.

Giancarlo conducting.

Soft focus of violinists in orchestra, that come into focus.

Giancarlo conducting.

John: Here’s the smoke line, and it’s given to one, two, three, four, five, six different instruments. And I tell those instruments, like the violin, "from nothing, slowly, as you’re playing this, make a little crescendo, get louder, and make it very soft and beautiful for three or four of five seconds, and then disappear again, but keep playing when you disappear."

Violinists.

WS of orchestra.

WS of audience.

John: The ultimate experience for me is that there is nothing, and that you can sit down and imagine something, and channeling that imagination through craft, into written notes for players, you can make something that is a real thing that’s invisible.

Orchestra concludes piece

Kenilworth kids’ faces

 

Scene of Ralph Pantozzi's AP Calculus class.

Ralph Pantozzi: And I’d also have to take this minus that…

Student: It’s just A pi — That’s what you get.

Jeff: I think almost everyone who ever reads any kind of math problem, the first thing that goes through everyone's head is, "I can't do this...like how...are you kidding?"

Jeff and Romina working together

Romina: Calculus is so much an abstract concept.

 

Jeff: Yeah, but that doesn't go to the right forever, it goes up forever.

Romina interview

Romina: You can't just like look at it and just be like oh, you have to really understand and see it.

Jeff and Romina working together

Romina: But doesn't this go forever and never touches zero?

Jeff: I tend to, I tend to believe that.

Kids working together in Mr. Pantozzi's class.

Narr: Now in their senior year of high school, the students in the Rutgers experiment are enjoying the chance to grapple with the concepts and techniques of advanced mathematics.

CU of paper with calculations

Carolyn interview.

Carolyn: The can express these ideas in very appropriate symbolic notations. But they’ve connected to those symbols, meaning.

Jeff and Romina working together

Romina: Can't you bring the two down?

Jeff: I think you can just...you can cross them out.

Romina: I don't know, is that legal in math moves?

Jeff: I don't know. But say you did, right...

Class hunched over desks, calculating

Narr: Everything they’ve learned in the sessions with Carolyn Maher has only helped them to succeed in their school work. Nationally, most students stop taking math as soon as they’ve met the minimum requirement. But every student in the Rutgers study is taking this advanced placement calculus class.

Scenes from AP Calc class.

Carolyn: That all of these students would end up with AP Calculus in their senior year is beyond my wildest expectations. I never would have dreamed that were possible.

Jeff and Romina working together

Jeff: Mr. Pantozzi, does x equal negative one in the absolute minimum of this?

Mr. Pantozzi: I think so.

Jeff: Oh, that is so dope. That is so dope!

Jeff interview.

Jeff: When you get to the end it's like really rewarding. Like all right, you know, we did this. We started with nothing and we worked for a couple of hours and we argued and we fought and this and that, but now we've got an answer and it's right. And it was just ‘cuz we did it and that's it. That's a very rewarding feeling.

 

Jeff pitching in baseball game.

Jeff’s mother Joanne and father Stan watching from stands.

Joanne: I wanted to shut my eyes and listen for the crowd noise, if they were cheering, yes it was a strike, if they’re not then I think, ok Jeff, you gotta settle down. And you know, I’m his mother, I can say that.

 

Jeff pitches, batter connects, mom yells.

Guy hits ball

Mother watches

Jeff: I know how to concentrate. I know, when there’s other things going on, how to focus on the problem at hand.

 

Sounds of the game.

 

Joanne: The Rutgers program has applications that have affected all parts of his life.

Umpire: 3-0 on the batter, no outs.

Joanne: He uses those kind of critical thinking skills in everything, in pitching, the duel between himself and the batter, what’s the best way to strike this guy out.

Jeff Gocel stand up interview.

Kid running bases.

Jeff: Physically you have to have a decent arm, you have to be able to throw different pitches. But for the most part you’ve got to be out-thinking the people over there.

Jeff, Romina and Brian on bench.

Jeff, Romina and Brian sitting on bench, then walking across the playing field.

Jeff: I don’t think he ever got a hit off me before today. The kid has always been a good hitter.

Romina: Do you know all of them?

Jeff: We play baseball with them.

Romina studying at desk in library.

Narr: Jeff attributes much of his confidence to his experience with the Rutgers experiment. His native math sense was allowed to express itself, and his ideas were given respect. Like Jeff, Romina saw her own math competence emerge, and she learned how to apply it to problems beyond math class.

 

Romina: It's provided you with confidence. You can go into anything. You'll give me an essay to write about and I won't have a clue but I'll just try. Like in my English classes my whole essays is just me taking something and working hard enough to prove it until it's right.

Romina D’Andrea interview.

Romina: If you have a good background in math you pretty much have a good background in anything. Because math is working with like through steps. Like thinking logically and working through steps and trying new things. And you work through steps and try new things in every area of your life. So if you're good at math you're pretty much, you're set in life.

Patagonia mountain climber scaling cliff face.

Narr: As they begin their climb into the world of work, more and more graduates discover what Romina has already learned: the mathematical thinking skills they develop in school can make them top prospects for today’s most attractive jobs.

Jib shot down on meeting through glass door.

CU of a hand writing notes.

Man in meeting 1: The goal was to get this on one page and at the beginning of the planning season say, here are the key things that the brand team has decided are important for building the brand…

Shots of people in meeting.

Michael Crooke interview

Michael Crooke: We have a very complicated business. We distribute our product all over the world. We have to get a lot done very professionally.

Woman preparing red fabric before cutting.

Screen shot of patterns.

Narr: The business is Patagonia, maker of upscale clothing for the outdoor lifestyle. The company is committed to helping protect the environment its customers enjoy. So it was eager to respond to a concern that came from some manufacturing plants.

Automated machinery cutting fabric markers.

CU of machinery making cuts.

Shot of woman walking behind machine cutting paper.

Adrienne Moser: They said the biggest thing you could do, is how can you deal with the amount of scrap that’s left over after we cut out your garments, because that to us is the biggest change you could make to really improve the environmental quality of your products.

Adrian Moser interview, she picks up a pair of pants.

CU of pattern laying on table.

Adrienne: You look at something like a pair of pants, and one of these legs would be this pattern piece, but all this white that you see, this is all what’s left over.

Shots of people picking scraps off table.

Narr: Design chief Adrienne Moser and her team might have struggled forever, trying to figure out new uses for leftover fabric.

Adrian interview.

Woman picking up scrap.

Adrienne: The scrap in previous times had gone into landfill, or had been recycled. So they would take a fabric and maybe it would go into insulation for houses.

Woman rolling out white paper.

CU of pattern in CAD computer.

Narr: Patagonia came up with a much better solution — by re-defining the problem. The fabric never becomes scrap in the first place. Instead, the designers created a whole new product, by fitting small pieces in between the adult patterns. The result: a quirky and colorful infant line, called Seedlings.

Woman's hand on mouse.

CAD marker on screen.

Judy: What efficiency do we have on this...on this marker?

Woman: We have 71.9%

Adrian interview

CAD scenes

Adrienne: We weren't going to make a regular adult garment marker have worse utilization so we could fit Seedling. We fit Seedling where there was an available hole.

Woman shopping at Patagonia store.

Narr: CEO Michael Crooke was worried how customers might respond.

Michael Crooke interview.

Baby being dressed in Seedlings.

Michael: You know, how is that going to look, all different scraps on a baby suit for instance might have six different types of patterns and fabric?

It was cute, it was fun, it was different, and it was the right thing to do.

Pan of Patagonia office.

Narr: Every business values people who can solve problems this way — weaving together know-how and imagination.

Michael Crooke interview.

Michael: The kind of people that we are looking for are the people that can stand up and challenge our ideas and what we’re doing and help us move forward, those are the kind of people we’re looking for.

Climber on rock face.

Narr: Skill, confidence, independence — habits of mind the Kenilworth kids are learning from math — are also sources of courage and innovation for artists .. like Frank Gehry.

Street scenes of Bilbao, Spain

 

Frank Gehry interview.

Frank Gehry: I’m searching through the ether for some response to the problems that are confronting me.

Bilbao street scenes

Narr: From testing first ideas to constructing the final building, architecture is a fundamentally mathematical discipline.

Steps of Guggenheim

Kid runs into museum

Frank: In math, you would speculate on what an answer would be, and you would go down a path that would lead to a series of calculations that would either prove right or wrong.

Guggenheim interior.

Frank: It starts to evolve into what it evolves into.

Children looking up.

Narr: In this case, Frank Gehry’s explorations evolved into the Guggenheim Museum in Bilbao, Spain… already considered a masterpiece of modern architecture.

Very blue, dusk, shot of exterior of front of Guggenheim.

Moving and static detail shots of Guggenheim exteriors.

Frank: Your visualization is like a dream. You imagine something. It’s ephemeral but it’s got some form, but it’s not clear.

Kid running along outside of museum

More exteriors

Frank Gehry interview.

Details of Frank’s drawings.

Frank: It’s an intuitive process.

Narr: An architect’s vision unfolds in stages. Gradually, pure imagination becomes a precise plan.

Frank: I do a lot of drawings, fantasizing. And I think of the drawing that way, that I’m sort of swimming through the paper to find the images. And they’re not coherent for lay people, but then if you see them afterwards, they make sense.

Pull back from large model in studio.

Shots of work on models.

Frank: My kids here know how to interpret them. The first models are built from those sketches.

Frank: I have some pretty rigid rules, if I’m going to get buildings built. If I made a bunch of arbitrary forms … that’s not how I design. These forms I do take months and months of study.

Shots of Guggenheim models.

Shot of Dennis at CATIA computer, with model on screen.

Motion effected shot of large scale model.

Dennis Shelden: Frank has obviously been trying to push the forms in his architecture for a long time. A lot of it is unconventional and is complex in terms of its geometry.

Narr: Dennis Shelden translates Gehry’s inventive flair into blueprints and engineering specifications.

Dennis Sheldon/CATIA interview, tilt down to Dennis bending paper.

Dennis: One of the issues that we have to grapple with is when we’re trying to build complex surfaces out of materials which behave like sheet metal or paper, or something like that, things which fold like this.

One guy leaning next to guy at computer, telling him what to do while pointing at computer screen. Cut to screen shot, then back to CU of guy leaning.

Architectural assistant: Grab an arc of this, and try like grabbing an n- point here ... I draw over the p-line with an arc, because the arc has the same radius all along.

Dennis Sheldon/CATIA interview.

Inter-cut shots of model on screen and building during construction.

Dennis: And the process is really about building all the pieces in the computer so that then they can be built and assembled in the physical world.

Shots of beams in main atrium of museum.

 
 

Kids in museum

Exteriors of Guggeheim

CU of kids faces looking at art.

WS of kids leaving a gallery, with shot of winding sculpture in foreground.

Frank Gehry interview.

Quick shots of interior details of atrium.

CU of children’s faces.

WS of children moving through gallery.

Frank: I was only 7 or 8 years old, my grandmother must have been in her 60s. And she played on the floor with me. We would make freeways and put structures on them. When I was scattering around, looking for a profession, or something to be, it's the memory of that that sort of led me into at least trying architecture.

Frank: I suspect everything I’ve done comes from that game on the floor.

Exterior of Guggenheim.

Clapping, sounds of graduation ceremony

   

Romina at head of line of graduating students.

Nancy Baton: Valedictorian, Romina D'Andrea.

Audience shots.

Shots of Ralph,Carolyn.

Romina receiving bouquet

Sync cheering.

CU of math notations in class.

Romina interview.

Romina: I think a lot of my confidence came from this whole Rutgers thing. Before that I was...I thought I had no, like, no talents in math. But I can do it. Like you can give me any problem and I will eventually do it. Like, it's not an issue.

"Over the Years" montage of Rutgers footage

Carolyn: We start with a random group of students in first grade, from a community that's very typical, and we have produced all these students who are very talented in mathematics.

Rutgers footage

Romina: We just went from point to point on the thing.

Jeff interview

Jeff: You start to do what you know how to do. And you start to get in a rhythm and you flow a little bit and you start to figure out what's going on, and the next thing you know it’s like two hours later...

Rutgers footage

Jeff: Don't tell me we’re out of time.

Carolyn: I know, isn’t that awful, Jeff?

Jeff: Oh!!

Jeff Gocel interview.

Jeff: The bell will ring at the end of the day and no one will move. You’re just really proud that you worked from nothing and build this huge thing that you did.

Joanne clapping at graduation

Nancy Baton: Jeffrey Gocel.

Babies in the lab

Pre-schoolers

Rutgers archival footage

Narr: We’ve seen that children are natural math learners. When teaching recognizes and exploits this … surprising things happen: Kids develop an enduring love for mathematics, and they gain skills and confidence which enrich all the things they do.

.

Jeff is handed his diploma and hugged by Joanne.

Romina stands up

Audience watches

Narr: The first step is understanding that kids come to school with math in mind. If schools greet their intelligence with respect, the Kenilworth story can happen anywhere.

Carolyn in stands.

Graduates stand.

Carolyn: Research has so often in the past said what students can't do–you know how unsuccessful they are. But these students, have shown the possibilities and the promises.

Graduates throw caps.

Nancy Baton: Congratulations to the David Brearly class of 2000.

 

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