Workshop Sessions

PROBLEMS AND POSSIBILITIES

Workshop 4: Thinking Like a Mathematician

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TRANSCRIPT

Part 1. "STRATEGIES FOR SOLVING PROBLEMS"

[Music]

NARRATOR: One of the key goals of mathematics education is building problem solving skills. How does a mathematician approach problem solving? What does it mean to think like a mathematician?

In this program, we'll look at a mathematician at work. We'll also visit students in a variety of problem solving situations to uncover parallels in the way that students and mathematicians solve problems.

[Music]

FERN HUNT: My name is Fern Hunt. I'm a mathematician at the National Institute of Standards and Technology. Typically, what I like to do is I like to try to do a little bit of creative thought in the morning. And my thoughts are of a more theoretical nature. I might be either doing some reading or I might be actually trying to figure something out. There are a couple of long-term problems that I think about. And so I do a lot of, you know, sort of - It's hard to describe. I have a notebook, and it's somewhat discursive. It's not quite daydreaming. It would be the equivalent, maybe, of an artist sketching something or improvisation, only it's based on some particular piece of mathematics, and you're trying to extend it.

The larger part of my childhood, I grew up in a housing project. I wouldn't describe us as poor - we weren't on welfare. But there many, many things that we could not afford. I think in elementary school, I think I was probably adequate. I couldn't say that I was really a fan of mathematics. I was decent, but I wouldn't say I was a fan, until the 9th grade. At that point I got Algebra, and that's when I was given a dose of abstract mathematics. And the fact that it made arithmetic so, -it looked to me like a shortcut. I wouldn't have to do all these tedious computations, I could sort of express things in terms of symbols, and then at the last minute do the arithmetic. So I wouldn't, -it was a very handy, compact way of thinking.

TED: Simply modeling the surface. If it was -

MARY: Morning! How are you?

FERN HUNT: Great.

MARY: Have a seat.

TED: Okay. We're in the middle of -

MARY: We're in the middle of a little discussion here, Fern. Okay?

MALE: Maybe changing this model.

FERN HUNT: Of metallics?

TED: Yeah.

LI-PING: Yeah.

FERN HUNT: I work with other NIST scientists, physicists, engineers, chemists on problems that are of importance to the United States business and industry, as well as scientific problems, in general.

LI-PING: ...yes, different.... far away.

TED: You were actually able to get a signal off of them.

MARY: So there's a particle down there, enough to reflect and see it. But we're looking, really, at normal, normal geometry.

FERN HUNT: One project involves properties of materials, especially as visible light falls upon them.

LI-PING: This is a very fine structure.

FERN HUNT: Yeah.

LI-PING: And it looks like....

FERN HUNT: You might wonder why we're working on a problem like that. Well, it turns out that this is a problem of great importance to industry; for example like the paint industry, the automobile industry, the paper industry. All of these, and others, are involved in the manufacturing of materials, and they need to know how they are going to appear in order to be able to better design them, in order to be able to continue to invent new materials.

FERN HUNT: Well, you have two parameters, at least, right?

MARY: Yeah.

TED: Particle size and particle spaces. And so as a function of angle, we should see this rainbow effect.

NARRATOR: In this project, Fern will eventually come up with new equations for how light is scattered by physical materials. This may help industry create computer simulations of materials that haven't been invented yet.

FERN HUNT: Is there anything of the measurements that we were discussing originally- there's nothing unexpected in either of these. So you're going

LI-PING: Right.

FERN HUNT: You're going to have a whole range of distances. So your choice is to exactly reproduce this particular structure, or you could use the distribution, and then generate another structure which would have exactly the same parameters.

FERN HUNT (VO): You can't come up with arbitrary explanations about how light interacts with materials. You have to find out within the laws of physics what actually is going on. And that could be pretty complicated- lots of variables.

NARRATOR: To Fern, solving difficult mathematical problems like this is a little like working on a puzzle or a game.

FERN HUNT: It turns out that many of the same skills, many of the same problems, many of the same insights that we need, in order to solve difficult problems, involve the principles that can be found in games and puzzles.

So, for example, suppose this is an example of a problem. Although it doesn't look terribly mathematical, it is, in a sense that I hope you'll see, and you'll agree. And this is a game- it's about 100 years- and it's known as Towers of Hanoi. The object of the game: focuses on this tower of eight disks. We want to move this tower of disks from the peg here on the right to the peg here on the left. And the rule is that I have to move each disk one at a time. So I can't do this. So that you can do.

And the second rule is that you can put one disk on top of another. But you cannot put a larger disk on top of a bottom disk.

Billions of people do games and puzzles of various kinds. And I think not only to refresh ourselves, but it engages the creative powers of the mind. And for any problems, no matter how serious, the creative powers of the mind are needed in order to solve them.

NARRATOR: When the Kenilworth students were in the sixth grade, the late Robert B. Davis led a research session based on the classic game, Towers of Hanoi. The researchers were interested in finding out how the students would solve problems involving exponential functions, even though at this point, their formal knowledge of exponents was very limited.

ROBERT DAVIS: You may know this puzzle. It's called The Tower of Hanoi. Do you know the story that goes with it?

STUDENTS: Yeah...

ROBERT DAVIS: They claim there was an order of monks in the City of Hanoi, who were religious men who lived by themselves. And they were concerned about when the world was going to end. And so they made a puzzle like this which has 100 disks in it. And they spent all of their time- plus they eat and sleep and things like that- but when they're not doing things like that, they spend all their time working to solve that puzzle. When they have it done, that's supposed to be when the world ends. Okay? And I thought it might be interesting to figure out when the world's going to end, so we'd know too.

JEFF: Yeah, but I'd be scared.

ROBERT DAVIS: Now let's agree on what the rules are. The rules are you can only move one disk at a time, and what else?

ANKUR: You can't move a bigger onto a smaller one.

ROBERT DAVIS: You can never put a bigger one on top of a smaller one. Okay. Now if we want to find out when the world is going to end- some safe way; we're not going to do a 100...

CAROLYN MAHER: Bob Davis came to mathematics education as a mathematician. He had three degrees from MIT in mathematics, but decided that he was really interested in how people learned mathematics. He was really interested in thinking. And so he was one of our very first pioneers to come into this field and lead the way.

ROBERT DAVIS: What could I do?

ANKUR: Do you think if we get all hundred, the world will really end?

MIKE: We'll probably be dead when you get it.

JEFF: Yeah. By the time we figure it out with a hundred, we'll be dead.

ROBERT DAVIS: I want you to do it.

CAROLYN MAHER: For them, this was an unsolved problem. For them, like the mathematician who's working on a problem, they don't know the answer. And even though we know there is an answer, they know that we're not going to tell them what that answer is. So for them, the conditions are very much like mathematicians doing original mathematics.

AMY-LYNN: We got the whole thing in....

FERN HUNT: It's hard when you've got a lot of disks flying around, to try to find a way to move these disks without breaking those rules.

STEPHANIE: ...twenty two, twenty three, twenty four.

FERN HUNT: I think if you were to start to actually carry this task out, one finds out pretty quickly that it can get pretty complicated.

ANKUR: One, two..

MICHELLE: ..Three, four..

ANKUR, MICHELLE: ..Five, six, seven.

ANKUR: Three is seven.

FERN HUNT: One of the first things that a mathematician often does is to simplify the situation. Rather than look at the problem in all its complexity, look at another problem. And that problem shares, perhaps, some of the characteristics of the original problem. But it has many fewer of the complexities. And you would work with that simpler problem to see what one could learn. Hopefully, what you learn in that situation, you can apply to the more complex situation.

ROBERT DAVIS: Okay. I want somebody to come- Suppose we had just one disk. Somebody come and solve that puzzle.

FERN HUNT: So a simplification is an important step.

STUDENT: It has to move at least once.

ROBERT DAVIS: Okay. And now I need to keep track. When there was one disk it took one move. Everybody agree with that?

STUDENT: Yes.

ROBERT DAVIS: And that's what we've got here. Somebody told me if there was two disks, it would take three moves. Is that right?

STUDENT: Yes.

ROBERT DAVIS: Somebody come and do that. Amy-Lynn, can you come do that?

ROBERT DAVIS: Great it took her three moves, okay? That looks all right. Is that okay? We need somebody to come down and do it with three disks.

JEFF: I'll do it with three.

ROBERT DAVIS: Two, three, four, five, six, seven. Looks like it's right, huh? Okay. Now is that all right? Everybody happy with that? Now how about four moves? Milin, have you figured out what it will be with four moves?

MILIN: That's nine.

ROBERT DAVIS: It's nine.

MICHELLE: I can do it in fifteen.

JEFF: Go Milin.

MIKE: Oh it's times 2 + 1. Oh we got it. I know , I know... Can I tell everybody? Is it the number times 2 + 1?

MILIN: Yes. The number times 2 + 1. It always works.

ROBERT DAVIS: Is that right?

MILIN: Yes.

JEFF: Can we still play the game now?

ROBERT DAVIS: Okay, we need to try this, I think, because we've got some disagreements here.

ROBERT DAVIS: Fifteen. You did it in 15.

STEPHANIE: I did it in 15.

ROBERT DAVIS: Okay, Stephanie just did it in 15. Can anybody do it in less than 15?

MATT: I found a pattern. I found the pattern with it.

ROBERT DAVIS: You found the pattern?

MATT: Yes.

ROBERT DAVIS: What is the pattern?

MATT: It's like ...look at from this way, two times two, times one is three. Three times three plus one. Four times four minus one. Then it would go five times five plus one..

STUDENT: We noticed a pattern.

ROBERT DAVIS: You know the pattern too.

MATT: Six times six minus one.

ROBERT DAVIS: Don't do that too quickly here.

MICHELLE: Like one and one is three, and then you add one more, and then it's- three and three are six. And then you add one more, and then seven and seven are 14, and you add one, is 15, and 15 and 15 are 30-

ROBERT DAVIS: Michelle and Ankur have found something very clever, but we may not end the world today.

FERN HUNT: Another thing that a mathematician does is look for patterns. They look at, perhaps, many instances. And from those classes of problems that the mathematician is solving, certain patterns may arise.

The idea is in some sense to try to understand or somehow summarize what that pattern might be.

MICHELLE: This is what we did. One plus one is two, and then one more is three. Three plus three is six, and then plus one is seven. And then ...

BRIAN: Wait. there's an easier way. See, there's two between there. It doubles becomes four? Four is between there, it doubles and becomes eight. Eight doubles..

ANKUR: But that doesn't ...

ROMINA: Yeah, I know.

BRIAN: But it's the easiest way to figure it out.

ANKUR: Oh please.

ROBERT DAVIS: Michelle- Could I get everybody's attention, please, for just a minute, because Michelle has something interesting to say. Can you show everybody what you're doing?

MICHELLE: Well, one and one is three, and then you add one more, and then it's- Three and three are six, and then you add one more. And seven and seven are 14, plus one is 15. So then the next one would be 15 and 15 is 30, plus one is 31. And then so on, and so on, and so on...

ROBERT DAVIS: Thirty-one. Okay. And now what is the one we really care about? The one that counts...

STUDENT: One hundred.

ROBERT DAVIS: .. is 100. So we want to know what number goes there.

STUDENT: Oh my God.

ANKUR: Maybe if we get ten, we can get like, 20, and then 30.

STEPHANIE: Ten is 1,023.

ANKUR: Ten is 1,023.

STEPHANIE: I already got down to ten.

ANKUR: Ten is..

MICHELLE: ... 1,023. Want to work with us Steph? If you guys are just off in la-la land.

STEPHANIE: Matt, come on.

BRIAN: And then do ten times ten. Not ten times ten. Ten times 1,023.

MICHELLE: Very quickly here. We've got to catch on.

ANKUR: Shelly, this is 2 to the tenth power.

MICHELLE: Oh my God. Duh, we had it right there.

ANKUR: What's 2 to the 100th power? That's the answer.

MICHELLE: 2 to he 100th power?...

ANKUR: We got it. We got it.

ROBERT DAVIS: You've got it. Okay, can we get one discussion so everybody can hear? Who's going to do the talking about this problem?

STUDENTS: All of us.

ROBERT DAVIS: All of you?

STUDENTS: We all did it.

ROBERT DAVIS: All right. Can you sort of face the rest of the people and tell them what you got?

ANKUR: We tried to figure out ten, right? And it was ..one hundred and twenty three ... so we found... that two to the tenth power also equals 123. So we figured that two to the hundredth power should equal the answer.

ROBERT DAVIS: Now, I'm not sure that I think two to the tenth is 1,023.

MICHELLE: We figured this out by going through the numbers.

STUDENTS: It's 1,024.

STEPHANIE: That's not right.

STUDENTS: It's 1,024.

STUDENT: Because you can't have an odd number as the last number.

ROBERT DAVIS: Thank you very much. That's a cleaver idea.

MICHELLE: And then we realized since it would work for this to this, why wouldn't it work, oh excuse me, from 2 to the 100th power.

STUDENT: Why can't it be 10,240?

ANKUR: It's 2 times 2 times 2 and so on to a hundred.

ROBERT DAVIS: Yeah. Instead of multiplying, instead of writing ten twos and multiplying them, you have to write 100 twos and multiply them. That's more than adding zeros.

ANKUR: That's the equation but we didn't figure out the answer, yet.

MIKE: I just saw something. What we're trying to do is 1 and 3, the difference is 2. Three and 7, the difference is 4. Seven and 15, the difference is eight.

ROBERT DAVIS: I'll write those numbers, too, if that helps.

MATT: Oh, I have it. I have it. You're multiplying everything by two. Two times 2 is 4 times 2 is 8 times 2 is sixteen, times two....

ROBERT DAVIS: Do we agree that we've got something very valuable here. Do we agree that that's a pretty good idea?

STUDENTS: Yes.

NARRATOR: Almost seven years after this session, Matt, currently a freshman at Virginia Polytechnic Institute, watched and discussed his work as a sixth grader with Australian mathematician, Gary Davis, Professor of Education at the University of Southhampton, in southern England.

MATT (VIDEO): You're multiplying everything by two.

GARY DAVIS: Have you got it?

MATT: Uh-huh.

GARY DAVIS: You got the pattern?

MATT: Uh-huh.

GARY DAVIS: Do you have any feeling when you look at that? Do you have a feeling of reconstructing what you were doing there? Because you're sitting there by yourself.

MATT: I just pretty much was sitting there, like, concentrating, just looking at numbers, you know? Like with these and what we did here, it's a lot of just trying to look at patterns and looking at different kinds of patterns.

GARY DAVIS: That's right.

MATT: And if you just see, like, 2 and 4, you automatically say, "All right. That's either adding 2 or multiply by 2." And you say "Four times 2 is 8, times 2 is 16, times 2. No wait." And you see- I just pick up on things.

GARY DAVIS: So you just try? You're trying different things in your head, and-

MATT: Yeah. Pretty much. Yeah.

GARY DAVIS: So in a sense, what's on the board's really important to you-

MATT: Yeah.

GARY DAVIS: -because that's what you were looking at.

MATT: Yeah.

ROBERT DAVIS: We know one thing we could do is we could keep extending this table. All right, is that what you were doing or not? What would go here for six? What would go here for six if I used I think it was Michael's rule?

BRIAN: Sixty three.

ROBERT DAVIS: Right, 63. So one way, you could come down and find out what goes according to this rule. Who made up that rule? It's a neat rule. Who said "take this number and double it and subtract, no- add one; double this and add one"- who made up that rule?

STUDENT: ...not me.

GARY DAVIS: Well of course his original question was "how many moves would it take for 100 rings.?"

MIKE: Probably a lot.

GARY DAVIS: A lot, it'll take a lot, there's no question about it. Is there any way you can figure that out from this?

MIKE: Yeah, uhhmm, when moving the four you got this to a point where there was seven.

GARY DAVIS: There was seven.

MIKE: When you move those three it took seven moves.

GARY DAVIS: It did. It did.

MIKE: So then it took another eight to move the rest of it. I'm trying to think.

GARY DAVIS: Sorry, eight?

MIKE: Yeah, it was fifteen, right?

GARY DAVIS: It was fifteen, yeah. yes it was.

MIKE: So, maybe to move three, take seven.

GARY DAVIS: Yes.

MIKE: Now you've got to move this guy somewhere.

GARY DAVIS: You do.

MIKE: Eight. And to move those three again is another seven.

GARY DAVIS: Oh right. So the 15 is 7 plus 1 and 7?

MIKE: Plus one, plus seven- That's a possibility. I don't know if that's-

ROBERT DAVIS: One of the questions is, I still don't know where this 2 to the 10th came from. What did you do to get the 2 to the 10th? But the other thing is, you're telling me 10 and you're telling me 100, and I'd like you to tell me how to do it with any number of disks. Okay? What would happen with 7 disks or 700 disks, or whatever? Because we really want to able to do it for any number. Okay?

ANKUR: I got ten million with just 26 digits.

MATT: Guys, I ran out of room at about 20.

STEPHANIE: Shell, do you get 2 times 50 on your calculator?

ROBERT DAVIS: Now let me tell you that I don't believe you're going to be able to use calculators for this. The numbers are bigger than calculators can deal with.

[simultaneous conversation]

ANKUR: I'm just multiplying by 2 by hand.

STEPHANIE: Real smart. Okay, what number are we multiplying by?

ANKUR: Seven, 7, 2, 8. This number.

[simultaneous conversation]

MATT: Thirty two... 64.

ROBERT DAVIS: Sixty four.

MATT: I figured it out.

ROBERT DAVIS: What?

MATT: I figured it out... plus 1 plus 1 plus 1 plus 1 plus 1 plus 1-

ROBERT DAVIS: That's a neat idea. Really neat idea.

MATT: Ooooohhhhhh.

MATT: I like to see things, kind of- more like a visual learner than sitting there and doing it in my head and saying, "Well, two times is this." Say, "Okay, put it down on a piece of paper. See what we have." Because then you can look for patterns.

MATT: That would be 127.

ROBERT DAVIS: That's what I get, too -127. See if you can make that table go through further than that.

MATT: Hey guys, I figured out the pattern.

ANKUR: Is Matt's right? Does Matt have a pattern?

MATT: ....Plus one, plus 1, plus 1.

MICHELLE: That's what we said before.

MATT: Just like this, not with all this. Just like this.

MICHELLE: Oh I get it. I see what you're doing.

JEFF: Oh these are going up.

GARY DAVIS: And as the end approaches, Ankur comes up and starts working with you on it, and Jeff comes over and starts working with you on it. It's like you're a magnet.

MATT: Yeah, maybe, I guess.

GARY DAVIS: Well, why does he come up to you?

MATT: I don't know. I guess he sees something, or he can see the same thing that I see- or sees something different than what I see, and can add on to what I've done.

GARY DAVIS: What allows him to do that?

MATT: Instead of just seeing a bunch of huge numbers on a paper, seeing more of a pattern to it, and seeing it written as, like, a pattern, instead of seeing it as 2 to the first, as 2 to the second- as this huge number, and keep on going in huge numbers- those that are easier to see a pattern between.

[simultaneous conversation]

AMY-LYNN: What are you doing?

BOBBY: When I go home I'm going to write "times 2" 100 times to figure it out. I'm just going to keep on putting "times 2" in my calculator. I'm going to figure out the answer.

CAROLYN MAHER: Fern outlined some of the things mathematicians do when they do mathematics. And I think it's very interesting to watch that the children and the tape do some of these same things. They do think of a simple problem, they do look for patterns, they look for finite differences- as you see. They notice the pattern and they notice that there's an exponential here. They posit two to the end. And that's what mathematicians do. They see these patterns, they pose a theory, that they have to go back and test it.

NARRATOR: A few days later the students were still interested in finding out how long it would be before the world would end.

Bobby reported that he knew how many moves it would take for 100 disks.

ROBERT DAVIS: Okay, let me show you -Bobby wrote something here which I think several of you had last Thursday. If you had a hundred disks, he says it would take this many moves. Okay, let's assume -Bobby and Amy-Lynn worked pretty carefully on this and they think they've got the right number. So let's assume this right: 28 comma, 458 comma, 001 comma, 530 comma, 100 they say. Okay. Suppose it takes that many moves -and I don't really believe that story about the world ending, but let's pretend we did. Let's figure out when the world would end. If it takes that many moves, how long is that going to take?

STUDENT: A long time.

MIKE: It could take a day. It could take a day.

JEFF: ... because if seven of them take ten minutes.

ROBERT DAVIS: Okay, I want somebody to come and solve the problem here with disks. Four disks. Milin will you time this carefully? Okay, Milin say go when you're ready... what?

STUDENT: Now.

ROBERT DAVIS: Go.

NARRATOR: The students performed a series of tests to find the average time per move.

STUDENT: Go Matthew.

ROBERT DAVIS: How long did it take?

MILIN: Thirty one seconds.

ROBERT DAVIS: It took thirty one seconds.

STUDENT: Yes, she's got it.

ROBERT DAVIS: How much time?

MILIN: Two Minutes and fifty seconds.

ROBERT DAVIS: Two Minutes and fifty seconds.

BRIAN: Oh yeah.

ROBERT DAVIS: So it's about..

ANKUR: ... It's two seconds per move.

ROBERT DAVIS: So it's about twice as many seconds as there are moves. Right? If we assume that Bobby as the right number of moves here, Okay. He says that many moves. So how many seconds will that be?

STUDENT: Oh boy.

ROBERT DAVIS: Well it's going to be twice as many. Would you all double this, multiply this by 2 and tell me what you get?

NARRATOR: Finally the students did a series of calculations to convert the units from seconds to years.

ROBERT DAVIS: So it's about, it's about that many years. Sot what is that? It's saying... 2 billion years. Isn't that what it's saying?

STUDENT: Oh my god, it's going to take that many years to do that?

ROBERT DAVIS: Somebody once said if you really knew the world was going to end you wouldn't be able to get on the telephone, everybody would be busy calling somebody to say "I love you."

STUDENT: I love you!

MIKE: I love you Jeff.

JEFF: Really, what do you think you would be doing if you were going to die?....

NARRATION: We've seen students using a variety of problem solving strategies to approach the Towers of Hanoi problem. What strategies have you observed your students using to solve difficult problems?

[Music]

NARRATOR: Timpview High School is located in Provo, Utah, a small city that has experienced rapid growth in recent years. The school has an enrollment of nineteen hundred, with an average class size of thirty students.

JANET WALTER: I see that you put in four for x and got e squared.

NARRATOR: Janet Walter teaches algebra, pre-calculus, and calculus. One ways that she helps her students build problem solving skills is by encouraging spirited discussions as students share their work.

JANET WALTER: Are you sure? What?

STUDENT 1: Yeah, because, well, I was like confused at first because I was like, what happened to the five z squared, but actually you guys factored out z at the end and the very beginning.

STUDENT 2: What if you don't factor out z though?

STUDENT 1: See I didn't factor out...

NARRATOR: Janet is trying to encourage her students to become more active learners, but she is aware that this transition can be difficult for students who are not familiar with her style of teaching. Let's watch as Janet works with an Algebra I class that she inherited just two weeks before this taping, when their previous teacher quit in mid-year.

JANET WALTER: My understanding was that the teacher was having problems with behavior with the students and unable to really get them engaged in learning the mathematics that they needed to learn. And so, when they came to my class they were not prepared for their experience with this new teacher. And they were not sure that my asking them what they thought was a legitimate thing. And for me to wait to hear from them, and to ask them to contribute was something they just were surprised by.

JANET WALTER: ...positive linear correlation

STUDENT: Negative.

JANET WALTER: -Or a negative correlation or- what else did we see?

STUDENT: No correlation.

JANET WALTER: No correlation at all, good. And so, the problems that you were supposed to do were clusters of points, and what were you supposed to do with them?

STUDENT 1: Find out-.. Like, the points of the slope, like what the slope is.

STUDENT 2: Points on the line.

JANET WALTER: Points on the line?

STUDENT 1: A general line for the points.

JANET WALTER: A general line for the points. Okay, is everybody ready?

STUDENT: Yes.

NARRATOR: In this unit, Janet's students are working with scatter plots, collections of data points in the form of a graph. They are trying to draw a line that makes the best fit for this data, and then come up with an equation for this line that minimizes the distances between the line and the data points.

JANET WALTER: Okay, so, somebody show us what you did. Who wants to volunteer? Go ahead.

MARY: It's a y equals three-fourths x plus- you graph it out and you get-

JANET WALTER: So if we wanted to sort of visualize where the line that Mary came up with is, you use negative three, negative two-

MARY: And one, negative one.

JANET WALTER: This point right there?

MARY: Yes.

JANET WALTER: And which was the other point?

MARY: Negative one, one.

JANET WALTER: So, those two points, pretty much?

MARY: Yeah.

STUDENT: It was somewhat more at an angle.

STUDENT: Yeah.

JANET WALTER: You think it would be better if we did it more like that?

STUDENT: Yeah.

JANET WALTER: So, would you do this differently, then?

JAYMEE: Does there have to be a point there?

JANET WALTER: What do you think?

MARY: Find one of the points on here.

JAYMEE: Or can you just- like -I came up with a point.

JANET WALTER: You just made up a point?

JAYMEE: Yeah, so the line would go through it, so half would be on half.

JANET WALTER: Do you want to show us what you did?

JAYMEE: I guess so.

JANET WALTER: Okay, so here was one line-

JAYMEE: The only problem is that, my answer, I did the line write, except the equation- I didn't like the equation I came up with. It was really weird, didn't kind of-

JANET WALTER: It was really weird?

JAYMEE: The equation- no- when I come up with like the n equation, you figure out the slope of the line and everything.

STUDENT: Y equals-

JANET WALTER: So it was weird?

JAYMEE: It was weird, yeah.

JANET WALTER: Show us.

JANET WALTER: I really didn't want to go to the board and give them some examples and just say, "Well, this one doesn't fit and that one doesn't fit," but there seemed to be a lot of confusion on what exactly are we trying to do.

JANET WALTER: Show us, use the ruler and maybe we can see. And then, did you have a question?

STUDENT 1: What is you went down the box on the Y axis you went down one more negative?

JAYMEE: So, move it down to, like, here?

STUDENT 2: So move, like, the whole down-

STUDENT 3: So it'll be on that dot over the line.

JAYMEE: Well now it looks kinda-

STUDENT 2: Looks a lot better.

NICOLE: But was the other answer right?

JANET WALTER: She came up with a line that she thought fit the points really well, right? And is that what we're trying to do?

NICOLE: Yes.

JANET WALTER: How are we going to decide if one line's better than the other? What do you think, Nicole?

NICOLE: I don't know. That's why I'm wondering. I want to, like, keep on doing more.

JANET WALTER: Okay, so let's draw this and see if we think this is pretty good or not.

JANET WALTER: That's okay?

GREG: Looks okay to me.

JANET WALTER: Right, so what are we supposed to do about this?

JAYMEE: Make the equation another line?

JANET WALTER: All right, how can we do that?

JAYMEE: You can do it by adding- taking one of these points in to get B, or you could just find the Y intercept where the line intercepts from the line, which would be easier, which would be negative one. So it would be y squared-

STUDENT 1: There's an answer in the back of the book, but I don't see how any of us got it. It's different than-

JANET WALTER: What does it say?

STUDENT 1: The answer in the back of the book is, "Y equals one half x minus one" and it's a positive slope.

JANET WALTER: But we got a positive slope.

JAYMEE: Yeah, I got a positive slope.

STUDENT: Yeah, but I guess they have two-thirds x, two-thirds instead of one-half. I don't know.

JANET WALTER: They did one-half and we've got two-thirds?

STUDENT 1: Yeah.

JANET WALTER: How does that make anything different?

GREG: On two-thirds, you go up two and over three, on one-half you go up one and over two, and that makes it either steeper or not as steep.

STUDENT 1: One-half will be steeper.

JANET WALTER: Does everybody agree with that, that one-half is steeper and two-thirds is not as steep?

STUDENT 1: No, two-thirds is steeper because it's over half, something like that.

JANET WALTER: Two-thirds is steeper? Now we've got a different opinion.

We're comparing the answer in the back of the book to the line that we came up with-

STUDENT 1: I want to see what the back of the book's answer is. Like, we should draw it-

JANET WALTER: Could you do that-

STUDENT 2: You could get like thirty different answers on this.

JANET WALTER: Could we sketch their line and- leave, let's leave Jaymee's line up there.

STUDENT 1: A line that's more mellow, going this way, I guess. Let me think, if I go one, two- right there, and one and two right there. This is their line.

JANET WALTER: Do you like their line better?

STUDENT 1: I think it's just the same thing, just about.

JAYMEE: Yeah, I think it's about almost the same. The line just a little bit not quite so slanted.

JANET WALTER: So, are you happy with your line?

JAYMEE: Yeah, I'm happy with both.

JANET WALTER: I think they're just unsure because they haven't been allowed to believe in what they know. So, now they're finding that they can have confidence in what they think is right, and they're being encouraged to voice what they understand and what they don't understand, and I don't think they've had the opportunity to do that.

The students wanted to know, "Okay, which one is it?" What's the best answer?

STUDENT: Would that be right too?

JANET WALTER: Well, let me ask you this: Would that be a right answer?

STUDENT: No.

STUDENT: No.

JANET WALTER: Well, it might not be too bad, but that's a better answer.

STUDENT: Is there a better answer than that, though? I mean, is there an answer that was just like, this is the final answer, this cannot be changed, this is it.

JANET WALTER: So, is there a better answer than that? Chances are, yes, and what it requires is a little bit more advanced math than we're doing here. There are lines that are called "best-fit" lines, but we're not going to be able to figure it out with what we're doing here; we're just trying to get an idea of what looks the best. Is that okay? And that's why the answers can be different. Terence?

JANET WALTER: So, I'm trying to help them get used to the idea that different approaches are valid, different perspectives are valid, different answers are okay. By the end of class today, they were much more comfortable with- there can be some ambiguity in what's going on, there can be some differences of opinion and still have it be okay. And so I'm really trying to have them build this community of support and a willingness to risk and to voice their ideas and to find out that it's okay and to have more freedom in how they think in math.

JANET WALTER: All right, for tomorrow- thank you, good job. You did an excellent job today.

CAROLYN MAHER: I think it's wonderful how Janet held back. Students will always push students- students will always push teachers to tell them how to think if they can get away with it. Students are very crafty; if they can get someone to do the work for them, if they can make it easier for themselves- remember, they don't know what that good feeling is yet. They know that they have to know something because they have to know it to get a grade, and they have to get a good grade; that's their responsibility of going to school. Some of them see that as their major responsibility. They haven't yet seen the wonderful reward of feeling good about themselves when they've really achieved building a solution that they own, that's theirs.

So, Janet not responding and holding back with all of this pressure is very, very admirable, and it's tough. I mean, teachers are told that their supposed to teach. Teachers are taught that they're supposed to explain things clearly, and Janet says, "No, it's your job to build up those ideas, and I'm not going to do it for you. It's your job to build up those ideas." And when they learn they can, they'll do it more and more and more.

JANET WALTER: ...Make sure you bring it up, Okay?

MARY: OK.

GREG: OK.

JANET WALTER: See you later

MARY: Thanks, Ms. Walter.

GREG: Thank you.

[Music]

NARRATOR: Caroline Maher invited a group of Kenilworth tenth-grade students to meet after school to eat pizza and work on math problems. In spite of their busy schedules, the students would continue to attend these sessions several times a year for the rest of their high school careers. Let's look at how the mathematical ideas that the students had built in earlier grades can reemerge- sometimes in surprising ways.

CAROLYN MAHER: This was the first time they had come together since they returned to the high school. Before that we were meeting outside of school because they had gone to the regional school. And we, as usual, brought food for them- it was after school- and just sort of in a carefree, friendly way, we made the analogy to, "Do you remember when you were younger and you had to solve the problem about pizzas?" Immediately they wanted to know and get into it, and while this was just supposed to be feeding time, they got to work and worked on the problem.

NARRATOR: The problem was, "How many different pizzas can you make when choosing from five different toppings?"

ANKUR: How many did you get?

JEFF: I got more than four. Name your things.

BRIAN: One-two-three.

JEFF: Hmm-hmm.

BRIAN: One-two-four-

JEFF: Hmm-hmm

BRIAN: Two-three-four

JEFF: What about one-three-four-

BRIAN: One-three-four-

ANKUR: That's four-

ANKUR: Two, three, four-

BRIAN: I want to write them down and then cross out the ones that would be the same.

ANKUR: No, just write this. Two-three-four, two-three-five, three-four-five-

NARRATOR: Just as they had done in fifth grade, the students made categories based on the number of toppings, and counted the number of combinations they could make in each category. By carefully counting they came up with an answer.

MICHAEL: Thirty-one plus cheese.

JEFF: Now, thirty-two plus cheese.

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

CAROLYN MAHER: There seems to be- some people weren't convinced about that-Mike-

MICHAEL: Without cheese it's what?

JEFF: It's thirty-one.

MICHAEL: With cheese, it's thirty-two.

CAROLYN MAHER: Okay, so if your finding pizzas when you could pick from five toppings, how many are there?

MICHAEL: Thirty-two.

CAROLYN MAHER: I think what's to be learned from this particular episode is what a payoff there is to revisit problems later in one's learning. I think it's unfortunate that we don't do more of this in school, because it enables us to make connections in the mathematics we're learning and the ideas we're building that we would not otherwise make. Michael made some very interesting connections here, and revisiting the problem provided the opportunity to do that.

NARRATOR: Michael surprised the group with an exciting, mathematically sound approach to the problem.

MICHAEL: You remember like, the binary system a little while ago? The ones and zeros, binary, right?

STUDENT: Hmm-hmm.

MICHAEL: Ones would mean the toppings, zeros no toppings. So, if you had a four topping pizza, you'd have four different places in the binary system. The first one would be just one, second one would be- That's the next number up. You remember what that was? One just was a two, and this was three.

JEFF: I don't remember what each one was.

MICHAEL: You don't remember the binary-

JEFF: No, I know exactly what you're talking about. It's the thing we looked at in Mr. Poe's class, with the zeros and the-

MICHAEL: Yeah.

JEFF: Yeah, I know how it is. I don't know how to add it or whatever.

CAROLYN MAHER: These children were introduced to binary numbers by their teacher in sixth grade, so they learned about this as an alternative system to count, and it seems that Michael remembered this idea, and picked up the idea of binary numbers to account for all the pizza choices, so he had the advantage of also keeping track of how many combinations he had at a particular point, because he could read it as a binary number.

NARRATOR: In Michael's solution, each place- or bit- in the binary number corresponds to one topping on the pizza. The students asked Michael, how did he keep track of all the combinations?

ROMINA: How did that equal to fifteen, Mike?

MICHAEL: Because, you have- I know how to write binary.

ROMINA: Did you work it all out.

MICHAEL: This is how binary- this is one, this is two, this is three, this is four, this is- and those are the different combinations, you know, of- and eventually you would come up to this. That is it for four different-

JEFF: Yeah, you see, he's just leaving out, like, these, right-

MICHAEL: Yeah, I'm not writing those.

JEFF: Which is what confused me at first. Now it looks a lot-

CAROLYN MAHER: Our system uses the digits zero through nine, and groups numbers in groups of ten. The binary system, which is not based on a base of ten, is based on a base of 2 and it uses the digits one and zero. So, if you're counting in the binary system, you are counting how many ones or how many groups of two, how many groups four, or two squared, how many groups of eight, or two cubed, and so forth. What the binary does is, it counts numbers, but it just groups them differently.

MICHAEL: We're just having one- up through one. Zero and one. So, the first place would be one, the second would be two, the third would be four, and then- this is if we only have four places-

RESEARCHER: Places as place values?

MICHAEL: Place values, yeah.

ROMINA: Like, do they double?

MICHAEL: Yeah, it goes by double. If you have to express the number one, it's just this. The number two would be expressed by this. Number three would be expressed by this, four, five-

JEFF: Then you skip to eight-

MICHAEL: And that would be six- no-

BRIAN: Seven.

MICHAEL: -Seven- you understand it, right?

CAROLYN MAHER: It turned out, as we look at our data and as we review our data, that this idea of binary notation introduced by Michael is one that the others find helpful. It's not just Michael now using binary notation; this idea traveled among the group and they use binary notation. Every pizza choice could be mapped into a binary number between zero and one-one-one-one, zero being the pizza with no toppings, one-one-one-one being the pizza with all toppings, and that would be enumerating his choices, giving him fifteen plus the zero or sixteen.

MICHAEL: Okay, so if you have four different toppings, this will show you how many different pizzas no matter if it's with one topping, pizzas with two toppings, pizzas with three. If you go by the number system, the lowest number you can get is this, the highest you can get is this. And, if the value of this number is fifteen. That's the number of pizzas, because you will have-

RESEARCHER: Wait, where are you getting 15 from?

MICHAEL: Add these numbers up, you'll get- You add them up.

RESEARCHER: You're saying it's one 1, one 2, one 4, and one 8?

STUDENTS: Yes.

MICHAEL: You can go from- the lowest number is just this, and the highest would be that. And all the numbers in between are the different combinations of pizzas. You can get one like this- the highest number you can get is this, which is fifteen; there are fifteen other combinations.

JEFF: If you go through all the other combinations-

MICHAEL: And if you have a five-topping pizza, you add the next place, which is a sixteen, and there's thirty-one. This is without the plain; you just add one for the plain pizza. The next place would be thirty-two, and the place after that would be like a sixty-four. All you've got to do is add- to do this binary is just find the value of that, you'll get the number of pizzas.

ROMINA: I got it Mike.

JEFF: Going from just the one line to all four lines, it has to go through each combination to give you each number in there.

MICHAEL: It has to go through every single ones and zeros combination possible.

JEFF: To move up every number it's got to go to a different on, so-

CAROLYN MAHER: When they have an opportunity to revisit an earlier problem or a piece of an earlier problem, provide a justification, what do they do? They map into one's and zeros. Later, Romina, as she is justifying her solution to Ankur's challenge what does she do? She starts coding with ones and zeros. The idea is introduced by one of the students, but then they all own it soon enough. And the evidence of that, of course, is that they use it, they apply it to solve other problems.

CAROLYN MAHER: So you have to convince me that you've found them all.

NARRATOR: About one month later, the students met again with Caroline Maher after school. On the agenda were variations of the towers problems. As a warm-up exercise they looked at the problem: Choosing from two colors, red and yellow, how many total combinations exist for towers five tall that each contain two red?

ANKUR: - the first number, it's a one there, and then puts a one here, and then the rest are zeros.

NARRATOR: Mike and Ankur quickly solve the problem by counting combinations.

ROMINA: You guys proved it already.

ANKUR: Yeah.

BRIAN: ...Throw it out.

CAROLYN MAHER: No, we're going to wait to hear what you do, and you're going to hear what they did.

BRIAN: - tell us now.

NARRATOR: While they were waiting for the rest of the group, Ankur started playing around with a new problem that he had just invented: How many combinations can you make with towers four tall, selecting from a choice of three colors, and using at least one of each color in every tower?

ROMINA: Let's use one's, zero's, and x's.

JEFF: One's, zero's, x's?

CAROLYN MAHER: Notice, Ankur's problem is not trivial; it is really rather complicated and it is challenging, I'm sure, for those who are watching these tapes- will find the problems, themselves, challenging. But mind you, this is a problem posed by one of the group. So, I think this notion of problem-posing of students is something that we ought to think about and ask ourselves: What are the problems that our students pose to each other to solve, and can they solve them?

NARRATOR: Building on their experience with counting towers, they worked on Ankur's new problem for fifteen minutes, arriving at the answer to the simpler problem: The number of combinations for towers four tall, choosing from three colors.

ANKUR: There's eighty-one total.

JEFF: Of these?

ANKUR: No, of, like, everything.

ROMINA: How did you get eighty-one?

ANKUR: Do it and you'll figure it out.

ROMINA: No Ankur.

JEFF: X to the y. X is three?

ANKUR: It's three to the fourth. Because look-

JEFF: Three times three is nine, times three is twenty-seven, times three is eighty-one.

NARRATOR: After calculating that there were eighty-one total towers when selecting from three colors, Mike and Ankur returned to the conditions of Ankur's problem and came up with thirty-nine combinations, which is close but is not the correct answer. Brian, Jeff, and Romina approached the problem differently.

ROMINA: You have to have two of the same color, right, in one of them, if we're going to have all three colors, right?

ANKUR: I have no idea what you just said.

JEFF: You need to have 2 of every one.

ANKUR: Okay, okay.

ROMINA: So you have to organize them so you don't have any doubles. You can have them next to each other, you can have them separating, one on the end, in the middle, then two in the fourth spot, and the third in the fourth spot, right?

ANKUR: Yes.

ROMINA: So that's six.

ANKUR: Yes.

ROMINA: Now, in the other spots, you have an O and an X- those are colors, these are three different colors- an O and an X and an X and an O. So you have to multiply each of these six by two.

JEFF: And you couldn't have, like, XX, because that wouldn't meet the requirements. So you multiply each one by two, so that would give you twelve, correct? Because that means you could have either the bottom or the top? So that's 12.

ANKUR: Hold up. I just want to think about it for a second.

ROMINA: Six times 2 is 12, 6 times 2 is 12, 6 times 2 is 12, 6 times 2 is 12, 6 times 2 is 12, 6 times 2 is 12.

JEFF: Why you keep crossing that out?

ROMINA: Because that's wrong.

BRIAN: Yeah, it is.

ROMINA: You multiply all this by two, right, and you multiply all that by three because of the three different colors. That's what we were trying to say, but we wrote it bad.

JEFF: She wrote it funny.

ROMINA: So you can multiply these all by two, right, because you have one-color or the other, right? And then you have to multiply all by the three because ones can be any color, can be the three colors.

JEFF: There's twelve this way and there'd be twelve if you took the Xs, put them here, took the 1s, put them there- that's twelve more, and there's twelve more if you took the 0s and put them here and put the Xs back over there with the ones.

ROMINA: So it's thirty-six.

NARRATOR: Romina visualized the possible set of towers being divided into six groups. since every tower would have two of one color, Romina focused on the placement of the duplicate color. This gave her six placements. Romina understood that for each placement of the first, or duplicate color, there would be two possible combinations for the second and third colors. She also realized that these combinations would have two opposite arrangements for the second and third colors. So then 12 is multiplied by 3 to represent every color, making a total of thirty-six.

MICHAEL: Explain the thirty-six one more time because I was not paying attention. [simultaneous conversations] Now I want to know. Was it a good explanation? I wasn't paying attention.

ROMINA: I knew you weren't paying attention. All right. We have all three colors, right?

MICHAEL: What's 1, what's 0, and what's X?

ROMINA: The three different colors.

MICHAEL: I understand.

ROMINA: We have three different colors, and then you know that they have to be paired up, like the fourth color being added has to be the same as one that's already there, right-

MICHAEL: The fourth color has to be the same, yes-

ROMINA: Because you have-

MICHAEL: Yeah. Okay, so what we did, we said you have, let's say these are your four different ones, and we came up with six different possibilities where the match could be. It could be here and here, here and here-

CAROLYN MAHER: Romina provides a solution that is a proof, and she uses, notice, some of the notations that were introduced earlier by Michael, and she accounts for all possibilities. This is a beautiful moment; this is spontaneously working for a very short period of time, the student provides a solution that is an elegant proof.

ROMINA: Okay, do you agree with me? And then each one, this is either going to remain 0 and an X or an X and an 0.

ANKUR: So there's two of each one. You can't have an X and an X.

MICHAEL: I get that.

ROMINA: You get that?

MICHAEL: Yeah.

ROMINA: So, so far we have six and we have to multiply the six by the two for all these, so you get twelve, right? You multiply the twelve times the three, to get thirty-six, you multiply it because it's three different colors. So each one can be- you multiply that to get 36.

[Music]

CAROLYN MAHER: You know, when we started working with the students, we never knew how far they would get. We just accepted what they did. So, teachers who are starting to do this with their students now in their classroom should be encouraged to know that it's helped the students enormously to have begun the investigations and tasks in their earlier years. It should also help them to know that our students, just as theirs, did not all get to the same place. All of them didn't come up equally with the same always-new, brilliant idea. That varied, and that's okay, because they do learn from each other and ideas travel within the community. The students are doing mathematics, they're working as mathematicians, they are building for themselves very powerful images, they are having an opportunity to use and develop and try strategies, and they're having such fun doing it. They're enjoying it. They feel good about that enjoyment, they feel good about themselves and their success.

[Music]

NARRATOR: Is Romina's argument convincing? Why, or why not?

[Music]

[End of program]