Workshop Sessions
PROBLEMS AND POSSIBILITIES
Workshop 4: Thinking Like a Mathematician
Watch the video:
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 longterm 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.
LIPING: 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.
LIPING: ...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.
LIPING: This is a very fine
structure.
FERN HUNT: Yeah.
LIPING: 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
LIPING: 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.
AMYLYNN: 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. AmyLynn, 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: Thirtyone. 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 lala 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: Uhhuh.
GARY DAVIS: You got the pattern?
MATT: Uhhuh.
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]
AMYLYNN: 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 AmyLynn 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, precalculus, 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 midyear.
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 threefourths
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 twothirds x, twothirds instead of onehalf. I don't know.
JANET WALTER: They did onehalf
and we've got twothirds?
STUDENT 1: Yeah.
JANET WALTER: How does that
make anything different?
GREG: On twothirds, you go
up two and over three, on onehalf you go up one and over two, and that
makes it either steeper or not as steep.
STUDENT 1: Onehalf will be
steeper.
JANET WALTER: Does everybody
agree with that, that onehalf is steeper and twothirds is not as steep?
STUDENT 1: No, twothirds is
steeper because it's over half, something like that.
JANET WALTER: Twothirds 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 "bestfit" 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 tenthgrade 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: Onetwothree.
JEFF: Hmmhmm.
BRIAN: Onetwofour
JEFF: Hmmhmm
BRIAN: Twothreefour
JEFF: What about onethreefour
BRIAN: Onethreefour
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.
Twothreefour, twothreefive, threefourfive
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: Thirtyone plus cheese.
JEFF: Now, thirtytwo plus cheese.
MICHAEL: Thirtyone plus one
is thirtytwo. That's with cheese. It's thirtytwo, that's the answer.
CAROLYN MAHER: There seems to
be some people weren't convinced about thatMike
MICHAEL: Without cheese it's
what?
JEFF: It's thirtyone.
MICHAEL: With cheese, it's thirtytwo.
CAROLYN MAHER: Okay, so if your
finding pizzas when you could pick from five toppings, how many are there?
MICHAEL: Thirtytwo.
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: Hmmhmm.
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 oneoneoneone, zero being the
pizza with no toppings, oneoneoneone 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 fivetopping
pizza, you add the next place, which is a sixteen, and there's thirtyone.
This is without the plain; you just add one for the plain pizza. The next
place would be thirtytwo, and the place after that would be like a sixtyfour.
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 warmup 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 problemposing 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 eightyone total.
JEFF: Of these?
ANKUR: No, of, like, everything.
ROMINA: How did you get eightyone?
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 twentyseven, times three is eightyone.
NARRATOR: After calculating
that there were eightyone total towers when selecting from three colors,
Mike and Ankur returned to the conditions of Ankur's problem and came
up with thirtynine 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 onecolor 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 thirtysix.
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 thirtysix.
MICHAEL: Explain the thirtysix
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 thirtysix, 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 alwaysnew, 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]
