PROBLEMS AND POSSIBILITIES
Workshop 3: Inventing Notations
Watch the video:
Part 1. "Putting it on Paper: Elementary Students Invent Notations"
[Video: Students presenting
NARRATOR: In mathematics, how
do we make visible an idea, or keep track of a line of thought?
From kindergarten to calculus, mathematics involves notations,
symbols as surrogates for abstract ideas.[Student: ..I saw
it like that. So that's A plus 2B.] [Student voices, computer
noises.] Often, the goal of math education is to give students
the standard notation: the written language of symbols and
equations that is commonly accepted in the math profession.
But, how often have you seen students come up with unique
and surprising ways to express a mathematical idea? It happens
more often than many teachers realize. In this program, we'll
examine how giving students a chance to create their own notations
can help advance learning.
STUDENT:.. starting with...
CAROLYN MAHER: What we've been
finding from our research is that students find a way to represent
their ideas. [STUDENT: ..and pepperoni I got again.] And these
ways of representing, I'm going to call notations. Notations
are a natural part of people's lives. It's a convenience.
It's a handy way of keeping track, of remembering, and for
sharing with other people.
ENGLEWOOD TEACHER: And then
at one point we realized that we hadn't ...
NARRATOR: Last time, we observed
a two-week summer workshop for teachers in Englewood, New
Jersey. These teachers presented solutions to problems and
tried to justify them to the group, using their own notations.
ENGLEWOOD TEACHER: ...and then,
we just pretty much play with them, and so that there are
no other possible combinations...
NARRATOR: In the fall following
the workshop, some of these teachers began to try out activities
from the workshop with their own students.
STUDENT: ...and these are the
STUDENT: I do.
NARRATOR: How might this new
approach to teaching and learning help these teachers discover
more about the notations their students invent?
[Student voices in classrooms.]
CAROLYN: Children surprise
us. They have wonderful ideas. They can represent their ideas
in very interesting ways, in ways that would not even have
occurred to us. So the teacher, in a sense, has to become
a learner. The teacher is learning about other ideas, the
ideas of all of those students. And they may differ, and they
will differ, from the way the teacher thinks about those ideas.
And they are often very brilliant, if we take the time to
listen to what children do, and what they say, and what they
write, and what notations they use.
NARRATOR: At the Lincoln Elementary
School, fourth grader teacher, Blanche Young, introduced a
challenge she brought from the workshop.
BLANCHE: Imagine yourselves
as a pizza owner. You've just opened a pizza shop and you
have four toppings...
NARRATOR: The problem asks,
starting with the basic cheese pizza, how many different pizzas
can be made by selecting from four additional toppings - sausage,
pepperoni, mushrooms, and peppers?
BLANCHE: ..Okay. What were
our four toppings, again?
NARRATOR: Before the class,
Blanche had expected each group of students to make a written
list of the different combinations they had found. However,
she gave the group free rein to come up with their own notations,
asking only that they share their results with the rest of
BLANCHE: ... toppings, and
this work together to try and get the toppings.
CHRISTINA: Oh. That looks like
JASMINE: That looks like a
STUDENT : Want me to write
it? ...Peppers and mushrooms...
CAROLYN: There are limitations
on what a teacher can do, given classroom time. What teachers
can do is encourage students to write. When their ideas get
put on paper and they're recorded in a particular way, in
the particular notations they choose, now there's an opportunity
to share. So the notations are very, very important; it becomes
the text, if you like, that is negotiated.
JASMINE: ...how plain? You
see plain on the board?
JASMINE: So you have to do
ALVIN: You can put plain pizza.
JASMINE: We can?
ALVIN: Yeah of course. Don't
you, when you order pizza from the store you can say, "I want
a plain pizza."
JASMINE: Yeah, but it was not
on the board.
ALVIN: So? When the professor
was here, it was on the board.
JASMINE: No it wasn't.
ALVIN: Yes it was.
JASMINE: No it wasn't!
CHRISTINA: OK...stop arguing.
ALVIN: Forget it. Anyway, I'll
just put the thing on it.
JASMINE: Make that whole half
over everything on it.
BLANCHE [handing out transparency
sheet]: Just in case...
JASMINE: Put all the mushroom
and pepper and stuff like that over each half
ALVIN: Now we can make the
ones with the two toppings....
ALVIN: Two toppings.
CHRISTIAN: That's good.
NARRATOR: Blanche is not working
to make these changes alone. After the summer workshop, the
facilitator, Arthur Powell, continued to support the Englewood
teachers by visiting classrooms and holding after school seminars.
ARTHUR POWELL: Can you guys
explain to me the list that you have?
BLANCHE: I think they need
something other than the words, to keep them categorizing...
ARTHUR POWELL: I was wondering
whether you thought some might want to do a report back now?
BLANCHE: Okay. All right.
BLANCHE: Is there a group that
would like to get started with sharing their work with us
as to what they did today?
NARRATOR: Each group that came
to the overhead had a slightly different way of presenting
LISVER: ...the fifth one? We
put sausages with pepperoni. The next one we put pepperoni
NARRATOR: Some students simply
made a list. .
BLANCHE: Is anyone looking
at her toppings? Do you notice anything that she has there?
CHRISTINA: She said something
BLANCHE: Which one is repeated?
CHRISTINA Sausage and pepperoni.
BLANCHE: Which pizza got repeated?
CHRISTINA: That one right there,
and then -
LISVER: And now you're saying
BLANCHE: Do you see it, Lisver?
BLANCHE: You have one of them
repeated, okay? Now let Christian explain his work.
CHRISTIAN: Right over here
I put pepperoni and pepper.
NARRATOR: This group made a
drawing that showed all of their possible combinations on
CHRISTIAN, CON'T: ... Over
here I put pepperoni and mushroom. Here I put sausage and
pepper, pepperoni and sausage -
NARRATOR: Categorizing by the
number of toppings, this group drew a different circle for
each category that included all the toppings that would be
STUDENT: OK, in this pizza,
in each slice, I put the different toppings. First, in this
one I put pepper and mushrooms, sausage and pepperoni...
BLANCHE: So am I correct in
asking you that this is the one topping choice?
BLANCHE: That's my one topping
choice pizza? Okay. That's my one topping choice pizza.
DAVID: I put pepperoni... Then
I put tomatoes, mushrooms...
STUDENT: This is the three
BLANCHE: This should have been
the three choice pizza...I think all of you have made very,
very good observations with this pizza problem, and you did
a tremendous job representing it in your drawings, putting
it on the overhead, and explaining it to the group. But again,
it's a problem that needs to be revisited again. We may see
this problem again. I'd like to save your overheads. And those
students that did not have an opportunity to put their work
on an overhead, I will make sure you get that opportunity
the next time we revisit this problem.
NARRATOR: After the presentations,
Arthur and Blanche met briefly to discuss the activity. Arthur
started by bringing up one of Blanche's previous concerns.
ARTHUR POWELL: ...there was
some disappointment, you had some disappointment about those
students who attacked the problem through drawing.
BLANCHE: Yes. When the problem
was first posed to them, we had asked them to come up with
a list of the different combinations of choices they could
offer. And when I walked around and children were making drawings,
I panicked, because that's not a list. In my mind, it wasn't
a list. Yet, in their minds, it was as clear to them as the
list I thought they would have, you know, come up with. And
the picture list was just as valid as the word list, which
has led a lot of them to come up with combinations.
CAROLYN: Children are natural
thinkers. If you give them something to think about, if you
give them an investigation, a problem to pursue, they have
ideas. All children have ideas. And unless you know what those
ideas are, you're not going to know what the appropriate intervention
is, what the next step is, what the question is that you should
be asking. Where to take that idea, to help the understanding
grow for that child.
NARRATOR: Back in the spring
of 1993, Carolyn Maher and her research team began work at
a new site, the Redshaw Elementary School, in the urban district
of New Brunswick, New Jersey.
ALICE ALSTON: What we're going
to do today, I had some ideas and the ideas were completely
changed because Dr. Davis and Dr. Maher were particularly
interested in seeing how you all would go about solving some
problems that some other students worked on....
CAROLYN: The Redshaw students
had just finished working on the towers problem: Building
towers four tall, selecting from two colors. The children
did not want to pursue building towers five tall and six tall,
as the children had done in other sites. They were more interested
in how can you build towers four tall when you can select
from three colors or four colors. So this is what they were
concentrating on before they worked on the pizza problem.
ALICE: This is a problem about
pizza and what I'm going to do first is hand it out, let everybody
read it -
CAROLYN: You still have to
be in tune for what the students are ready to do, or what
they're patient with doing at that time, particularly if they
have something else in mind that's also valuable and important.
For them, it was the exploration with several colors.
PATRICK: "PM" means pepperoni
NARRATOR: Understanding the
students' interest in selecting from multiple choices to make
combinations, the researchers presented the pizza problem
- How many different pizzas can you make when selecting from
four toppings? One of the questions they wanted to address
was: In what ways would these 5th graders use notations to
represent their ideas?
ARTESHIA: We have this one.
NARRATOR: Arteshia and Desiree
used blocks, with yellow to represent cheese and different
colors to represent each of the additional toppings.
DESIREE: ..see if my solution
matches with your solution.
NARRATOR: Eboni and Kersa simply
wrote out the names of the toppings.
AMY MARTINO: You have boxes
around these. What's in this first box here?
KERSA: One cheese pizza, one
pepper and cheese pizza, one sausage and cheese pizza, and
one mushroom and cheese pizza, and one pepperoni and cheese
pizza. And the total is five pizzas.
NARRATOR: Frederick wrote out
the names while his partner, Marcel, put the symbols inside
circles. Patrick and Benny used a series of abbreviations
to represent the toppings.
PATRICK: Look. "PM" means pepperoni
and mushrooms. Peppers and mushrooms. Pepperoni and sausage,
mushroom alone, and just sausage alone.
CAROLYN: We deliberately wrote
the problem so that two of the toppings would begin with the
letter "P": peppers and pepperoni. That was deliberate, because
that, in essence, forces students to think even beyond "Let's
just use the first letters. Now we have to make a modification
here." And they do. They do. There will be, of course, some
students who write out and spell out the toppings. But students
do not like to do more writing than they have to. Just as
we adults look for abbreviations and shorthand, so do they,
in very natural ways.
NARRATOR: Like most of the
groups, Patrick and Benny organized the combinations they
were finding into categories, based on the number of toppings
ALICE: What are these?
BENNY: A pizza with 2 toppings.
ALICE: And how many of them
CAROLYN: You see so many different
kinds of notations coming from individual students. And it
shows the power and the potential of the students. They like
very much being asked to be creative, and they responded.
They were creative.
LATIMA: This is cheese - regular
cheese pizza, and this is cheese with peppers. This is cheese
EBONY: Just say we had cheese
with all of them.
ALICE: Sure. Yeah, I understand
that. And so these are the four single toppings? And what
about those six?
LATIMA: These two have the
two toppings; the rest of them have three. And this one has
NARRATOR: After about an hour
of working in small groups, the students presented their findings
to the class. The researchers found that each of the 9 pairs
of children had at least one representation that differed
from all the others, including this unique drawing, produced
by Marcel and Frederick.
MARCEL: ... And this is a cheese
with the pepperoni and sausage,[ALICE: That's 2 toppings.]
and cheese, pepper and pepperoni, cheese, pepper and mushroom,
cheese pepperoni and mushroom, cheese, sausage and pepperoni...
CAROLYN: It's so fascinating
to see what these children can do. And they naturally do invent
their own notations, and they naturally do invent their own
ways of communicating to each other. What we learned is that
their mathematical thinking was very parallel to the mathematical
thinking in the other sites.
BHARPUR: ..we tried to make
another in the "3's", we got a duplicate.
CAROLYN: Maybe with little
twists to interpretations of the problems or with interest
in pursuing it or the problems in different directions, but
the findings were very parallel in all of the sites.
NARRATOR: In March of 1993,
the researchers brought their investigations to a 4th grade
class at the Conover Roads School in Colts Neck, a small town
in rural New Jersey.
NARRATOR: When the researchers
gave them the pizzas with four toppings problem, most of the
students made lists of toppings and counted their combinations.
But researcher Amy Martino noticed that one student, Brandon,
used a highly unusual and insightful method of keeping track
of his combinations. Brandon first made a chart with the toppings
arranged vertically in columns. Moving down the page, he worked
methodically row by row to create his pizzas. He wrote a one
in each column to represent the inclusion of a topping and
a zero to indicate when a topping was not present.
BRANDON: ...I'm making a graph.
AMY MARTINO: What does that
mean, one-zero, one-zero?
BRANDON: Well, instead of using,
like, you have pepper down, or sausage down, I'm just going
to put, like, a one, for like, "Yes, it's going on," and zero
for "No I'm not."]
NARRATOR: One month later,
in an interview with Amy Martino, Brandon was asked to recreate
his chart and account for all possibilities.
CAROLYN: The interview was
to validate what we already found in the classroom, and Amy
wanted to push it further. We did not expect Brandon to do
what he did. It was spontaneous.
AMY: Okay. You want to tell
me about what you're doing here, and how these turn out to
be pizzas, these zeroes and ones?
BRANDON: Well, since there
are three, four toppings, that is. Nothing on the pizza. And
you could have one pepper on the pizza with nothing else,
one mushroom on the pizza with nothing else. Then you could
have a couple sausages on the pizza with nothing else, maybe
a couple pepperonis. And if you don't want to have that, you
could start getting fancy and go into twos. So have a pepperoni
and mushroom, nothing else, then a pepperoni-sausage, nothing
BRANDON: Pepper and pepperoni,
nothing else, and so on. Then, since we're all done with pepperoni,
you could have a mushroom and sausage with nothing else.
AMY: What do these zeroes and
ones mean? Like what does the zero represent here?
BRANDON: You have nothing on
that - that's nothing. I don't know why I chose to use zeroes
AMY: Mm-hmm. I was going to
ask you about that, where you got this idea from?
BRANDON: I don't know how I
got it. It just popped into my head.
CAROLYN: Some of my colleagues
were saying to me, at the time, "Maybe his father is a computer
scientist, and he is exposed to binary numbers, and that's
how he knows his ones and zeroes." Well, his father is a businessman.
His mother was a homemaker. And as we pushed that, nope, we
eliminated that possibility. Brandon didn't have a computer
at home. He wasn't a person who worked on the computer all
of the time. Literally, the idea of zero and one popped into
his head, just as he said it.
AMY: Can you show me what -
you have them in groups here - can you show me what those
groups are on here?
NARRATOR: Brandon divided his
chart into groups, organized by the number of toppings.
AMY: Okay. And what group is
BRANDON: Okay. Here's the "ones"
AMY: Okay, and what does that
mean, the "ones" group?
BRANDON: You only have one
topping in the group.
BRANDON: Then you could have
the "twos" group, which will go about - The "twos" group is
like the most.
AMY: What do you mean, "the
BRANDON: You get the most out
of two, because you get more choices than one, and you get
more choices: pepperoni and mushroom, pepperoni-sausage, pepper-pepperoni,
and that so on ... So the "two" group is, like, the biggest.
AMY: Can you convince me that
there aren't any more in the "twos" group, that there aren't
seven or eight?
BRANDON: You go, pepper-mushroom,
that's one. Pepper-sausage, that's two. Pepper-pepperoni,
three. Then you can't do any more, because you already used
sausage once and mushrooms once. And to tell that you already
- And to see that you made duplicate, look over there, and
"one." Because if you just look there, you'll see another
one. But if you see a zero there, that means it's not a duplicate,
because you've got nothing there.
BRANDON: So if there's a "one/one",
then that would be the same as there. Then you get into mushrooms...
CAROLYN: He decided to keep
track of his pizzas by saying it either had a particular topping,
or it did not. And he did it in a very systematic way. And
as his chart reveals, he accounted for all possible pizzas,
and he had 16. It was the notation he used that helped him.
BRANDON: So then your only
choice left is having an "all" pizza, with everything.
AMY: Interesting. And what
are we calling this group?
BRANDON: The "all"...I don't
know what I call that. The "total."
AMY: Okay, the total. You call
these the "zeros," the "one toppings," right?
BRANDON: Yeah. "Two toppings,"
"three toppings," "four toppings."
AMY: You call it four toppings,
right? Sure. Does this problem with pizzas remind you of any
other problems we've done this year?
BRANDON: It kind of a little
reminds me of the blocks, because you ...
NARRATOR: When Amy asked Brandon
if this problem reminded him of any other problem. He asked
for manipulatives, and started making towers. He showed how
each topping column in his chart corresponded to one position
on the tower, with a "one" on his chart representing yellow,
and orange represented by a "zero." Brandon organized his
answer by categories, based on the number of blocks of each
BRANDON: It's kind of like
the pizza problem. You start off with the group. Like this
would be the "ones" group. Oh yeah, I see this now. This is
like the "ones" group. You only have one of the opposite color
in there. This isn't how I did it, but I just noticed this.
AMY: This is fascinating to
BRANDON: I just noticed it.
Then you would have - that would be the "ones" group - you
only have one...
CAROLYN: He did exactly the
same rebuilding of towers at that interview session as he
did in the classroom. He found the tower and an opposite,
the tower and an opposite. And he found all 16. But something
happened; something happened in his head. Because he said,
"Wait, I just thought of something. Just a minute." And he
had these tower models right in front of him, and he reorganized
them in a way that they mapped into his chart for pizzas.
BRANDON: ... you have one pepperoni.
That would be like - one pepperoni is like. Since we were
looking at yellows, a yellow would be "one", the reds would
be "zeroes." You could have one pepper, like I chose here,
and right there. Then it's like stairs. If I draw a line down
AMY: You need a pen?
BRANDON: If I draw a line down
here like this, it would go like - sort of look like stairs.
AMY: I see.
BRANDON: Then you'd go across,
draw a line down there, go across, draw a line down there,
across, draw a line down there - across - So you would have,
like, "one," "one," "one," "one." It's sort of like here.
You have one pepperoni, one mushroom, one sausage, one pepper.
AMY: Oh! Is what you're saying
to me then that, like, the yellow cube here is like a number
one on your chart?
BRANDON: Yes. If we were focusing
on red, a red would be a number one.
AMY: Okay. Well let's continue
with yellow. This is interesting. I think this is really neat.
Now, what would come next, with what we have here, if we want
to reorganize. You said these would be like the one - yellows.
BRANDON: Yeah. These are the
AMY: Okay. What about -
BRANDON: Now you would start
with the "two" yellow group.
NARRATOR: Brandon referred
to his notations, and demonstrated an exact correspondence
between each tower he had built and each pizza on his chart.
BRANDON: Yellow-yellow, red-red.
Same here. Because if you wanted to stand them up, it would
be harder to have to stand up the paper. So it's yellow-yellow,
AMY: I understand.
BRANDON: That would be a "two."
Then you could have 'em
AMY: Yeah, what would the tower
be that would like this pizza?
BRANDON: Right here you would
have yellow stand for "one." So you would have a yellow "one,"
red "zero", yellow "one," red "zero."
AMY: I see.
BRANDON: That would be another
NARRATOR: When two problems
that might look different on the surface, like towers four
high and pizzas with four toppings, have the same underlying
mathematical structure, this is called isomorphism.
CAROLYN: Brandon recognized
the isomorphism after working on pizzas. What students sometimes
do is they think of one problem one way, they think of the
other problem the other way, and don't see the equivalence
in structure. So to recognize the isomorphism is to disclose
that equivalence by looking at both problems in very deep
BRANDON: If we're just focusing
on yellows, then the pizza with everything.
AMY: Oh, I see. Okay. And are
we missing any?
AMY: You know what I'm wondering?
We have this guy left, right?
BRANDON: Yeah, because we're
AMY: Because he's the opposite
of this guy?
BRANDON: Yeah, we're not focusing
AMY: If we had to call him
a name, though -
BRANDON: Oh, this will be the
"zero." Oh yeah. Since the reds would stand for "zero," this
would be a "zero" guy.
AMY: This is neat. This is
really neat, Brandon.
BRANDON: I finally found out
what the red would be. Red: "zero" guy.
AMY: I wanted to ask you. Could
we have done it the other way around? Could we have focused
on red and gotten it to work the same way?
BRANDON: Same way. It would
just look like this. Here's the "ones" group, "twos" group
AMY: One red. Okay.
BRANDON: The "twos" group would
be the same. And then all you'd do is -
AMY: What would these be? What
would these things be?
BRANDON: That would be the
"threes" group. And just switch those around. Same thing.
AMY: Neat! Now, would we be
changing the number names for red and yellow? In other words,
when we did this -
BRANDON: Yeah. Now the reds
would be "one" and the yellow would be "zero."
AMY: This is really nice. Are
you convinced that you found all the towers and all the pizzas?
BRANDON: Yeah. All the towers,
all the pizzas. Yeah.
AMY: They both come out to
BRANDON: It's 16. Two, four,
six, eight, ten, twelve, fourteen, sixteen.
AMY: Are you convinced of this
AMY: Yeah? This is really very
CAROLYN: Brandon had an opportunity
to think deeply about a problem. And he had an opportunity
to talk to someone about his ideas. I think we have to remember
- We see Brandon and we all so impressed with what he did.
And what he did was very impressive. But at that time, the
schools grouped students according to math ability. They don't
do that anymore. This was many years ago. And Brandon was
in the lowest group. And when later we went to the teachers
with what we found, with our interview of Brandon, and we
said, "Look. Look at this! This is just absolutely brilliant.
This is wonderful; this is amazing!" And they hadn't seen
anything like that, they told us.
Well, I think we don't see
these things because we don't give students an opportunity
to show us their thinking. I think the world is full of Brandons.
We just don't take the time to find them and to listen to
them. We don't have mechanisms to pull them out. I think they're
NARRATOR: What do these three
examples --- Englewood, New Brunswick, and Colts Neck ---
have in common in terms of how notations help students justify
STUDENT: In this pizza, in
each slice I put...
NARRATOR: We've seen some of
the thoughtful and creative approaches the students used as
they uncovered the mathematical similarity between the towers
and the pizza problem. [Brandon: ...red- zero; yellow-one;
In mathematics, just because
everyone agrees on an answer, it doesn't mean they're right.
How can you teach students the difference between feeling
you're right and proving you're right?
CAROLYN: Okay. Hi, everybody.
There's a problem on your table. And if you'll all take a
copy - and you night want to read it yourself; then we could
be sure you understand it. You might talk among yourselves.
CAROLYN: It seems, on the surface,
like a very simple problem: how many different pizzas can
you make when you select from two toppings? However, as in
many restaurants, you're allowed to order a different topping
on half of the pizza, if you choose. So how many choices do
you have? So, this was a very real problem. It was something
that they would encounter in their normal, everyday life.
And they never thought about all possibilities before. So
it was not difficult to engage them in this problem. And they
quickly saw that it got complicated very fast.
NARRATOR: The students were
divided into two groups, and worked on the problem for about
ANKUR: Wait. O.K. Look, the
plain pizza that's one. [BOBBY: Half a plain.] Then half sausage,
and half pepperoni.
BRIAN: One whole plain. [ANKUR:
No, wait.] One whole sausage, one whole pepperoni. There's
ANKUR: Now, half plain and
BRIAN: One plain, one sausage.
ANKUR: OK, Mike you draw the
pizza, and then Amy will write underneath ... Make a pie,
and make it whole plain. Just put, like cheese - a cheese
BRIAN: Here, Ankur. Half pepperoni
and sausage, half pepperoni. Half sausage and pepperoni and
ANKUR: No, half plain and sausage,
BOBBY: What are you doing?
ANKUR: Forget the flames, Mike.
Okay? There. Now make - now put one sausage, like a sausage
NARRATOR: While one group of
students tried to write or draw all possible combinations,
the students at the other table argued over the best way to
organize their answer.
STEPHANIE: Matt, that kind
of graph isn't right, because it's a cheese, pepperoni, and
sausage. All you're going to get there is cheese, pepperoni
and sausage. You cannot put - Because it's not organized.
You can't put cheese and sausage in a group. You'd have to
put the cheese over here and the sausage over here. So why
don't you just make - OK a little graph like this.
JEFF: Because you're going
to put all in one column, and then you're going to put the
same amount in the next column, and then you're going to put
the same amount in the next time, and then you're going to
be crossing out two column's worth. It's a waste of time.
JEFF (VO): We didn't know if
we were right, most of the time, you know? I would have an
idea of how to get to a certain point, and you might have
the same idea how to get to it, but we'd have to - Getting
there was the hardest part, and that's what we were arguing
about - the right way to get there or the right way to make
sure that you'd covered all the bases. You know, anyone could
pick up a pen and get the right answer. But knowing how to
get there, that was what we were arguing about: the right
way to get there and how to make sure, how to prove. That
was a big question at the time, how to prove what we needed
MATT: Why don't you just draw
it, like -cheese and.. But that isn't organized. Keep it organized,
it'll be easier.
STEPHANIE: Well, that's not
- Well, how can you organize it? How do I know whether to
put this under cheese or sausage? How do I know whether to
put this under cheese or pepperoni
JEFF: Your graph was great.
Like, you said, we should make a graph with the one toppings
and the two toppings and the threes.
MATT: But it's not organized.
JEFF: It's more organized than
going like this!
STEPHANIE: Yeah, because Matt
[JEFF: Nobody knows what that means.] - how do you know? How
do I know? You know, how do I know whether I put this under
cheese or sausage? Or how do I know whether I put this under
cheese or pepperoni?
MATT: Put it under the column.
STEPHANIE: But, yeah, but there's
not going to be a cheese and pepperoni column, I mean, or
a cheese and sausage column. That's a pizza. You don't have
to make a column for that one little pizza. Do you know how
many graphs that is? You know, you'd have to make, like, tons
of little, separate, eeny-weeny [JEFF: Eeny-weeny.] [Laughter]
MATT: Steph! I'm just talking
STEPHANIE: Yeah, but you can't
put that under a column, because you don't know which column
to put them under. If you tell me how to...
MATT (VO): Maybe you took your
idea, and put this on it. Okay. So then you go around - another
person. "What do you think about this?" "Do that and that."
And he'd say, "Well, what if you put this on it?" And it kind
of comes into one big, whole thing that you use to solve your
BRIAN: I'm saying one plain
ANKUR: Do what Brian says!
BRIAN: One sausage and pepperoni
BOBBY: We already have that.
ANKUR: We have that.
BOBBY: That is mixed, almost.
BRIAN: It's half and half!
I mean mixed.
ANKUR: I know what he means.
NARRATOR: Even though they
had some disagreement over their methods, by the end of this
session, both groups had come up with a preliminary answer:
ANKUR: Ten. Now that's seven,
eight, nine, ten, ... .
CAROLYN: Okay. I think that
JEFF: Don't tell me we're out
CAROLYN: I know. Isn't that
CAROLYN: It's really kind of
disappointing to me that we do get out of time so fast.
JEFF: Why don't we eat lunch
here and come back after lunch?
CAROLYN Can we come back tomorrow
CAROLYN (VO): They're so committed
to working these problems out that they don't want to be disturbed,
and that they say "Let us have the time." Isn't it lovely?
I mean, schools aren't structured to do that. But isn't it
so nice when we can do that?
CAROLYN: This is a real problem,
by the way. In fact, we have here Mrs. Weir, who's given the
same problems for a college class. So we're not really giving
you things that aren't important and the kinds of things we
want you to do in the future. So think hard about this. You
know, it's one thing to find them - "I think I have them all."
Remember the towers, "I think I have them all?" But then there's
the next question. How could you convince us that you have
all possible ones?
JEFF: Why do you always have
to ask that question?
CAROLYN: Because that's the
mathematics of it; that's when you become mathematicians.
That's when you become real problem solvers.
MILIN: IF everybody agrees,
then - if everybody agrees in this whole class, then can you
JEFF: Yeah, but this is just
a class of 12 kids. If you go to ask another class, they might
not all agree.
STEPHANIE: Besides that, you
know, the person that doesn't agree could be right.
CAROLYN Let me say it another
way. I have you on film in certain grades where you've all
agreed, and you've been wrong. So that's the challenge to
you now. That's what it is to do mathematics. That's what
mathematicians do. You've taken it to the level of trying
to convince, and that's what we're asking you to do. So kind
of put your names on your papers, and leave them there, and
we'll see you tomorrow.[BELL]
SHELLY : Like, with the Rutgers,
a lot of times, we found an answer. And that usually wasn't
good enough. They wanted to know, well, how did you know it
was the right answer? And because there was no teacher there
to tell us, "Yeah, that was right" or "That was wrong," and
they didn't just tell us how to do it, you had to look at
it and look at it over and over again, and compare it to everybody
else's answers, and see how they came about their answers
and how it compared to how you got your answer. And you went
through the whole process over and over again, and then you
started to branch out to different answers to see if they
were right. And a lot of times, in the end, you ended up with
your original answer, but you were more secure, knowing that
was the right answer.
NARRATOR: The next day, the
students returned to the same problem for another 45 minute
ALICE ALSTON: Would you all
mind if we sort of worked together, if some how we worked
out a way of checking your lists and your pictures and each
other's list and making sure that we all agree that everything
we got is right?
BRIAN: Here, a person can read
out one of them, and we could say if [ANKUR: check them] we
wrote them or not.
ROMINA: One plain. [Wait.]
[Check.] One sausage. [Check.] One pepperoni. [BRIAN: Check.]
Half pepperoni- half sausage.
MATT: What are we doing?
STEPHANIE: Figuring out our
MATT: Here's what we'll do.
NARRATOR: The students spent
a few minutes negotiating their justifications, and preparing
charts to help them present. By now, both groups had confirmed
that there were ten possible combinations.
CAROLYN: Can you sort of, in
a very general way, tell me why you think ... ? You know,
you really were -
STEPHANIE: We can't get any
more. We've been working, we've been -
CAROLYN You should be able
to have a picture in your head ... of why ... -
STEPHANIE: We've proved everything
to everybody in this group. All right. What we did is we put
them into columns of one - which is a whole pie, [JEFF: I
just wrote mine out.] two - which is two toppings on a pie,
[JEFF: Put that in you key.] and three - which is three toppings
on a pie. Okay?
NARRATOR: Stephanie's group
made notations to account for all of their combinations. Notice
that they treated the plain, or cheese pizza, as a topping.
They listed three categories of pizzas, based on the total
number of toppings that were used.
STEPHANIE: Now for a whole
pie, you can have cheese, you can have pepperoni, and you
can have sausage. You can't have it any other way. There's
no other way you can get a one topping whole pie. [MICHELLE:
Why!?] Because there's only three toppings.
JEFF: Explain why.
STEPHANIE: Because there's
only three toppings.
JEFF: How are you going to
JEFF: I'm not convincing you.
I'm convincing her. Are you convinced?
STEPHANIE: See, she's convinced.
Okay. [CAROLYN: Jeff, you're convinced, too, aren't you? JEFF:
Yeah.] Two, we have halves and two toppings. Two toppings,
okay? Plain old two toppings. And we have pepperoni, and then
on the other side, sausage.
TEACHER: Now is cheese on there
STEPHANIE: Yeah, cheese is
automatically on there.
STEPHANIE: Then we just put
cheese on there to show you that there's, like, cheese on
it, you know?
JEFF: Yeah, she was sitting
there crying before, "that's not cheese, why do we call it
STEPHANIE: Leave me alone.
You can put cheese, and then on the other half, pepperoni;
cheese, and then on the other half, sausage. Or all together,
mixed, no one-half the other, sausage and pepperoni.
JEFF: You sure that's it?
TEACHER: All right.
MILIN: Jeff if you ask another-
STEPHANIE: Are you convinced?
Okay. Then for three, we have sausage and pepperoni on one
side, and sausage on the other.
TEACHER: Oh, so you're allowed
to mix the sausage and pepperoni on one side?
STEPHANIE: Yeah. Okay. And
then we have sausage and pepperoni on one side and pepperoni
on the other. Then we have sausage and pepperoni on one side
and cheese on the other.
MICHELLE: Or half of the side
TEACHER: All right. I think
I got it.
CAROLYN ... OK you're convinced?
You all convinced? Okay, that's great.
NARRATOR: Brian's group also
divided the pizzas into categories: whole pizzas with single
toppings, halves with different single toppings, and mixed.
Pizzas with two toppings, both sausage and pepperoni.
BRIAN: We know that there's
no more wholes, there can't be any more.
ANKUR: There can't be wholes.
We know there's no more halves. And no half and mixed.
ALICE: How do you know there's
no more halves?
ANKUR: In halves, because we
used all the, like, ingredients in the pizza.
ANKUR [VO]: When the Rutgers
program comes over here, they always ask us to convince them
or they always ask us to convince the other people in our
group. While we convince, we realize that we're actually learning
more, we understand the concept better, and we help others
understand the concept, and everyone in the group learns together.
BRIAN: Because, uhhmm, plain,
that's like considered like a topping.
BRIAN: Yeah, plain, you can
only use two other toppings, because that's all they give
BRIAN: So you use pepperoni
as half and half, or half pepperoni and half plain. And then
you use the other topping, which is sausage to put on half
a pizza. Not mixed on whole ones, like half pepperoni and
ALICE: Okay. And you had one
ANKUR: Yes. We had...
BRIAN: Half pepperoni and sausage,
ANKUR: Half one side and the
other side mixed. One side is half, the other side is mixed.
ALICE: OK, now say that again.
ANKUR: One side is like, mixed
and the other side is, like, a whole -no, wait.
ROMINA: Like with colors.
BRIAN: Like with colors. Like
one side could be all different colors.. [ANKUR: and another
side the same color.]
ALICE: So one side is the mixed
sausage and pepperoni?
ANKUR: And the other side is
[BRIAN: just, like, one thing.] Just one thing. And so how
do we write that?
BRIAN: ..it could be sausage
ALICE: And that's all it could
BRIAN: Right. [ALICE: Why?]
At the end, the one's that are non-mixed. That's all the toppings.
ALICE: Because one side is
either sausage or pepperoni?
CAROLYN: Always, we try to
push students a little beyond where they were. It was never
about solving a particular problem. It was about looking at
other problems, maybe, in this class, and seeing if they could
come up with a generalization. So very early on, they were
doing this. They might not have had the - quote - "standard
notation" to do this. They sometimes did it in words. And
when we thought they had the idea, we thought that would be
the opportune time to now bring in the standard notation and
see if they now re-represent their idea with the standard
NARRATOR: One month after working
on the problem of pizzas with two toppings and halves, the
same group of 12 students met for an extended session, lasting
approximately 2-1/2 hours. This time, the researchers began
with the simpler problem: How many different combinations
could be made when selecting from four toppings, with no half
ALICE ALSTON: We have to make
a decision. Did they say anything about halves or is this
JEFF: Oh, wait there's no halves.
ALICE: Read it, what do you
think it says?
ANKUR: Wait, but it says how
many different choice does...
JEFF: I don't think they make
halves there. [Wholes! Wholes! Wholes!]
ALICE: I think it's just whole
JEFF: Thank the Lord!
MATT: Cross it out!
ALICE: OK, do you all want
to work for a couple minutes, and see if you can come up with
[Student conversation and writing]
ROMINA: ... and the plain,
ANKUR: What about the mixed?
JEFF: The plain!
ALICE: And then what was this
ANKUR: I started with the first
one, and mixed it with the second one. That's "P" slash "S."
Start with the first one mix it with the third one: "P" slash
"M." And then "P" slash "PE." And then start with "S": "S"
slash "M", "S" slash "PE," then "M" slash "PE."
NARRATOR: Approximately 15
minutes later, the students were confident that they had found
all possible combinations.
ALICE: Did everyone come up
with a solution to this one?
ALICE: Okay. If you're going
to do 16, who's going to convince me of it? [ANKUR: I will.
I already did.] Stephanie and Matt?
STEPHANIE: All right, uhhmm.
Well, we have whole and then we have a mixed column.
MATT: Well, we have - They're
thinking we have [STUDENT: Sub-titles.] the whole column and
the mixed column. The sub-title. [STUDENT: That's what we
ALICE: Okay. Whole and then
mixed, and then sub-titles? Is that what you're saying?
STEPHANIE: And when we started
out, we did, like, ... And then cheese, we did pepperoni,
we did sausage, we did peppers and we did mushrooms. And each
one of them was all by themselves. You know, nothing was ...
ALICE: Okay. This was in your
ALICE: How many were in that
ALICE: So that was an easy
one, wasn't it?
ALICE: Okay. Now, Stephanie
and Matt, you're saying that your second category was sub-divided?
Tell me what your first sub-division was.
MATT: Our subtitle was "the
mixed ones." And what we did for the mixed ones was we started
with the topping, and we added a topping. So we had -
ALICE: Ankur, this is sounding
a little bit like the way you described it to me, too. How
did you do it?
ANKUR: I had a pattern.
ALICE: What was your pattern?
ANKUR: I started with the first
one and mixed it with the second. Like, so my first one was
peppers and sausage. So I took peppers slash sausage. So I
skipped the second - I started with the first one again, skipped
the second one, and took the third one, "P" slash "M". And
then I put peppers and skipped the second and third, and I
went with the fourth one, "P" slash "PE." And then I started
with the "S" and -
ALICE: And then you're sure
you were finished then. And what did you do?
ANKUR: And then I started with
the next, the second one. I started with S, sausage, and mixed
it with mushrooms. And then sausage and pepperoni. Then I
went down to the next one, mushrooms - mushrooms and pepperoni.
NARRATOR: Ankur's idea of holding
one topping constant and changing the others is a strategy
that Matt noticed and will use again in the next problem.
MATT: We started with peppers
and pepperoni, and added.
ALICE: Okay. You say peppers
MATT: And then we added.
ALICE: And you added -
MATT: Sausage. Peppers and
pepperoni, with mushrooms. Then we had - then we couldn't
do any more with peppers and pepperoni. So then we figured
out a peppers, sausage and mushrooms.
ALICE: Peppers, sausage, and
mushrooms. Yeah. Is that all?
MATT: No. And there was no
more for peppers. We were convinced there was no more for
ALICE: That was all you could
do with peppers? Yeah.
MATT: There was only one thing
you should do with pepperoni -
ALICE: Which was?
MATT: Pepperoni, sausage, and
ALICE: And then you were done?
MATT: And then you have the
big one, the four topping pizza, which was the pepperoni,
the peppers, the sausage and the mushrooms.
CAROLYN: In most of our other
sessions, and even with adults, and even with college students
and high school students, if you give students the pizza problem
and you ask them to account for all possibilities, that takes
at least a session to do. These students did it in a matter
of 10, 15 minutes, if that long. And I suspect that the complexity
of the pizza with halves made this a very trivial problem
for them. Just had to write it up and tell us what it was.
They had to think about the idea of whole pizzas in solving
the pizza with halves. And generalizing it to four toppings
was very easy.
ALICE: Pizza Hut feels like
they didn't get their money's worth from their consultants,
and so [Student: Another pizza problem.] they're saying, OK,
[Groans from Students] now I want to see if...
NARRATOR: About half an hour
into this session, the researchers introduced a final problem,
one that included half pizzas.
ALICE: Sure, Robert, would
you read it for us ?
ROBERT: At customer request,
Pizza Hut has agreed to fill orders with different choices
for each half of a pizza. Remember that they offer a cheese
pizza with tomato sauce. A customers can then select from
the following toppings: pepper, sausage, mushrooms, and pepperoni.
There's a choice of crusts: regular, thin or Sicilian, thick.
How many different choices with pizzas does the customer have?
List all the possible choices. Find a way to convince each
other that you have accounted for all possible choices.
ALICE: Is this going to be
more? Or is this going to be less?
STUDENT: It's going to be more.
STUDENT: What you do is you
times it by two.
NARRATOR: The researchers deliberately
chose this problem to stretch the students' thinking. The
number of combinations is much larger than in the previous
problems, too large to accurately count out, using trial and
error. The students built on their past work, and Matt immediately
came up with a system that could find the answer.
ALICE: Before you start working
on it, Matt, you have an answer?
MATT: Well, I'm going start
with - What you could do is you start with the cheese, and
then you put a half, then you add all the rest of the toppings,
the peppers, all the rest of the toppings, the pepperoni,
all the rest of the toppings, the mushrooms, all the rest
of the toppings.
ALICE: OK, you all want to
work on it for a little while? Remember...
NARRATOR: The other students
ignored Matt's solution at first, and attempted to find their
NARRATOR: A few minutes later,
the researchers asked Matt to explain his strategy in more
MATT: We got 120 pizzas. I
figured it out. I figured it out. Some way I thought I might
have been right. What I did was I got the half cheese, the
half cheese- divided it in half; then I took each topping
and I put it in the half. Then I went to the peppers, each
topping, put it on that, put it on the side. Then to pepperoni,
ALICE: Okay, Matt, explain
to me what you're saying. You're saying that you started with
your cheese, and it could be with all of the others? Okay,
that was how many?
MATT: That was 15. It's like
Ankur, it's like Ankur did ..with the last problem. He moved
down the line, and added all the other toppings as he went.
So it was like this.
CAROLYN: If you think about,
you know, Matt's solution, and if you think about Matt's reference
to the idea that he gives credit to Ankur for presenting in
the two topping choice of the earlier problem, think of what
he does. You know, he makes use of all of the ideas, from
the more complex problem to the simpler problem, to, again,
a more complex problem, and he introduces a strategy of controlling
for variables. Now he says "Well I have all the sixteen, you
know?" But he talks about holding one topping constant. And
then you can, on the half, you have all your choices.
NARRATOR: Matt knew, from the
previous problem, that there are 16 possible combinations
of toppings for whole, undivided pizzas. Matt next considered
all the possible pizzas that are made up of two different
MATT: So it's half cheese,
and half -
ALICE: And half each of those
NARRATOR: He started with a
pizza that is half cheese and half other toppings. Since he
had already counted a whole cheese pizza, he couldn't use
cheese on the other half, and so he had to count only 15 possible
ALICE: And on this page, what
do I have here?
MATT: Half pepperoni.
NARRATOR: Then he moved on
to his second topping, pepperoni, holding that constant on
one side of his pizza. Since he couldn't repeat either cheese
or pepperoni, he counted the remaining 14 toppings.
MATT: ...pepperoni and sausage
- like that...
NARRATOR: Going through his
list, he eliminated the toppings that would have made duplicates,
eventually accounting for each of the possible remaining combinations.
Finally, he added up the numbers in each column: 16 plus 15
plus 14 plus 13, and so on, all the way down to one.
ANKUR: Is it possible to write
out all different combinations?
MATT: Well, if you wrote out
all the different combinations that I had -
MILIN: You'd die!
MATT: - your hand would be
BRIAN: All right Matt.
ALICE: Are there any duplicates
in Matt's approach?
ALICE: Is everybody convinced
that you've got a solution?
CAROLYN: Matt's notation was
particular to Matt. You know, he had his elaborate lines to
show the detail of the possibilities. He said, "Well look,
you know, if you keep this constant you could have it with
this topping, with this, with this - Notice the care. Now
an adult might say "You could have it with any of those 15
toppings," or "Now you have 14 left." Now Matt eventually
said that, but Matt, remember, was part of a group, and he
had to express his idea to others. And in order to do that,
he had to provide detail. And the detail was provided in the
notation he used.
NARRATOR: We've seen students
spontaneously creating ways of keeping track of their solutions
to a problem. What notations are students using to represent
their ideas and organize the pizzas?
[End of program]