PROBLEMS AND POSSIBILITIES
Workshop 2: Are You Convinced?
Watch the video:
PART 1. "TEACHERS BUILDING PROOFS"
NARRATOR: Englewood, New Jersey is a small city located just
across the George Washington Bridge from Manhattan. Englewood
students have recently been scoring poorly on the statewide
standardized assessments. Superintendent Joyce Baynes, a former
math teacher, was hired with a mandate to improve student
achievement in the district's public schools.
JOYCE BAYNES: I really felt very confident that I'd be able
to make a difference in the teaching and learning here in
Englewood. And I really feel that we're making some impact
in that way, really concentrating on student learning and
concentrating on teachers understanding students so that they
can learn much better than they had been learning in the past.
The city of Englewood has a very diverse picture to it. In
fact, it is a very interesting city. We have families who
are homeless; we have families who are "middle class;" we
have families who are extremely affluent. We have youngsters
who may go to private schools or other non-public schools
such as parochial schools outside of the city.
Approximately 55% of the eligible school-age youngsters actually
attend the public schools, so that gives us cause to be concerned
about the other 45% who are not in our schools.
When I came on board, I didn't see a solid staff development
plan for the teachers. The teachers, I felt, were open to
new ideas. I wanted to bring on board a philosophy about teaching
and learning so that I could trust when a teacher went, you
know, into his or her classroom and closed the door, that
there was a certain feeling about teaching and learning that
would take place throughout the district.
As a former math teacher, I actually felt encouraged about
my ability to move the mathematics achievement to a higher
level. So, seeing mathematics as a challenging subject for
many teachers and students here, I began to feel I could put
myself into this; I can really share what I've done with others,
and bring in people and resources that I've used to move the
NARRATOR: To raise achievement in mathematics, Joyce is implementing
a professional development initiative that encourages teachers
and students to think deeply about problems and to justify
and convince others about their solutions.
ARTHUR POWELL: "Englewood signal" [raises his hand, laughter].
I think we're probably ready to return to some work that we
left off with yesterday...
NARRATOR: To begin the initiative, Joyce Baynes has invited
Arthur Powell, Associate Professor at Rutgers University -
Newark, to work with Englewood teachers, grades K through
ARTHUR POWELL: I'd like to ask that you spend seven minutes
reflecting and writing on the following two questions...
NARRATOR: During the summer, Arthur begins the project by
engaging 30 teachers in a two-week professional development
workshop. Arthur has defined clear goals for this initial
ARTHUR POWELL: My goals for the workshop are many-fold. One
has to do with introducing to the teachers a new way of working
in the classroom whereby they pay very careful and close attention
to what students actually do and say. The other goal of the
workshop is to get teachers to rethink how they view themselves
in front of mathematics - that is, if they use their powers
of perception and action while they're working with materials
and then make statements about what they're doing and what
they see, that eventually that will lead them to making all
sorts of discoveries about mathematical relationships and
ARTHUR POWELL: ...so, spend some time in your groups talking
about where you are. Finish up preparing for your presentations,
and then we'll have presentations probably in about 10 or
15 minutes. OK?
NARRATOR: During the workshop, Arthur asked the teachers
to work on combinations problems such as - How many different
towers four blocks high can you make by selecting from blocks
of two colors? For each solution, teachers explained and attempted
to convince the others that they had found all possible towers,
ARTHUR POWELL: In terms of teaching mathematics and in terms
of mathematics itself, it's very important to get involved
in looking for justifications because at the heart of mathematics
is the idea that we can look at patterns and relationships
and try to understand the underlying reasons why those patterns
and relationships exist, given the particular mathematical
objects. And in reasoning - in understanding why they exist
- one is developing ideas of proof.
NARRATOR: This group is developing a justification that they
have found all possible towers four-high by building up from
towers one-cube high.
ANITA: ...so, the only reason we did this was in order to
demonstrate how we knew there were no more, and also the reason
why we set it up so organized like this, was to show how exactly
you can double things, where the doubling comes from. And
the way to convince others is this - is the one tower, two
tower, three tower, four tower.
NARRATOR: After discussing the problem in the small groups,
the groups presented their findings to the other teachers.
ARTHUR POWELL: I'd like to invite this group here to present
what you found. Now remember - what we're going to be after
- we're after trying to provide for each other some way of
convincing each other that what you found is, in fact, all
that there are.
GROUP 1 TEACHER 1: How do you know if you have all possible
duplications? Well, pretty much we went through observations.
This is pretty much what we based everything by. I made a
little chart of how many possible blues and yellows we could
NARRATOR: The first group began by making a table of all
possible numbers of blocks of each color. They made a table
of towers four cubes high, showing five different cases for
the combinations of blue and yellow cubes.
GROUP 1 TEACHER 1: I arranged the towers this way according
to this chart, and then we just pretty much played with them
so that there are no other possible combinations. We end up
with 16 possible combinations.
ARTHUR POWELL: Yeah, you say all possible combinations. I
don't quite understand how you found, how you are justifying
that you found all possible combinations.
GROUP 1 TEACHER 2: It's all based on this chart, and then
we just did it through trial and error. If we got a combination
that was a duplicate, we knew we already had it, so it didn't
work, and we just basically went on to the next one, and that
GROUP 1 TEACHER 1: In this one, two, three, we kind of use
a checkerboard to kind of just check and balance ourselves.
And then we did the same thing here and then here and then
here and then the same thing here. And these two are opposites,
and that was it.
ARTHUR POWELL: Any questions for them? Okay. Robin, Tobey,
and Helen, yes?
NARRATOR: Organizing by cases, this group approached the
problem differently. They arranged towers by ìopposites,î
pairing the cubes in each tower with cubes of the opposite
color in a second tower.
TOBEY: We chose to do this in pairs, so we took four of one
color and then took the opposite of it. And after we did that,
we knew we'd done every possible combination of four of one
color together. Then we went to using three of color together.
So we took three blues and we matched it with the one yellow,
and then we immediately took the opposite of it and took the
three yellows and matched it with the blue, and these are
the only possible combinations that you can - well, no, they're
Then we took - we put the three blues on the bottom and put
the opposite color on top and the same in pairs; everything
was done in pairs. So, we have the single color on the bottom
and the single color on top. And these were the only possible
combinations that we have found where you can use three single
We then went to try and use two single colors together and
this row shows all the possible combinations that you can
make. And after we've exhausted all the combinations of two
single colors together, we were left with using alternating
colors together. We made one; we turned it opposite and matched
it and felt that when we finished it, we were pretty well
convinced that we have made all combinations possible.
ARTHUR POWELL: Any questions for Tobey? Comments?
ARTHUR POWELL: They're beginning to understand that in order
to develop a convincing argument that they have to say more
than, "Well, I've considered all the possibilities, so there
can't be any others." Their level of justification, the way
in which they're justifying themselves, is far more sophisticated
than what they were doing before.
ARTHUR POWELL: Caroline and Patricia.
CAROLINE: Okay, what we did is we started off with four yellow
cubes to make the tower. And when we realized that there was
no other possibilities, we decided to go to three yellow cubes
and one blue, and then what we did-
NARRATOR: Reporting for their group, Caroline and Patricia
justified their solution by combining strategies used by the
first and second groups. They looked carefully at the number
of cubes in each category to be sure that there were no more
combinations before they moved on.
CAROLINE: We started off with the yellow at the top. Is that
what we did?
PATRICIA: We had blue on the top.
CAROLINE: We started with blue at the top and then we had
our three yellows and then we moved down...
TEACHER: There are only four towers, cubes, so there are
only four possible positions for that blue to be in, so you
can only come up with four different patterns.
ARTHUR POWELL: So, you've now talked about the case of towers
that contain four yellows...
CAROLINE: Three yellows and one blue.
ARTHUR POWELL: And now you're talking about the case of towers
containing three yellows and one blue.
ARTHUR POWELL: Okay, so you're taking a kind of case analysis
here. And then where did you go from there?
CAROLINE: After three yellows? We went with two yellows and
two blue. We started off with the yellow/yellow/blue/blue
and that was..
ARTHUR POWELL: As you have there.
CAROLINE: Right. And then we just, you know, we switched
this around to make the next one. And we had them all lined
up so we knew that there were no other combinations but the
ones that we listed.
ARTHUR POWELL: You're saying because you had lined up you
knew that there weren't others?
CAROLINE: Yes, just looking at them, once we had all six
of them in front of us, we looked and we saw what - in ascending
order going one way and you know, there were no other combination.
ARTHUR POWELL: Yeah, I don't feel as convinced as I do in
your other two cases.
ARLENE: If you had yellow/yellow/blue/blue and then blue/blue/yellow/yellow,
if you had to try to find another combination, to move the
two again, it would give you the same thing; it would give
you the same possibility. Because there are only two colors,
so you have, you do the rotation another time and you know
that you would get the same color again, same combination
ARTHUR POWELL: So, Maria, you've helped me see that if I'm
looking at towers four tall in which two adjacent cubes are
to be yellow and the other two to be blue that those are the
only two. But you're claiming that there are six towers four
tall with two of each color. The third one that you have there?
ARLENE: ... trying one blue and still keeping the same, you'd
ARTHUR POWELL: So, you're keeping two yellows together?
ARLENE: Still keeping two yellows together.
ARTHUR POWELL: Okay. So, is it the case that you thought
about this - the first three that you have there - as keeping
two yellows together, looking at the different ways of keeping
two yellows together? Is that how you thought about it?
CAROLINE, OTHERS: Yes.
ARTHUR POWELL: Teachers often times, at least in this workshop
and in others, feel uncomfortable with the way in which I'm
conducting the workshop. And it raises anxiety in them, it
raises frustration, and in some case, anger, because they
come to the workshop with an expectation of being told: "This
is how you should teach. Here's a mathematical task; you've
worked on it; now let me confirm for you that you're right.
Or, If you're not quite right, let me tell you how to get
on the right track, or, Here's the answer." So, that's their
But instead, they're getting something very much to the contrary.
I'm asking them questions; I'm asking them: are they sure
when they make a response? I'm asking that they listen to
each other, that the conversations that we have are not just
different atoms put out into the air, but that there is some
connection, there is some link in what's being said. And so
they feel uncomfortable.
ARTHUR POWELL: ...Yeah, I was confused. So, that helps me
through the first three of those.
CAROLINE: When we did the blue/yellow/yellow/blue, the fourth
one became a reversal of that one. And then when we did the
alternating, the sixth one became a reversal of the fifth
ARTHUR POWELL: Ah...so that's a different scheme, a different
way of thinking about how you organize this. See, what we're
trying to do is I'm trying to understand how did you think
about the construction of these four tall towers? All right?
And I'm sometimes hearing different things. Maybe you thought
about them in both of these ways, I'm not sure. And then:
for the case where there's one yellow and three blues underneath?
TEACHER: Like I said before with the one blue and three yellows,
there are only four possible places for the one yellow cube
to be in. It can be at the top, the second, third, or fourth
position. Once you've done all those positions, it cannot
be in any other, so you know you've done all of the possible
ARTHUR POWELL: And then?
CAROLINE: And then when we had no more yellows, we went with
the four blues.
ARTHUR POWELL: And there could only be one -
CAROLINE: - way to do the four blues. And then if you look,
you know, going up, you'll see that the same -
ARLENE: We get the same pattern as we did with the yellow;
we used all four blues, then three blues, then two blues,
then one blue, then going to no blues.
ARTHUR POWELL: Oh, interesting. So, you're saying that you
could read your chart either from top down or bottom up.
ARTHUR POWELL: Uh-huh. When you worked, which way did you
ARLENE: We started with the yellows.
ARTHUR POWELL: You started top down with the yellows. Uh-huh.
Does anyone else see any other patterns or have any comments
to make about the chart that they have up? Does anyone see
a different way of talking about that chart than what has
been described? Martha?
MARTHA: They had up at the top zero blue, then they had one
blue in each case, but in a different position. And that pattern
occurred again. They kept, say, the blue constant, the two
blue together, change the yellow. Whatever way they did it,
the way they were explaining it, that's what came to mind,
the concept of keeping something constant and changing another.
ARTHUR POWELL: Does any other group care to present what
they have which is different from any of the presentations
that we've heard so far? Yes, Anita?
ANITA: We went through a whole series of, like a chapter
book... So I'll start at the beginning. When we first made
all the combinations, we arranged them in eight pairs, like
a lot of people did, each with its opposite. And then, when
we wanted to answer the questions, we wanted to demonstrate
how we knew there were only 16. And we knew that from prior
experience, we wanted to connect that to the doubling effect,
and that if we went to the lower tower number possibilities,
we could show that it was doubling in a pattern. So we did
that. We showed all the combinations of three towers and all
the combinations of two towers and the combination of one
ARTHUR POWELL: I'm sorry. When you say you showed all the
combinations of two, you mean -
ANITA: We took yellow and blue and made two tower combinations.
ARTHUR POWELL: So, choosing from two colors, you made -
ANITA: The same two colors.
ARTHUR POWELL: - towers two tall.
ANITA: Right, and then we made towers three tall.
ARTHUR POWELL: All the different ones.
ANITA: And then we began trying to arrange the four tower
combinations, the three tower combinations, and the two tower
combinations in some sort of a way that would make it clear,
visually, that we were doubling each time, and why we were
doubling each time.
Basically, you can make all the combinations by reproducing
this and then putting either a yellow or a blue on top. So,
you have yellow/yellow or yellow/blue. This one, you can reproduce
this, put down two blue cubes, and then on top of each one,
just put either a yellow or a blue, because those are your
only two other choices. And then you can do the same thing
NARRATOR: Anitaís group first arranged towers in pairs,
each tower with its opposite. However, to justify that they
had found all possible towers, they investigated what they
called a ìdoubling effect.î They started with
towers one-cube high and then systematically built towers
two-high and so forth.
Because each time a new cube is added to a tower, it can
be either of two colors, the number of different towers doubles
with each new tower height.
The resulting set of towers can be placed in an arrangement
that resembles a tree.
ARTHUR POWELL: So, you're going from towers two tall to towers
ANITA: Towers two tall to towers three tall. So, each tower
that you have in the two cube row becomes two more - because
you have two colors. So that's how it doubles.
ARTHUR POWELL: Is anyone confused by this? [simultaneous
DORIS: I'm thinking, wow, you seem to be going through a
lot of work making these shorter towers to get to the bigger
ones, when the ultimate goal was a four tower, four cube tower.
ANITA: In reality, we started with the four. We made the
16 four tower combinations. But then when the question was,
"Well, how do you know? how can you prove this? how can you
demonstrate that there's only 16, no more than 16?," We knew
that was related to the fact that it was doubling, that this
was a pattern, and it wouldn't suddenly become 17. But the
question then is, "Well, how do you know? Where does this
pattern come from? How do you show that?" And we rearranged
our little cubes so many times, and this was an easy way to
see how each one in the lower level turns into two at the
next level up if you have one new cube allowed to be added.
BLANCHE: I want to thank Anita's group because your explanation
clarified it for me. It became a lot clearer. And we did take
the doubling concept into mind and dealt with the 16 from
ARTHUR POWELL: Are there any comments or questions that you
have? ... What you can do is would you create towers 10 tall
with one color? [Laughter] So that I can then put them back
in their boxes...
PAULA HAJAR: It's not about fun activities. It's about a
whole way of looking at mathematics and mathematics teaching.
And I know some of the tensions we have, even in seminars,
is between coverage and going deeply. One teacher, she was
so dear, she said, "This is lovely, but we don't have time
for it." And he said, "Well, what's the it?" And she said,
"Well, you know. Deep understanding." And then she caught
herself. And by the end of the seminar, she was very much
an advocate for what we were doing, and tried very hard to
incorporate some of the activities. But also, she knew it
was a whole way of looking at things. It wasn't just about
ARTHUR POWELL: I have a question: Do you think that this
problem could effectively be worked on by young students?
What do you think?
TEACHERS: Yes. Yes.
ARTHUR POWELL: What do you think they might find? Do you
think they might come up with the kinds of justifications
and organizing schemes that you came up with?
ARTHUR POWELL: I think that, in fact, the kindergarten teachers
in the workshop are the most comfortable ones with this particular
approach. In some ways, kindergarten teachers see the importance
of allowing students to talk, getting students to express
themselves, asking students to work with each other. It's
as you get into first and second and third grade that teachers
begin to think of the classroom and students as working more
individually and focusing their attention more on the teacher,
less on students working in groups.
What we're talking about is a different pedagogical outlook,
a different way of approaching the teaching of mathematics.
The content is different. But the way in which you want students
to engage their minds, the way in which you want students
to verbalize and to externalize what's in their minds, are
I would expect to see, teachers using one or two activities
that we've worked on, and look into the curriculum and try
to find one or two kinds of lessons that they might redirect
in this new way. They will sort of incrementally begin to
see that they can add on to their repertoire more and more
open-ended activities, more and more tasks that have students
working in groups, more and more opportunities to be asked
to develop convincing arguments, more and more opportunities
that they'll give to students to reflect, using writing or
Hopefully each month and each semester as we go along, that
they will slowly, incrementally, increase the repertoire of
lessons that they give, which has a change to it.
NARRATOR: We've seen teachers presenting a number of arguments
for finding all the combinations of towers four high when
selecting from two colors. Which arguments are convincing,
ANITA: ...how can you demonstrate that there are 16, no more
PART 2. "STUDENTS BUILDING PROOFS"
NARRATOR: The "Towers" problem that the Englewood teachers
were working on came out of the Rutgers long-term study. The
researchers originally presented this problem to the Kenilworth
students in October, 1990.
AMY MARTINO: All right. We're going to do something a little
different today. We're going to build towers today that have
four stories to them. You're going to get two colors of Unifix®
cubes. Your job is-
NARRATOR: The question was - How many different towers four
blocks tall can you build when selecting from blocks of two
TEACHER: ..and again, it's like the shirts and pants. You
have to be convinced that you've found them all...
NARRATOR: On this particular day, the students spent about
an hour working on the problem. The researchers wanted to
find out how the focus group of students would build mathematical
ideas, not just today, but over a long period of time. And
this was the first in a series of carefully linked activities.
CAROLYN MAHER: We start with at least four tall. The students
keep trying to do the problem until they can't find any more,
even if they haven't come to organize their findings in a
way that would account for all possibilities. We do not believe
that you start them building towers one tall, two tall, three
tall, four tall, and then they see this pattern. That was
not what we were trying to do. That, to me, is a programmed
way of proceeding.
NARRATOR: The researchers always tried to challenge the students
with problems that would force them to invent new strategies.
CAROLYN MAHER: For instance, in the four tall tower problem,
when you have two of a color inside that tower, and you produce
all possible towers with two of a color, making the argument
that you have them all is demanding of some interesting reasoning,
like controlling for variables, keeping one row constant and
changing the other. So, it pushes them to invent other approaches,
heuristics - methods of solving problems - like "guess and
check" is a heuristic; a random method is a heuristic; working
backwards is a heuristic.
AMY MARTINO: You guys working together? Do you have any of
the same towers as each other?
AMY MARTINO: Yeah? Which ones are the same?
STUDENT: This one...
NARRATOR: After spending less than five minutes making random
combinations, the students started to compare their towers
and eliminate duplicates.
STEPHANIE: Everything we make, we have to check. Everything
we make... Let's make a deal. Everything we make we have to
DANA: All right. I'll always make it and you'll always check
STEPHANIE: Okay, you make it and I'll check it.
AMY MARTINO: How's it going guys?
JEFF AND BRIAN: We're done.
AMY MARTINO: Okay, you- What'd you get?
BRIAN: We found 17 towers.
AMY MARTINO: 17? Is there a way that you can check to be
AMY MARTINO: Is the way that you could, you know-
BRIAN: We like laid them down and we saw if they're the same
or not, and they weren't. They weren't the same.
AMY MARTINO (OFF CAMERA): Stephanie, what makes you sure
that you got everything?
STEPHANIE: I don't know.
DANA: Well, we just test it, like we used all of our blocks
and then we had matches and the ones that matched - because
one of them that matched, and we eliminate them.
AMY MARTINO: Could you have missed one?
AMY MARTINO: How come? How do you know?
DANA: Because we double-checked about four times.
STEPHANIE: Okay, Dana, I'm going to try and make one more.
NARRATOR: The students recorded their findings, and most
agreed that there were 16 combinations. The following day,
the researchers returned to the problem. In a whole group
discussion, they asked the students whether there would be
more, fewer, or the same amount of combination for towers
ALICE ALSTON: So, do you think there'd be more than 16 or
fewer than 16?
ALICE ALSTON: You think there'd be more?
CAROLYN MAHER: It is very interesting to say to students
- "Now you've built all towers four tall; you've convinced
us, or you haven't, that you've found them all. What about
three tall?" The three tall is interesting because very young
children predict that there will be more towers three tall,
and that surprises many teachers, many researchers, who expect
them to think there are fewer.
NARRATOR: Matt suggested taking off one cube from every tower.
MATT: Take one block off each pattern. And then count up
how many of these things...
ALICE ALSTON: Could everybody with your partners see if you
can figure out how many there would be if there were only
three. Remember that each one is to be different from all
the others at your place...
CAROLYN MAHER: If you really listen carefully to what the
students are saying and ask them why do they think there are
more, well, what they're imagining is removing one of the
rows of cubes from their four-tall towers and having more
cubes to create new towers. And until they try to do that,
they are really unaware that they end up duplicating towers
of three that are already built. And that's very important
for them to recognize why there are fewer rather than more
or the same. It's important for them to understand that reversibility
in their thinking.
DANA: Amy, we figured out that it's less.
AMY MARTINO: You think so? Why?
STEPHANIE: Because if we took one away, we had these. If
we took one away from these, so-
AMY MARTINO: And you can't have those?
STEPHANIE: Well, we can't have them the same because-
AMY MARTINO: That's right. They have to be different.
BRIAN: ...'cause, if you take like one off the bottom or
one off the top, you might have another one that's the same
as that. And then you have to make like - then you can't use
that because it's a match, and they have to be different...
NARRATOR: Jeff shared an interesting way to look at the problem.
JEFF: You could link - first of all, you could do this as
a math problem, because you could do 16 minus 8 is 8.
ALICE ALSTON: You mean, there's something about math to it?
JEFF: Yeah, because 16 minus 8 or 8 plus 8 equals 16. And
when you took the one away from each, it would be minus 1,
so instead of saying minus 1, minus 1, minus 1, you could
have just said 16 minus 8 or 8 plus 8.
ALICE ALSTON: I see. So it has something to do with 16 minus
CAROLYN MAHER: ...and so let's talk about what they should
be. First of all, how tall should they be...
NARRATOR: 16 months later, in the fourth grade, the Kenilworth
students investigated towers five cubes tall.
CAROLYN MAHER: ...and you have to be able to convince us
that you have found all possibilities - that there are no
more or no less. Got the problem? Have fun!
STEPHANIE: Okay, we'll start out with the easiest one. One,
two, three, four, five reds and five yellows.
DANA: One, two, three, four, five.
STEPHANIE: I only have four. Okay, well, stand them up straight
so we know what we have.
CAROLYN MAHER: In towers five tall, to make a convincing
argument that you found them all is harder except for when
you have all of a color or one of a color.
SHELLY: Now we take one of these, one of these.
NARRATOR: Building towers five tall offered a richer, more
complex challenge for the students to investigate. Students
spontaneously invented strategies, such as making a tower
and then building its "opposite".
BRIAN: ...this one matches with this.
ROMINA: Put the pairs.
BRIAN: Like the opposites.
DANA: And then I got another idea.
STEPHANIE: Well, tell me it so I can do the opposite.
DANA: I'm going to do the red - this, that-
STEPHANIE: Show me. Oh, okay, and I'll do the red - and I'll
do it with the red at the top.
CAROLYN MAHER: They were holding one variable fixed, constant,
and then varying the other. It was exciting that these children
at a very early age were showing evidence of controlling for
variables. It's lovely. And they were being exhaustive.
BRIAN: I have to do the opposite. I'll do this-
STEPHANIE: We made a pair!
DANA: No, look. Look, that's fine. That goes with this one.
STEPHANIE: No it doesn't because if you turn it around, it's
the same, so that doesn't go with that one.
DANA: That one goes with that one.
STEPHANIE: Wait, let me check. Let me make sure...No that
ROMINA: I think we have them all.
CAROLYN MAHER: Do you think it's possible to have an odd
CAROLYN MAHER: They have an odd number - 35.
MIKE: You're not supp- You can't because when you have a
number, you could have the opposite. And if you would have
one of this, you have another one because it's the opposite.
And if you have 10 of these, you have another one that's opposite,
so that makes 20.
SHELLY: We found 32.
CAROLYN MAHER: You found 32? How did you do that?
JEFF: Easy. You just go this way and then-
CAROLYN MAHER: You're tired, Jeff. Jeff, how do I know that
you don't have duplicates?
JEFF: You can check: all you want.
CAROLYN MAHER: Because you checked it. How? ... That's how
you checked it, you compared? How do you know you there're
JEFF: I can't make any more. My brain is tired!
CAROLYN MAHER: Your brain is tired?
CAROLYN MAHER: So you might ask us - Why did we ask them
to convince us? Why do we ask them to justify? Well, we do
that because beginning when they start, they solve their problem
randomly. It's sort of guess and they try something. When
you don't know what to do, you try something, so you'll build
something. And maybe you'll notice certain kinds of patterns
in your building; maybe you won't. You might just do trial
and error, trial and error, trial and error. We want students
to get past trial and error.
CAROLYN MAHER: Okay, let's take another set and try to convince
me the same way.
JEFF: We'll show you the other set.
CAROLYN MAHER: Okay. I believe this one, too - you can have
one red, right? And you have the other possibilities. I buy
JEFF: There's only two kinds of these because there are alternates.
CAROLYN MAHER: Okay, I buy that. All right. You're convincing
me. That's great.
JEFF: This, we just ... How are we going to convince her
about this one?
CAROLYN MAHER: You've got to convince me about this one.
Why don't you think about this? I'm convinced about these
that there are no other possibilities when you have one of
a color - either one yellow or one red. Okay? I'm not so sure
I'm convinced if there's two reds or if there's two yellows,
so why don't you work on convincing me of that? You think
about it and I'll be back, and you can call me.
CAROLYN MAHER: But they're thinking was still very, very
exhaustive and it was very organized - when they had to justify
their solutions. What it does, then, is it enables them to
look at what they have, that they did just by hard work and
drive, which we skip in school; we skip that piece of it.
How awful - because we don't have time. You know, we skip
that drudgery of that going through this hard, hard work we
might not see the point of. We don't look enough. Because
as they're going through this real intense, hard work, they're
noticing things about the structure of the problem - maybe
not seeing it overall, but they begin to notice relationships,
they begin to notice sub-patterns, and they invent names for
these. They really get to know the task well. This is what
we expect mathematicians to do in their work.
JEFF: ...we tried to make like patterns...
NARRATOR: The following day, the researchers interviewed
the students about their thinking on the towers problem.
JEFF: First we did this - we started out by moving the block
up in each one.
CAROLYN MAHER: What did you come up with as a solution to
the problem? Did you finally decide on how many?
JEFF: We decided on 32. And we kept on going up -
CAROLYN MAHER: Right.
JEFF: - And then we did the opposite of it.
CAROLYN MAHER: So, you decided on this pattern, that there
were how many like that?
JEFF: It would go up to five.
CAROLYN MAHER: The interviews served multi purposes for us.
It validated some of what, to us, were just theories. We had
certain theories about what they were doing and what they
were thinking based upon what we observed, what the cameras
caught. But we weren't sure how aware they were of that thinking.
STEPHANIE: Well, 'cause this one, we had the pattern, the
two, and then the two blocks up, and then the two blocks up.
CAROLYN MAHER: Yes.
STEPHANIE: And then we had the two in the middle, and we
had the two here, and the two here.
CAROLYN MAHER: Okay. So, that was the other pattern you had.
I'm confused though. How do you know that some of these aren't
STEPHANIE: Oh, thatís right. This one is this one.
So, this one's -
CAROLYN MAHER: How did you deal with that yesterday? Did
you end up counting things?
STEPHANIE: We ended up counting a lot over. We had 34 and
we had it so we subtracted, I think, three groups, because
we were down to 28 and then we added two groups.
CAROLYN MAHER: So, so, you think that that's what was happening
STEPHANIE: Yeah, we kept, we kept finding different patterns,
but we didn't check it with the other patterns.
CAROLYN MAHER: Uh-huh, okay.
STEPHANIE: It's really the same pattern in different places.
CAROLYN MAHER: Right.
STEPHANIE: It's taking one - building on one pattern. It's,
okay, say I started with the pattern at the top. You're taking
that pattern and you're moving it down one and then you're
moving it down another and another.
CAROLYN MAHER: I'm just wondering if we can come up with
a way that would make it easier to remember, because it's
a nice way of trying to find them. I like your patterns, but
I worry about if we're missing some or counting some twice.
That's tricky. You might want to think about that: a way of
trying to come up with a way to do that.
STEPHANIE: Yeah. How could you be absolutely, positively...
A guess, a very lucky guess.
CAROLYN MAHER: Yeah, but math isn't a guess. Math you should
be able to figure it out and be convinced. And you promise
you're going to work on this?
CAROLYN MAHER: Okay, that's going to be great. I can't wait
to talk to you about it some more. So, imagine the four towers
and imagine the five towers and if you have time, imagine
STEPHANIE: All right, and I'll work on a way to be definite
about your answer.
CAROLYN MAHER: Yeah, that would be exciting. That would be
really fun. Terrific. Well, thank you so much.
STEPHANIE: You're welcome.
CAROLYN MAHER: You have a good weekend. I enjoyed this very
STEPHANIE: Thank you.
CAROLYN MAHER: It was good thinking, very good.
NARRATOR: About one month later, the researchers held a group
interview with fourth graders: Jeff, Michelle, Milin, and
Stephanie. In this session, they were especially interested
in what made the students' reasoning convincing?
CAROLYN MAHER: The last one you did in class - remember what
that was about?
JEFF: Robin Hood? That was the last one we did.
STEPHANIE: Towers of five.
CAROLYN MAHER: Towers of five. Do you remember what you did
with those towers of five?
CAROLYN MAHER: And tell me about it. What was the problem?
MICHELLE: You had to figure out how many towers you could
make, different towers you could make from five blocks up.
CAROLYN MAHER: Any five blocks?
STUDENTS (ALL): They had to be two colors.
CAROLYN MAHER: Two colors, okay. And did you figure out?
CAROLYN MAHER: And what is it? Do you remember?
CAROLYN MAHER: You're sure of that?
CAROLYN MAHER: How can you be so sure?
MILIN: 'Cause we checked!
CAROLYN MAHER: How can you be so sure?
JEFF: Remember we did all those - the chart, the thingies,
for like all the different patterns? Remember I convinced
you up in the whatch-a-ma-callit?
CAROLYN MAHER: Yes, in the room.
JEFF: So, I don't feel like convincing you again.
CAROLYN MAHER: You don't feel like convincing me again. [LAUGHTER]
NARRATOR: This session was a performance assessment to learn
more about the students' reasoning.
CAROLYN MAHER: And Stephanie did try to work on towers of
six and I asked all of you if you -
MILIN: So did I.
CAROLYN MAHER: You did too? If you were building towers of
six, how many would there be?
MILIN: Probably 64.
CAROLYN MAHER: Why do you think 64?
MILIN: Well, because there was a pattern.
CAROLYN MAHER: What's that?
MILIN: You just times them by two.
CAROLYN MAHER: Times what by two?
MILIN: The towers by two, because one is two, and then we
figured out two is two, I mean four, and then -
JEFF: You're not making much sense!
MICHELLE: See, if you had only one block up and two colors,
then you would have two towers, and we figured out that the
other day -
JEFF: Everything is opposite!
MICHELLE: - that you keep on doing, like two times two would
be four and then...
CAROLYN MAHER: So four would be for what?
STEPHANIE: All you have to do -
MICHELLE: - four for, there would be four towers for two
CAROLYN MAHER: Okay.
JEFF: They're always opposite, though.
CAROLYN MAHER: Wait, let me hear what Michelle is saying.
MICHELLE: And then for this three high, you would eight towers.
CAROLYN MAHER: Do you agree with that?
JEFF: I don't know what you're talking about.
CAROLYN MAHER: Okay, let's get a piece of a paper and write
down what you're saying and see if you all agree. I think
Jeff hasn't been with us for a while and he doesn't know what
we're talking about. But let's take one at a time.
CAROLYN MAHER: There are a couple of ways of approaching
the problem. There's the notion of you can start with towers
one tall, each color. And once you have the tower - let's
say red and blue - you start with a red tower and now you're
making a two tall tower. Well, you can either put a red on
the top or a blue on the top. So, from that one, you get two.
Likewise for the tower with the blue on the bottom. You could
either put a red on the top or a blue on the top. From that
So, from the two towers - one tall - you generate four towers
two tall. Each of those towers you can choose to either put
a red on the top or a blue on the top, red on the top or blue
on the top. From the eight, two more, two more, two more,
two more. You could begin to generalize this idea and, later
on, you can come up with a nice way of expressing your justification,
which leads you to a kind of way of reasoning that we call
NARRATOR: For about half an hour, the students shared their
different approaches. When it was Stephanie's turn, she presented
a version of Proof by Cases to justify her solution for the
number of towers three high.
CAROLYN MAHER: How about you, Stephanie?
STEPHANIE: I found it like this. I drew my lines and then
I went red/red/red, blue/blue/blue, blue/red/blue, red/blue/blue,
blue/blue/red, red/red/blue, red/blue/red, blue/red/red.
CAROLYN MAHER: So, what I'm hearing you say is that you're
CAROLYN MAHER: - you believe they are eight, but you say
guessing. Now, why does that sound like guessing?
MILIN: Because what if you can make more?
STEPHANIE: Okay, this is the three high, right? And you're
convinced you can make eight?
STEPHANIE: I'm convinced I can make eight.
MILIN: But could you convince her?
STEPHANIE: Convince who? Michelle? Him?
MILIN: No. Her.
STEPHANIE: Yeah. All right. I've done this before. OK.
CAROLYN MAHER: Take another piece of paper if you want to.
You've got to convince me there are eight and only eight,
and no more or fewer.
STEPHANIE: All right, first you have without any blues, with
CAROLYN MAHER: Okay, no blues.
STEPHANIE: Then you have with one blue - blue/red/red or
red/blue/red or red/red/blue.
CAROLYN MAHER: Anything else?
MICHELLE: And you would do the same pattern for -
STEPHANIE: No, not with the blue, not with one blue -
MICHELLE: You would do it, you would do it with one red and
JEFF: You would alternate-
MICHELLE: You would do it the other way around.
CAROLYN MAHER: Let her finish. That's what you would do.
You would alternate. Let's see what Stephanie does. Maybe
STEPHANIE: Well, there's no more of these because if you
had to go down another one you'd have to have another block
on the bottom. Then you have with three blues - well, not
with three blues. I'll go like this.
CAROLYN MAHER: You have no blues and now you have exactly
STEPHANIE: Now you have exactly two blue. You could put blue/blue/red;
you could put red, blue and blue.
MILIN: You could put blue, red and blue.
STEPHANIE: Yeah, but that's not what I'm doing. I'm doing
it so that they're stuck together.
CAROLYN MAHER: Okay.
JEFF: There should be one - there could be one with one red
and then you could break it up and there's one with two reds
and there's one with three reds.
MILIN: Ah, but see - you did the same thing, but there's
JEFF: There's all reds and then there's three reds, two reds.
There should be one with one red. And then you change it to
STEPHANIE: Well, that's not how I do it.
CAROLYN MAHER: Let's hear how Steph - We'll hear that other
way; that's interesting. Okay, now, so what you've done so
STEPHANIE: One blue, two blue.
CAROLYN MAHER: Okay, no blues-
STEPHANIE: One blue, two blue.
CAROLYN MAHER: One blue, and two blues, but Milan just said
you don't have all two blues, and you said - Why is that?
STEPHANIE: All right, so show me another two blue. With them
stuck together, because that's what I'm doing.
MILIN: In that case, here.
CAROLYN MAHER: Okay, so now what are you doing, Stephanie?
MICHELLE: What if you just had two blues and they weren't
STEPHANIE: But that's what I'm doing. I'm doing the blues
stuck together. Then we have three blues, which you can only
make one of. Then you want two blues stuck apart - not stuck
apart - took apart.
CAROLYN MAHER: Separated?
STEPHANIE: Yeah, separated. And you can go blue/red/blue
CAROLYN MAHER: The pattern that Stephanie chooses in justifying
that she found all possible of three tall towers selecting
from two colors is an organization by cases. Now, her organization
by cases does not map into the organization that some of her
classmates want. She is adamant about her organization, and
it works. It's not "the most elegant" organization by cases,
but it's an organization by cases.
JEFF: I have a question. Do you have to make a pattern?
JEFF: So, then why is everybody going by a pattern?
MILIN: Because we liked it.
STEPHANIE: It's easier.
STEPHANIE: It's easier to find than just going, Ooh, there's
MICHELLE: 'Cause if you, 'cause if you just keep on guessing
like that, you're not sure if there's going to be more.
STEPHANIE: It's easier, maybe, like Shelly and Milan's pattern
was to go put this in a different category.
JEFF: We know their pattern.
STEPHANIE: Okay, but what I'm saying is it's easier, it's
just easier to work with a pattern since it's like -
MILIN: Oh, here's another one. Let's see -
MICHELLE: Because you might have a duplicate, and then you
may not know.
STEPHANIE: It's harder to check. It's harder to check just
having them like come up from out of the blue.
MILIN: Then just going like this and getting two -
JEFF: How do you know there's different things in the pattern?
CAROLYN MAHER: We have naturally, in the solution of these
problems, thinking of these ways of organizing that account
for all possible towers that can be built.
I'm building towers of a particular height, let's say four
tall. In building these four tall towers, I could say, "How
many towers can I build that have no red? Only one. How many
can I build that have exactly one red?" Well, students will
argue, eventually, that there are four of those, if they're
four tall, one in each position. "How many can I build that
have two red?" That's a little tougher. That kind of reasoning
pushes students to begin to have to control for variables.
Is anything else possible? Can I build five? The students
will say, "No, I can't do that because you told me the towers
had to be four tall." In that kind of reasoning, they're beginning
to do proof by contradiction. We see that kind of reasoning
also, which is an indirect method of proof.
MILIN: And you can't make any more in this one, so you go
on to the next one.
JEFF: How do you know you can't make any more?
MILIN: Because there's not any more colors.
STEPHANIE: Look, okay, start here. Start here - okay, you
have the three together. The one, one blue, right? You have
the one blue. How could I build another one blue?
JEFF: You can't.
STEPHANIE: All right. So, I've convinced you that there's
no more one blue. All right -
MICHELLE: But if you didn't have that pattern, it would be
harder to convince you.
STEPHANIE: Okay, so I've convinced you that there's no more
MICHELLE: Then you have to go to two blue.
STEPHANIE: Two blue. Here's one - all right? Two blue - you
have one, blue/blue/red, then we have red/blue/blue. How am
I supposed to make another one?
STEPHANIE: No, this is the other. Milin gave me that same
JEFF: But the thing is does it matter that they're together?
MICHELLE: She means stuck together.
STEPHANIE: Stuck together, that means like - Okay, so can
I make any more of that kind?
MICHELLE: Then you have to move the three, which you're trying
to make one.
STEPHANIE: Yeah, you can only make one and then you can make
without blue with the three red.
MICHELLE: And then you can make two split apart.
STEPHANIE: Two split apart, which you can only make one of,
and then you can find that you can - you can find the opposites
right in this. All right, so I've convinced you that there's
CAROLYN MAHER: When Jeff says, "Why do you have to have a
pattern?" that's really a very important question. My theory
is that in Jeff's own exploration with the towers and his
finding of many patterns and the opposites of those patterns,
they got him in trouble when he was eliminating duplicates.
And so all of his pattern finding became really an enormous
task of trying to keep track. And I have always wondered if
when he says, "Why do you have to have a pattern?" if he doesn't
mean that in frustration. But there are other interpretations
of what Jeff means. I wonder what other people think. Why
do you have to have a pattern?
CAROLYN MAHER: How many if you're making towers of four?
MICHELLE, MILIN: 16.
CAROLYN MAHER: Do you agree, Jeff?
JEFF: Yeah. [laughter]
CAROLYN MAHER: Jeff, why do you agree? Don't let them get
by so easily. This could be pressure here.
MICHELLE: Just because say you add a red or a blue, you can
add a red or a blue here -
CAROLYN MAHER: Make a "Y" or something to show me...
JEFF: I understand because you can only -
MICHELLE: You can put two colors here, two colors there,
two colors - and keep on going.
JEFF: Yeah, you can keep on doing two colors for each one.
And that's two, four, six, eight, ten, twelve, fourteen, sixteen.
CAROLYN MAHER: And so that's the towers of-?
CAROLYN MAHER: You made towers of five in class and what
did you get?
CAROLYN MAHER: Does that work the same way?
MICHELLE: If you get towers of four -
STEPHANIE: The hard part is to make the pattern. Like, from
now, we know how to just - Oh, you could give us a problem
like how many in 10 and we could just go -
CAROLYN MAHER: How many in 10, and you'd know the answer?
STEPHANIE: Yeah, I know the answer. I figured it out. It's
CAROLYN MAHER: 1,024.
ALICE ALSTON: Are you sure?
JEFF: Don't try to convince her.
CAROLYN MAHER: Try to convince me. [laughter]
STEPHANIE: Just give me a couple thousand Unifix® cubes!
CAROLYN MAHER: You can do that later.
NARRATOR: We have seen fourth graders naturally develop two
mathematically sound methods of proof-making:
One, Inductive --- building taller towers from shorter towers,
one at a time, and using known results about the shorter towers
to derive results about the taller ones;
and Two, organizing towers of a given size into different
cases which could be studied separately.
What are some of the similarities and/or differences in the
mathematical reasoning by the teachers and students that you
have observed in this program?
[End of program]