Workshop Sessions
PROBLEMS AND POSSIBILITIES
Workshop 1: Following Children's Ideas in
Mathematics
Watch the video:
TRANSCRIPT
PART 1. "THE YOUNGEST MATHEMATICIANS"
[Student voices]
Narrator: As teachers, we rarely
get to see what happens to the mathematical ideas our students
have built after all the hard work we we've put in together
in the course of a year.
Student voices:...3, 4, 5, 6!
Narrator: Are these ideas extended
in subsequent grades? Will the mathematics my students have
learned help them in their careers?
Teacher: OK, 6 times 6..
Narrator: In this workshop
series, we'll have an opportunity to follow a group of children
 beginning in first grade and continuing through graduation
from high school  to see how students build mathematical
ideas over time.
Student: ..you're going to
get one more, because when we did the pattern with six, you
got one more..
Student: ..and there's 2 with
a b...
Narrator: Over the course of
the six workshops, we'll follow stepbystep as these students
connect ideas that they have built in earlier years to new
thinking of everincreasing complexity.
Narrator: And we'll visit teachers
who are changing classroom practice to help their own students
make the same kind of connections.
[Bell ringing]
Student: .. until the end of
class.
Teacher: Until the end of class?
I'll see what I can do.
[Music]
[Preschool]
Narrator: In this program,
we'll set the stage for this longterm progression. Let's
start by looking at very young children  before they begin
their formal education. The child psychologist Lawrence Frank
said, "Play is a child's work." Is there mathematical thinking
in this play?
[Music]
[Voices of preschool children]
Narrator: Prof. Herbert Ginsburg,
a psychologist at Columbia University Teachers College, spends
a lot of time in preschools.
Herb: I don't know how to do
these, so you have to tell me how to do them, O.K. ?
Narrator: He investigates what
children, at the age of 3 or 4, do during free play.
Herb: So I went to visit a
preschool in Manhattan. If I watch the kids when they are
in their free play, they seem to be doing some really interesting
things. They got involved in who has more. They got involved
in building beautiful block designs.
Herb: They engaged in counting.
What are they really thinking about? What are they really
learning? So I started to do a detailed study of this.
Herb: We made video tapes of
these kids as they're engaged in free play. We break down
the 15 minute segments into one minute segments. We carefully
code each of those segments.
Herb: When we do that we find
that half the time they're doing something that we think we
can legitimately call mathematical. By mathematical I mean
everyday activities that involve numbers, that involve size
comparisons, that involve dealing with patterns, with shapes.
When kids are playing with blocks, they will line up a number
of blocks in parallel, and they'll be all blocks the same
size. They will be very careful to make a square. They'll
be very careful that the right angles are correct. So we think
when they're doing activities like this, they are learning
some very fundamental geometric ideas.
Herb: With playdough, they
will often start out with something like a sphere. They will
start systematically to cut this thing up. Okay? They cut
it into pieces, which may be more or less even. Now what are
they doing here? Well, there are ideas about division, there
are ideas about taking away. Well, what happens when you take
this thing off? It gets shortened. Cutting apart allows you
to reverse the transformation, too. Say, "All right. I can
cut this apart, but I put it together."
Herb: So, in effect, they're
dealing with ideas about equivalence, about conservation of
volumes and quantities. It's like our basic mathematical idea
of changing things  when you add, when you take away, when
you divide it up, when you put it back together. These are
really fundamental mathematical ideas. Later on, when we deal
with calculus, for example, we're dealing with ideas about
change.
Herb: So they're doing little
scientific experiments, they're learning how to observe, they're
interested in physical phenomena, or how fast something turns,
how much water do you need to put in. There's the measurement
ideas. So in that kind of play there's an awful lot going
on that kids don't encounter in school until much later. When
these kids in preschool are doing it, it's real; it's literally
handson. It's something you do. It's not just reading about
some of this in textbooks.
[Music]
Narrator: We've seen that students
enter school with the ability to do some surprising mathematical
thinking. Is it possible for students to keep up this early
enthusiasm for math?
[Kenilworth,
NJ]
[Band playing, voices cheering]
Narrator: Part of the answer
might be found in Kenilworth, a small New Jersey town. Some
of these students, now in their senior year in high school,
are part of a focus group of children whose mathematical thinking
has been followed in depth, year after year, for 12 years.
What kind of a town is Kenilworth?
[Voices cheering]
Joanne Gocel: It's the quintessential
small town. Everybody knows everybody. Lots of school spirit,
lots of town spirit, lots of pride. We have a lot of different
nationalities which compliment each other. If there's something
at school, if there's a show at school, everyone comes, the
senior citizens come, the grandparents come. We get huge crowds
for football games. It's a great place to raise kids.
Mike Aiello: Kenilworth is
the same as it was probably 20 years ago. It's basically the
same thing. Like, everybody knows each other in this town.
It's a little boring, there's not much to do. All there is
is restaurants and banks. That's all we really have in this
town.
Narrator: Kenilworth, with
a population of under 10,000 and a median household income
under $50,000, has a mixture of housing and light manufacturing.
Narrator: In 1985, the principal
of the Harding Elementary School, in Kenilworth, invited Carolyn
Maher from Rutgers University to work with his teachers in
mathematics. At that time, no one could have guessed that
what started as a project with teachers would open the door
to a 12year study about the development of mathematical ideas.
Carolyn Maher: I began my career
as a mathematics teacher. I taught high school for several
years, and then taught mathematics at the 2year college level
for another six or seven years. Then I moved to mathematics
education at Rutgers University and discovered that I really
didn't know about little children. My own work was with older
students, high school students, and collegeage students,
and I had to learn more. So I began to go to classrooms and
work in classrooms with younger students.
Carolyn: When I visited Harding
elementary school in Kenilworth in 1985, I really had not
fully realized how rigid the math instruction was, it was
very rote; and it was sort of sad because the principal at
that time was so aware that the students who moved on to the
high school  their very best students  were barely making
C's in mathematics. And he was very concerned. This is really
before standards, this is before the reform movement in mathematics.
The curriculum was very rigid, very atomistic, and very much
computationdriven. And there was really no attempt to develop
understanding on the part of those students.
Carolyn: We worked in the elementary
school with the teachers in K8 for 3 years. That was really
a wonderful time. Now we worked with the teachers to think
more deeply about the mathematics they were teaching. And
we were aware that there was really no research about how
students developed mathematically, when they had an opportunity
from a very young age to do thoughtful mathematics.
[Voices of students in classroom]
Narrator: In 1989, just as
small, reasonably priced video cameras were becoming available
to researchers, Carolyn's group began to videotape the Kenilworth
students working on mathematical activities. These activities,
which were separate from the standard school curriculum, were
designed to reveal the students' mathematical thinking.
Carolyn: Video cameras became
an important tool for us  became not only a tool for us so
that we could share what was happening in certain instances
with the kind of exciting discoveries children were making,
and the inventions they were doing, and how much fun they
were having, and how deeply their thinking was. But it was
also important for me to share that with my own students and
my colleagues so we could talk about this and learn from it
ourselves.
[Voice of student]
Carolyn (on camera): Oh did
anyone else here get 34?
Narrator: The Rutgers team
has collected over 2000 videotapes following the same group
of students, from first grade through high school. This is
the longest study about the longterm development of mathematical
ideas ever conducted.
Amy Martino: This is the question.
How many of the little tiny blocks make up the big block?
Narrator: One of the researchers
was Carolyn's former student, Amy Martino:
Amy: My affiliation with Rutgers
started back in 1988. I started out as a graduate student
of Carolyn Maher's. And I was currently a middle school teacher;
currently a middle school math teacher. And I came back to
school fulltime. I just felt, after teaching  going out
in the field and teaching for a year  that there's got to
be more. I've got to know more in order to be able to do this
better.
Amy [to students]: The next
one was the one with the,.. where Stephen had 3 different
colored shirts...
Amy: I was thrown into classrooms
my first year, doing research.
Amy [to students]: ... And
I was curious as to how you did this, because you drew a nice
picture here. Can you explain the picture to me?
Amy: I went from being more
of an assistant or a helper to actually going on and doing
my own research in the classrooms . Things were pretty primitive
back then, 12 years ago. You had to beg, borrow, steal to
get people to come that day to film. You know, they had to
make time in their schedule. And, you know, we basically had
these very primitive microphones. I didn't even have microphone
holders.
Dana: [tapping on microphone]
Testing. Testing.
Stephanie: Hi!
Amy [to students]: ...What
was the order? You put down the number of pencils and the
number of erasers...
Narrator: To study how mathematical
thinking develops, the research team began by asking: How
might children solve problems that use mathematical ideas,
before the procedures to solve them were formally introduced
in school? In the second grade, the researchers introduced
the problem: Shirts and Pants.
[Shirts and
Pants. May 1990  2nd grade]
Narrator: Stephen has a white
shirt, a blue shirt, and a yellow shirt. He has a pair of
blue jeans and a pair of white jeans. How many different outfits
can he make?
Let's see how the students
in the Rutgers' study approached this problem.
Stephanie: I'm going to make
a shirt and I'm going to put a "W" for white.
Michael: Yeah, white shirt,
white pants.
NARRATOR: The students spontaneously
began to make drawings.
Dana: Blue and... a yellow
shirt
Stephanie: He has a pair of
blue jeans and a pair of white jeans. How many different outfits
can he make?
Michael: Well, he can only
make two outfits.
Stephanie: No, how many different
outfits? He can make a lot of different outfits. Look, he
can make white and white...
Dana: He can make all three
of these shirts with that outfit.
Stephanie: Shh! You can do
it in a lot of different ways. You can do white and white
and that's one. By doing "W" and "W".
Stephanie: Two. Blue. Blue
jeans and a white shirt.
Shh! Yellow shirt...number
three could be a yellow shirt
Dana: It can't be a yellow.
A yellow shirt can't go with the white.
Stephanie: Yeah but how many
outfits can he make? It doesn't matter if it doesn't match,
as long as it can make outfits.
Stephanie: It doesn't have
to go with each other , Dana.
Dana: What outfits can it be?
It can make more if you put them mixed up. Look, I'm on my
fourth one. Number four, it could be a blue shirt and blue
pants.
Stephanie: Number five. It
can be a white shirt, and...It can be a blue shirt...wait,
did I do blue and white?
Dana: What's two?
Stephanie: It can be a yellow?
Dana: What's two?
Stephanie: Two's a blue shirt
and white pants. Wait a yellow shirt... Wait did I do yellow
and white? A yellow shirt and blue pants. Yellow shirt and
blue pants...
Michael: Well I'm gonna do
it the way you want.
Stephanie: 'Cause look. There's
five combinations. There's only five combinations.
Dana: Lemme see your paper...
Stephanie: You can do this.
Listen, Michael. Michael, will you listen for once? Five combinations:
Number 1  white and white. Number 2  Blue and white. Number
3  Yellow and white. Number 4  Blue and blue. Number 5 
Yellow and blue.
Michael: (inaudible)
Stephanie: You can do four
combinations, I'm sure of it.
Dana: Five!
Stephanie: I mean five.
Dana: Amy. Amy, we're done.
We made five combinations.
Amy: What did you do? what
are your combinations?
Stephanie: [to Amy] I've got
white and white...
Narrator: Let's look at this
segment again and see what the researchers found out about
each of the students' thinking:
Narrator: Very early in the
tape, Dana drew lines to show combinations.
Narrator: At that time, the
school's curriculum did not include teaching multiplication
in the second grade.
Narrator: But Dana's approach
to the problem shows that she is capable of the kind of mathematical
thinking that underlies multiplication.
Narrator: Dana's initial graph
shows three shirts, with all but one, the yellow shirt, connected
to two pairs of jeans. At this point Dana was influenced by
her sense of fashion!
Dana: It can't be a yellow.
A yellow shirt can't go with the white.
Stephanie: Yeah but how many
outfits can it make? It doesn't matter if it doesn't match,
as long as it can make outfits....
Narrator: She believed that
yellow and white don't "go together."
Stephanie: They don't have
to go with each other, Dana.
Narrator: As teachers, how
often have we seen cases like this  where students come up
with a logic that makes sense to them but is completely different
from what we expected?
Narrator: Stephanie made drawings
and wrote letters and numbers to keep track of her random
attempts to find outfits. Using these representations, she
found five outfits. Notice that Stephanie, on her fifth combination,
first wrote "W" over the "B", then wrote a "Y". Does this
notation stand for one or two outfits?
Narrator: For Michael, perhaps
his sense of style required that outfits have matching colors.
He created a new color of pants  yellow  that wasn't included
in the problem.
Narrator: The researchers were
wondering: How were these students influenced by each others'
ideas?
Amy: What was fascinating was
that nobody seemed to be truly bothered by the fact that they'd
come up with different numerical answers to the question.
To me that was very important, because it really did say you
do need to give children time to build, just to build and
to think. And that's what that session was for. That was really
the purpose. So we really kind of left it open.
Amy (to students): Is that
all the ways you can make it?
Dana: ... Yes, that's all the
ways you can make it. I have the same thing.
Amy: I see that. That's really
good! Okay, Stephanie, put your name on this, okay? And write
to me that you found five ways, okay?
[Shirts and
Pants Revisited: October 1990  3rd Grade]
Narrator: The first opportunity
to revisit the problem came four and a half months later,
when the students were in third grade.
Amy: Are we ready to start?
Students: Yes!
Narrator: In the meantime,
the student's classroom teachers, who cooperated with the
study, were careful not to tell the students how to solve
the problem.
The wording of the problem
was identical. Let's see how their thinking has grown.
Stephanie: Want me to read
it out loud?
Dana: No, I'll do it... he
has a pair of white jeans and a pair of blue jeans? How many
outfits can he make?
Stephanie: Why don't we draw
a picture?
Dana: He has a white shirt,
a blue shirt and a yellow shirt.
Stephanie: He has a pair of
blue jeans and a pair of white jeans.
Narrator: First the students
made drawings and used letters to show the colors.
Stephanie: All right, let's
find out how many different outfits you can make. Well, you
can make white and white, that would be one...
Narrator: When they started
counting the outfits, it was Stephanie, not Dana, who started
drawing lines to connect the different shirts and pants, and
both of them solved the problem this way.
Dana: ...the blue and the white...
Stephanie: You mean the blue
pants and the white shirt?
Dana: Now we could have three
with the blue and the blue pants. And the yellow could go
with the....
Stephanie: Hold on Dana, you're
going too fast. And we could have the blue and the white.
That would be three. The blue and the blue, that could be
four. We could have the yellow and the white, that would be
five...
Dana: One, two, three, four,
five, six. I have six so far.
Stephanie: I've got one, two,
three, four, five. What are your other combinations? I have
white and blue. I've got white and white. I've got blue and
white. I've got yellow and white. What were your two other
combinations?
Dana: I mean I have six. Six.
Stephanie: What were your two
other combinations?
Dana: You mean one other combination.
The yellow and the blue.
Stephanie: The yellow and the
blue.
Dana: And the yellow and the
white.
Stephanie: I put the yellow
and the white.
Dana: All right.. Then the
blue and the white?
Stephanie: Four, five, six.
I have six too. I have six.
Dana: Amy! We're done. It's
six.
Amy (off camera): Are you both
convinced of that?
Dana and Stephanie: Yes
Amy: Can you explain it to
me?
Stephanie: Well OK. If we have
three shirts and two pants
Dana: and we just drew lines
they all made combinations. Six combinations.
Stephanie: We made white and
blue. We made blue and white.
Dana: We made yellow  blue.
and Yellow  white. Six.
Amy: Is that six
Dana: Yes
Amy: What are these lines that
you drew? You drew lines between the shirts and the pants.
Stephanie: So that we could
make sure; so instead of we didn't do that again and say,
"Oh, that would be seven, eight, nine, 10." We just drew lines
so that we can count our lines and say, "Oh we can't do that
again, we can't do that again."
Amy: Oh. That's very nice...
And you're sure that there are six?
Dana and Stephanie: Yeah.
Amy: Positive that there aren't
seven or five?
Dana: No six!
Amy: OK, now what I want you
to do then, as a pair, is to open this up and take a fat marker
and take either the blue or the red and draw this for me.
Dana: How about we each do
one thing?
Stephanie: Dana, you can write
and I'll draw, 'cause you're a better writer...
Narrator: Michael solved the
problem with lines too. But he didn't make drawings. Rather,
he drew lines between the words. This is interesting because
last time, when he came up with a completely different answer,
it wasn't clear whether Michael was hearing or noticing the
solutions of his partners.
Amy: The second time around
is interesting. Say, a strategy that one child had come up
with in second grade, in third grade, another child had modified
that strategy and was making use of it. And it's fascinating,
because most of the class, that time, many children did not
even recall having done that exact problem. The ones who did
recall it all remembered having come up with the exact, correct
number of combinations, which goes to show that, really, that
first session was just so important for getting them to build
strategies and to modify. But those two tapes had a heavy
impact on me.
Cups, Bowls,
and Plates
April, 1991
 3rd Grade
Narrator: Since the students
were able to come up with a way to solve this problem on their
own, would they be able to use this strategy to solve a more
complex problem?
Six months later, when the
students were in April of the third grade, the researchers
presented an extension of the Shirts and Pants problem.
Amy: ...first, we're going
to pretend that there's a certain situation. We're going to
pretend that today is somebody's birthday in your class.
Narrator: Called "Cups, Bowls,
and Plates," the problem adds another choice to be considered.
Narrator: "Let's pretend that
there's a birthday party in your class today. It's your job
to set the places with cups, bowls, and plates. The cups and
bowls are blue or yellow. The plates are either blue, yellow,
or orange. Is it possible for 10 children at the party each
to have a different combination of cup, bowl, and plate?"
Amy: ... it's supposed to be
finger food, so we're not going to have forks and knives today.
And what your job is going to be to do is that you're to make
different combinations...
Dana: B, yellow, yellow.
Narrator: Stephanie and Dana
worked together on the problem .
Stephanie: B, B, Y.
Dana: No, it isn't "B, yellow,
yellow..."
Stephanie: B, Y, Y. B, yellow,
yellow.
Dana: B, yellow, blue?
Narrator: Using their own system
of notation based on letters, they came up with the answer
that there are at least 10 different combinations.
Stephanie: ...and we found
10 by recording our answers. We found 12 by looking at the
recording from the first problem...
Narrator: The next day, researcher
Alice Alston interviewed Stephanie to explore her thinking
in more depth. After reviewing her previous work with Shirts
and Pants, Alice went on to ask how Stephanie could justify
her total number of combinations  12. Stephanie used the
same strategy that she had used with Shirts and Pants  linking
choices by drawing lines, then counting the lines.
Stephanie: ..so then, I guess
what you'd do is that you'd go like this...
Narrator: Then she had an idea
for a different way to solve the problem: multiplication!
Stephanie: ...and then I just
thought of it at the end when I was done figuring out that
that was 12 I thought, "Hey, there's three fours and four
times three equals 12."
Alice: OK, what does the three
represent?
Stephanie: OK. The three represents
the three fours that we got. Or the three plates
Alice: OK, the three represents
the one, two, three plates? What does the four represent?
Stephanie: The four represents,
see each one could have could have four different ways, so
you then multiply three times four and then you got 12.
Alice: ah...
Carolyn: Students, we have
found, in a very natural way , like to represent their ideas
symbolically. Algebraic thinking must begin early. In our
first task, shirts and pants, the origins of algebraic thinking
are there. The notion of the generalized solution is there.
The notion of controlling for variables is there. The notion
of extending that particular solution structure to more complicated
problems like cups, bowls, and plates is there. So that's
very, very important.
Narrator: We've just seen the
teacher/researchers repeating the same, or similar problems,
three times over the course of almost a year.
What can we say about the changes
in the students' methods over time?
Amy (to student): ...why, why
couldn't you find some more, couldn't you just look for more?
Dana: Well...
Stephanie: Before we had recorded
what we had found already...
End of Part
1.
PART 2. "FROM TOWERS TO HIGH SCHOOL"
Narrator: Right after the shirts
and pants activity, the Rutgers team introduced a new challenge:
Towers. Students were asked to select from stacking cubes
of two colors and assemble ìtowersî of a given
height. This problem is from a branch of discrete mathematics
called combinatorics which is usually taught in high school
or college as part of probability.
Narrator: Towers would become
one of the central activities of the study.
Prof. Carolyn Maher: The towers
problem is a nice introductory counting problem because you
don't really need to know anything to start it.
Towers of Five
 February, 1992  4th grade
Brian (off camera): Look at
these cameras. They look they're TV cameras.
Romina (off camera): Look at
that one.
Brain (off camera): They're
huge.
Romina (off camera). We're
going to be on TV!
Narrator: When the students
were in the 4th grade, Rutgers received funding from the National
Science Foundation and began an ambitious program of regular
taping sessions, beginning with towers.
Carolyn: Okay, I wonder why
you have these? Any idea? What do you think?
Jeff: We're gonna build with
them.
Carolyn: You're going to build
with them. Okay we're going to try to build towers that have
certain characteristics...together. Now let's talk about what
they should be. First of all, how tall are they going to be?
The rules here say they are going to be five cubes high. And
they also said what else about them? They're going to be
Student (of camera): As many
different towers.
Carolyn: As many different
towers. And what colors are you allowed to use to build your
towers? What do you think, Sebastian?
Sebastian: Red and yellow?
Carolyn: Red and yellow. Okay.
Can somebody make a tower that satisfies those conditions
and hold it up?
Narrator: The researchers referred
to this task as "Towers of Five": How many different fivetall
"towers" can the students make by selecting from blocks of
two colors?
Carolyn: We choose four tall
to start with because we thought it was not obvious. Then
we moved to three tall. We move up to five tall because it
forces them to need other strategies that they didn't need
for four tall. They have more to manage and they have to organize
their thinking more carefully in order to keep track.
Carolyn: Many of the problems
in the strand of combinatorics came from my earlier teaching
of high school and college finite math and probability. In
those days, teaching those students, I found it very frustrating
that they didn't find these problems easy. That they looked
for formulas and they were not very self reliant in terms
of building solutions. And yet I believe they should have
been able to. So my earliest work with those problems began
in my own teaching at the secondary and post secondary levels.
So, as you can see, it was really very risky of me to pose
these same tasks in a variety of different forms where children
would have materials available where they could actually build
and explore.
Carolyn: So it's not that we
chose topics that were not school math we just chose them
earlier looking to see how students did.
Carolyn: See these two? Are
we going to call these the same or different?
Jeff: Same
Carolyn: We are going to call
these the same  even though I am holding different blocks.
I really am holding different blocks. For our purposes we
are going to call these the same. And we didn't say that.
Jeff, what do you think is going to be another condition about
our towers?
Jeff: That you can't turn it
around.
Carolyn: You can't turn it
around. It has like a chimney on top. You see that? OK. How
many do you think you can do under those conditions? Do we
have any good estimates of how many we can build? Dana, what
do you think?
DANA: Ummm.... 13.
Carolyn: Dana thinks 13. Okay.
Do we have any other guesses?
Jeff: 25.
Carolyn: Jeff thinks 25. Anybody
think something else? Steve?
Steve: Ahhh.. 10.
Carolyn: Steve thinks 10. What
do you think Jamie?
Jamie: 10.
Carolyn: Jamie thinks 10.
Student: 15.
Student: 12.
Student: 26.
Carolyn: Sebastian?
Sebastian: 12.
Carolyn: But were not agreeing
on this. Jennifer?
Student: 15.
Jennifer: Matthew?
Jennifer: 8.
Carolyn: Well I think what
your going to have to do, is work on it and see how many,
but remember: You have to be sure that you have no duplicates.
Can't have two of the same and you have to be able to convince
us that you have found all possibilities. That there are no
more or no less. Got the problem? Have fun!
Stephanie: Okay we'll start
out with the easiest ones. 12345 Reds and five yellows.
Dana: 12345. Okay well
stand them up straight so we know how many we have.
Narrator: The students were
asked to attack the problem in their own way, and the researchers
simply recorded what they did. At the end whatever strategy
they used the students would have to justify their answers
and convince the others that their thinking was correct.
Jeff: First we'll do the four
big ones then we'll do the four easy ones.
Shelly: Two of these and three
of them.
Jeff: Then we'll take three
of these and two of these.
Shelly: And we can switch these
all different ways. Then we'll take one of these and four
of these.
Narrator: At the end, whatever
strategy they used, the students would have to justify their
answers and convince the others that their thinking was correct.
Carolyn: When the decision
was made that students ought to be challenged in their thinking
about math, that it shouldn't just be rote computation, memorization,
the interest on our part was to see what was possible, because
we didn't know. And the driving question for us was to see
how particular mathematical ideas developed. We chose key
mathematical ideas that made sense because it wasn't too far
away from what ultimately we would expect kids to learn in
school. The vision was for the students to be successful.
And success  and I think that's still true today, but certainly
it was at that time  was to be successful in algebra. Algebra
is the first great separator. Students who don't succeed in
algebra, they usually don't continue their study of mathematics.
And then that's a path for them, they don't go on an academic
track, and so they are eliminated from other programs as well.
So, we were interested in the beginning of algebraic thinking.
Brian: Blue. White. Blue. Blue.
These things stick.
Romina: Do we have them like
this we only have white on the top?
Brian: Four blues and a white.
Oh I got it I got the four blues.
Romina: You got the blue.
Brian: Oh God, this is a lot!
Narrator: The researchers found
that after the students tried creating random towers, they
quickly switched to using more sophisticated approaches 
like pairing towers with opposites.
Allston: Can you tell me what
your thinking about?
Brian: Well once when we find
one we do just do the opposite of it.
Allston: what do you mean the
opposite?
Brian: Like when we found this
one out we just put two blues on the top and two whites in
the middle
Teacher: Do they always have
an opposite?
Brian: Yes, well not  yeah.
Let's see if we have any one without an opposite.
Romina: Yeah, that's a good
idea.
Brian: I found one already.
Romina: Which one?
Brian: Three blues one white
and one blue.
Brian: ìTowers" is my
first memory. It seems like we've been doing "Towers" since
I was about eight. Back then, I thought it was pretty much
fun. I enjoyed building little towers. It was very childlike,
but I really pretty much had more fun than thought of it as
math.
Brian: We don't have it.
Romina: Are you crazy? We have
to have that!
Brian: We don't have it, we
don't have it... That ain't it. Oh.
Romina (older): We called ourselves
your guinea pigs. (laugh)
Brian: This one better fit
in there.
Romina: Well give me some of
these whites and blues. In case that we find another.
Romina: We thought that you
thought that we were smart. And when you came up I thought
you just wanted to experiment with what our capabilities are.
And that was very scary when you're young, because none of
us had any confidence.
Brian: Let's see if we have
some of the same. Let's put the sames in order.
Romina: Ummm.... no. Well some
of them could be the same.
Brian: Oh good idea.
Romina: This matches this.
Brian: Put the pairs like the
opposites.
Narrator: Stephanie and Dana
used a strategy of making pairs they called "duplicates".
Dana: We just did this one
Stephanie: And here's the duplicate
,see?
Carolyn: What do you mean by
duplicate?
Stephanie: Well it's not the
duplicate, it's the upside down version.
Dana: It's like if we turn
this upside down it matches this one.
Stephanie: See, it goes with
this.
Carolyn: I don't quite understand,
would you go through that one more time?
Stephanie: Instead of going,
okay look. Okay we take the design and instead of just like
making a new design we take one and make the same design upside
down.
Carolyn: Yeah, but how did
you get this one?
Dana: This one goes to this
one.
Stephanie: We were just fooling
around when we came... and it does that. See?
Stephanie: So it's not actually
the same but when it is turned upside down it is the same.
Carolyn: Aha. Is that how
you did all of these?
Dana & Stephanie: Most
of them, yeah.
Carolyn: Most of them? Show
me another.
Carolyn: We had certain ground
rules in our research, because we were trying to study the
development of mathematical ideas. But there were consequences
of what we were doing. We said we would not show them we
didn't intend to show them how to think about a problem. That
was not in our set of rules for being researchers. We did
not believe that if they worked on a problem for one or two
hours, now that when they were all finished we had to tell
them if they were right or wrong. We didn't have to follow
that rule. We could say, "Well, maybe you need to think about
that some more."
Student (off camera): 22, 24,
26, 28, 32
Narrator: By building the towers
and checking for duplicates, most of the students agreed that
there were 32 different combinations.
Carolyn: Now how many do you
have?
Child: 33
Child: 32. Oh yeah, 33.
Carolyn: We've been interested
in creating conditions whereby we can give children an opportunity
to show us how they think about mathematics. So the only thing
we're doing is making conditions possible. The children are
doing the rest.
Carolyn: And you're all convinced
you could take a stranger and tell them there are no more.
Right? Now remember when I walked around and you were convinced
cause you worked so hard doing this problem. Remember you
really worked hard. I don't think I have ever seen a 4th grade
class work so long on a math problem, ever  as you worked
without stopping. Your incredible. I think your great. Let's
give you a round of applause?
[4th  8th grade]
Narrator: The study continued
from 4th to 8th grade. The research team was able to videotape
the same group of students year after year, solving a variety
of math problems in combinatorics, probability, and algebra.
Jeff: ...man, you're wasting
your time doing that. Because you're going to put all in one
column, and then your gonna put the same amount in the next
column, then the same in the next column, and then, of course,
you have two columns worth...
Jeff: We got so indepth on
topics that that leaves an impression you know. We can talk
about, you know doing the blocks in first grade. We could
almost go through the problems we did: Shirts and Pants in
second grade. How many people can tell you the math they were
doing in second grade?
Amy: I need you to write your
reasons for why you picked that. Okay? Why you picked seven
yellow and three blacks?
Jeff: I have no idea.
Amy: She left you space, Jeff,
yes you do.
Jeff: I don't know.
Amy: You told me why.
Jeff: Oh yeah.
Mike: It was basically like,
what we thought. Like, you know, they were giving us a probability,
like, questions. And those I liked those kind of questions.
Those really get you to think.
Mike: Well since there are
16,  umm to make with those toppings you put a Sicilian crust
on it and that's 16 plus...
Carolyn: These students were
always up to the task. We'd give them a problem, they would
work hard on it and would often take it to higher levels themselves.
Romina: Then you're going to
go two three oh two three...
Romina: We thought we were
real weird. What fourth grader's interested in math and arguing
with their own friends about it?
Ankur: That's how she told
me I still don't agree with her.
Student (off camera): It is,
it's half and half.
Romina: Look okay, you have
a whole pizza and you split it in half...
Amy: And we'd come in four
or five times a year for a couple days each time and just
film what went on. Children working in small groups, encouraging
them to discuss and to argue and to justify.
Romina: First when you had
the pepper you got four and then when you had the sausage
you got three and the mushrooms you got two and the pepperoni
you had one.
Carolyn: Here is this group
of students, just, you know, kids going to school, doing what
was expected. And now we're challenging them with problems,
and they were taking them, and solving them, in ways that
were very impressive to us. We pulled back and gave them more,
and they responded to more. And we pulled back and gave them
more, and they responded to more. ... I hope that we can show
that under certain conditions, and these conditions were not
in place all of the time, remember we went in six times a
year in the first few grades and then maybe four times a year
in the middle grades and so on and so forth. That that kind
of an intervention seemed to be very very important for this
group of students And it wasn't  We did not change all of
the curriculum, we did not change, we didn't throw out textbooks.
We didn't say teachers don't do what you do. What we tried
to do is bring into the curriculum thoughtfulness and the
notion it's so very important to understand fundamental ideas.
The building blocks really have to begin very early.
[Fall, 1996]
Narrator: When the students
began 9th grade the Kenilworth high school was closed.
Nancy Baton: The regional district
served six different communities and had four high schools.
Because the cost per pupil was about $15,000, the taxpayers
were very concerned about it, and the superintendent had a
very difficult decision to make, but he decided to close one
of the four regional schools. And they picked on Kenilworth,
which was the newest building in the regional district. And
so I closed the building. But the residents of Kenilworth
were very upset and concerned. After all, this school is really
the hub of the community.
Narrator: The study continued
during this period in living rooms and kitchens. The students
attended 9th grade at the regional school. The subject was
geometry, the approach: strictly textbook based.
Mike: I'd sit in class. My
name starts with an A, so I'm in the front. The teacher would
just tell us, "These are the laws of geometry." I don't remember
half the stuff she'd say.
Brian: In the first year of
high school, I had geometry. And the way we learned was sick,
I guess you can call it, because it was not taught, it was
read from a book. And we never got involved. It was "Read
this and see if you can do it yourself, and then I'll tell
you the answers the next day." Which I don't think anybody
learned too much from it.
Jeff: She had, my math teacher,
had a plan to get through the entire book in one year. That
was her goal. That's what we were going to accomplish. She
gave us a sheet the first day of school, saying, "All right,
on day 154, the homework from this chapter is due." And, you
know, you totally  you could do all your homework in one
night if you had the time to do it.
Carolyn: Particularly in high
schools, deviations from the curricula are not encouraged
for teachers. So I think we have to be careful  but I wonder
what we can learn by listening to what those students are
telling us. I wonder what we can learn about their own assessment
of their own learning over that year.
Romina: Ask me one question
about geometry, because I won't know it.
Narrator: When the students
were in 10th grade, protests in the town were successful in
reopening the local school. A core group of students  Romina,
Mike, Ankur, Brian, and Jeff  continued to meet in the evenings
and after school.
Loretta Malina: As the time
went by a lot of this was done after school and it was done
on their own time. And they weren't always always just dying
to get there. But they went and they had the option of dropping
out whenever they wanted to, but they didn't. They kept going
and kept going and going.
[Romina and
Parents  October 1999  12th Grade]
Romina: Daddy, I'm sending
this one out. This week. I need a check actually. (Laughter)
Romina: I want to go to college.
As far as I can go in schooling I want to do that and everything
necessary. I want to probably major in economics in something
math related. Hopefully in four years graduate, get a good
job, and live well. I guess that's everyone's dream though.
Romina [to parent]: Rutgers
I'm done, I'm almost done with. They're going to tell me.
This is the first school I'll find out. They are going to
tell me by January first, but if I don't make it into that
one...
Romina: My parents never were
able to actually be like, "Oh this is how you or let me correct
your paper or let me read this." But they're always so encouraging.
They don't think I have any, I can't be held down in any way
but they give me the moral support I need to do whatever I
can.
Mario D'Andrea: She likes to
study, you know she goes hard on the books and you know I
want the best for her.
Yolanda D'Andrea: It's important
that kids today learn mathematics. It's the future of this
country. It is essential for everyone.
Romina: And I have to call
1800SATSCORE. They took them last week, and my guidance
counselor told me I needed to break at least 600 in math,
because I'm going into economics.
Romina: The SAT's... Everyone
knows the SAT's are the most important test you'll ever take.
And I've been in classes with people that have teared during
the test. Because they understand how important it is. And
they've come out, like  hey, I've never seen anyone look
sicker. And it's just like they were put through the worst
thing of their life. And it was only a test. It was a three
hour test.
Romina: Daddy get out a credit
card.
Mario: What, you need a credit
card?
Romina: Yeah.
Narrator: Romina's score on
the math section of the SAT, a test where 30% of the problems
cover geometry, was a little under 600.
Mario: How much do you need?
1170?
Romina: 1170.
Mario: 1170?
Romina: That's not good Dad.
I just can't take those tests. I just can't do them.
Mario: 1170.
Yolanda: Yeah.
Romina: Yeah. They told me
I should be getting at least 1250  1300. I'm a little bit
off.
Romina: It's upsetting, because
I work so hard in every other aspect of my life. And then,
taking a test, one test, that takes up four hours  not three
hours  of my time, on a Saturday, and that's going to decide
whether I get into college or not? That's really upsetting.
Because those three hours, like, reflect over, like, 100,000's
of hours that I put into everything else.
Mario: So what are you going
to do now?
Romina: I'm going to hope they
don't look at my SAT scores? (laughter)
Mario: You better hope.
[The next day
at school]
Narrator: There are many different
skills involved in being successful in mathematics. Experienced
teachers, like Romina's AP Calculus teacher, Ralph Pantozzi,
know that not all math skills get measured on standardized
assessments.
Ralph Pantozzi: Doing well
involves being able to actually do mathematical inquiry. To
approach a problem and then say "all right, I'm approaching
it this way and I am going to try different ways of going
about a problem; or I'm going to look for patterns  or I'm
going to look for whether the change in this number creates
some change in some another number." Looking for things that
might be related to the general question that their asking.
And when you see kids looking for that and generating those
questions themselves, before you ask them those questions,
that is really the key. And that is what I've seen this class
do.
Romina: ...the area at zero
is zero, so you don't actually subtract anything, so your
answer is "D"..
Brian (off camera): Oh man!
Narrator: Nationally, only
a small percentage of students take four years of mathematics,
culminating in AP Calculus. Yet, all of the students who participated
in the Kenilworth study, that remained in the district, chose
to take advanced placement calculus in 12th grade.
Ralph: ...OK, the area from
"A" to "B", between the graph of f of x and the x axis. In
between those two points. OK...
Narrator: Today, Ralph is reviewing
problems involving integrals, including one that centers on
finding the difference of the two areas under two similar
curves.
Ralph: Draw what that says...
Romina: This year I am doing
calculus and I'm actually liking it only because I feel that
I understand it and we just talk about it in class like it's
just, we don't talk about like it's math. We talk about it
like it's an issue.
Ralph: That area in the bottom
that's already shaded in is A + 2B. If we add on that top
area which is always 5 above the bottom one, what's the new
total going to be? The integral from A to B of F of X plus
5?
Romina: B minus A times 5,
all plus A plus 2B. Only because...okay, if we move it by
five up, there is going to be another rectangle there. The
length of the rectangle is five and the width is B minus A.
Ralph: Romina, I didn't see,
I don't see any rectangle there. All I see is that curvy thing
on top. How do you know that that is b minus A times 5?
Romina: I didn't take it, I
didn't do it like that, I kinda just assumed, like your just
moving it...
Ralph: Go ahead.
Romina: This part, like, like
this part moved up right? so this all moved up, so this part
right here takes up this much. Like I saw it like that so
that's A plus 2B. Because this part, the top of the graph
just did that
Ralph: Oh oh oh all right,
I didn't see that at all.
Romina: And underneath here
is where you added the plus five and you added just a regular
because like you go straight down so there's no curves and
there's just a straight block.
Ralph: Fascinating.
Romina: No?
Ralph: I've never seen anyone
do that before. Yeah that's right
Romina: Yeah, okay
Ralph: Do you understand what
she just did?
Students: Yep. Yes. I got it.
Jeff: I was on top of everything.
Ralph: I was thinking that
you would say the area of this shape up here was B minus A
times 5. But you saw it in a completely different way. And
just as valid.
Ralph: They end up suggesting
the questions that I have in my lesson plan. And that tells
me that they're doing well, because they're actually doing
mathematical thinking. And that's the best evidence you can
get.
Romina: So I'm thinking I'm
going to have to use a lot of my background to figure out
a lot of the problems that I'm going to have to deal with.
I'm not sure what they're going to be. I actually have no
clue. Pretty sure my career is going to revolve around mathematics.
But I think it's going to revolve around other things too,
but mainly mathematics. And maybe not the math everyone thinks,
but the math that I've learned, in the last 10 years.
Romina: I don't even know what
I can go into without math. So anywhere, any path I choose
is probably going to have to deal with math..but we'll see.
[On screen text: In April,
Romina found out that she was accepted by all of the colleges
to which she applied, including top choices Boston College
and University of Pennsylvania.]
Carolyn: It's so fascinating
to see what these children can do when you give them the opportunity.
When you don't tell them how to think. They naturally want
to think. They naturally want to solve problems. They naturally
want to make sense of their mathematics.
Narrator: We've had a chance
to look at an overview of the Rutgers' longterm study. In
what ways are the students in the study similar to your students?
How are they different?
[End of program]
