Teacher: What type of angle is this?
Students:
[In unison] A right angle.
Kathie: But if you turned it the other way, it would be a left angle.
Teacher:
If this is a right angle [fig. 1a], and this is a left angle [fig. 1b],
what is this [fig. 1c]?
Kathie:
That's not an angle.
The
power of mathematical conversations was epitomized during a series of
lessons in which fourth-grade students were encouraged to make conjectures
and defend their conjectures to their classmates. The opening excerpt
illustrates that the students had had limited experiences with angles
in various orientations. Consequently, they had internalized a very narrow
definition for a right angle. However, this limited definition did not
become clear until we asked them to elaborate on their thinking.
Driver and others (1994) state that making meaning involves "persons-in-conversation."
The Professional Standards for Teaching Mathematics (NCTM 1991) highlights
the teacher's and students' roles in mathematical discourse. Both students
and teachers are called on to listen carefully to each other, respond
to each other, and pose questions to each other (NCTM 1991). Student-student
interactions are as important as teacher-student interactions in making
mathematical meaning. Teacher-student interactions also take a different
form. The teacher no longer assumes the role of the transmitter of knowledge
but becomes a part of the mathematical learning community. Mathematical
conversations must become an integral part of classroom activity--teacher
with students and students with students conversing about mathematics.
In such a setting, students can reflect on, and clarify their thinking
about, mathematical ideas (NCTM 1989).
In general, classroom interactions about mathematics should, from this
perspective, be characterized by a genuine commitment to communicate in
which the teacher assumes that a student's mathematical actions or explanations
are reasonable from his or her point of view even if that sense is not
immediately apparent to the teacher. (Cobb et al. 1991, 7)
The
foregoing excerpt suggests that the classroom culture should be renegotiated
so that students respond to one another's comments instead of having their
comments filtered through the teacher. This approach significantly changes
the mathematical tone in the classroom. The teacher becomes a participant
in the conversation and focuses on the mathematical understandings of
her or his students as revealed in the conversations. Ball (1993) states
that the teacher needs a "bifocal perspective--perceiving the mathematics
through the mind of the learner while perceiving the mind of the learner
through the mathematics" (p. 159). That is, the teacher seeks to infer
the mathematics that the student has constructed and the reasoning behind
those constructions. The teacher poses tasks that offer insights into
the student's mathematical reasoning.
What
follows is the story of conversations that took place in four fourth-grade
classes. Two of the conversations are about finding the volume of a cube.
The first conversation shows not only what these fourth graders understand
but also the ways in which the physical positioning of the students affected
the quality of the conversation.
The
second conversation shows the adaptations made by the teachers in posing
the task and in physically positioning the students. A conversation about
integer subtraction is included to illustrate the power that these conversations
have in revealing misunderstandings about the subtraction algorithm. The
final conversation, about angle measure, shows the students' discussing
mathematical ideas without the teacher's orchestrating the conversation.
Each conversation offers insights into the ways in which these children
had constructed mathematics. Without fail, the adult participants were
surprised by what they heard and learned a great deal about how their
questioning and physical positioning affected the quality of the mathematical
conversations.
VOLUME
CONVERSATIONS, PART 1 I sat on the couch with all the children
gathered in front of me. Some of the children sat on the floor. Others
sat in chairs. One child leaned over the arm of the couch to my left.
The classroom teacher and the principal sat in chairs on the perimeter
of the group. Very few of the students were able to see the faces of the
other students. Each child had a calculator and a clipboard with paper
and pencil. I held up a six-inch-by-six-inch-by-six-inch cube, which I
had made with 216 snap cubes. It had an accordionlike construction that
allowed me to open the cube to show the interior cubes.
I began the conversation by asking the children how many small cubes I
had used to make the large cube. Several children began pointing at the
air as if counting the cubes. Some began writing on their paper while
others pushed the keys on their calculators. After a short time, students
began to raise their hands.
Teacher:
Susan
Susan:
Two hundred and sixteen.
Teacher: How many of you agree with Susan? Thumbs up if you agree, thumbs
down if you disagree, sideways if you aren't sure. [All students show
thumbs up.]
Teacher:
How did you decide that there are two hundred sixteen cubes? Steven.
Steven: There are six rows of six cubes. That gives thirty-six. And there
are six sides. So I timesed thirty-six by six and got two hundred and
sixteen.
Peter,
Sherri, Angela: Yep. That's what I did.
Teacher:
Did anyone do it differently?
Carlos:
I just said there were six down and six across and timesed them, and that
gave thirty-six. And then six sides made two hundred and sixteen. At this
point I was trying to formulate a question that would require the students
to reconsider their thinking without telling them how volume is calculated.
Teacher:
Let me see if I understand [I hold up the cube]. You are saying that there
are thirty-six cubes on this face [I moved my hand across the front face;
see fig. 2a] and thirty-six cubes on this face [I moved my hand across
the lateral face; see fig. 2b]. Did you count these cubes twice? [I pointed
to the column of edge cubes; see fig. 2c]. I thought that I had formulated
the perfect question to cause the students to discover a flaw in their
thinking, but I quickly found that I had not. My repeated asking whether
they were counting the edge cubes twice was not problematic to these students.
Teacher:
Okay, let me see if I understand. There are thirty-six cubes on this face
[as I moved my hand across the front face]. And there are thirty-six cubes
on this face [I turned the cube to the left and moved my hand across the
new front face]. And there are thirty-six cubes on this face and this
face and this face and this face [as I continued rotating the cube to
show the remaining four faces]. [All students shook their head in agreement.]
Teacher: So that gives two hundred sixteen?
Students:
Yeah.
Teacher:
Then what about these cubes? [I opened the cube, which revealed the cubes
in the center.] A spontaneous look of wonder appeared on the students'
faces. I knew that what I had asked was problematic for most of the students.
I had finally found the right question. Now it was up to them to figure
out the solution. After a while, the boy leaning over the arm of the couch
spoke up.
Brandon: There are six layers. [He moved his hand horizontally showing
the six layers.]
Teacher: What do you mean? Class, please listen.
Brandon:
There are thirty-six cubes [as he pointed to one of the faces], and there
are six layers [he pointed to each of the layers]. That gives two hundred
and sixteen cubes.
I
was shocked when I looked at the clock and realized that these students
had been engaged in talking about mathematics for an hour. Reflecting
on this conversation, I realized why counting the edge cubes twice was
not problematic to the students. Although I said cube, the students heard
square. They were counting the faces of the small cubes--the small squares--not
the cubes themselves. Therefore, they were not counting a cube twice because
they were counting each face only once. The importance of asking how they
arrived at the answer became clear. The students did give the "correct"
answer in saying 216. Had I not asked, I would not have known that the
students were thinking in terms of surface area, not volume.
As
I reviewed the physical arrangement of children and the discussion, I
realized that the conversation was directed to me and filtered through
me and that very little student-to-student interaction had occurred. I
wanted to structure the physical environment to encourage better conversations
in which the students would interact with one another.
VOLUME
CONVERSATIONS,
PART 2 Learning from the previous conversation, I repeated
this activity in another fourth-grade classroom. A large braided rug covered
the wood floor in one section of the room. Students sat on the perimeter
of the rug so that each could see his or her classmates. The teacher and
I sat on the rug with the students. Again, I asked students how many small
cubes I had used to construct the six-inch-by-six-inch-by-six-inch large
cube. The conversation began as had the conversation in the other class.
The
students once again said that the cube was made from 216 small cubes because
36 cubes were on each face and the cube had six faces. Once again my question
about counting the edge cubes twice did not cause the students to see
a flaw in their thinking. This time I had a five-inch-by-five-inch-by-five-inch
cube, which I held up, and asked the same question: "How many small cubes
did I use to make this cube then?" Students once again began working with
their calculators and writing on their clipboards.
David:
One hundred fifty.
Teacher:
How many of you agree with David? [About two-thirds of the thumbs turn
up.] Anyone disagree? [No one shows thumbs down.] Anyone unsure? [About
one-third of the students turn their hand to the side.] How did you get
that, David?
David:
There are twenty-five cubes on the side times six. That's one hundred
fifty.
Adam:
But there are only one hundred twenty-five cubes.
The
class went on to discuss that the large cube has 125 cubes because it
had five layers of 25. My choice to use another cube example to motivate
them to rethink the method that they had used to find the number of cubes
in the first cube was successful. The students' original method would
not generalize to a second example. They needed to find another explanation--layers
instead of faces.
Interestingly,
in this class the students began to discuss the problem with one another
rather than direct all comments to me. Whether this discussion reflected
the physical positioning of the participants or the classroom culture
that had been negotiated prior to this lesson was unclear. The classroom
was furnished with tables instead of traditional student desks. The teacher
did not have a teacher's desk in the room. She chose to sit at one of
the tables with the students. In addition, this teacher had been able
to create a classroom environment that offered the students opportunities
to take ownership of their learning. For example, the teacher and the
students developed scoring rubrics for many of the class assignments.
Students were also given opportunities to grade themselves. This sampling
illustrates how this teacher had reconceptualized what it meant to become
involved in the teaching and learning process.
CONVERSATIONS
ABOUT NUMBER AND ANGLE I asked the fourth-grade teachers if
I could try this instructional strategy for an extended period of time.
The teachers and I agreed that I would work with two new fourth-grade
classes for two weeks. One teacher was doing data-analysis activities
that would include the construction of a circle graph.
INTEGER
SUBTRACTION CONVERSATIONS In the middle of the data-analysis
unit, I began to wonder about the students' understanding of the four
arithmetic operations. We pushed all the desks to the perimeter of the
room and sat in a circle on the floor. Each student brought a clipboard
to the circle. The assistant principal and I sat on the floor with the
students. The classroom teacher sat at a table on the perimeter of the
circle so that she could take notes.
I began the discussion by asking students to add 5 and 7 and 7 and 5.
Then I asked the students to multiply 2 and 4 and 4 and 2. I asked the
students what they noticed about these problems. They stated that "two
times four and four times two give the same answer." Next, I asked about
5 minus 2 and 2 minus 5. Although these students had no difficulty with
the initial questions, the students disagreed on the answer to 2 minus
5. Some students said that the answer to 2 minus 5 was 0, some said that
the answer was -3, several said that the answer was 3, and one student
said that the answer was 8. We had an interesting and enlightening conversation.
Teacher:
How did you get eight?
Walter:
You can't take five from two. So you borrow one from the five [fig. 3a]
and give that to the two [fig. 3b]. Twelve minus four, eight.
Teacher:
Let me see if I understand. You can't take five from two, so you borrowed
one from the five. Gave it to the two, so twelve minus four equals eight?
Walter:
Right.
Teacher:
What do the rest of you think?
Several
students indicated that they understood what Walter had done. It was very
challenging to pose a problem that would lead Walter to see a flaw in
his thinking. I asked how he was able to turn a 1 into a 10 when borrowing
one from the 5 and turning the 2 into 12. This question did not create
a conflict. I asked the students to comment on Walter's solution. To refute
Walter's argument, one student said, "You don't do that with those problems.
You do that when they're up and down." Wow! What now? Finally, the classroom
teacher reached into a desk drawer and pulled out two spools of thread.
Classroom teacher: Look! You have two spools of thread. Your mother asks
to borrow five spools of thread from you. What do you need to do?
Serena:
I know. You sell each of the spools for like a dollar each and then go
buy what you need.
All
the adults in the room were struck by the power of algorithms to diminish
the sense-making process in mathematics. Walter had become so focused
on the algorithm that he no longer was giving meaning to subtraction.
At that point, we had been conversing for almost an hour. Since I was
uncertain of the individual students' understanding of this problem, I
asked each student to describe his or her solution to 2 minus 5 in writing.
Some of the students provided solutions that seemed more reasonable to
the adults in the room. We understood the reasoning of the student who
argued that 2 minus 5 equaled 0. We inferred that this student had not
had many experiences with negative numbers. We were intrigued also by
the counting-down strategy used by several students in obtaining -3. We
realized that most of the students in the room did have a sense of subtraction
and that a few did have some understanding of negative integers. Nevertheless,
we could not forget Walter's mathematical construction.
ANGLE-MEASURE
CONVERSATION The second excerpt from the data-analysis unit
is a conversation about angle measure. Of the four conversations presented
in this article, this one is the closest approximation to my vision of
how I wanted these conversations to sound when I began these lessons.
Again
we pushed the tables aside and sat in a circle on the floor. We placed
two large sheets of newsprint and several markers on the rug in the center
of the group. On one sheet of newsprint, a 90 degree angle was drawn;
a 60 degree angle was drawn on the other sheet.
Teacher: [I pointed to the sheet with the 90 degree angle.] What can you
tell me about this angle?
Kadeejah: It's 90 degrees.
Peter:
It's a right angle.
Teacher:
Does everyone agree with Kadeejah and Peter? [All students showed a thumbs-up.]
Teacher:
[I pointed to the sheet with the 60 degree angle.] Is this angle smaller
or larger than the right angle?
Kadeejah:
It's smaller.
Teacher:
Does everyone agree with Kadeejah? [Again, all students showed a thumbs-up.]
When I originally drew the two angles, I unknowingly made the rays of
each angle the same length. The assistant principal wondered whether the
students were judging the size of the angle by the lengths of the rays.
She reached over and extended one of the rays of the 60 degree angle.
Assistant
principal: How about now? Is this angle smaller or larger than 90 degrees?
Kadeejah:
It's larger. Michelle and Adam: No, it's still smaller.
Teacher:
How can we find out which is larger? How do we measure a angle?
Sonia:
We could measure it with a ruler in centimeters. [I handed Sonja a centimeter
ruler.] Sonia [She took the ruler and placed it across the 90 degree angle.]
It's forty-four centimeters.
Adam:
It's supposed to be ninety.
Kathie: Not there. Measure the lines.
Sonja:
[She measured the "lines," using the ruler.] It's eighty centimeters.
Peter:
Oh, you don't use a ruler. I think you use that curved thing. It looks
like half a circle.
Michelle:
Yeah, a protractor.
Teacher:
[I placed a chalkboard protractor on the carpet.] Is this what you're
thinking of? How do you measure with this? [Michelle moved to the center
of the carpet, took the protractor, and placed it on the angle, turning
it several different ways. Peter joined Michelle on the center of the
carpet.]
Kathie:
Oh, yeah. I remember Mrs. Daniels showing us how to use that once.
Michelle
and Peter: [They placed the protractor on the angle so that it showed
that the angle was 90 degrees.] See, it's 90 degrees.
Sonja:
But that [other] one is still bigger.
Adam:
What do you mean, it's bigger?
Sonja:
[She took the ruler and closed in the third side to make a triangle.]
See, it's bigger.
Michelle:
[She placed the chalkboard protractor on the angle.] Look, it's 60 degrees.
[She placed a desk protractor on the angle.] Look, it's still 60 degrees.
The lines could go for miles, but it's still 60 degrees. The corner doesn't
change. Do you see?
Sonja: No. It still looks bigger.
Adam:
[He moved the paper and protractors in front of where he was kneeling.
He placed the protractor on the angle.] What does that say? [He pointed
to the protractor.]
Sonja:
Sixty.
Adam:
Right! [He placed the ruler on the paper to form the third side but did
not touch the protractor.] Look, it still says sixty. Do you see how it
hasn't changed?
Sonja:
No.
Adam:
It's changed?
Sonja:
Yes.
Adam:
[He slid the materials toward Sonja.] Then you show me how it has changed.
In
this conversation we were delighted that we were able to "fade into the
woodwork." The students began to challenge one another, to ask one another
questions, and to offer explanations. These students began to talk to
one another rather than to the adults in the room.
REFLECTIONS
ON THIS PROCESS I found that I was exhilarated but tired after
these conversations. If I wanted to facilitate the conversations and let
students lead us where we needed to go, then I needed to listen to what
they were really saying, infer the mathematical meaning that they had
constructed at that point, and formulate a question or encourage the other
students to formulate questions. I did need a "bifocal-perspective" (Ball
1993).
As
I went through this process, my own mathematical understandings were quickly
surfacing. It was unavoidable. I could not understand the connections
that these students had made without thinking in terms of my own mathematical
understandings.
My
role as a teacher was redefined in the process. Although I was responsible
for finding a rich mathematical task, I had to work to renegotiate the
classroom culture from turn taking to engaging in a mathematical conversation.
I believed that I had accomplished this task when I was no longer involved
in the conversation.
CONCLUSIONS
Engaging in mathematical conversations is an evolutionary process. The
tone and quality of our conversations change as we change and learn from
the students and our own interactions. Mathematical conversations provide
a tool for measuring growth in understanding, allow participants to learn
about the mathematical constructions of others, and give participants
opportunities to reflect on their own mathematical understandings. The
selection of the appropriate task and the questioning techniques used
by the teacher are vital to this conversational approach. In addition,
the physical layout of the classroom affects the quality of the conversations.
In the conversations described in this article, positioning the students
so that they could see one another encouraged richer conversations among
them and increased the likelihood that the teacher could become a member,
rather than leader, of the mathematical community. Students of that community
began to form a collegial relationship in which they challenged, modeled,
and reconstructed one another's ideas.
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