A BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY IN A MATHEMATICS CLASSROOM
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MIRIAM GAMORAN SHERIN
A BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY
IN A MATHEMATICS CLASSROOM
ABSTRACT. This article examines the pedagogical tensions involved in trying to use
students’ ideas as the basis for class discussion while also ensuring that discussion is
productive mathematically. The data for this study of the teaching of one middle-school
teacher come from observations and videotapes of instruction across a school year as well
as interviews with the participating teacher. Specifically, the article describes the teacher’s
attempts to support a student-centered process of mathematical discourse and, at the same
time, facilitate discussions of significant mathematical content. This tension in teaching
was not easily resolved; throughout the school year the teacher shifted his emphasis
between maintaining the process and the content of the classroom discourse. Neverthe-
less, at times, the teacher balanced these competing goals by using a “filtering approach”
to classroom discourse. First multiple ideas are solicited from students to facilitate the
process of student-centered mathematical discourse. Students are encouraged to elaborate
their thinking, and to compare and evaluate their ideas with those that have already been
suggested. Then, to bring the content to the fore, the teacher filters the ideas, focusing
students’ attention on a subset of the mathematical ideas that have been raised. Finally,
the teacher encourages student-centered discourse about these ideas, thus maintaining a
balance between process and content.
KEY WORDS: class discussion, discourse community, student-centered discourse, teacher
cognition, teacher’s role in discussion
A central goal of mathematics reform is for teachers to develop classroom
learning environments that support doing and talking about mathematics
(National Council of Teachers of Mathematics [NCTM], 1991, 2000).
However, creating and maintaining these environments is a complex
endeavor for teachers. In particular, two key tensions are apparent. On
the one hand, teachers are expected to encourage students to share their
ideas and to use these ideas as the basis for discussion. At the same time,
teachers are supposed to ensure that these discussions are mathematically
productive. The tension comes in trying to find a balance between having
a classroom environment that is open to student ideas and one whose
purpose is to learn specific mathematical content.
These tensions are explored through an investigation of one middle-
school teacher’s attempts to implement mathematics education reform.
The teacher, David Louis, worked hard to establish and then maintain
a discourse community in his mathematics classroom. In doing so, he
Journal of Mathematics Teacher Education 5: 205–233, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.206 MIRIAM GAMORAN SHERIN
struggled to facilitate class discussions in which student ideas were at the
center and in which mathematics was discussed in a deep and meaningful
way. David explained this dilemma in a journal in which he reflected on
his teaching:
Today I was forced to consider an interesting issue. The issue is, ‘Do I sacrifice some . . .
content in order to foster discussions during class?’ . . . There were several different places
today where discussion arose . . . I should have expected that considering I’m trying to set
a culture of expressing one’s ideas, but it caught me by surprise a little. At first I tried to
press on [to the content he had planned to cover], but students still had [new] ideas. In fact
at one point, their ideas [about the content] were quite different than mine . . . [and] when
I wanted to move on, they didn’t. (Louis, 1997a, p. 10)
Unable to resolve this tension, David moved back and forth in his emphasis
on student ideas and on mathematics learning – sometimes striking an
excellent balance, and sometimes finding his efforts less successful. The
purpose of this article is to characterize how the tension played out in
David’s classroom by contrasting the teacher’s focus on the process of
mathematical discourse with his focus on the content of mathematical
discourse. In brief, the process of mathematical discourse refers to the way
that the teacher and students participate in class discussions. This involves
how questions and comments are elicited and offered, and through what
means the class comes to consensus. In contrast, the content of mathe-
matical discourse refers to the mathematical substance of the comments,
questions, and responses that arise.
This research advances both our theoretical and practical understand-
ings of the nature of the teacher’s role in a discourse community. Prior
research on teacher cognition has explored the process through which
teachers learn to elicit and to monitor student ideas. This article extends
such work by examining the tensions involved in this process, and
the manner in which teachers manage competing goals. The research
described here can also provide teachers and teacher educators with one
vision of a discourse community, and with a model for interpreting class
discussions and the teacher’s role in such discourse.
BUILDING A DISCOURSE COMMUNITY
When researchers speak of classroom discourse, or discourse more gener-
ally, they are referring to the processes through which groups of individuals
communicate (Cazden, 1986; Pimm, 1996). Analyses of discourse decom-
pose these processes and underlying structures in different ways. Some
researchers attempt to enumerate norms that define aspects of classroom
discourse. For instance, norms can govern who can speak and whenA BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY 207
(Mehan, 1979; Sinclair & Coulthard, 1975). Other researchers look to
identify discursive strategies used to support instruction. One example is
the work of O’Connor and Michaels (1996), who describe “revoicing” as
a technique used by teachers to restate a student’s idea for the class. In still
another approach, researchers examine the meaning of particular words
and phrases in the context of instruction (Lampert, 1986; Lemke, 1990;
Pimm, 1987).
Recently, mathematics educators and researchers have placed increas-
ing emphasis on fostering classroom discourse that has certain properties
(Elliott & Kenney, 1996; NCTM, 1989, 2000). Specifically, students are
expected to state and explain their ideas and to respond to the ideas of
their classmates. Teachers are asked to facilitate these conversations and to
elicit students’ ideas. In this article, classroom environments where such
discourse flourishes, are referred to as a discourse community. Further-
more, the use of the term mathematical discourse community emphasizes
that this communication concerns mathematics in particular.
To examine the development of a mathematical discourse community,
two related perspectives are examined. The first perspective examines what
a mathematical discourse community might look like and evidence that
such a community can exist. A second viewpoint considers the teacher’s
role in developing a discourse community and the teacher learning that is
often required as part of this process. Together, these lenses serve to frame
the current study.
Visions of a Mathematical Discourse Community
Recent research demonstrates that a discourse community can exist in the
mathematics classroom. For example, Ball (1993) and Lampert (1990)
share vignettes from their own classrooms in which students defend and
argue for mathematical ideas. In these examples, students build on the
thinking of their peers and the class works to come to consensus on the
meaning of important mathematical ideas. Models such as these are crit-
ical if we want to help teachers and researchers have a vision of what a
discourse community might look like in practice.
Additional research seeks to characterize the key components of a
discourse community. For example, in looking at how such a community
develops, Yackel and Cobb (1996) describe the importance of classroom
norms. In particular, they argue for the existence of sociomathematical
norms, norms that are specific to participating in discussions of mathe-
matics. Thus, while norms for justification and explanation might apply to
discourse in any subject matter, they argue that “what counts as an accept-
able mathematical explanation and justification is a sociomathematical208 MIRIAM GAMORAN SHERIN
norm” (p. 461). In their view, then, becoming a member of a mathematical
discourse community involves learning to talk about mathematics in ways
that are mathematically productive.
The Teacher’s Role in a Discourse Community
Considerable evidence shows that moving from teacher-directed
classrooms to more student-centered classrooms places complex demands
on teachers (Fennema & Nelson, 1997). First, teachers have a very
different role to play in student-centered classrooms than they do in tradi-
tional classrooms. In the past, teachers often relied on presenting facts and
procedures for students. Today, however, teachers are encouraged to move
away from this format of instruction and “telling” is seen as only one of
several ways in which teachers can communicate and interact with students
about mathematics (Chazan & Ball, 1999). As a result of this shift, teachers
need to develop a new sense of what it means to teach mathematics, and
of what it means to be an effective and successful mathematics teacher
(Smith, 1996).
Second, leading a discourse community requires that teachers develop
new understandings of content and pedagogy. For example, in studying
changes in her own mathematics teaching, Heaton (2000) found that it was
relatively easy for her to get students talking and sharing their ideas about
mathematics. However, it was quite another matter to understand, from the
teacher’s point of view, what to do with those ideas – where to go next,
when to pursue an unexpected digression, and when to head off a poten-
tial misconception. Heaton claims that she needed new understandings of
the mathematics that she was teaching in order to facilitate the discourse
effectively.
Despite these obstacles, developing a discourse community in one’s
classroom can be a powerful form of professional development. Specific-
ally, in a discourse community, it is not just the students who learn,
but the teacher who learns as well (e.g., Fennema et al., 1996; Hufferd-
Ackles, 1999; Schifter, 1998). And the fact that students are sharing
and explaining their ideas seems to be a key factor in this learning.
For example, previous research demonstrates that novel student ideas
prompted teachers to rethink their understandings of mathematics and the
pedagogical strategies that they use in teaching such ideas (Sherin, 1996).
Because of the critical role that teachers play in the implementation of
mathematics education reform, exploring ways to support teacher learning
is of great importance to the mathematics education community.A BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY 209
Supporting the Process and Content of Classroom Discourse
Before proceeding to a discussion of the research design, it is necessary
to elaborate on the tension that is the focus of this article – the diffi-
culty that teachers face in trying to use students’ ideas as the basis for
discussion while also ensuring that discussion is productive mathematic-
ally. This challenge can be characterized as a tension between supporting
the process of mathematical discourse on the one hand, and the content
of mathematical discourse on the other hand. The term process refers to
how the teacher and the students interact in discussions – who talks to
whom, when, and in what ways. An important component of the process
of discourse involves the expectations for participation. For example, are
students expected to share their ideas with their classmates? Is the norm
that all comments are to be directed to the teacher or to one’s classmates?
These questions concern the process of the classroom discourse.
The content of the discourse, in contrast, refers to the mathematical
substance of the ideas raised, to the depth and the complexity of these
ideas in terms of the mathematical concepts under consideration. Further-
more, the content of the discourse concerns how closely the ideas that are
raised in discussion are aligned with the teacher’s curricular goals and with
mathematics as it is understood by the mathematical community that exists
beyond the boundaries of the classroom.
A number of researchers discuss this tension (Ball, 1996; Jaworski,
1994; Nathan, Knuth Elliott, 1998; Schifter, 1998; Silver & M. S. Smith,
1996; Wood, Cobb & Yackel, 1991), and some make similar distinctions in
terminology. For example, Wood (1997) discusses the form and the content
of classroom discourse, where form refers to “knowing how to talk,” and
content refers to “knowing what to say” (p. 170). Similarly, Williams and
Baxter (1996) describe two types of scaffolding that teachers provide for
classroom discourse. First, teachers offer social scaffolding that helps to
establish and support classroom norms for how students should talk about
mathematics. Second, teachers provide analytic scaffolding for structuring
how and what mathematical ideas are discussed in class. Both Wood’s
form and Williams and Baxter’s social scaffolding are similar to what is
defined above as the process of discourse. Furthermore, Wood’s content
and William and Baxter’s analytic scaffolding are related to what this
research considers the content of discourse.
In discussing the tension between supporting the process and the
content of classroom discourse, some researchers suggest that teachers
manage this tension by first turning their attention to the discourse process,
and later, once classroom norms have been established, turning to issues
of the content of the discourse (Rittenhouse, 1998; Silver & Smith, 1996;210 MIRIAM GAMORAN SHERIN
Wood, 1999; Wood, Cobb & Yackel, 1991). This research illustrates a
somewhat different situation that occurred with David Louis. Though he
did lay a foundation of process in the first few weeks and then moved
onto to content issues, maintaining the integrity of both the process and
the content of the mathematical discourse was a continuing struggle.
Throughout the year, David moved back and forth in his emphases, always
struggling to balance what proved to be competing goals. The purpose of
this article is to characterize this struggle and to explain how and why
David ended up shifting his focus between the process and content of
discourse in his classroom.
CONTEXT AND RESULTS OF LARGER STUDY
This research took place in the context of the Fostering a community of
teachers as learners project (FCTL) (L. Shulman & J. Shulman, 1994).
The central goal of the FCTL project was to examine how middle and
high-school teachers from different subject areas might implement the
pedagogical reform outlined by Brown and Campione (1992, 1996) in
their Community of Learners (COL) research. In addition, the researchers
explored the design of professional development and teacher education
activities intended to support teachers’ efforts to implement the COL
pedagogy.
The Teacher
The teacher, David Louis, taught middle-school mathematics in an upper-
middle class suburb of the San Franciso Bay Area. During the 1995–
96 school year, he explored how specific COL participant structures and
principles might apply to a mathematics classroom. To do so, the teacher
designed and tested curriculum units that incorporated many of the COL
participant structures. For example, groups of students worked together to
become experts in a specific area and were then organized into “jigsaw
groups” comprised of experts in each of the different areas.
Context for the Study
In the summer of 1996, two researchers from the FCTL project (Edith
Prentice Mendez and the author) met with David to discuss his experience
thus far with the COL pedagogy. David explained that despite imple-
menting COL units, he did not believe that the COL principles had come
alive in his classroom and he had yet to feel that he had successfully
developed a community. Furthermore, David had come to believe that
encouraging students to talk about their ideas was the critical elementA BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY 211
in developing community in the mathematics classroom. Thus, for the
coming year, he planned to focus on developing a “mathematical discourse
community” rather than adhering strictly to what he thought was the struc-
ture of typical COL units. He imagined a classroom in which students
were “enthusiastic about sharing their ideas with their classmates” and
in which “students would comment on and critique each other’s ideas”
(Louis, 1997b, p. 4).
Data Collection
During the following school year, 1996–97, we observed and videotaped
in the teacher’s classroom, choosing one eighth-grade class as the focus of
the data collection. This class met four days a week. From September to
December, an average of three of the four classes were observed. And from
January through June, two classes a week were generally observed. In all,
78 classes were observed and videotaped throughout the school year. In
order to capture much of the discourse that took place in the classroom,
we used multiple microphones and an audio mixer. The teacher wore a
wireless lapel microphone, and two additional microphones were placed
around the room on students’ tables. The sound was then fed through
an audio mixer to the video camera. In addition, we made copies of all
assignments given in class and of all the overhead transparencies that were
used.
Field notes also were collected for the days observed. For all lessons,
a lesson-structure summary was created during the observation. This
summary listed the various activities that comprised the lesson on that day,
gave brief descriptions of each and the times at which each activity began.
In addition, for over 60% of the observations, more detailed notes were
taken during class. One focus of these notes was to track the mathema-
tical ideas that were discussed in class and to record how these ideas were
represented and by whom. The notes often contained snippets of transcripts
from class discussions. Similar notes were made for the rest of the lessons
using the videotaped data.
In addition to the classroom data, the teacher kept a written journal in
which he reflected on his teaching approximately three times a week from
September to December, and twice a month after that. David, Edie, and I
also met once a week to discuss what was happening in David’s class and
to watch video excerpts from the class. Furthermore, the teacher was inter-
viewed four times across the year. In these interviews, David discussed his
goals for the coming year, his impressions of the discourse that existed
in his classroom, and his perspective on what he and his students were
learning. The meetings and interviews were audiotaped and transcribed.212 MIRIAM GAMORAN SHERIN
Data Analysis
The research described in this article seeks to understand teaching by
looking closely at classroom interactions across one school year. In
general, the research reported is qualitative in nature, based on analysis
of videotape data and interviews with the teacher. Furthermore, the teacher
whose classroom is the focus of this study, David Louis, was a collaborator
throughout the project.
Analysis for this study focuses on class discussions. First, using video-
tapes and observation notes, those lessons in which a class discussion was
one of the primary activities of the day were identified. This included
a total of 68 lessons across the school year1 . Preliminary analysis then
involved coding these lessons on a coarse scale (high vs. low) for the
extent to which David focused on the process and the content of the
classroom discourse. Discussions in which his focus was rated high on
process were those in which David consistently elicited students’ ideas and
asked students to comment on each other’s ideas. In contrast, discussions
rated low on process were mainly teacher-centered with little room for
students to contribute their ideas. Discussions in which David’s focus was
rated high on content were those in which his comments were intended
to move the discussion along mathematically. For example, David might
compare and evaluate the mathematical substance of ideas that arose or
ask the students to do so, or he might direct their attention to a relevant
mathematical issue. Discussions rated low on content were those in which
David allowed extended discussion of non-mathematical ideas or ideas
which were only superficially mathematical.
Initially, one researcher coded the entire data set in this manner. A
second researcher then reviewed the coding of each lesson. Agreement
between the two researchers for each lesson was 91% and above. Cases
of disagreement were reviewed together until the researchers reached
consensus. Table 1 displays the results of this coding.
TABLE I
Distribution Across Lessons of David’s Focus on Process and Content
Low Process High Process
Low Content 28% (19 lessons)
High Content 15% (10 lessons) 57% (39 lessons)A BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY 213
A number of questions arose based on this preliminary analysis. First,
what happened in those lessons in which David apparently focused on both
process and content? Was he able to use the students’ ideas to discuss the
key mathematical concepts in the lesson? And what affected whether and
how this was achieved? In addition, why was it that at some times David
chose to focus on either process or content, but not on both? And how did
those lessons play out in class? Investigating these questions formed the
research study that is reported in this article.
STUDYING THE TENSION BETWEEN PROCESS AND
CONTENT
In order to investigate these issues, a subset of 20 lessons from across the
school year were selected for more detailed analysis. In general, one lesson
was selected every other week from September through May. Because of
various school holidays that occurred throughout the year, the 20 lessons
were comprised of two lessons per month from September through May,
with a third lesson included from the months of September and February.
No lessons were selected from the month of June because school ended
during the first week of that month.
Class discussions from these 20 lessons were transcribed and a fine-
grained analysis of video (Schoenfeld, Smith & Arcavi, 1993) was then
used to analyze the teacher’s role in these discussions. In particular,
based on prior research on the role of discourse in the mathematics class,
specific areas of discussion were identified to be the focus of the analysis
(Ball, 1991; Brown & Campione, 1994; Mendez, 1998; NCTM, 1991;
Silver, 1996). These areas included the questions raised by the teacher, the
teacher’s responses to students’ questions, the mathematical content intro-
duced by the students, and the mathematical content introduced during
discussion by the teacher. In addition, analysis examined the different
mathematical representations used during discussion. The results of this
analysis are described in the next section.
RESULTS
The tension between process and content in David’s classroom played out
at two time scales: (a) at a macro-level across the year, and (b) at a micro-
level, within class discussion in individual lessons. First at the macro-level,
David’s efforts to balance process and content across the school year are
discussed. These results draw from the coarse ratings of all 68 lessons as214 MIRIAM GAMORAN SHERIN
being high or low on process and content as well as from analysis of the 20
lessons selected for more detailed study. Following this, David’s efforts to
balance process and content within particular class discussions, the micro-
level, are examined. The focus here is exclusively on the analysis of the 20
selected lessons.
Process and Content at the Macro Level
Across the school year, David shifted his efforts between supporting the
process and the content of the discourse community that he desired (Figure
1). Initially, David’s interest was in process; the first seven lessons of the
year were coded as high on process and low on content. Two of those
lessons were the focus of detailed analysis. From the observer’s perspective
they reveal that during this time, David’s goal was to establish the struc-
ture for class discussion. In general, he did this by brainstorming with
the class about appropriate roles for the students during class discussions
and by experimenting with these roles in the context of non-mathematical
activities. Thus, students were explaining and comparing ideas, however
these ideas were not mathematical in nature. For example, on the first day
of class, groups of students worked together to make shapes with a loop
of yarn. The class then came together to discuss the activity and David
encouraged the students to comment on working as a group rather than on
the different shapes that students had been able to make and why2 :
D. Louis: How did it feel to [work in groups] today?
Jason: Fine.
D. Louis: Expand.
Jason: It was easier because when we had a problem, it was easier to
work through if you had someone to talk to about it.
D. Louis: What do other people think about what Jason said? Do you
agree or disagree?
Ben: I agree.
Julie: I agree too. Without group members you couldn’t hold the
corners [of the yarn].
In writing about this lesson in his journal, David was explicit that his goals
for the day were to “debrief with attention to questioning techniques,” and
to “comment on discussion skills.” Furthermore, he was not concerned that
“we didn’t discuss too much mathematics [today]” (Louis, 1997a, p.2).
Instead, David had chosen specifically for the start of the school year, to
focus on establishing the process of discourse in his classroom.A BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY 215
Figure 1. A sketch of David’s emphasis on process and content across the year.
After a few weeks, David was satisfied that the norms for discourse he
had envisioned were established and he was ready to add content to this
process. In an interview he explained:
[T]he students learned the protocol for talking to each other . . . and listening to ideas, and
they learned expectations for giving ideas . . . The basic skeleton of norms are there. Now
they [need to] move past that and talk much more about [mathematics], to use the protocol
that I’ve tried to establish to learn math.
David began to prompt the students to talk about mathematics and
the students responded accordingly. For example, in the following lesson,
students worked in groups to determine a method for estimating the
number of dots placed randomly in a 9 × 14 cm rectangle (Lappan, Fey,
Fitzgerald, Friel & Phillips, 1997). Several students then explained their
group’s method to the class:
Julie: We divided [it] up by one centimeter by one centimeter . . .
and then we’d have 126 little squares. So we counted [the
dots in one of] the little squares and there’d be about 17 little
dots in there. So then we multiplied 17 by 126.
D. Louis: Okay. What do people think about this group’s method?
Robert: I think it’s a good idea but bigger squares would have been
more accurate.
D. Louis: Why do you say that?
Robert: Because . . . there may be a bunch of dots packed into a small
area. In just that particular area. Or, there might be not a lot
of dots.
Amy: I agree . . . because there are not the same amount of dots in
the same place.
D. Louis: And why would that make a difference?216 MIRIAM GAMORAN SHERIN
As seen in the above excerpt, in discussing the activity as a class, David
focused on the mathematics of the problem. He asked students to comment
on and to compare the different groups’ methods. Furthermore, he encour-
aged Amy to explain why the two different methods would produce
different results.
Achieving balance. A balance had now been achieved with David
focusing on both the process and the content of the classroom discourse.
Students were asked to share their ideas and to comment on the ideas of
others, and they were expected to do so in the context of the mathematics
of the given activity. This balance lasted at the macro level for several
months.3 In fact, during the months of October, November, and December,
over 85% of the lessons were rated as high on both process and content.
David was pleased with the level of discourse that the class had achieved.
He wrote:
There are several interesting things happening here. First, the [discourse] norms are hard
at work. Students are building on each other’s knowledge and work. . . The second . . . is
the mathematics. I never would have expected to discuss [the mathematics] in such detail
and depth. (Louis, 1997b, p. 21)
Shifts from balance to process or content. Mid-year however, the
balance shifted. In January and February, over 50% of the 68 total lessons
were rated high on process but low on content, similar to the beginning of
the school year. This shift appears due to the fact that David had become
concerned with the level of justification that students offered in support
of their ideas and methods. In writing about his goals for the second half
of the school year, David stated that he wanted to improve the classroom
discourse by “focusing with students on what counts as justification for
a mathematical idea” (Louis, 1997b, p. 5). In discussing this goal in an
interview, David explained that he wondered if he had made it clear to the
students that “you shouldn’t let things by without a justification . . . and
[that] it’s the class’ responsibility to judge this.” He was also concerned at
times that “students would agree with each other, but without expressing a
different viewpoint than the one first given.” For example, a student would
respond by saying, “I agree because of the same reasons that Amy gave.”
With this in mind, David once again turned his focus to process
and encouraged students to take on new roles in the structure of class
discussions. In particular, students were expected to contribute to class
discussions not only by sharing their ideas, but also by providing the
reasoning behind those ideas and by judging whether their classmates
had given sufficient justification for an idea. For example, David explic-
itly discussed with students whether the statement, “I agree because
that’s what I got,” is a “good” mathematical argument. Other promptsA BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY 217
that David used included “How could you verify that Jin’s conclusion is
correct?” and “Does that make you more convinced?” In a sense, David
was attempting to renegotiate the classroom norms for participating in
discussion to include a sociomathematical norm for justification (Yackel &
Cobb, 1996). However, David’s emphasis on justification occurred partly
at the expense of the mathematical content of the lessons. He worked hard
to help students justify their ideas and was less concerned with the direc-
tion that the discourse took in terms of the mathematical concepts under
discussion. Thus, during this time, his main emphasis was on process, with
less attention given to content.
A final shift occurred late in the year when David began to teach
a unit on algebra. In contrast to lessons earlier in the year, during the
algebra unit over 85% of the coded lessons were rated low on process
and high on content. At the end of the year, David chose to emphasize the
content that he wished students to learn. Furthermore, David set aside the
pattern of discourse that had developed in his class, and relied on more
teacher-directed instructional techniques to introduce the class to algeb-
raic methods. For example, David wanted the students to create a table
showing how the price of a pizza depended on the base price plus the
cost per topping. Rather than asking students how they might represent
this information, David gave the class explicit instructions for making a
T-table and filling in the columns. David himself recognized this shift in
pedagogical style and wrote about it in his journal:
[Today’s] lesson was quite different than what I was used to . . . What happened today was
much more directed instruction than usually exists in my classroom . . . I was telling the
students what I wanted them to know about [algebra] . . . I just showed them what to do
and why to do it. I didn’t provide a forum for discussion about student ideas or check for
understanding via discourse. (Louis, 1997a, p. 39)
David’s reasons for focusing on content at this point in the year were two-
fold. First, David was influenced by his beliefs about the nature of algebra.
David explained to his class at the beginning of the algebra unit that he
believed algebra was a highly structured domain and that learning algebra
required a structured approach. Thus David believed that he needed to
sacrifice the open-endedness of the discourse in order to help students learn
a set of predetermined algebraic procedures. In his journal he claimed that
the shift in his pedagogical style was due to “the [math] that I wanted to
discuss today” (Louis, 1997a, p. 39). Second, David taught in a community
that was highly political and was in the midst of a controversy concerning
mathematics instruction. While there were many proponents of mathe-
matics reform in this community, support came mainly for reform at the
elementary and middle school levels. At the same time, a very vocal group218 MIRIAM GAMORAN SHERIN
of parents and teachers at the high school level argued for more emphasis
on skill and computation in order to help prepare students for high school
and college mathematics. As a result of this controversy, David was partic-
ularly sensitive about his teaching of algebra – a topic that is typically
taught at the high school level, or only in an honors class at the eighth
grade. He wrote, “I could not help but think, what if a parent were to view
this videotape?” (Louis, 1997c, p. 3).
Summary. This analysis shows that the tension between process and
content was not something that David easily resolved as the school year
progressed. It was not the case, as one might have imagined, that David
began the year struggling to find a balance between process and content,
but once a comfortable balance was reached, it was maintained for the rest
of the school year. On the contrary, Figure 1 illustrates that this dilemma
was ongoing throughout the year as David continued to shift his emphasis
between the process and the content of classroom discourse.
Process and Content at the Micro-level
Fine-grained analysis of 20 lessons from across the year show that the
tension between process and content existed not only at the macro level
as described above, but also arose within individual class discussions.
Specifically, in the context of a single discussion, David shifted his
emphasis between the process and the content of the discourse. Further-
more, it appears that moving back and forth in his emphasis at times
helped David to facilitate meaningful discussions about mathematics in
which students’ ideas were a key component of the discourse. How did
this occur?
Prior to the beginning of the school year, David identified three ques-
tions that he planned to use to guide his comments during class discussions.
Based in part on his viewing of a videotape of Deborah Ball teaching
mathematics to third grade students (Ball, 1989), David planned to ask the
following questions: 1) “What do people think about this idea?” 2) “Why?”
and 3) “What do other folks think about that?” David hoped that using
these questions repeatedly would encourage students to share their ideas
and to build on each other’s ideas. These questions represented a pattern of
discourse that was quite different from traditional classroom discourse in
which discussion followed a pattern of IRE – 1) Initiation by the teacher,
2) Reply from the student, followed by 3) an Evaluative comment from
the teacher (Mehan, 1979). Instead, David planned to respond to students’
comments with additional questions, either asking the student to elaborate
or for other students to comment on the idea.A BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY 219
As a result of using these three questions, a structure for class discus-
sions emerged. Specifically, many of the class discussions followed a
similar format involving three main components: (a) idea generation, (b)
comparison and evaluation, and (c) filtering (Figure 2). In the first part of
discussion, idea generation, David elicited ideas from students concerning
whatever topic was being discussed. He used the three questions extens-
ively to facilitate this initial brainstorming of ideas. David would elicit an
idea from a student by asking, “What do you think?” After the student
responded, David would ask for elaboration: “Why?” or “Can you explain
that?” David would then turn to the rest of the class and ask, “What do
other people think?” Following this trio of questions, David would cycle
back to the first question, “Okay. Other ideas on this?” and the cycle
continued. As can be seen from this description, ideas were not only gener-
ated, but were also preliminarily elaborated and evaluated by members of
the class. At this point in the discussion, David was not particularly worried
about taking control over the content that was being raised. Instead, as he
explained in an interview, he used the three questions at the beginning of
a discussion to “draw out kids’ ideas,” and to give the students a sense of
ownership over the discourse.
Figure 2. Components of class discussion.
Once several ideas had been raised, the class generally shifted into
a second phase of discussion: comparison and evaluation. The shift was
somewhat subtle. Rather than asking for one new idea and then another
new idea, and then another, David’s questions focused more on asking
students to consider one idea in light of another, “So, is what you’re saying
the same as Tina? What do you think?” Students’ comments also reflected
this shift. Students were less likely to introduce new ideas at this stage, and
were more likely to state whether they agreed or disagreed with a particular
idea that had been suggested.
The final structure was filtering. Here the class narrowed the space of
consideration and developed a plan to investigate a few ideas in detail.
Some ideas that had been raised were highlighted and pursued further,
while others were set aside for the moment. This occurred as David
focused the class overtly on two or three specific ideas. In addition, David220 MIRIAM GAMORAN SHERIN
introduced new mathematical content intended to help the class sort out the
issues under consideration. David’s emphasis here was on content issues.
He explained that he would “look for strategic, timely entries into the
conversation to push the mathematics to a higher level . . . to tie together,
or help make conclusions” (Louis, 1998, p. 5). The term filtering is used
to emphasize that any new content raised by the teacher is based on a
narrowing of ideas raised already by the students. Other researchers also
identify this seeding of ideas as an important component of mathematics
instruction. For example, Chazan & Ball (1999), argue that substantive
mathematical comments on the part of the teacher can be a valuable cata-
lyst for class discussions. Similarly, Wood (1994, 1995, 1997) talks of
teachers using a series of “focusing” questions that serve to direct students’
attention to the key elements of a particular solution strategy. Yet after
asking these focusing questions, the teacher did not take an active role
in discussing the ideas with the class. In contrast, during filtering, David
worked with the students to examine the narrow set of ideas that were now
under consideration.
The three components shown in Figure 2 appeared quite regularly
in class discussions. Of the 20 lessons selected for detailed analysis 19
involved idea generation, 16 involved comparison and evaluation, and
18 of the lessons included filtering. Furthermore, 15 of the 20 lessons
contained all three structures. Despite the frequency with which these
structures appeared, David’s class did not adhere rigidly to a prescribed
format for class discussions. On the contrary, class discussions proceeded
in a rather fluid manner. Although in general, the class progressed through
the three structures in the order presented here, it was not always the
case. In particular, a single discussion might involve cycling through
these components more than once, or repeating the first two components
a number of times before moving to filtering. Furthermore, further idea
generation or additional comparison and evaluation of ideas often followed
filtering.4
Taken together, the three components can be thought of as a framework
that highlights the ways in which different processes were used by the
teacher to make progress on content issues. As such, the framework is
particularly useful in exploring the tension that David faced in supporting
both the process and the content of classroom discourse. In particular, it is
possible to consider how control over each of the three processes shifted
among the teacher and the students and the affect that this shift had on the
mathematics that was discussed.A BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY 221
AN EXAMPLE FROM THE CLASSROOM
To examine this framework more closely, consider the following example
from the classroom. This example comes from a lesson that was coded
as high on both process and content. Thus, at the macro level, David
was trying to support the process and the content of the mathematical
discourse. However, at the micro level, David continued to shift between
these two goals throughout the discussion. In doing so, David was able to
draw out student ideas and to use these ideas to pursue what he believed
to be the mathematical content of the lesson. Far from being an anomaly,
this example is representative of many class discussions that took place
throughout the school year. In discussions such as these, the claim is
that David achieved an effective balance between his goals of supporting
student discourse and facilitating the learning of mathematical content.
Furthermore, examining David’s use of the discourse structures outlined
in the previous section helps to explain why this is the case.
Background on the slingshot lesson. The slingshot lesson took place
during a unit on functions in the second month of school. The lesson
lasted for two and a half class periods. An important goal of the unit was
for students to explore the relationship between changing quantities. This
lesson followed a format that was similar to several other lessons in the
unit. Students would first collect some data, they would then graph the
data, and finally they would write an equation to represent the relationship
involved. For example, the previous week, the students had measured the
changing height of the water level as one, two, and then three cubes were
added to a cup of water. To be clear, these students were not in a pre-
algebra or algebra class, and the goal of the unit was not the standard y =
mx + b material. Instead, the unit was intended to give students experience
exploring data, interpreting graphs, and writing simple linear equations.
During the slingshot lesson, small groups of students were given an
apparatus that resembled a slingshot. The apparatus, which consisted of a
rubber band strung between two nails, rested on the floor. Using the rubber
band, students were to measure the distance that a small ball made out
of tinfoil traveled along the floor after being released from the slingshot
(Figure 3). The groups were to begin by pulling the rubber band back one
centimeter and letting the ball go. They would then repeat the experiment
for two and three centimeters. Students were encouraged to take more
than one measurement for each of the three distances, and to average their
results.
Unlike the cubes in a cup lesson, here David did not expect the class to
produce uniform data. While he believed that in an ideal physical world,
increasing the stretch of the rubber band by a constant amount would result222 MIRIAM GAMORAN SHERIN
Figure 3. The slingshot apparatus.
in a constant increase in the distance the ball traveled, he recognized that
the classroom was not an ideal physical world.5 Thus David did not expect
the students’ data to exhibit a linear relationship perfectly. He explained
that the class “was entering the wide world of a data collection” where
“you never know what you’re going to get.” In particular, David believed
that it would not be a simple matter for the students to find an equation that
corresponded to their data.
Fire away: The slingshot lesson in action. On the first day of the
slingshot lesson, the students worked in groups to complete their data
collection. Following this, David held a brief discussion before the end
of the period. During this time, the students raised a number of ques-
tions regarding the procedures they had used for collecting their data: “We
couldn’t make the ball go straight,” and “Our rubber band broke so we
stapled it. Does that matter?” Before handing out the homework, David
encouraged the students to begin looking for patterns regarding how far
the ball traveled. He asked, “For every centimeter you pull it back, the ball
goes how far?” David explained that the class would pick up the discussion
on the following day.
David began class on the second day of the lesson by reviewing the
students’ homework. For homework, the students were asked to complete
a worksheet with six questions concerning the slingshot activity. The first
two questions, which are listed below, formed the basis for much of the
class discussion (Figure 4). This discussion is the focus of analysis.
Figure 4. The slingshot homework assignment.
The class quickly agreed that, in Patrice’s equation, y corresponded to
the distance that the ball traveled and x corresponded to the amount that
the rubber band was stretched. Furthermore, it was clear that in Patrice’sA BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY 223
case that for every centimeter that you pulled the rubber band back, the ball
traveled another 120 centimeters. David then asked, “Was it pretty accurate
to say that it’s about 120 centimeters?” In response, students introduced
a number of factors that they believed would affect whether or not the
ball traveled 120 centimeters. The following interaction is typical of the
conversation that took place:
D. Louis: What do you think?
Jeff: Depends on what floor it is.
D. Louis: Okay, depends upon what floor it was. Why do you say that?
Jeff: The more, the less, the less friction, the further it goes.
D. Louis: Okay, what do other people think?
The students recognized that their data did not demonstrate the constant
increase suggested by Patrice’s equation. Thus, they suggested other
factors as possible reasons for some variation within each group’s data.
Additional variables mentioned included human error and the fact that the
balls did not always travel in a straight line.
After a few minutes, Ben joined the conversation, raising an issue that
was somewhat different from the types of comments made up to this point.
Ben explained that while the factors that students had named already would
account for some of the variation the groups encountered in collecting their
data, there might also be another issue in play. Specifically, Ben wondered
if the increase in distance might not actually be constant. Another student,
Robert, then explained that if this were the case, graphing the data would
produce a curve rather than a line.
Ben: I also think it depends like on how far you pull it back.
D. Louis: What do you mean?
Ben: Like if you pull it back to the one centimeter, and you do
that like three times, like it might be 120 centimeters. But
then the first time that you pull it back it, say the second
one, it might be farther than 120 centimeters. It might just
keep going at a steady rate, but . . . it might be larger than
120 centimeters apart.
D. Louis: Does anyone understand what Ben is saying because I don’t
quite exactly understand . . .
Robert: I think he means that the graph might not be linear. If you
make a graph out of it, it might not go at a constant rate.
D. Louis: Is that what you’re saying?
Ben: Yeah.
D. Louis: What do other people think about that?224 MIRIAM GAMORAN SHERIN
As the conversation continued, Jeff responded in agreement with Ben,
“The change between zero cm and one cm will be less than the change
between one cm and two cm.” In contrast, Sam argued that the variation
was due to human error and was not because the difference in the distance
traveled was increasing. At this point, David highlighted these two issues
for the class:
D. Louis: So I hear people saying two things. One group of people [is]
saying that you pull back a certain amount, and then it will go
that much farther each cm you pull it back. So each time it
goes 120 centimeters farther . . . the same amount farther each
time. I hear another group of people saying that possibly, the
further you pull it back each time, it goes a little farther. So if
you pull it back the first time it goes 120, and you pull it back
the second time, or 2 cm back, it might go 140. You pull it 3
centimeters back, it might, well the first was 120, then 140, and
then maybe 160. So it goes a little farther each time you pull it
back. So what do you guys think about that idea?
To respond to David, students began to look at their data to see which
pattern fit most accurately. Soon David suggested that the class pursue
this issue using the graphing calculator. David introduced the notion of a
“scatter plot” as a graph whose values do not make a perfectly straight line.
With the students’ help, David entered one group’s data into a graphing
calculator that worked with the overhead projector. David selected the
scatter plot function so that the data was now displayed in view of the
entire class. The students discussed how to visually estimate which one
line would most accurately represent the data. In addition, they used the
graphing calculator to determine a “line of best fit.” In this way, the class
was beginning to deal with different ways to interpret the complex set of
data that had been collected. In fact, students began to offer a number of
different ideas about why the notion of scatter plot was useful for them,
and how they could determine whether one of their own estimates was a
line of best fit.
Analysis of the slingshot lesson: A filtering process. In this example,
David achieves multiple goals. In particular, he achieves a balance between
process and content by first allowing a great deal of open-ended discourse,
and then by focusing the discussion himself, and thus taking more control
of the content. The beginning phase of the discussion is a typical example
of idea generation. David uses the three questions to draw out students’
ideas and to keep the conversation moving. The content of the discussion
is clearly in the students’ hands at this point, as they are the ones suggesting
which factors affect the data.A BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY 225
After students raised several ideas concerning why the distance might
not consistently be 120 centimeters, there is evidence that the class
shifts into the comparison and evaluation structure. In particular, Ben’s
comments indicate that he has classified the ideas raised thus far as being
about physical factors that affect data collection. In contrast, Ben had a
different kind of argument to make. Following Ben’s comment, David
encouraged further comparison among the students’ ideas including eval-
uation of Ben’s proposal. Thus, in this phase of discussion, the students
and David appear to share responsibility for the content of the lesson – yet
open-ended discourse is still a prominent feature of the discourse.
The beginning of the third phase, filtering, is much more obvious. Here,
David shifts his position in the conversation somewhat and brings the
students’ attention to two particular ideas. In addition, he seeds the ensuing
discussion with the notion of a scatter plot and of finding a line of best fit.
For a time then, David has taken control of the content of the conversation,
and has narrowed the space of ideas being raised and discussed. Open-
ended discourse is not closed off completely, in fact David often asks
for student input to explain the ideas he is presenting. However, this part
of discussion resembles teacher-directed discourse more than that which
occurred earlier.
The class then uses this filtering by David to redirect their attention, and
return again to idea generation. Specifically, they began to discuss what a
“line of best fit” would look like (e.g., “There must be the same number
of data points above and below the line.” “Should some data points pass
through the line?”) It is important to note that considering how to interpret
a scatter plot and how to determine the features of a line of best fit, consti-
tute significant mathematical content for these students. In the past, they
had explored data intended to represent linear functions more precisely –
the difference between data values was often consistently the same. Here
the students were dealing with a very different set of data and they needed
a new set of mathematical tools to do so. The combination of the graphing
calculator with the notion, not of a line that fit perfectly, but rather of a line
of best fit, had the potential to help them explore these issues productively.
Examining the flow of ideas in the class, a pattern is evident. First, in
terms of the process of mathematical discourse, many ideas are encouraged
early on, a few are chosen for more focused attention, and then the class
returns to soliciting many ideas. This is a view of the process of mathe-
matical discourse because it describes how and when ideas are solicited.
Furthermore, this particular process involves a great deal of open-ended
discourse in which students are encouraged to have control of the ideas
being raised.226 MIRIAM GAMORAN SHERIN
Figure 5. A representation of the process of the classroom discourse.
Second, this approach serves a very different purpose for the content of
mathematical discourse. Each time that the teacher narrows the scope of
ideas that are considered during filtering, he takes control of some of the
mathematics that is discussed. And even though many ideas are then gener-
ated about this filtered topic, the mathematical content has nevertheless
been redirected and narrowed. As this cycle is repeated, the mathematical
content of the lesson moves from a broad arena to one that is more focused.
The initial question or topic that the teacher raises is certainly an important
factor in determining the direction of the content of the discussion. Yet
in addition, the filtering process allows the teacher continually to refocus
the content of discussion in areas that he or she feels are mathematically
significant and that will be productive for the class to pursue (Figure 6).
Figure 6. A representation of the space of mathematical content.A BALANCING ACT: DEVELOPING A DISCOURSE COMMUNITY 227
Taken together, these two perspectives demonstrate how David was able
to balance the process and content of mathematical discourse in conversa-
tions such as the one discussed here. Furthermore, this example illustrates
that this balance was achieved in part because David shifted his emphasis
between process and content in the context of the discussion. Thus, rather
than hindering his goals, at times, the ongoing tension between process and
content was an important factor in enabling David to facilitate classroom
discourse successfully.
DISCUSSION AND IMPLICATIONS
Teaching with open-ended discourse poses a problem for the learning of
content. On the one hand, students are expected to learn specific content,
but on the other hand, students’ ideas are supposed to direct the discus-
sion. How do teachers respond to the need to support both the process
and the content of classroom discourse? Under what circumstances are
they able to manage both of these goals simultaneously? Based on the
analysis presented in this article, two issues are proposed as being at the
core of teachers’ efforts to meet these competing demands. For each issue,
both theoretical implications and considerations for teacher education are
discussed.
New Structures for Classroom Discourse
First, the teacher does find ways to structure class discussion in order to
support both the process and the content of classroom discourse. Specific-
ally, a filtering approach involving a combination of three discourse
processes is used to make progress on content issues. In this approach,
multiple ideas are solicited from the students in the initial phase. Students
are encouraged to elaborate their thinking, and then to compare and eval-
uate their ideas with those that have already been suggested. The filtering
part of the discussion comes next, as the teacher focuses the students’
attention on a subset of the ideas that have been raised. In addition, the
teacher may introduce a new mathematical idea or approach that the class
can use to consider the focused content. This focusing on the part of
the teacher is then followed by additional idea generation on the part of
the students. A single class discussion may involve several cycles of this
pattern.
This filtering approach can serve both process and content goals. In
terms of process, the students have a great deal of opportunity to share
their thinking and the teacher’s “filtering of ideas” is based on the ideasYou can also read
