SYMBOLS AND MEANINGS IN
TEACHER-STUDENT INTERACTION DURING MATHEMATICAL PROBLEM SOLVING
Olive Chapman
University of Calgary
Teacher-intervention during problem-solving
instruction was investigated for 6 inservice elementary teachers. The study was
framed in the perspective of symbolic interaction, i.e., a focus on the
teachers’ personal meaning and supporting symbolic systems. The findings
indicated that the teachers constructed a symbolic system to guide their
interactions with students in a way that emphasized an autonomous role for the
students during problem solving. Their approach to intervention consisted of a
5-action sequence associated with particular symbols and embodied a process of
separation and connection between teacher, student, and/or problem. The
findings suggest the importance of helping teachers to understand such patterns
in their own teaching as a way of facilitating changes to it.
In the current reform movement in mathematics
education, problem solving is being emphasized as a basis of mathematics
thinking and a basis of learning mathematics (e.g., NCTM 1989). Implicit in
this perspective is that problem solving is an open-ended process that requires
flexibility in the thinking and behaviors of both teachers and students. This
view of problem solving forms the basis of this paper. The paper reports on a
study of teacher-student interaction during mathematical problem-solving
instruction from the perspective of symbolic interaction, i.e., a focus on
teachers’ personal meanings and symbolic systems that formed a basis for how
they conceptualized and facilitated students’ problem-solving behaviors.
There is a large body of literature on problem
solving, but very little deals with problem solving instruction from the
teacher’s perspective. Historically, a number of differing viewpoints regarding
instruction on problem solving have been proposed, the most common of these are
based on Polya’s four-stage model (Polya, 1957). Charles and Lester (1982)
identified two contexts in which teacher behavior during problem solving has
been described in the literature. One context dealt with the types of teacher
behavior that should be used at each of Polya’s stages. The other considered
teacher behaviors in terms of 10 teaching actions grouped into 3 distinct time
periods that make up a problem-solving session: before, during and after the
problem is solved. While these prescribed behaviors form useful guidelines to
problem-solving instruction, they ignore the humanistic aspect of classroom
interaction, e.g., they involve surface or observable behaviors in isolation of
the teachers’ thinking. Since it is the teacher who must interpret such
guidelines in the context of real classrooms, then her/his perspective becomes
a necessary lens through which to understand classroom behaviors during problem
solving as opposed to the prescribed behaviors by themselves.
Symbolic Interaction
Symbolic interaction was
used as the basis to interpret teacher-student interaction from the teacher’s
perspective. The central idea of symbolic interaction (Blumer, 1969) is that
human interactions are carried on through the medium of symbols and their
meanings. Reality is not disclosed directly, but is experienced through symbols
and activities mediated by language and culture. As Blumer (1978) explained:
Human beings interpret or “define” each other’s
actions instead of merely reacting to each other’s actions. Their “response” is
not made directly to the actions of one another but instead is based on the
meaning which they attach to some actions. Thus, human interaction is mediated
by the use of symbols, by interpretation, or by ascertaining the meaning of one
another’s action. This mediation is equivalent to inserting a process of
interpretation between stimulus and response in the case of human behavior [p.
97].
In relating symbolic interaction to the
classroom, the assumption is that teachers and students rely on symbols,
whether consciously or not, both to create and “read” the learning environment.
Teachers and students do not typically respond directly to each other’s actions
as stimuli, but assign meanings to the actions and act on the basis of the
meanings. Such meanings are socially derived through interaction with others
rather than inherent in the actions themselves or idiosyncratically assigned by
the teacher or student. Thus, when viewed through symbolic interaction, the
mathematics classroom can be seen as interweaving symbols and signification
systems that students and teachers use, whether consciously or not, as texts of
mathematics learning and teaching. These symbols emerge from a variety of
situations in the classroom. During a lesson, who can talk, when, how, and
about what are examples of symbols. Other examples are: the way a teacher
structures a lesson (e.g., focus on drill); the way the teacher uses time
(e.g., time spent on a particular concept) and space (e.g., arrangement of
desks); the location of the teacher (e.g., circulating among students). These
symbols convey what should be valued in the mathematics classroom and about
mathematics.
Research Process
This study is based on a larger project that
investigated the effect of a humanistic approach to teacher development as a
basis for facilitating change in teachers’ thinking and teaching of problem
solving (Chapman, 1999). The approach, the problem solving inservice [PSI]
program, was found to be effective in allowing the participants to make
significant shifts in their thinking and teaching that were consistent with the
reform perspective of teaching mathematics in terms of recognizing the active,
social, and constructive nature of the learning process. This study is an
analysis of the participants’ teaching after the PSI program with a focus on
teacher-student interaction during students’ problem solving.
The
participants were six inservice elementary school teachers (Grades 3 to 6) who
volunteered for the study. They participated in the inservice program over a
4-week period during their summer break. Prior to the PSI, they had little or
no experience solving non-routine problems as learners. The PSI activities
involved non-routine problem solving, role-play, and narrative reflection of
personal meaning of past, present, and future experiences with mathematical
problem solving.
Data included
transcripts of open-ended interviews on the teachers’ thinking and teaching of
problem solving and all oral aspects of the PSI activities (e.g., group
discussions and narrative reflections). Copies of all written work during the
PSI program (e.g., solutions of problems, journals of individual reflections,
summaries of group discussions) were also obtained. The teachers were observed
in their classrooms while conducting lessons involving problem solving, and teacher-student
verbal interactions during these lessons were audio-taped and transcribed.
An interpretative research approach (Creswell,
1998) was used to determine meanings associated with the teachers’ actions. The
data were scrutinized to identify recurring themes of how the teachers viewed
and practiced intervention. Themes from interviews were triangulated with
themes from the teachers’ group discussions and their actual classroom
discourse to determine the final set of themes. The symbols and descriptors
(e.g., separation and connection) to reflect the essence they embodied for the
teachers were deduced from the data based on these themes. The analysis built
on the findings of the PSI study in terms of the personal meaning the
participants constructed during and after the PSI experience as a basis for
interpreting the themes identified in relation to their symbol systems.
Teacher
Intervention
The outcome is
presented here only in terms of when and how the teachers intervened and their
bases for intervention. In addition, teacher intervention is considered only in
terms of what the teachers did when and after a problem was assigned to
students to solve. The teachers’ behaviors and thinking reflected one general
pattern of intervention consistent with the knowledge they constructed during
the PSI program. The general theme of this knowledge was that intervention
should be both passive and active. During passive intervention, the teacher
should only listen to the students to become aware of their thinking and to
give them time to think on their own. Active intervention, however, required
that the teacher communicate with the students, not to tell them how to get the
answer, but to stimulate their thinking to get beyond obstacles and to make
sense of their processes. Based on their PSI experience, the teachers selected
a set of terms they felt specified the essence of the problem-solving
experience that was relevant to the classroom context. Although this was done
individually after the PSI experience, for the most part, the teachers selected
the same terms, e.g., obstacle/barrier, stuck, off-track, challenge, make
sense, interpretation/meaning, strategy, listening to. These terms became
symbols that helped to facilitate mutual interpretation of when and how the teacher
should intervene in order to create a learning environment that allowed the
students to be more autonomous during problem solving.
During the PSI program, the teachers had
constructed for themselves particular meanings for the terms. Of particular
importance were stuck, off-track and lost, all of which were considered to be
important indicators for active intervention. The teachers considered stuck to
be when students tried everything they could by themselves and were about to
become frustrated. Here the student should initiate the intervention and the
teacher should intervene by asking open-ended questions and/or make an
open-ended suggestion of something to try. Off-track was considered to be when
students were doing something incorrect based on how they interpreted the problem or on the strategy they were using to solve the problem.
Here the teacher should initiate the intervention and intervene by asking
open-ended questions or making an open-ended comment. Lost was considered to be
when students were confused and disoriented, lost control of the problem, and
could no longer make sense of the problem or any help provided. Here the
teacher should take control of the situation in order to re-orient them to a
specific solution. This could involve the teacher explaining the problem and a
possible solution directly or with the help of students who were able to do it.
Another common theme that emerged from the
teachers’ behaviors and thinking was that intervention was a process of
separation and connection. Separation involved a form of decontextualization in
which the teacher consciously removed herself from the student’s experiences
with the problem or from her experience with the problem. Connection involved a
form of contextualization in which the teacher participated in the student’s
experiences with the problem or relived her own experience with the problem.
The teachers conceptualized the problem-solving lesson in 3 stages for a
problem: problem presentation, problem solution, and solution sharing. A
summary of the pattern of intervention, connection and separation for each
stage follows. Only common patterns are discussed instead of individual
situations/differences.
Problem-Presentation Stage: The dominant goal of this
stage was to let students own the problem. As Susan explained:
I am more
aware of things like [the student] owning the problem, like is this really a
problem for me [the student] or is this just something that I have to do to
make it through the next 20 minutes.
In order to own the problem,
students were required to interpret if for themselves. For example,
[My]
focus the whole year was on meaning and thinking through things. ... I give
them a problem, which is written, they read it themselves, then I ask for their
interpretations.
There were variations in how this stage evolved. For example, Mary led
her Grade 3's in a large group discussion while Pam had her Grade 3's work,
first, individually, then in partners to arrive at their own meaning of the
problem. Rose told her Grade 5’s, “Tell what you think the problem means to
you.” Most of the teachers required that the students write their
interpretations.
In this stage, the
teachers became detached from the problem in order to allow students to connect
to it. Thus the teachers did not intervene with any predetermined
interpretation of the problem. The teachers, for a few minutes, also became
detached from the students in order to allow them to connect to the problem.
They only reminded students of their task, e.g., write your meaning, discuss
your meaning with a partner. Finally, the teachers helped the students to
connect with each other and the problem as they shared their interpretations.
For example, the teachers would ask for volunteers to respond to any queries
raised and remind students to listen to each other’s interpretations. Most of
the teachers chose passive intervention at this point. They listened to the
interpretations but did not try to make corrections.
Problem-Solving Stage: The dominant goal in this stage was for
students to become decision-makers in deciding on a strategy and testing it.
The teachers continued to be detached from the problem to now allow students to
develop and work on a strategy. However, they became connected to the students
and the learning environment by circulating and constantly interacting with the
students either passively or actively. During passive intervention the teachers
tried to see what the students saw, thus trying to connect with the students’
perspective. During active intervention, they focused on when students were
stuck, off-track, or lost. For stuck and off-track, Pam, for example, first
intervened with questions like, "What have you tried?" "Why did
you add?" "What part of the problem asked you to do that?"
"What else do you think you can try?" "Why don't you try drawing
a picture or using a chart?" "Is that what you really want to
do?" She gave them time to resolve difficulties on their own. For lost,
she provided more direct guidance by telling them what was wrong or how to get
started. But in general, she allowed them to arrive at a solution in their own
way even if incorrect.
Solution-Sharing Stage: The teachers remained detached from the problem
in order to allow students to share and justify their solutions. Only if the teachers’
solution was very different from the students’ that it got presented as an
alternative and not the solution. The
teachers intervened to encourage reflection and discussion of the solutions in
a variety of similar ways. For example, Mary asked questions like: "What
do you think of ....?" "Which of the answers do you think is/are
correct and why?" "Why does it make sense?" Pam asked questions
like: "What do you think about their method?" "Does it make sense?"
"What doesn't make sense?" "How can they fix it?" Students
were also encouraged to talk about what they thought about the problem, e.g.,
what they liked or did not like about it. In general, then, intervention during
this stage was not simply to check solutions but to connect students, teacher,
and problem in meaningful ways.
Discussion
In this study, the symbols the teachers used to
frame their interventions (e.g., stuck, off-track, lost) were triggered by
particular actions of the students in relation to getting to a solution for a
problem. These actions were students’ questions, oral and written responses,
and physical expressions. They provided cues for the teacher about when
students were successful, stuck, off-track, and lost. These cues were dependent
on the teacher’s judgement or personal meaning. For e.g., what was considered
stuck for one student could be considered lost for another based on how the
teacher perceived the student’s ability to solve problems. Thus the teacher’s
personal meaning of the cues and not the cues in themselves guided
intervention. This interpretation often evolved from communicating with the
students. When considered necessary, the teachers were able to suspend their
interpretations until after communicating with students to make sense of the
context embodying the cues. For e.g., a student was perceived to be off-track
only after the teacher got a sense of what the student was trying to do after
questioning him or her. The goal of this communicating was to listen to the students as opposed to listening for specific behaviors.
In general, intervention involved a sequence of
related teacher behaviors: awareness of students’ actions, identification of
cues in the actions, communication with students about context embodying cues,
relating cues to symbol, enacting symbol. Intervention also involved a process
of separation and connection with students and/or problem. Separation appeared
in the teachers’ behaviors as a distancing from, a low level of awareness of,
or a decontextualizing of an experience, while connection appeared as the
opposite. For the most part, during teacher-student interaction, the teacher
tried to separate from the problem while the students were encouraged to stay
connected to it. The teacher also tried to stay connected to the students and
thus to the problem through the students. Each teacher’s commitment to the
students’ interpretation and solution of the problem was the focus of when and
how separation and connection with students’ experiences occurred. The teacher
[T], students [S] and problem [P] existed as a triad of two-way relationships
in which the teacher recognized a T-P connection (i.e., T specified P and P
specified T), a S-P connection (i.e., S specified P and P specified S) and a
T-S connection (i.e., T specified S and S specified T) i.e.:
T
S P
The specifying between two entities refers to how each allows the other
to perceive it/him/her (Merleau-Ponty, 1962). For e.g., in the T-P and S-P
situations, the specifying is dependent on the problem providing opportunities
for alternative solutions and the teacher and students being able to see one or
more of them.
Conclusion
The pattern of intervention discussed in this
paper involved a sequence of 5 related teacher behaviors associated with
particular symbols for teacher-student interaction. This pattern also embodied
a process of separation and connection between teacher, students and/or problem
that characterized their interactions. Helping teachers to understand such
patterns in their own teaching could be useful in facilitating changes in their
teaching. In particular, teachers would need to understand the symbols and
meanings they use that influence intervention and what alternative symbols they
could adopt to change their approach to it.
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