Revealing Discourse in PBL Mathematics Classroom · a part in that instruction. Secondary school...
Transcript of Revealing Discourse in PBL Mathematics Classroom · a part in that instruction. Secondary school...
Dialogue in a PBL Classroom 1
Revealing Dialogue in a Problem-Based Learning Mathematics Classroom:
The Perspective of a Pedagogy of Feminist Relation
Carmel Schettino
University at Albany/SUNY
Author Note
Carmel Schettino, Department of Educational Theory and Practice, University at
Albany/SUNY
I would like to acknowledge with gratitude the teachers and students who
participated in this study without whom this work could not have been possible. I would
also like to express my appreciation to Alan Oliveira of the University at Albany for his
guidance and suggestions with this manuscript.
Dialogue in a PBL Classroom 2
Abstract
In this paper, I explore the discursive nature of Problem-Based Learning (PBL) in a
secondary mathematics classroom situated in a Pedagogy of Feminist Relation (PFR).
Because the pedagogical approach calls for an intersection of discussion-based and
student-centered teaching techniques, dialogue is an important aspect of instruction. The
values of PFR influence classroom practice and hence, it is worth asking how these
values are manifested in the classroom dialogue and in what ways they are revealed to the
classroom community through the discourse. In this study, transcription of dialogue from
two integrated algebra and geometry mathematics classes were coded for evidence of
teacher dialogical techniques, pronominal use, and signs of politeness that dissolved
hierarchical structures of authority, empowered student agency, and encouraged student
voice. Findings indicated that classroom characteristics were consistent with the
theoretical framework they purport to follow.
Dialogue in a PBL Classroom 3
Since the National Council of Teachers of Mathematics published the new
Principles and Standards for School Mathematics (NCTM, 2000) there has been a lot of
interest in the new process standard regarding communication in the classroom. This
standard asked teachers to “help students use oral communication to learn and to share
mathematics by creating a climate in which all students feel safe in venturing comments,
conjectures, and explanations”(NCTM, 2000). Previously, the importance of classroom
communication in mathematics was not explicitly stated. However, now mathematical
discussion in the classroom is necessary for improvement of 21st century skills, and
teachers need to help students learn to defend their opinions and utilize incorrect answers
as conversation starters to explore misunderstandings. The question of safety and risk-
taking for students, however, is one that has been ever-present in the traditional
mathematics classroom even before communication became a focus for teachers to try to
encourage.
Some studies found safety and equity in mathematics classes especially an issue
for underrepresented groups like females and racial or ethnic minorities, or those with
lower ability level (Boaler, 2008; Gilbert & Gilbert, 2002; Kellermeier, 1996). So it was
even more revolutionary for the NCTM to include the “Equity Principle” in their 2000
publication which states that “excellence in mathematics education requires equity – high
expectations and strong support for all students”(NCTM, 2000). This helped propel the
already forward-moving gender equity and social justice movements in mathematics
education even farther ahead into the 21st century. Research is coupling these two ideas
with how discussion and more relationally-based teaching methods are often preferred by
and improves learning for marginalized groups in mathematics classrooms (Boaler, 1997;
Dialogue in a PBL Classroom 4
Ladson-Billings, 1995; Lubienski, 2000; Mau & Leitze, 2001). Because both
communication and equity have been recommended as priorities in mathematics
education, it would behoove our community to take a closer look at the connections
between implementing instructional methods that utilize discussion and how equity plays
a part in that instruction. Secondary school mathematics teachers remain challenged by
the idea of creating a classroom climate that serves the needs of diverse learners and
addresses the issue of communication skills as a priority in mathematics education.
Research Questions
Because a discursive mathematics classroom is such a novel concept to many
teachers, the call to standards of communication and equity raise many interesting
research questions. How can classroom practitioners know if a classroom that claims to
create a safe environment for students to take risks and “venture conjectures” is actually
fulfilling its claim? What ways of talking can help fulfill the ideals of the type of
pedagogical practice that supports this type of classroom climate that engages the
marginalized groups in need of support for equity? In this study, I plan to describe such a
pedagogical practice and address the question of the role that dialogue plays in that
classroom practice to foster empowerment of student agency. This study is framed by the
following research questions:
1) In what ways does a teacher who claims to attempt to empower students and
create equity in the learning process use discursive methods to attain that
goal?
2) Specifically, what aspects of classroom discourse define a Problem-Based
Learning mathematics classroom situated in a pedagogy of feminist relation?
Dialogue in a PBL Classroom 5
Literature Review
As these questions became part of the research arena, the postmodern view of
mathematics education, influenced by feminist and critical pedagogical theories, took
shape moving constructivism and student-centered teaching into new arenas. The
concern for creating more equity in the mathematics classroom spurred a range of
theoretical and research writings on Feminist Mathematics Pedagogy (FMP) from the
mid 1990’s. A review of the recent literature found major themes of a feminist
mathematics classroom of collective and individual empowerment, ownership and
authorship of material, dissolution of hierarchy in the classroom community and a
movement to work for social change (Anderson, 2005; Jacobs, 1997; Meece & Jones,
1996; Solar, 1995). It is clear that the intersections between feminist pedagogies and
constructivist and student-centered ideologies are many (Meece & Jones, 1996;
Noddings, 1993; Spielman, 2008). More importantly, FMP emphasizes “connected” and
relational learning that many females desire in the classroom experience and are missing
in other pedagogical approaches (Becker, 1995; Maher & Thompson Tetreault, 2001;
Zohar, 2006). The valuing of emotion, risk-taking, belonging and prior mathematical and
personal experience are all parts of the facets of FMP that allow students to gain voice
through self-representation in the classroom. The goal through this pedagogical approach
is to support not only females, but also other underrepresented or marginalized groups in
need of voice in the mathematics classroom with a student-centered dialogue that seeks to
dissolve the traditional hierarchy that is generally present in a mathematics classroom.
Some research has found that with a focus on and commitment to respectful
learning and discourse in the classroom, mathematical achievement can improve across
Dialogue in a PBL Classroom 6
gender, race and low socio-economic status (Boaler, 2008). From a feminist perspective,
belonging and becoming, in terms of ‘learning in community’ are key agents in an
individual’s practice in that community (Griffiths, 2008). In other words, how one enters
that community of practice, helps not only define who they are individually, but it also
defines the practice of that community. Using a FMP and focusing on the respectful
learning sets the tone for individuals to be who they are and to support one another as a
community of learners.
However, this pedagogical approach can only be successful when accompanied
by a curriculum and instructional practice that also supports the ideology of the theory
behind it. In my classroom practice, I have been lucky enough to find such a curriculum,
which seemed to integrate many of the desired outcomes of both of these
recommendations. Problem-Based Learning (PBL) is a teacher facilitated approach to
learning where complex problems are discussed by students using their prior knowledge
and enabling problem solving skills (Hmelo-Silver, 2004). This extremely student-
centered approach relies greatly on discursive practice that is generated by student
solution presentation. The discussion is often student directed, but the teacher always has
the broad goals of the problem in mind, at least when PBL is used at the secondary level.
Because the PBL method requires students to eventually become more and more
responsible for their own learning, the teacher’s scaffolding of the learning and discourse
fades as students become more expert in their discourse strategies and capability to move
forward in discussion (Hmelo-Silver & Barrows, 2006). In many ways, this type of
instructional approach is a model of cognitive apprenticeship, as the teacher is constantly
modeling problem-solving, conjecturing and risk-taking, while coaching the student
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learning (Hmelo-Silver, 2004). Although some research has been done about the
effectiveness of PBL in teaching problem-solving skills and self-regulation, (Savery,
2006), it is true that much more research needs to be done, especially at the secondary
level (Strobel & van Barneveld, 2009). Most recently, research has shown that problem-
centered approaches improve both achievement and attitudes of students regardless of
perceived ability level (Ridlon, 2009).
At the same time, it is clear that a PBL teacher needs to be a facilitator of
discussion and utilize strategies that allow for student learning in this complex situation.
Some master PBL facilitators have been found to utilize strategies such as (a) probing
students for deep explanations (b) using open-ended metacognitive questions (c)
revoicing and (d) summarizing (Hmelo-Silver & Barrows, 2006). The discourse in a
PBL classroom has been found to be very different from the typical teacher-directed
instruction and more than half of the questions are generally student-initiated stemming
from the ideas under investigation (Hmelo-Silver & Barrows, 2008). In my experience, a
PBL classroom can be run in many different ways but to foster the values of the equity
and social justice, if that were one’s goal in the classroom, this must be done deliberately
and with a pedagogical philosophy in mind. One instructional method known as
Dialogue, Participation and Experience (DPE) (Chow, Fleck, Fan, Joseph, & Lyter,
2003), states facilitator strategies such as voicing student views, placing learners on
center stage, focusing on interdependency and reducing frustration to diffuse tension as
means to help create a classroom climate that helps students participate in active
dialogue. Orchestrating that dialogue is a major component of the role the instructor
plays in the classroom community, making sure to “build on student’s ideas…and
Dialogue in a PBL Classroom 8
develop an understanding of increasingly powerful perceptual operations that
underscore” the students’ thinking and construction of knowledge (Ridlon, 2009). The
instructor must also be cognizant of student engagement at all times and the social
interactions that are occurring that are allowing (or not allowing) mathematical meaning
to be made in the community. Situating PBL within the context of FMP has done just
that, in my experience, and allows students the inclusive and relational learning
environment within which a diverse group of learners can learn and thrive.
Theoretical Framework
To frame this exploration, I have chosen to combine the theories of Feminist
Mathematics Pedagogy with a recent theory of education based on the interhuman
connectedness of learning, a pedagogy of relation. I will discuss the conceptual themes
that stem from both of these theories namely relational authority, relational equity, the
dissolution of classroom hierarchy, and other feminist values. This introduction will lead
to a discussion of the main goals of the pedagogical theories including empowering
student voice and agency in learning, as well as authentic participation and inclusion of
all members of the community of practice.
A Pedagogy of Relation
This study places mathematical discourse in a setting where learning is part of a
greater relational approach to knowing – where “knowers are social beings-in-relation-to-
others”, and these relationships must be built on respect and care, not oppression and
power (Thayer-Bacon, 2004). According to this view, education has a relational
character and it is just that relationship between the teacher and the student, and even
possibly the student and her classmates, that affords the community the opportunity for
Dialogue in a PBL Classroom 9
the interaction of education (Biesta, 2004). The communication in these interactions
between individuals is not about the transport of meaning but about the participation in
and co-construction of meaning between individuals and those members of the
community in relationship to each other which in turn allows “education [to] exist only in
and through the communicative interaction between the teacher and the learner” (Biesta,
2004, p.21). This relational view could also be expanded to be seen in the collaborative
learning experience between learner and learner. This statement places high priority on
the communication skills and interaction between the members of the classroom
community of practice as well as the ability of those members to feel comfortable in
those relations.
Relational Authority and Relational Equity. There are many types of authority
to consider in classroom discourse – expert authority, legal authority, traditional
authority, charismatic authority (Amit & Fried, 2005). In all of these types of authority,
it is described as something that one single person holds and possesses. Although many
authors describe the concept of “sharing” authority, it is difficult to get away from the
concept of authority being held by one person who is the sole leader and wielder of the
“influence over another” (Bingham, 2004). Gadamer’s philosophy of authority is
elaborated on here:
For authority to succeed in its aim of educating the student, the student must
acknowledge that there is an important insight to be gained from the teacher.
The student has an active role of authorizing the teacher by following the
teacher’s pedagogical lead. To learn thus entails the authorization of the
teacher by the student. (Bingham, 2004, p.31)
This concept of relational authority is at the heart of a pedagogy of relation. If education
happens relationally in the interactions between individuals in the community of learning,
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then there must be an acceptance that all members of the community have authorized the
learning to take place. It is that respectful and reflexive relation that allows for the
opportunities to arise for education to happen. Connected to this construct of authority is
a similar view of equity. The term relational equity in the classroom (Boaler, 2008) has
been used to describe classroom relations between students, and I would extend that to
teachers and students, where respect for others’ ideas is held as priority, as is treating
different viewpoints fairly. There is also a commitment to learning from others’ ideas,
and this mutual respect and common commitment leads to positive intellectual relations
(Boaler, 2008).
Voice and Agency. In theory, reflexivity in authority and equity in relations in the
classroom is a very idealistic notion, but those of us who strive for these ideals in our
practice know the realities of the obstacles that encumber the development of student
voice and agency. They are all too aware of the hidden curriculum, the unspoken social
prescriptions that govern the classroom and the habits of learning that have been
subconsciously taught for years through their educational process. Especially for those
students who consider themselves in underrepresented groups because of gender, race,
ethnicity, sexual orientation or other categorization, including opportunities for dialogue
in the classroom by itself might not be enough. Taylor and Robinson state:
Student voice…may not currently have the practical or theoretical tools…to
explain, or to contend with, the multifarious ways in which power relations
work within school…processes. As a consequence, it may find itself
implicated in reproducing, rather than unsettling or transforming, the
hegemonic-normative practices it sought to contest. In addition, it may
remain bound by the presumption that…such dialogue is itself a
manifestation of a classed, gendered and ‘raced’ form of cultural capital.
(2009, p.169)
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In other words, if not done in a deliberate and careful way, dialogue, even when
attempting to be emancipatory, can simply perpetuate the hierarchy that already exists in
the community of practice. Voices that were silenced can remain silenced and those that
have been heard will continue to be heard. One view of student voice work is geared
towards action, participation and change (Taylor & Robinson, 2009). These are worthy
goals that need to be focused towards allowing the individual student to use that action,
participation and change to move towards their own agency in their learning process.
Taylor and Robinson (2009) discuss the focus of postmodernist theory on reflexivity and
the production of knowledge in the context of student voice. It is important that the
dialogue move individuals towards growth in their agency in the educational process.
Keeping in mind the multiplicities of identities that students construct as they move
through the process of belonging to a community of practice (Maher & Thompson
Tetreault, 2001), which can make the formation of student voice even more complex.
Therefore, any empowerment that is promoted in the dialogue needs to also have these
realistic goals in mind as well. Empowerment can be attained in the learning process, as
in the realization of how much prior knowledge a student has presently, and it can be
used in conjunction with their agency to construct further knowledge in relation to their
community.
A Pedagogy of Feminist Relation in Mathematics
The theoretical framework that includes relational authority, relational equity,
voice and agency resembles the one that structures the Feminist Mathematics Pedagogy
with which I began this discussion. The intersections and overlaps of these constructs are
not coincidental. Solar (1995) posited an inclusive pedagogy based on postmodern
Dialogue in a PBL Classroom 12
epistemology and identified concrete attributes that characterized the “four dialectical
aspects” of feminist pedagogy : (a) passivity and active participation, (b) silence and
speech, (c) omission and inclusion, and (d) powerlessness and empowerment. The
framework is also corroborated by another model of a feminist mathematics classroom
(Anderson, 2005) in which empowerment, agency, development of authority, valuing of
intuition, and honoring of voices were the key components of the structure of this model.
In summary, the characteristics listed are the main tenets of the theoretical framework of
the pedagogical approach in which the discourse in a mathematics classroom should be
situated if the goals are to dissolve a hierarchical structure of authority, empower student
agency in learning and encourage student voice.
To obtain a unified structure of this framework and move forward with the study,
I viewed it on three levels of understanding – theoretical, conceptual, and observational.
This allowed for easier association and relation from the theoretical construct to the
observational data in the classroom practice.
Table 1 Theoretical Framework Organizational Structure
Pedagogical Theory
Level Pedagogy of Relation Feminist Mathematics Pedagogy
Theoretical Relational Authority Relational Equity
Dissolution of Hierarchy Ownership in Learning Inclusion Empowerment
Conceptual Student Voice (Speech) Agency
Sharing Power Valuing Intuition Participation
Observational Withholding, Pronoun Use, Politeness Teacher Questioning Methods
Naming Differences, Teacher Self-Correction Nonjudgmentalness
Dialogue in a PBL Classroom 13
Methodology
Looking for evidence that support these pedagogical theories, and thus
student equity and empowerment in the mathematics classroom, would require
certain observable behaviors for all participants in the learning community. What
follows is an outline of a framework for the observable characteristics of a
Pedagogy of Feminist Relation (PFR) in the mathematics classroom. These focus
on discursive traits that denote student agency and empowerment as well as equity
being built by specific actions on both the students’ and teacher’s part in the
classroom community. The diagram below summarizes the connections between
the observable behaviors identified that are associated with the characteristic and
theoretical traits of each of the pedagogical theories utilized in the theoretical
framework.
Figure 1. Relationship between observable characteristics and theoretical framework concepts.
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What follows is a discussion of this organizational framework, after which I will discuss
the specific methods of the design of this study.
A Framework for the Observational Level of Dialogue
Because PFR supports creating a classroom community that encourages discourse
as a means to its ends, it is important to identify characteristics that iconify the attributes
that theoretically would reveal the feminist perspective. As previously stated, the
feminist perspective can be viewed by its four dialectical aspects which are (a) passivity
and active participation, (b) silence and speech, (c) omission and inclusion, and (d)
powerlessness and empowerment (Solar, 1995) - either end of the spectrum which can be
observed in almost any classroom. However, in a classroom that claims to be motivated
by PFR, an observer would expect to find the characteristics from the more positive end
of the continuum such as active participation, speech, inclusion and empowerment. Thus,
these help form the foundational attribution level of the framework.
Observable characteristics of discourse. Teacher dialogic techniques that
follow from actions helping to promote these characteristics would include explicit
statements and direction that allow for turn-taking, wait time, opinion stating, sharing of
different solutions and other respectful methods of discourse. These behaviors would be
evidence of the instructor valuing intuition and naming differences. Evidence of the
creation of a warm and supportive climate and the allowance of self-solution of problems
would be shown in the amount of risk-taking and self-explanation that happens within the
dialogue. These could also be categorized as helping students to share power with
authority in the solution process and knowledge construction. Teacher self-correction
Dialogue in a PBL Classroom 15
would be a sign of sharing of power, as well, in order to create unity and a more balanced
power relationship with students in the learning community.
Nonjudgmentalness. Evidence of nonjudgmentalness in the dialogue would
support the idea of creating a classroom based on respect and relational equity.
Nonjudgmentalness manifests itself in classroom discourse by the teacher encouraging
and modeling a role of active listening and truly believing that each member of the
classroom community has the potential to add something important to the dialogue
(Fisher, 2001). Although students may become impatient with each other, it is key for the
instructor to model the importance of taking responsibility for how, as members of a
learning community of practice, our own statements effects others and the right we all
have to share our ideas freely. The concept of consciousness-raising of social justice
issues from a political perspective might be seen as foreign in a mathematics classroom,
but from a pedagogical view it can be seen as providing a “platform for individuals to
describe their experiences, feelings and ideas" and allowing for and valuing a
collaborative process through which individuals are supported as “speakers and actors”
(Fisher, 2001, p.39). This would be evidenced in the dialogue by moments where the
teacher would allow students to continue to explore their ideas, allow others to question
them, and have them come to conclusions collaboratively or when students are freely
expressing their disagreement or agreement with solution methods that are presented.
This behavior shows encouragement of student voice growth and empowerment of
student agency in learning.
Pronomial Use. The use of personal pronouns makes statements about inclusion
and exclusion in dialogue, therefore creating certain implications about classroom culture
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with respect to sharing of power. The use of the exclusive ‘we’ in conversation denotes
generality and a more authoritative presence (Pimm, 1987). The reverse then, student
self-mention, and the teacher’s own role-modeling of self-mention with additional
encouragement of the use of ‘I’ in students, followed by the repeated use of ‘I’ in
statements, conjectures and hypothesizing by students, would signify less generalization
and more individual agency in the communication being made. Also, the use of the
pronoun ‘you’ functions to qualify generality in a statement as well (Rowland, 1999),
again arguing that the use of ‘I’ by a student is making the statement less general and
more personal, showing ownership for the communication. Similarly, common use of the
inclusive ‘we’ by classroom members, including the teacher, would be signifying
dissolution of the authority of a single person and increased agency on students’ part.
Another way of promoting student agency is teacher use of the pronoun ‘you’
when talking about student work or in student questioning. This also follows a theory of
pronominal use in mathematics discourse (Pimm, 1987) as indicating the student herself
by pronoun creates more of a relational connection with the action or question at hand, as
well as the person with whom the student is speaking, as opposed to generalizing or
excluding the student. Both forms of pronominal use also encourage student voice as
they connect the student directly with their action and forming identity and the dialogue
that is occurring at the moment.
Teacher Questioning & Politeness. The methods of teacher questioning which
are commonly used by PBL facilitators can also be seen as empowering agency (Hmelo-
Silver & Barrows, 2006), since the questions are attempts at allowing the students to
hypothesize, take risks and learn from mistakes. However, in a classroom situated in a
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feminist relational pedagogy the community is also focused on the respect and safety that
each member should afford to each other. Signs of teacher and student questioning that
reveal politeness must be afforded to each member of the community of practice in order
to uphold these values and in turn build trust in the learners. The use of hedges in
discourse can be a means of observing politeness in the mathematics classroom. This is
seen for students through the use of rounders and plausibility shields (e.g. about, around,
approximately, I think, probably, maybe) to save face and for teachers through the use of
shields and adaptors (e.g. I think, a little, sort of, kind of, somewhat) to save face for
students (Rowland, 2000, p.140). However, more importantly, when students perceive a
more balanced power relationship (i.e. power sharing) it is often the case that there is a
“relevant absence of hedging” because students are “not coming to know the matter
[they] articulate[s]; rather she knows it” (Rowland, 2000, p.141). This is an important
characteristic in a classroom of feminist relation and would signify student agency and
voice work if there were a lack of hedging on the students’ part.
Withholding. In many classrooms, instructors struggle with the assistance
dilemma (Koedinger & Aleven, 2007) of when to give information and when to withhold
information so that students have the time to construct their own knowledge. This is also
true for instructors in the PBL classroom, but those following PFR goals would err on the
side of allowing students to move through their learning at their own pace. This is
consistent with the PFR attribute of allowing students to define their own learning
process, and hence sharing power in that process. Withholding also allows the instructor
to send the message to the students that their intuition about the problem is just as
important as what the instructor might have to say. In fact, it may be more important.
Dialogue in a PBL Classroom 18
However, frustration is also a major cost of withholding in the problem solving process
for students as they seek concrete information to further their methods. It would be
important to see if teacher withholding in the feminist relational PBL classroom caused
the same reaction, or if the classroom culture fostered by the feminist relational pedagogy
allowed for other possible reactions.
Methods
In this study, I chose to analyze discourse from PBL classrooms situated in a
pedagogy of feminist relation. This discourse was taken from mathematics classes from a
single-sex female private high school that utilizes an integrated algebra and geometry
PBL curriculum as the required second year mathematics course for all students. This
PBL approach had been implemented at the school three years prior to the research study,
and the instructors and I had been integral to the writing, development and
implementation of the problem-based curriculum. The curriculum is rather different from
a traditional textbook oriented curriculum in that the daily lessons are motivated by
problems that are not bound by a chapter-by-chapter context. The curriculum spirals and
topics resurface throughout the course of the year as students review topics within the
context of new ones. Students are expected to attempt new problems with their prior
knowledge as a foundation and problems are carefully written with the prior knowledge
in mind. The purposes of the problems range from review of material from previous
courses to introducing new ideas, but all are presented by students individually or in pairs
at the board. Classwork also consists of students working in groups at the board or at the
table on problems that extend their knowledge and apply concepts already learned. The
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curriculum also includes dynamic geometry labs and problems including use of handheld
graphing calculators.
This specific course is offered with five or six sections per year. In this given
year, there were four teachers available for the study. The two teachers chosen were the
teachers who had been involved in the writing of the curriculum and had taken part in a
summer professional development workshop I had run, three years prior, that focused on
the pedagogical approach of the PFR. In the training, we read articles on FMP, did role
modeling scenarios of class discussion of problems, discussed facilitation of class
discussion and many other issues concerning fostering a classroom climate that met the
goals of FMP and enhanced PBL instruction. The other two teachers were not chosen for
the study because of inexperience with the curriculum or conflict of interest with the
research. Pseudonyms are used in discussing the teachers involved in the study. Ms.
Williams is a teacher in her late twenties with undergraduate and graduate degrees in
physics, and five years of teaching experience. Ms. Munson is a mid-career teacher and
administrator with a background in mathematics and education, with over ten years of
experience. Both instructors are committed to the process of PBL and its goals, and are
experienced in the facilitation and cultivation of the PFR and relational ways of the
classroom climate the pedagogy hopes to foster.
Ms. Williams’ class included 17 female adolescents, with four students unable to
participate in the study, and Ms. Munson’s 14, all in the ninth through 11th grades.
Classes are of mixed ability and include a diverse range of racial, ethnic and socio-
economic status. In total, 27% of the participating students were African American,
Asian American, or other ethnic minority status. The classrooms are physically arranged
Dialogue in a PBL Classroom 20
with students and teacher sitting around an oval table or with desks facing each other to
facilitate conversation. Black or white boards are on three of the four walls in order to
maximize presentation space and there is also computer projection capability. All
students have tablet or laptop computers to facilitate and encourage the use of dynamic
geometry software use in problem solving. All students are encouraged to be a part of
the discourse throughout the class period as much as possible. Traditional classroom
practice of hand-raising is not expected at all times, but happens as a matter of politeness
or to suggest to the facilitator an idea or thought to share.
Data Collection and Analysis
Two classes from each instructor, one of 75 minutes and one of 50 minutes, for a
combined 250 minutes, were video recorded, however due to technical difficulties only
234 of the recorded minutes were able to be transcribed. The transcripts ranged from
student presentations of solution methods, student group work, teacher facilitation of
whole class discussion and student pair presentation of ideas. The transcriptions were
imported into the qualitative coding software MaxQDA to facilitate the analysis. At
times the unit of analysis was a whole excerpt of dialogue, a single personal utterance, or
a single word, in the case of pronoun analysis. I entertained both qualitative and
quantitative analytic methods and decided to see what patterns emerged.
Using a form of grounded theory analysis, I looked through the data to come up
with initial codes and then decided which of those made the most analytic sense and
created focused codes (Charmaz, 2006). I was looking for which observable behaviors of
FMP or pedagogy of relation appeared in the classroom discourse. For example, initial
codes for questioning were from student-to-student, teacher-to-student and student-to-
Dialogue in a PBL Classroom 21
teacher. However, more focused coding led me to identify the excerpts involving
teacher-to-student questions as those most related to behaviors indicating FMP
association, as it would be the teacher’s deliberate choices that would most readily
connect with pedagogy. I also found that subtleties in teacher questioning helped to
create a culture of respect, safety and attending to experience of the students which in
turn, led me to bring teacher questioning in with politeness as a category to consider.
Also, initial coding led me to look at the use of both student and teacher
pronouns. It was important to look at both frequency of use and context of use to make
connections back to FMP and relational pedagogy goals. More focused coding of
pronoun use allowed for those connections to take both teacher and student pronominal
use to student agency, while teacher use generally connected to dissolution of hierarchy.
See table below for a detailed example of the coding structure.
Table 2 Example of Coding Method for Classroom Discourse
Code Type
Name Example
Initial Teacher Question: traditional Teacher Question: FMP, higher order Teacher Question: eliciting or guiding
“The two sides of the isosceles trapezoid are which ones?” “What’s different about the formulas for the transformation that would indicate that one might be a mirror and that one might a vector translation?” “When can we set up equivalent8like a ratio between sides? What has to be true about those triangles?”
Focused Teacher Questioning for Agency “How did this one differ from the transformation in #9?”
A quantitative analysis of the frequency of specific pronouns resulted from initial
coding of student and teacher pronoun use. The teacher and student pronoun use (‘I’,
inclusive ‘we’, exclusive ‘we’, generalized ‘you’, and specific ‘you’) were counted from
Dialogue in a PBL Classroom 22
all 234 minutes of videorecording. The distinction between generalized ‘you’ and
specific ‘you’ was coded by the context of the dialogue, but in general the specific ‘you’
was used when speaking of a certain students’ work (“So did you apply the Pythagorean
Theorem?”) and the generalized ‘you’ was used when a student was explaining how to do
a problem (“yeah, so basically you find the slope of the line BC”). Inclusive ‘we’ was
used in the context of talking about the class as a whole (“Ok, so can we actually set up a
proportion?”) and the exclusive ‘we’ was used more for self-reference of the group
speaking (“So we did it using C is equidistant from both A and D”).
Although I was the sole coder of the transcriptions, this method was used on a
pilot study previous to this larger one, on which I enlisted the help of colleague with
many years of professional experience in discourse analysis. His guidance was essential
to our collaboration resulting in the final coding structure on which my coding was based.
Findings
Utilizing the framework previously stated, the transcripts were analyzed for the
observable behaviors that indicate teaching situated in a pedagogy of feminist relation. In
this section, specific excerpts from the classroom discourse will be discussed that
exemplify each of the main observable behaviors.
Promoting Relational Equity & Ownership
In many of the parts of the transcript there were times when the group work
centered on discussion of an error that the presenting student had made. Instead of
simply revealing the error to the student and showing her how it was wrong, in PFR, the
instructor sees it as more prudent to make use of that moment and allow the student to
speak, have her share in the process of learning and thinking, values her intuition and
Dialogue in a PBL Classroom 23
allows her to speak for herself. In this excerpt, the class is discussing a problem that asks
them to show that two angles are congruent, given six coordinates for points labeled A,
B, C, P, Q, and R. The student, who is presenting the solution, Carrie, had made an error
in her work while calculating side lengths in her attempt to show the triangles ABC and
PQR congruent:
Ms. Munson: Alright, Carrie, now what page are we on?
Carrie: Um, we are on page 18, um ok so, it asks to prove that angle
ABC and PQR are the same size, and so it said ABC to PQR are
the same size, I said they are the same size because of the SSS
thing. All of the side lengths are equal, so the corresponding
angles are equal. So this is equal to this [pointing to the board]
because this is (5,2), wait.. hold on…[5 secs]..what am I talking
about…I wrote it wrong…
Jenna: You have two…um, sides the same…
Carrie: Yeah, wait I wrote it wrong.
Jacey: Oh yeah, you have two… the square roots of 65.
Carrie: I think this is um...wait…this one...that would be oh, three, I
think I copied it down wrong.
Ms. Munson: …so let’s help Carrie out…
Arianna: Just switch…
Alexa: The (3, 5) should be where the (5, 2) is on the bottom, just
switch them.
Carrie: Oh this one? OK, so yeah, they’re corresponding because all
these others are equal.
Dialogue in a PBL Classroom 24
Even though it appears clear to the other students in the class that Carrie has made an
error in her work, she has picked up on the fact that others disagree with her, and Ms.
Munson can sense that. She hesitates in her statement that “I think I copied it down
wrong” so Ms. Munson calls on the group to create unity and support within the
classroom community. In this way, Carrie is being given the opportunity to explore her
own work even though her classmates have pointed out possible errors. With the
instructor giving permission for her peers to lend support, she is also showing her sense
of value of the cooperative nature of the classroom. However, allowing Carrie to be the
one to move forward with the correction is a way of not only naming differences for
students, but also valuing their intuition and sharing power.
Valuing Intuition & Naming Differences. Another way of supporting the
dialogical aspects of PFR is to make it clear to the students that if their solution is based
on valid, solid mathematics and leads them to an answer that makes sense and is correct,
they are the judge of which method they should and can use. This is empowering and
validates their agency in their learning. The instructor will ultimately be the one
responsible for telling them which methods they will need to be held responsible for, but
when given a choice they can make that decision for themselves. In this excerpt, the
class had just finished discussing two different solutions for proving two angles are
congruent. The students were given the coordinates of five points A, B, C, D and E (see
figure 2) and were asked to prove that angle CAB was congruent to angle EAD.
Dialogue in a PBL Classroom 25
2
-2
-4
5
E
D
CB
A
Figure 2 Original Diagram
The student presenting the problem, Jenna, used a method that showed that CAB
and EAD were both complements to CAD, after showing that BAD and CAE were both
right angles, therefore making them congruent to each other. However, in the middle of
Jenna’s description of her method another student, Alexa, mentions another option when
she interrupts by saying, “You can see if you shorten C, half of C, to the midpoint of CA,
you just have the exact same as B.” Alexa’s insight here that the length of AC is double
the length of AE, comes at an inopportune moment while Jenna is attempting to explain
her thought process about complements of the same angle which many students in the
class are confused about:
Carrie: So wait…
Ms. Munson: Go ahead, ask that question
Carrie: Oh, I don’t really…Can you go write complementary angles
there? So like on this one where would the complementary
angles be?
Jenna: So the complementary angles would be CAD and DAE and um,
that’s like to form CAE.
Carrie: Oh, and then, BAC and BAD would also be complementary?
Dialogue in a PBL Classroom 26
Jenna: Yeah,…No, no, not BAD, that’s the right angle,…it’s CAD and
BAC.
Carrie: Oh, OK.
Jenna: ‘Cause the two angles add up to 90.
Lee: I just wanted to say that I did it differently just because I
couldn’t be sure if they could be 90. So how could you know if
they could be 90?
Jenna: Um, I found the slopes of like AD and AB and then they were
opposite reciprocals, so that’s the perpendicular lines, which is
90 degrees and I did the same for AC and AE.
However, it appears that Ms. Munson was aware that there were other methods that might
be brought up, since immediately following this exchange another student shares an
insight that allows Ms. Munson to return to Alexa’s previous idea:
Ms. Munson: Jacey?
Jacey: OK, well, I did mine on my computer and I made them triangles
which was easier for me to see. Um, and I just clicked the points
in order of the angles, and went off and said ‘Measure Angle’
and they both came out to 45 degrees.
Ms. Munson: Now, I heard earlier somebody said that if you take the midpoint
of AC, you’ll form congruent triangles.
Jacey: Yeah, it’ll be…it’s the exact same triangle only rotated.
Ms. Munson: Can you do that?
Jacey: So if we…[manipulates in dynamic geometry software on her
tablet computer while class watches on projection screen] From
this point right here to right there,
Dialogue in a PBL Classroom 27
2
-2
-4
5
C'
E
D
CB
A
Ms. Munson: …and if we know the two triangles are congruent, what can you
say about the corresponding angles?
Students: Oh!
Jacey: …they’re congruent.
Ms. Munson: Really nice, nice, three different ways of solving that problem.
Very, Very nice.
Student: Cool!
The more geometric approaches, that both Jacey and Alexa were visualizing using
congruent triangles or transformation, matched up corresponding sides in one of two
ways. Alexa halved side AC to make it congruent to AE, showed all three sides the
same, and using the SSS congruence criteria, stated the triangles were congruent,
therefore since the angles were corresponding, they were also congruent (see figure 3).
Figure 3 Alexa’s Solution
Jacey used her dynamic geometry software package on her computer to plot the points
and rotated triangle ADE 90 degrees onto angle BAC, and showed the class that the
rotation resulted in angle D’A’E’ having the same coordinates as BAC’ (where C’ was on
the line AC – see figure 4).
Dialogue in a PBL Classroom 28
2
-2
5
C'
D'
E'
E
D
CB
A
Figure 4 Jacey’s Solution
Allowing students to define their own learning process here and to take ownership
for the work they did, which other students are confirming as unique and interesting,
was very empowering for them at this moment. Ms. Munson attempted to value not
only hers but the others’ opinions about their processes – most especially the student
who had originally presented the problem – while at the same time allowing for all
voices to be heard. Creating an environment where this appreciation is possible,
where Jenna, Alexa, and Jacey can feel comfortable sharing their feelings about
mathematics in an open way, is part of dissolving the hierarchy of the classroom and
sharing the authority in decision making.
Teacher Self-Correction. One method of supporting PFR that was exhibited
in the transcripts that demonstrates well the concept of dissolution of hierarchy is the
idea of teacher self-correction. In this excerpt, Ms. Williams is trying to help a
student presenter, Heather, with a problem in which she had to write the equations of
two separate altitudes of an isosceles triangle ABC when given the coordinates (see
figure 5).
Dialogue in a PBL Classroom 29 8
6
4
2
-2
5 B
C
A
Figure 5: Diagram for Heather's problem
Heather had had some difficulty finding the equations on her own in preparing the
presentation and when at the board, realized that she had made some errors. She was
making corrections to her work in front of the class when Ms. Williams attempted to
help her out:
Heather: …and then (c) says, to find an equation of the line BC, so you
just find the slope of the line BC and then you can plug in the
point B or C, and we plugged in point B.
Ms. Williams: Oh actually, wait a minute, you only have one choice here.
Heather: Oh yeah, sorry, you could only plug in point B, cause it’s from
line B to C
Ms. Williams: Oh no… oh, sorry… it says find the equation of line BC, I’m
sorry, … go ahead you can use either choice.
Heather: OK, you can use either point B or C.
As Ms. Williams attempts to help Heather, she is cognizant of the fact that Heather is
in a vulnerable position as she is trying to make corrections to her work in front of
the class. This takes a great deal of courage and comfort with her abilities. One
instructional technique for the teacher at such a moment is to turn the attention on the
Dialogue in a PBL Classroom 30
instructor and how all members of the classroom community as a whole are all
capable of error. By apologizing to Heather, not once, but twice, for misjudging her
initial statement, Ms. Williams is making sure that the class is aware of her own
awareness of her mistakes. This action brings the class together as they share the
experience of Heather feeling empowered by her original correct statement. The
sharing of the experience and creation of unity of the group helps in the dissolution
of the hierarchical structure of the classroom authority and the perceived authority
that Heather may have been giving Ms. Williams in the moment of correction.
Observations of Withholding. In the PBL classroom, there seemed to be some
contradictory evidence to typical student response to teacher withholding. In general,
student reaction to withholding in the PFR classroom is not one of frustration. Students
seem to respond to teacher withholding as an invitation to engage. As Ms. Munson’s
class grapples with the question, “Is the statement 4 9 13x x x+ = true or false?” she
rarely answers the question directly withholding that specific answer until the end.
Instead she entertains the students’ hypotheses about what other algebraic statements
with radicals might be true. Fiona, the student presenting, proposes that the original
statement is false, but before she allows the class to respond, proposes a question to the
class:
Fiona: But I actually have a question for you all, would this [writing
4 9 13x x x+ = on the board]…would that be true or false?
Students answer:….that’d be false…
Jacey: That would be the same thing as what you had there
Dialogue in a PBL Classroom 31
Fiona: So what if you um…so even if it was like this: [writes on board]
Jacey: wait isn’t it because…is it because it’s multiplying by
something?
Ms. Munson: ..well, you can’t add over radicals…the square root of what you did
before…that would be 2√x…
Fiona: Right, so if I had done this [points to board] in this form would it
be false then?
Ms. Munson: unlike radicals…I see three hands up.
In this case, students are not waiting for the instructor to give Fiona the correct answer;
they are helping their classmates out with their own conjectures. Frustration does not
seem to be taking over the class, just the opposite is happening. Students are feeding into
the fact that no definite answer has been given to the original problem as of yet. More
students are actually coming up with other possible equations with radicals and x’s, like
when Lee questions, “ Are you allowed to square the quantity (4x+9x) and then the
quantity 13x? You know the square root?” More students eagerly chime in with their
opinions, and eventually they all come to consensus about the algebra properties of the
addition of radicals. The withholding has added a great deal of student engagement and
sharing of ideas to the conversation.
It may be that in coming to a place of comfort with PBL, students learn that
withholding is expected in the teacher’s instructional method and it becomes more of a
habit that they can respond freely with their ideas as opposed to the requested “correct”
answer as in a more traditional classroom. Even when a student initiates a question and
Dialogue in a PBL Classroom 32
the instructor’s answer is withholding, students see it as an opportunity to conjecture and
hypothesize, as in the following excerpt. Here, in a problem, the class is being
introduced to the transformation notation for reflections for the first time. They had
utilized the transformation notation for vector translations and had seen the product of the
transformation on a set of coordinates, but had not actually made note of why such a
function would produce such a transformation. Ms. Munson is attempting to allow Carrie
to answer this question herself:
Ms. Munson: But what was different, like, Carrie I see you have your book,
can you write down on the board what the transformation was?
Carrie: Well, it was +2 and -1.
Ms. Munson: What was the formula?
Carrie: Oh the formula? [writing on board] It was
( , ) [ 2, 1]T x y x y= + − …and so you can actually…oh my gosh!
that’s so weird, so that’s the vector [2…is the vector it’s
transformed by…
Ms. Munson: So what’s so different about [x+2, y-1] and this transformation
that created a reflection where you had [y+2, x-2]?
Carrie: What?
Ms. Munson: What’s different about the formulas for the transformation that
would indicate that one might be a mirror and that one might a
vector translation?
Carrie: The numbers are the same?
Dialogue in a PBL Classroom 33
Mary: The x and y are switched?
Lee: Instead of [x+2, y-1], and now it’s y plus or minus something
and then x plus or minus something.
Carrie: Oh so, it would be a reflection if the y and x are switched?
Within that entire excerpt, Ms. Munson did not give any information, but students shared
their ideas and corrected each other. They introduced new ideas, which the class now has
the opportunity to define and discuss with respect to the problem at hand, but the
instructor’s withholding of information did not seem to be frustrating the students at any
turn. It appears that the framework of the pedagogy creates an environment that forces
students and teachers to be susceptible to being uncomfortable, and living in that
environment on a regular basis in regular public risk-taking activities. This open
vulnerability in a sufficiently safe atmosphere supports the security needed to foster
required trust to endure the uncertainty of teacher withholding. It is just the relational
idea of education that couples the construction of knowledge and dialogue together as
almost reliant on one another.
Promoting Empowerment, Agency and Student Voice
Pronoun Use. At many times in the excerpts there was evidence of the feminist
pedagogical ideals of empowerment and inclusion manifested in the use of personal
pronouns by both the students and the instructors. Many times the teachers would model
for the students the importance of personal empowerment by focusing the utterances on
the self with pronouns, as in the following excerpt. Arianna was up at the board
describing her attempt at a solution for a problem that asked to find the point on a given
line that was closest to the origin. Arianna had just explained her method of finding the
Dialogue in a PBL Classroom 34
equation of the line through the origin that was perpendicular to the one they were given,
and then finding the intersection of that perpendicular line and the original line, but since
the class had not done solving systems of equations in a while many students were
confused. Carrie asked Arianna where she got the equation of her line:
Arianna: Um, I got that one because I knew it had to be perpendicular to
the other line, so that’s how I took that other slope and found the
reciprocal and knew it had to go through the origin, because
we’re trying to find the point closest to the origin, so I knew the
line had to go through the origin, and the y-intercept had to be 0.
Carrie: OK, that makes sense. Great, so then where’d you go from
there?
Arianna: And then I just graphed it. I took these two lines and put them in
the calculator and found the intersection.
Carrie: Yeah, but what are you doing over there? That’s what I don’t
get.
Ms. Munson: You’re solving the problem algebraically.
Carrie: I mean, but where do we start, like why do they equal each
other?
Arianna: Oh, because they both equal y – so therefore they can be equal to
each other.
Ms. Munson: You’re finding the point of intersection.
Carrie: I know that.
Ms. Munson: =Your two lines, you’re finding where they are equal…
Dialogue in a PBL Classroom 35
Carrie: I just didn’t understand why they were equal, but now I
understand. Thank you.
In describing their own work or questions, the students in this excerpt make use of the
pronoun ‘I’ eleven times. Arianna was very explicit in her answering and discussing her
work that the methods she used were her own and had no problem taking responsibility
and ownership for not only the choices she made, but the attempts, which might possibly
have been wrong. In Carrie’s questions, she is very clear that her questions and
confusions, and eventual clarity, is also her own. Ms. Munson deliberately personalizes
the statements she says focusing her responses to Carrie with the specific use of the
pronoun ‘you’ a total of four times. This type of pronoun use attaches ideas to the people
that have authored them. These dialogical techniques create a culture in the classroom
where students feel more comfortable to take ownership on a regular basis thereby taking
on more authority, self-representation and also agency. By creating discursive focus on
the student who has done the risk-taking in the problem solving, the teacher is modeling
interest, curiosity and respect for their own intuition, prior knowledge and experience in
the problem-solving process.
In another short excerpt from Ms. Williams’ class, students are working in pairs at
the blackboard on a problem with translating a triangle with vertices U, V and W, by a
given vector:
Sonora: Here….We are going over 2…
Maura: No, we are going U to V..
Sonora: They’ll be the same thing.
Maura: OK, so this is right here, right?
Sonora: 9 to 11 is 2, 0 to 4 is 4.
Dialogue in a PBL Classroom 36
Maura: How do we …do we have to scale the vector?
Sonora: Yeah, we do desired over actual.
Here, students are able to claim ownership in both statements of assertion of knowledge
and statements of uncertainty or question. Within the pair discourse, there appears to be
a sense of comfort with making such statements of relation to knowledge or lack thereof
and the personalization of the possession of the statement is apparent. Of course, when
students are working collaboratively, the self-reference comes in the form of the use of
the pronoun ‘we’ as opposed to the use of ‘I’. There is also no sense of preoccupation or
hesitation about making claims of individual or pair ownership about utterances – be they
positive (“Yeah, we do…”) or negative (“No, we are…”).
The table below displays the results of the quantitative analysis of the pronoun
use in all 243 minutes of discourse transcripts.
Table 3 Total Teacher and Student Pronomial Use
Individual I Inclusive We Generalized You
Specific You
Teacher 65 80 183 73
Student 185 150 207 75
In the class periods that were video recorded, Ms. Williams’ class presentations were all
done in pairs, so I separated out the number of times those classes use the inclusive we
for self-reference by students, and added that to the number of times Ms. Munson’s class
used ‘I’ for self reference. Students were found to be using the personal pronouns for
self-reference a striking 335 times in approximately 234 minutes of dialogue
transcription. That is approximately a student use rate of the pronouns (‘I’ individually
or ‘we’ in pairs) of 1.4 times per minute in class discussion, or almost one and a half
Dialogue in a PBL Classroom 37
times a minute. The students and the teachers were referring to each other by the
pronoun ‘you’ individually at practically the same rate (73 vs. 75), which may imply a
sharing of power and authority. The frequency with which these classroom communities
made use of personal pronouns in discussion to serve as a means of claiming agency in
their learning, their questions and their feelings was dramatic.
Politeness and Hedges. When looking at the role that teacher questions played in
fostering the classroom culture, initially it was clear that many of the teacher questions
were procedural and rather traditional, merely looking for information or confirmation
(“What’s number 25?” or “Is the height 160?”). Other teacher questions may be
attributed to PFR or could also be found in any student-centered classroom that proposes
to use guiding questions and less teacher-centered instructional approaches (Ms. Munson:
Which angles did she prove by CPCTC are congruent?, Ms. Williams: …but be careful,
what was the mistake that you made that you fixed?, Ms. Munson: So how does that
prove that they are congruent?). These types of questions are not totally exemplary of a
PFR because although they may set the student on the right track for constructing their
own knowledge, there is not direct agency for the learning that is an integral part of this
type of pedagogical style.
However, questions that invite student input and engagement could be designated as
illustrating PFR in some ways. Questions like:
1) Ms. Munson: OK, what did we just learn from Carrie’s problem?
2) Ms. Munson: Do we agree with that?
3) Ms. Williams: uh huh…why would that be?
These types of questions invite student response and participation because they are
requesting the students’ opinion and require that individual voices to be heard. The
Dialogue in a PBL Classroom 38
teachers are sincere in their questions because it is important to the mechanics of the class
to know the differences in their opinions and the choices they will make in their problem
solving processes. It is the diversity in the methods and their valuing of the experiences
that helps create an environment that both dissolves the authority of which process is
“best” and whose voice gets heard.
There was evidence that the instructors were making an effort to create an
environment of politeness in the discourse for the students, which can be interpreted as
role-modeling or sensitivity to the students’ risk-taking and vulnerability in problem-
solving. One example of teacher hedging for politeness is when Ms. Munson’s class was
discussing a problem about solving the sides of a right triangle. The students presenting
the problem had erroneously assumed that they could set up proportions to the sides of
the triangles.
Ms. Munson: When can we set up equivalent…like a ratio between sides? What
has to be true about those triangles? We haven’t really talked much
about this…They have to be similar, and these aren’t going to be
similar figures. So I see a few hands up, first of all let’s make sure,
Fiona, that we… understands that scenario, so can we maybe…
Ms. Munson’s hedging is being sensitive to the fact that the students have taken a
risk and attempted this problem, albeit erroneously, but she wants to foster the
feeling of safety in that context. In fact, although she points out a student, Fiona,
who has misunderstood the question, she then hedges by generalizing with ‘we’ to
make sure that Fiona does not feel singled out as the sole student that misunderstood
the question. In moving forward with the correct solution, Ms. Munson uses
‘maybe’ to soften the momentum towards the correct discussion and perhaps open
Dialogue in a PBL Classroom 39
the risk-taking process once again, even for those students that misunderstood
initially.
Ms. Williams also is found hedging for politeness when students are
confused about the differences between transformations.
Ms. Williams: Yeah, I think we actually just got a little mixed up with
glide reflection. So, a glide reflection…
Although a pair of students is at the board and incorrectly identified the transformation as
a glide reflection, Ms. Williams uses the inclusive pronoun ‘we’ to help hedge the error
of the two students and take the focus off of their misunderstanding. The added hedging
with ‘I think’, ‘actually’ and ‘a little’ also softens the fact that the transformation
identification was incorrect, and in that softening, she is valuing their risk-taking.
Here both teachers feel it necessary to hedge to protect students’ emotional well-
being as they either question someone’s process or point out misunderstanding. In
allowing students to follow their process through to find their own mistake, their self-
representation must be preserved at the same time. Another consideration from the
teacher’s perspective is a student’s safety in self-disclosure of her differential
vulnerability (Fisher, 2001, p.150). This concept is another consideration of both
politeness and care in dialogue for it encompasses the students’ awareness of not only
their intellectual risk-taking, but also social and emotional risks they take as well. The
instructors in a PBL classroom situated in a PFR create a climate that lets students
understand that they have the luxury of not only a second chance in the construction of
knowledge but multiple chances to be a part of that construction.
Dialogue in a PBL Classroom 40
However, the more powerful message that revealed itself in these transcripts was
an apparent lack of student hedges. According to Rowland (2000), this corresponds with
the manifestation of student agency and student voice work. Students do not feel the
need to hedge their ideas or questions throughout the discourse and can stand firm in their
hypotheses and conjectures. In the entire 234 minutes of transcription there were only 21
student hedges, three of which were plausibility hedges which generally indicate
uncertainty or lack of confidence. This confirms the assertion that PFR helps to
encourage student voice and empowers student agency in learning.
Discussion
In this study, classrooms that claimed to promote specific aspects of a pedagogy
of feminist relation were examined in order to analyze which observable discursive
characteristics revealed the intended pedagogical and philosophical goals. It was clear
that some of the observable behaviors could be connected to more than one of the
theoretical concepts. For example, the effect of the instructor naming differences in
student solution methods appeared to be many. Students heard the different voices of
their classmates when differences were named thereby creating a sharing of power, which
helps to dissolve the traditional hierarchy of authority. Naming differences also allows
for individual ownership of each method or idea, which in turn empowers students in
their learning. Through the analysis of the observable behaviors it slowly became clear
that the connections and direct effects to each of the main theoretical frameworks were
not as direct as originally conceptualized.
However, the need to create a classroom community that allows for all students to
feel able to communicate their ideas as freely as possible is important. These
Dialogue in a PBL Classroom 41
characteristics of self-mention in pronominal use, teacher self-correction,
nonjudgmentalness and withholding to encourage student active learning appeared to
enhance student communication within the classroom. Students were not hindered in
their communication about mathematical ideas, asked questions and clarified ideas while
taking ownership for those statements. Instructors fostered a learning community of
belonging and safety in risk-taking with all of these observable behaviors as well, which
enhanced communication ability in the discourse.
The Equity Principle was further upheld by these observable behaviors by the
encouragement of student voice with significant evidence of inclusive pronoun use. The
further significant lack of student hedging is an indication of a strong sense of student
agency in the classroom community. The fact that this evidence was rather general in all
of the classroom discourse analyzed and not limited to certain students shows that the
goal of equity in the discussion and dissolution of the hierarchical structure is starting to
be realized within this pedagogical practice. The larger goals of relational authority and
relational equity appear to be on their way to being realized as well. It is striking that
students are comfortable sharing alternative solutions, even when they might possibly be
incorrect, as the instructor has created a classroom climate that values the differences and
voices that exist. It is clear to all members of the learning community that the authority
of accepting those differences and valuing the voices is relational and exists between the
members.
This study has allowed for the creation of a collection of observable behaviors in
instructional discourse methods that denote the use of a PFR. In turn, these instructor and
classroom member behaviors would indicate an encouragement of the NCTM principles
Dialogue in a PBL Classroom 42
of communication and equity. However, my attempts to design a study that does all of
this in a rigorous way had its limitations. Conclusions are drawn solely on a descriptive
basis and no comparative conclusions are drawn about discourse characteristics regarding
non-PBL classroom practices or classroom practices situated within other pedagogical
contexts. The classrooms studied were in a specific context (i.e. an all-girls’ private high
school) limiting conclusions about discourse characteristics of coeducational and large
public settings situated in this type of pedagogy. Also, in deciding whether or not the
behaviors and dialog indicate a level of equity in the classroom, only observation of the
video and discourse analysis was part of the design of this study which allowed for
interpretation of social interaction as the only means for decision-making and discourse
coding. To address this question further, future designs might include student interview
or survey regarding equity or classroom authority. It is important, as in many forms of
qualitative research, that readers are clear about the descriptive nature of this research
study.
Other future implications for research include dialogic comparison of PFR
classrooms with traditional mathematics classrooms (including comparative studies of
pronominal use), further development of demonstration and organization of instructional
approaches and their advantages and disadvantages, professional development
opportunities in support of underrepresented students in mathematics, and a study of the
effects of its uses in the classroom.
Conclusion
In this study, I attempted to reveal distinguishing attributes of a problem-based
learning mathematics classroom that is situated in a pedagogy of feminist relation.
Dialogue in a PBL Classroom 43
Facilitated by this relational pedagogy, the PBL environment creates a climate of
discovery and discourse that enables community of learners to share in a dialogue and co-
construct meaning in many ways. A relational FMP has within it the goals of dissolving
the traditional classroom hierarchical structure, empowering student agency in learning
and encouraging student voice in construction of mathematical meaning. Through
describing aspects of utterances in textual context from two separate classrooms, I was
able to analyze specific techniques that correspond with the goals and outcomes of the
theoretical framework of feminist and relational pedagogies, as well as student voice
work. It is often difficult to find an environment in which mathematics is taught in a
truly feminist and relational setting. In fact, part of the reason that it is so difficult to find
a classroom in which to research this type of pedagogy is because of the traditional
methods with which mathematics is generally taught and viewed in U.S. schools. This
study gives an initial thorough description of the facilitation of these methods of
discourse. Teachers who have habitually resorted to traditional, Initiation-Response-
Evaluation, triadic dialogue (Lemke, 1990) in a lecture classroom in mathematics are
often distressed by the idea of trying something new, even when recommended to do so.
It will be necessary to do further, more structured research on discourse practices in the
feminist relational classroom to clarify the interaction further in the hope of any type of
transferability.
In looking toward the future where inclusion is the goal and the “Equity
Principle” states that mathematics teachers will strive to create strong support and uphold
high expectations for all learners, it seems most prudent at this time to find instructional
methods that fit the needs of all learners. Some may say that creating a classroom based
Dialogue in a PBL Classroom 44
on open dialogue, where students feel empowered to become agents in their learning or
safe to take risks and eventually can believe that their voice will be heard is an idealized
situation. However, if there are true techniques that can bring us closer to that ideal in
order for communication to be facilitated, this should in turn facilitate that interaction that
is at the heart of education. After all, it is in that communication between those in the
community of learners and the relationship between them, which is the place where
education happens.
Dialogue in a PBL Classroom 45
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