Revealing Discourse in PBL Mathematics Classroom · a part in that instruction. Secondary school mathematics teachers remain challenged by the idea of creating a classroom climate

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.


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.


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;


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?


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


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


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


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


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,


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)


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


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


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.


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


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


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


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.


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


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


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-


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


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


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.


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.


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?


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,


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).


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


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


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


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?


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


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…


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.


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


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


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


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.


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


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


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.


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


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.


References

Amit, M., & Fried, M. (2005). Authority and authority relations in mathematics

education: A view from an 8th grade classroom. Educational Studies in

Mathematics, 58, 145-168.

Anderson, D. L. (2005). A portrait of a feminist mathematics classroom: What adolescent

girls say about mathematics, themselves, and their experiences in a "unique"

learning environment. Feminist Teacher, 15(3), 175-193.

Becker, J. R. (1995). Women's ways of knowing in mathematics. In P. Rogers & G.

Kaiser (Eds.), Equity in mathematics education: Influences of feminism and

culture (pp. 163-174). Washington, D.C.: Falmer Press.

Biesta, G. (2004). Mind the gap. In C. Bingham & A. M. Sidorkin (Eds.), No education

without relation (pp. 11-22). New York: Peter Lang.

Bingham, C. (2004). Let's treat authority relationally. In C. Bingham & A. M. Sidorkin

(Eds.), No education without relation (pp. 23-38). New York: Peter Lang.

Boaler, J. (1997). Reclaiming school mathematics: the girls fight back. Gender and

Education, 9(3), 285-305.

Boaler, J. (2008). Promoting 'relational equity' and high mathematics achievement

through an innovative mixed-ability approach. British Educational Research

Journal, 34(2), 167-194.

Charmaz, K. (2006). Constructing grounded theory: A practical guide through

qualitative analysis. Los Angeles: Sage.


Chow, E. N.-L., Fleck, C., Fan, G.-H., Joseph, J., & Lyter, D. (2003). Exploring critical

feminist pedagogy: Infusing dialogue, participation, and experience in teaching

and learning. Teaching Sociology, 31, 259-275.

Fisher, B., M. (2001). No Angel in the Classroom: Teaching Through Feminist

Discourse. Lanham, MD: Rowman & Littlefield Pulishers, Inc.

Gilbert, M. C., & Gilbert, L. A. (2002). Challenges in implementing strategies for

gender-aware teaching. Mathematics Teaching in the Middle School, 7(9), 522-

527.

Griffiths, M. (2008). A feminist perspective on communities of practice [Electronic

Version]. Retrieved 8/4/09 from

http://orgs.man.ac.uk/projects/include/experiment/morwenna_griffiths.pdf.

Hmelo-Silver, C. (2004). Problem-Based Learning: What and how do students learn?

Educational Psychology Review, 16(3), 235-266.

Hmelo-Silver, C., & Barrows, H. (2006). Goals and strategies of a Problem-Based

Learning facilitator. The Interdisciplinary Journal of Problem-Based Learning,

1(1), 21-39.

Hmelo-Silver, C., & Barrows, H. (2008). Facilitating collaborative knowledge building.

Cognition and Instruction, 26, 48-94.

Jacobs, J. E., & Becker, J. R. (1997). Creating a gender-equitable multicultural classroom

using feminist pedagogy. In Yearbook (National Council of Teachers of

Mathematics) (pp. 107-114). Reston, VA: NCTM.

Kellermeier, J. (1996). Feminist pedagogy in teaching general education mathematics:

Creating the riskable classroom. Feminist Teacher, 10(1), 8-12.


Koedinger, K., R., & Aleven, V. (2007). Exploring the assistance dilemma in

experiements with cognitive tutors. Educational Psychology Review, 19, 239-264.

Ladson-Billings, G. (1995). But that's just good teaching: The case for culturally relevant

pedagogy. Theory into Practice, 34(3), 159-165.

Lubienski, S. T. (2000). Problem Solving as a means towards mathematics for all: An

exploratory look through a class lens. Journal for Research in Mathematics

Education, 31(4), 454-482.

Maher, F. A., & Thompson Tetreault, M. K. (2001). The feminist classroom. New York:

Rowman & Littlefield Publishers, Inc.

Mau, S. T., & Leitze, A. R. (2001). Powerless gender or genderless power? The promise

of constructivism for females in the mathematics classroom. In J. E. Jacobs, J. R.

Becker & G. Gloria (Eds.), Changing the Faces of Mathematics: Perspectives on

Gender (pp.37-41). Reston, VA: NCTM.

Meece, J., & Jones, G. (1996). Girls in mathematics and science: Constructivism as a

feminist perspective. The High School Journal, 79, 242-248.

NCTM. (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.

Noddings, N. (1993). Politicizing the mathematics classroom. In S. Restivo, J.P. Van

Bendegem and R. Fischer (Eds.), Math worlds: Philosophical and socials studies

of mathematics and mathematics education (pp. 150-161). Albany, NY: SUNY

Press.

Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms.

New York: Routledge.


Ridlon, C. (2009). Learning mathematics via a problem-centered approach; A two-year

study. Mathematical Thinking and Learning, 11(4), 188-225.

Rowland, T. (1999). Pronouns in mathematics talk: Power, vagueness and generalization.

For the Learning of Mathematics, 19(2), 19-25.

Rowland, T. (2000). Hedges. In Rowland, T., The pragmatics of mathematics education:

Vagueness in mathematical discourse. (pp. 115-144). London: Falmer Press.

Savery, J. R. (2006). Overview of problem-based learning: Definitions and distinctions.

The Interdisciplinary Journal of Problem-Based Learning, 1(1), 9-20.

Solar, C. (1995). An inclusive pedagogy in mathematics education. Educational Studies

in Mathematics, 28(4), 311-333.

Spielman, L. J. (2008). Equity in mathematics education: Unions and intersections of

feminist and social justice literature. ZDM - International Journal of Mathematics

Education, 40, 647-657.

Strobel, J., & van Barneveld, A. (2009). When is PBL more effective? A meta-synthesis

of meta-analyses: Comparing PBL to conventional classrooms. The

Interdisciplinary Journal of Problem-based Learning, 3(1), 44-58.

Taylor, C., & Robinson, C. (2009). Student voice: Theorising power and participation.

Pedagogy, Culture and Society, 17(2), 161-175.

Thayer-Bacon, B., J. (2004). Personal and social relations in education. In C. Bingham &

A. M. Sidorkin (Eds.), No education without relation. New York: Peter Lang.

Zohar, A. (2006). Connected knowledge in science and mathematics education.

International Journal of Science Education, 28(13), 1576-1599.


Documents

Revealing Discourse in PBL Mathematics Classroom · a part in that instruction. Secondary school mathematics teachers remain challenged by the idea of creating a classroom climate