Revealing Dialogue in a Problem-Based Learning Mathematics ... · Dialogue in a PBL Classroom 2 Abstract In this paper, I describe the discursive nature of a Problem-Based Learning

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

ETAP 751

November 2009


Abstract

In this paper, I describe the discursive nature of a Problem-Based Learning (PBL)

secondary mathematics classroom that is situated in a Feminist Mathematics Pedagogy

(FMP) in the context of an integrated Algebra and Geometry curriculum. Because the

pedagogical approach calls for an intersection of discussion-based and student-centered

teaching techniques, dialogue is an important aspect of instruction in this classroom.

However, the values of Feminist Mathematics Pedagogy are infused in the classroom

practice as well, with the goals of collective and individual empowerment, student

agency, dissolution of hierarchy, and other aspects of student voice work. Because of

this, 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 the

analysis of the data, evidence was found of teacher dialogical techniques, pronominal use

and politeness that dissolved hierarchical structures of authority, empowered student

agency and encouraged student voice. These characteristics were revealed as part of the

nature of a feminist relational pedagogy in the mathematics classroom.


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. Before

that, 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. It is clear in this publication that a message is being sent to

secondary mathematics teachers to make it a priority 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).

This question of safety and risk-taking, 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. The other major breakthrough with this publication was the

revolutionary statement of the NCTM’s “Equity Principle” 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 actually is coupling the two of these ideas with how discussion and

more relationally-based teaching methods are often both preferred by marginalized

groups in mathematics classrooms, and improve their learning as well (Boaler, 1997;

Ladson-Billings, 1995; Lubienski, 2000; Mau & Leitze, 2001). It still remains a

challenge to many secondary school mathematics teachers just how to create a classroom


climate that serves the needs of diverse learners and addresses the issue of

communication skills as a priority in mathematics education.

In fact, because a discursive mathematics classroom is such a novel concept to

many teachers today, these standards raise many interesting research questions. How

would we as classroom practitioners know if a classroom that claimed to create a safe

environment for students to take risks and “venture conjectures” was actually fulfilling its

claim? What ways of talking would help fulfill the ideals of the type of pedagogical

practice that would support this type of classroom climate that would engage the

marginalized groups that are 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. With such

pedagogical goals as the formation of student voice, dissolving the hierarchical structure

of the community of practice and attempting to empower students in the learning process,

it begs the question of how a teacher might attempt to use discursive methods to attain

them. The study that I undertook looked at this research question to examine which

aspects of classroom discourse are characteristics that 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, including 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 was clear

that the intersections between feminist pedagogies and constructivist and student-

centered ideologies were many (Meece & Jones, 1996; Noddings, 1993; Spielman, 2008).

More importantly, FMP emphasized “connected” and relational learning that many

females desired in the classroom experience and were 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 the 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). 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 1) pushing students for deep explanations 2) using open-ended

metacognitive questions 3) revoicing and 4) 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. 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

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. The theoretical concept of reflexivity in authority and

equity in relations is all well and good, 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, 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). I would agree with those goals, but focus them 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 in order to encourage the more positive aspects of what she called the

four dialectical aspects of feminist pedagogy – 1) passivity and active participation,

2) silence and speech, 3) omission and inclusion, and 4) 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.

Methodology

In this study, I chose to analyze discourse from a PBL classroom situated in a

pedagogy of feminist relation. This class was comprised of a single-sex female high

school integrated algebra and geometry course including ninth through eleventh graders.

This PBL approach had been implemented at my school three years prior to the video

recording, and I, as the instructor had been teaching with this approach for over 14 years

at two different schools. This was my own classroom and I had created a video recording

for my own professional development information. 15 students and myself sit around an


oval table and the classroom has black and whiteboard on three of the four walls. The

class is run generally by student presentation of problems and discussion ensues.

The PBL classroom climate is one on which I tend to spend a great deal of time

working from the beginning of the year to develop. Students come into the classroom

with very different backgrounds and habits of mind that inhibit their ability to participate

freely in discussion. Not only their experiences in discussion vary, but so does their prior

knowledge with the course material and technology, so their familiarity and comfort with

the understanding of PBL and faith in the instructional approach takes a great deal of

time and trust to foster. The curriculum book consists of a list of deliberately constructed

problems that allow students to build upon prior knowledge and co-construct new

knowledge in a social, discussion-based way. However, this takes care and attention to

the discourse and dynamic of the dialogue in order to be successful. The techniques that

I have found helpful and purposeful have happened only through finding congruence with

the FMP theoretical framework.

I chose to transcribe three separate sections of the video for this study in order to

focus on different discursive aspects of the classroom. In choosing these excerpts, I was

looking for looking for high student involvement in discussion, not just one student

explaining their solution, which often happens in a PBL classroom. The excerpts

selected were chosen as an example of group discourse since there were more than four

students talking for more than a six-minute period of time. At times the unit of analysis

was the whole excerpt of dialogue, a single personal utterance or in the case of pronoun

analysis, a single word. After transcription, I coded the text for dialogical signs of


Feminist Pedagogical Ideals according to the analytical coding system I will now

describe.

Data Analysis

Because FMP 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. Solar (1995) states that

the feminist perspective could be viewed in four dialectical aspects of the pedagogical

approach which, as previously stated are 1) passivity and active participation, 2) silence

and speech, 3) omission and inclusion, and 4) powerlessness and empowerment. In

almost every classroom, characteristics of these aspects can be observed. However, in a

classroom that claims to be motivated by FMP, an observer would expect to find the

more positive end of the continuum characteristics of active participation, speech,

inclusion and empowerment. I have adapted Solar’s (1995) extensive list of attributes of

an inclusive mathematics classroom and list here the discursive dimensions.

Dialogical Aspects of Feminist Mathematics Pedagogy Encourage collaboration and participation Making explicit and valuing all voices and thinking processes Using Inclusive language Valuing intuition, emotion and experience Naming differences, explaining them and learning from them Asking women cognitive-level questions Letting women (and other underrepresented groups) solve problems by themselves Sharing power and decision-making Creating a warm and supportive climate

Figure 1. Adapted from Solar (1995)

Teacher dialogic techniques that follow the above framework 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. 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.

Also, 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, I must show them the importance of taking responsibility for

how what we have to say 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 text by moments where I would stop and allow students who I knew

were wrong 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. These excerpts

would be showing encouragement of student voice growth and empowerment of student

agency in learning. In analyzing the dialogue from a class discussion, I would need to

know what aspects of the dialogue would reveal the pedagogical approach and in what

ways.


Many analysts state that the use of personal pronouns make statements about

inclusion and exclusion in dialogue, therefore creating certain implications about

classroom culture with respect to sharing of power. Pimm (1987) posited that the

intention of the use of the exclusive ‘we’ in conversation is generality and a more

authoritative presence. The reverse then, self-mention, the teacher’s encouraged use of

‘I’ by 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. Some researchers say that 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 a

dissolution of the authority of a single person and perhaps 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 Pimm’s (1987)

theory of pronominal use in mathematics discourse 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.

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. When analyzing the dialogue, I looked for signs of teacher and student

questioning that revealed the politeness that would be afforded to each member of the

community of practice in order to uphold these values and in turn build trust in the

learners. Rowlands (2000) states that 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 (Rowlands,

2000, p.140). However, more relevant to the FMP discourse data analysis is the fact that

when students perceive a more balanced power relationship 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” (Rowlands, 2000, p.141). This was an important

characteristic to look for in the text and would signify student agency and voice work in

the classroom if there were a lack of hedging on the students’ part.

Findings

Using Dialogical Aspects of FMP. In many of the excerpts in the text 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, I thought it would be more prudent to make use of that moment and allow the

student to speak, have her share in the process of learning and thinking, value her


intuition and allow her to speak for herself. In all transcribed excerpts, student

pseudonyms have been used, but I am the teacher. In this excerpt, the class is discussing

a problem that asks them to find the area of a quadrilateral given only the four

coordinates of the vertices. The student whose turn it is to present the solution had used a

very algebraic approach and has attempted to write the equations of the altitude and the

base and found the proper intersection points in order to use the distance formula to get

the desired lengths. The student presenting, Stephanie, is discussing the method she used

to find the area of the quadrilateral:

Figure 2 Quadrilateral ABCD

Ms. Schettino: [addressing student who is doing presentation] Ok, Stephanie, so what did you

do?

Stephanie: Ok uh, well I thought that because like, a quadrilateral would be like base x height,

so I just took, the [inaudible] and I got the distance from D to B and then the distance from A

to D and then I timesed them together…[3 sec]

Ms. Schettino: Ooooohhhhhh

Annie: =I definitely did not do that….

Laura: =yeah

Stephanie: =But I’m not sure if you needed to get the altitude

Meghan: =oh that’s what I did

Ms. Schettino: So let’s…here’s the…[3 secs]..How is that different from what Annie just said

in number 9? [5 secs] What’s the difference between just multiplying AD and AB together…

the sides together and multiplying base x height?

6

4

2

5

C

D

A

B


Even though it appears clear to the other students in the class that Stephanie has made an

error in her approach, she has picked up on the fact that others disagree with her, and I

can sense that. She hesitates in her statement that “I’m not sure if you needed to get the

altitude…” so I continue to let her go on with a leading question about what might be the

difference between her approach and the way another student did a similar problem

previously. In this way, she is being given the opportunity to explore her own idea even

though I am pointing out the difference for her. Allowing her to be the one to move

forward with the exploration is a way of following Solar’s (1995) idea of naming

differences for students, but also valuing their intuition and sharing power.

Another way of supporting the dialogical aspects of FMP 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. I 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 finding

the area of a quadrilateral given only the four coordinates of the vertices. In this previous

example, Stephanie attempted to use a very algebraic approach to finding the area of the

quadrilateral. Other students had addressed the problem in a very geometric manner and

drawn a rectangle with vertical and horizontal sides that surrounded the quadrilateral (see

figure 3). They then found the area of a rectangle and subtracted off the four right

triangles at the corners. Many students found this method much more straightforward

and to be less “work”, meaning less algebra. However, there was a difference of opinion.


Figure 3 Quadrilateral ABCD with rectangle outline

Ms. Schettino: so how many of you think that’s easier than writing the equation of the

altitude and finding the intersection point, and finding the length of the altitude and

multiplying it by the base?

Meghan: this is so much easier

Gretchen: it’s so much harder!

Ms. Schettino: really? Ok, ok so you can continue finding the altitude..

Gretchen: that one just took so much time..and I got so confused

Ms. Schettino: really?

Gretchen: my work is like that big..[shows space for work on her paper]

Ms. Schettino: ok, did you get 19?

Gretchen: yeah, I did…I found the base using the Pythagorean theorem, square root of

3, then I wrote the equation of that line, y= 2/7(x-8)=5 and then I plugged it in using the

point (7,2) - the other point. Then I plugged it in and found the intersection point and

used the distance formula. That took less time for me.

Ms. Schettino: Yeah, well either way. You do what you want to do.

Gretchen: So do we have to do it that way on a test?

Ms. Schettino: no, I will never make you do it this way.

Gretchen: perfect

Allowing Gretchen to define her own learning process here and to take ownership for

the work she did, which many of the students were naming as more difficult, was

6

4

2

5

C

D

A

B


very empowering for her at this moment. I 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 allowing for the all voices to be

heard. Creating an environment where this is possible, where both Meghan and

Gretchen 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.

One method of supporting FMP that was exhibited in the text that

demonstrates well the concept of dissolution of hierarchy is the idea of teacher self-

correction. In this excerpt, I am trying to help Stephanie with the alternative method

of finding the area of the quadrilateral as discussed in the previous excerpt, but she is

having trouble catching onto my direction:

Ms. Schettino: Stephanie, do you remember the hint I gave yesterday?

Stephanie: Ummm, to do this? [draws on board]

Ms. Schettino: Yeah…no, to go straight down to the x-axis from the point, go straight

down from the horizontal, don’t make a diagonal,[pause]…no that’s not straight, well,

you’re right it is straight, I mean go vertical down, sorry, I’m not choosing my words

right,…There you go.

As I attempt to describe what Stephanie should do, I am cognizant of the fact that she

is in a vulnerable position as she is trying a new method in front of the class. This

takes a great deal of courage and comfort with her abilities. My technique is to turn

the attention on myself and how I am not fully giving her the best direction (which in

fact I wasn’t), and be sure that the class is aware of my own awareness of my

mistakes. This brings the class together as we share the experience of Stephanie’s

seeing the geometric representation of the area problem, as opposed to the algebraic

method she had originally attempted. The solidarity of the experience helps in the


dissolution of the hierarchical structure of the classroom authority and the perceived

authority that Stephanie was giving me in the moment as I dictated direction.

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 me at times in my PBL classroom, but I do err on the side of allowing students to

move through their learning at their own pace. This is consistent with Solar’s (1995)

attribute of FMP of allowing women to define their own learning process. However,

Koeding and Aleven (2007) also state that frustration is also a major cost of withholding

in the problem solving process for students as they seek concrete information to further

their methods. In the analysis of my text, however, I have found some contradictory

evidence within the relational ways of the FMP. In general, student reaction to

withholding in the FMP classroom is not one of frustration. Students seem to respond to

teacher withholding as an invitation to engage. As I spoke about the “hint” to a

geometric method to calculate the area of the quadrilateral to the alternative algebraic

method, but withheld the method itself, I heard nothing but excitement from the group.

Ms. Schettino: Well, do you remember the hint I gave you at the end of class yesterday?

Nancy: that’s what I did!

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 my answer is

withholding, students see it as an opportunity to conjecture and hypothesize, as in this


excerpt where a student is curious about other types of polygons that might be

equiangular but not equilateral.

Emma: what um, other ones are?

Sylvia: what’s the definition of a polygon?

Ms. Schettino: A polygon?

Sylvia: something with straight sides

Kim: no it’s enclosed

Nancy: Is a totally...a pentagon might be equiangular but not equilateral [getting back to

the original problem]

Ms. Schettino: she’s thinking that this [drawing on the board] is equiangular but not

equilateral

Nancy: no wait – Octagon?

Ms. Schettino: do you think that it is?

Shelley: No, there, there, there…

Sylvia: if it’s regular….

Laura: There’s rules about regular and not regular and if there’s lines…

Within that entire excerpt, I did not give any information, but students shared their ideas

and corrected each other. They introduced new terminology, which we now had an

obligation to define and discuss with respect to the problem at hand, but my withholding

of information did not seem to be frustrating the students at any turn. Perhaps the

framework of the pedagogy is such that creating an environment that forces students and

teachers to be susceptible to being uncomfortable, and living in that environment on a

regular basis, 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.

Pronomial Use

At many times in the excerpts I was able to observe the Feminist Pedagogical

Ideals of empowerment and encouraging student voice manifested in the use of personal

pronouns both by the students and myself. I noticed that many times I would model for

the students the importance of personal empowerment by focusing the utterances on the

self with pronouns, as in the following excerpt. Nancy was up at the board describing her


attempt at a solution for a problem where we were asked to find the distance between two

lines that happened to be parallel. However, students were not given the definition of

“distance” in the question and it was up to their own interpretation of what that meant in

the question. Nancy had just explained her method of finding distance, which included

making approximations for some points that looked like they were on the lines, but she

hadn’t checked her conjecture mathematically and I wanted her to clarify her method not

only for my understanding but also for the class as a whole.

Ms. Schettino: OK, uh, Now..I’m actually curious Nancy, where did you get the 8 and

9 from?

Nancy: Um, I did, I went like, You know how you do the Pythagorean theorem,

[pointing to her work on her notebook] you do this like vector over 8 and up 9, so I did

the Pythagorean Theorem, you know?

Ms. Schettino: but where did you get the 8 and the 9 specifically coming from?

Nancy: Um, I just counted

Ms. Schettino: from where to where?

Nancy: from point to point

Ms. Schettino: but where did you get the points from?

Georgia: [inaudibly tries to explain for her]

Nancy: yeah, I just..

Ms. Schettino: =You just picked lattice points?

Nancy: Yeah, I chose points that were like the same distance apart from each other

Ms. Schettino: [to Nancy]did you take a ruler and go [sound effect] and say ‘that looks

perpendicular’?

Georgia: No, cos you go down three and over four..

Ms. Schettino: Oooohhhh, I see..OK, OK,

Nancy: yeah, yeah, yeah, I did the slope and I drew a line…

Ms. Schettino: so you took the negative reciprocal slope and kind of [pause] counted

something that was approximately like, over three and up four, over three and up four,

and thought that that was the same distance?.. because I think it’s just off by a little

bit…

Nancy: off?

Ms. Schettino: Yeah, I don’t think those lattice points are exactly on the line.

In describing her own work in response to my questions, there were seven instances of

Nancy making use of the pronoun ‘I’. She 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 my discussion of her work, I deliberately

personalized the questions and statements by making the discussion about her and her

ideas by specific use of the pronoun ‘you’ a total of six times. 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, I am modeling interest, curiosity and respect for their own intuition,

prior knowledge and experience in the problem-solving process. In another short excerpt

students are discussing a question that was discussed previously:

Ms. Schettino: OK, what was the question?[3 sec]

Laura: Um, I think it was to, [pause] OK, give an example of an equiangular polygon

that is not equilateral. So, I did a rectangle because all the angles are the same but not

all of the sides are the same. [2 sec]

Megan: I did a rectangle too

Sylvia: I did a parallelogram

Ms. Schettino: Hmmm, does it work for any other ones? [2 sec] So that’s good, it’s

equiangular, [2 sec] but it’s not equilateral…right?

Megan: Yay

Ms. Schettino: So what about a parallelogram? Is..are the angles the same?[3 sec]

Sylvia: Well, I think they would be [5 secs]

Megan: no they’re not all the same

Nancy: I wasn’t sure, but is it like one that’s like, that’s like a triangle on top and a

square

Here, students are able to claim ownership of both statements of claim of knowledge and

statements of uncertainty and question. Within the group 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. There is also no

sense of preoccupation or hesitation about making claims of individual ownership about

utterances – be they positive (“Yay”) or negative (“I wasn’t sure”).


Once I noticed this trend, I totaled all of this type of pronominal use in all three of

the dialogical excerpts analyzed. The results are in the table below.

Table 1

Total Teacher and Student Pronomial Use

Individual I Inclusive We Generalized You

Specific You

Teacher 12 4 8 18

Student 43 2 8 1

Students were found to be using the personal pronoun of ‘I’ a striking 43 times in

approximately 26 minutes of dialogue text. That is approximately a student use rate of

the pronoun ‘I’ of 1.6 times per minute in class discussion, or almost twice a minute.

Alternatively, I was using the pronoun ‘you’ to talk about the students themselves at a

rate of .69 times per minute, or almost once a minute. The frequency with which our

classroom community made use of personal pronouns in discussion to serve as a means

of claiming ownership of our work, our questions and our feelings was dramatic.

Teacher Questioning & Politeness

In analyzing teaching questioning, I considered the overall ideal of creating a

culture of respect, safety and attending to experience. I needed to see if these were

evidenced in the text. Initially it was clear that many of the teacher questions were

procedural and rather traditional (“Who did number 9?” or “OK, did you get 19?”).

Other teacher questions may be attributed to a feminist relational pedagogy or could also

be found in any student-centered classroom that proposes to use guiding questions and

less teacher-centered instructional approaches.

Ms. Schettino: Now go horizontally across from C. Now what are the dimensions of

the rectangle that surround the parallelogram ABCD?


Ms. Schettino: OK...that 40 is not the area of that inner parallelogram – it’s the area of

what?

Ms. Schettino: =of the huge…of the rectangle. but what do we have to take away

now? Right,…to get rid of…?

These types of questions are not totally exemplary of a feminist or relational pedagogy

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 FMP in some ways. Questions like:

Ms. Schettino: Do you remember the hint I gave you at the end of class yesterday?

Ms. Schettino: So how many of you think that’s easier than writing the equation of the

altitude…?

Ms. Schettino: How many people did it the way Nancy did it? Kind of this slanted

way?

invite student response and participation because they are requesting the students’

opinion and individual voice to be heard. I am sincere in my question 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.

Conversational politeness is also a form of fostering a culture of respect and

safety. As previously stated, Rowlands (2000) posits that discursive hedges are a means

by which politeness is created in conversation and the lack of student hedges can be a

powerful indication of student agency in learning. I considered “I think” to be a shield

hedge, “a little”, “sort of”, “kind of” and “somewhat” to be adaptor hedges. For the

student analysis, I considered “I think”, “I guess”, “probably” and “maybe” to be the


plausibility hedges and “about”, “around” and “approximately” to be the rounders. In the

three text excerpts analyzed, I gathered evidence of teacher and student hedges to see if

the occurrence of hedges was indicative of polite behavior in the discussions.

Unfortunately, because of the limited size of the data set, and the relatively small amount

of hedges it was difficult to make any conclusions in relation to hedges and politeness.

Although not conclusive, this data might show that I am making much more of an effort

to create an environment of politeness in the discourse than the students, which can be

interpreted as my role-modeling or sensitivity to the students’ risk-taking and

vulnerability in problem-solving. Here is an example of my use of hedging from the

discourse:

Ms. Schettino: so you took the negative reciprocal slope and kind of [pause] counted

something that was approximately like, over three and up four, over three and up four,

and thought that that was the same distance?.. because I think it’s just off by a little

bit…

Here I feel it necessary to hedge to protect the student’s emotional well-being as I

question her process, that I knew was incorrect. In allowing her to follow her process

through to find her own mistake, her 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. I must create a climate that lets them 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 here was the lack

of student hedges. According to Rowlands (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. This confirms the assertion that FMP helps to encourage

student voice and empowers student agency in learning.

Conclusions

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 meaning. Through describing aspects

of utterances in textual context from my own classroom, 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. Although this study

was limited in that it included only one small class of all female students at a private

school, 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. Next

steps for future research include 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. However, without an initial thorough

description of the facilitation of these methods of discourse, teachers who have habitually

resorted to traditional, Initiation-Response-Evaluation, triadic dialogue 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 and

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.


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Revealing Dialogue in a Problem-Based Learning Mathematics ... · Dialogue in a PBL Classroom 2 Abstract In this paper, I describe the discursive nature of a Problem-Based Learning