Post on 12-Sep-2020
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
Dialogue in a PBL Classroom 2
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.
Dialogue in a PBL Classroom 3
Since the National Council of Teachers of Mathematics published the new
Principles and Standards for School Mathematics (NCTM, 2000) there has been a lot of
interest in the new process standard regarding communication in the classroom. 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
Dialogue in a PBL Classroom 4
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
Dialogue in a PBL Classroom 5
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
Dialogue in a PBL Classroom 6
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
Dialogue in a PBL Classroom 7
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
Dialogue in a PBL Classroom 8
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
Dialogue in a PBL Classroom 9
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
Dialogue in a PBL Classroom 10
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
Dialogue in a PBL Classroom 11
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
Dialogue in a PBL Classroom 12
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
Dialogue in a PBL Classroom 13
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
Dialogue in a PBL Classroom 14
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.
Dialogue in a PBL Classroom 15
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
Dialogue in a PBL Classroom 16
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
Dialogue in a PBL Classroom 17
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
Dialogue in a PBL Classroom 18
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.
Dialogue in a PBL Classroom 19
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
Dialogue in a PBL Classroom 20
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
Dialogue in a PBL Classroom 21
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
Dialogue in a PBL Classroom 22
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
Dialogue in a PBL Classroom 23
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
Dialogue in a PBL Classroom 24
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”).
Dialogue in a PBL Classroom 25
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?
Dialogue in a PBL Classroom 26
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
Dialogue in a PBL Classroom 27
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
Dialogue in a PBL Classroom 28
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
Dialogue in a PBL Classroom 29
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.
Dialogue in a PBL Classroom 30
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