Hollenbeck Dissertation Draft
Theoretical Framework
Theoretical Perspective
Emergent Perspective
In Principles and Standards for School Mathematics, the NCTM (2000) recognizes
that establishing a challenging and supportive learning environment is an important
component of teaching. In this document, the NCTM espouses that a significant aspect
of the teacher’s role is “establishing norms within a classroom learning community that
support the learning of all students” (p. 270, emphasis added).
The notion of norms specific to students’ mathematical activity is a relatively recent
construct in mathematics education literature (i.e., Lampert, 1990; Simon & Blume,
1996; Voigt, 1995; Yackel & Cobb, 1996). For Yackel and Cobb (1996), these norms
arose as the result of finding a cognitive perspective limiting when attempting to develop
accounts of students’ mathematical learning at the classroom level. To account for
student learning as it occurs in the social context of the classroom, Yackel and Cobb
(1996) found it necessary to develop an interpretive framework in which psychological
and social processes were given equal emphasis. Yackel and Cobb’s (1996) approach
reflects the view that “mathematical learning is both a process of active individual
construction and a process of acculturation into the mathematical practices of a wider
society” (p. 460). This point of view, which blends neo-Piagetian psychological
constructivism with Vygotskian sociological perspectives, is often referred to as social
constructivism or the emergent perspective (Cobb & Yackel, 1996).
The emergent perspective involves the explicit coordination of constructivism (von
Glaserfeld, 1990) and symbolic interactionism (Blumer, 1969). From a radical
constructivist lens, mathematical knowledge development is fundamentally a cognitive
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Theoretical Framework
process. Although social interaction can stimulate individual development, it is not
integral to the cognizing individual’s constructive activity (von Glasersfeld, 1990). A
defining principle of social interactionism is that individuals have to interpret what the
other is doing, and each person’s actions are formed, in part, on the actions of others
(Blumer, 1969). According to Blumer (1969), “symbolic interactionism sees meaning as
social products, as creations that are formed in and through the defining activities of
people as they interact” (p. 5).
Cobb and Yackel (1996) make clear that a reflexive relation exists between the
interactionist and psychological constructivist perspectives. Cobb et al. (2001) explains
that this reflexive nature “implies that neither perspective exists without the other in that
each perspective constitutes the background against which mathematical activity is
interpreted from the other perspective” (p. 122). This reflexive relationship assumes that
“neither an individual student’s mathematical reasoning nor the classroom microculture
can be adequately accounted for without considering the other” (Cobb, 2000a, p. 155).
Thus, from a sociological perspective a student’s reasoning is located within an evolving
microculture, and from a psychological perspective that microculture is treated as an
emergent phenomenon that is continually regenerated by the teacher and students in the
course of their ongoing interactions (Cobb et al., 2001). The emergent perspective
provides a valuable theoretical framework that accounts for the contributions of
individuals as it occurs in the social context of the classroom.
In Cobb and Yackel’s (1996) framework of the emergent perspective, three pairs of
categories are reflexively linked across social and psychological dimensions (figure 1).
Hollenbeck Dissertation Draft
Theoretical Framework
Social perspective
Classroom social norms
Sociomathematical norms
Classroom mathematical practices
Psychological perspective
Beliefs about own role, others’ roles, and
the general nature of mathematical activity
in school
Mathematical beliefs and values
Mathematical conceptions
Figure 1: Cobb and Yackel’s (1996) interpretive framework
Social norms characterize regularities in collective classroom activity jointly
established by the teacher and students as members of the classroom community and are
themselves continually being (re)generated by and through interactions (Cobb & Yackel,
1996). At the same time, the teacher and students reorganize their beliefs about their own
role, others’ role, and the general nature of mathematical activity through these same
interactions (Cobb, 2000b). Cobb (2000b) posits that:
[I]t is neither a case of a change in social norms causing a change in students’ beliefs,
nor a cause of students first reorganizing their beliefs and then contributing to the
evolution of social norms. Instead, social norms and the beliefs of the participating
students co-evolve in that neither is seen to exist independently of the other. (p. 69).
Classroom social norms, such as expectations that students persist in solving
challenging problems, listen to and attempt to make sense of other’s solutions, and ask
questions and raise challenges in situations of misunderstanding or disagreement, are not
specific to mathematics. Norms that are specific to the mathematical aspects of students’
activity are referred to as sociomathematical norms (Yackel & Cobb, 1996). Normative
understandings of what counts as mathematically different, mathematically sophisticated,
mathematically efficient, and mathematically elegant are examples of sociomathematical
norms (Yackel & Cobb, 1996). According to Cobb and Yackel’s (1996) framework,
what becomes mathematically normative in a classroom is enabled and constrained by the
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Theoretical Framework
students’ changing mathematical beliefs and values. At the same time, these beliefs and
values are themselves influenced by what is legitimized as acceptable mathematical
activity.
The third aspect of the interpretive framework concerns the mathematical practices
established by the classroom community and their psychological correlates, individual
students’ mathematical interpretations and actions. Cobb and Yackel (1996) explain,
Students actively contribute to the evolution of classroom mathematical practices as
they reorganize their individual mathematical activity, and conversely that these
reorganizations are enabled and constrained by the students participation in the
mathematical practice. (p. 180)
Although the emergent perspective was developed with a focus on students’ learning,
Cobb and Yackel (1996) note that their framework can be adapted to guide analyses of
teachers’ socially situated activity. According to Boaler (2002), social and
sociomathematical norms “offer a lens through which to examine and describe the colors
and contours of mathematics classrooms, giving names to some of the important choices
to which teachers and students attend in the activity of mathematics teaching and
learning” (p. 243). Using the notion of norms to analyze mathematics teaching allows
one to capture differences in mathematical activity in the classroom. For example,
Kazemi and Stipek (2001) use the construct of sociomathematical norms as a useful
framework for understanding what teachers need to do to promote meaningful
development of students’ mathematical ideas.
Identifying Social and Sociomathematical Norms
For Yackel and Cobb (1996), a fundamental feature of mathematics classrooms is that
they are characterized by certain social and sociomathematical norms. That is, in
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Theoretical Framework
mathematics classrooms, there are normative understandings regarding expectations and
obligations for social interactions and for specifically mathematical interactions (Yackel,
2000). Yackel (2000) explains that understandings are normative if there is evidence
from classroom activity that students’ interpretations are compatible or taken-as-shared.
Norms are not predetermined criteria set out in advance to govern classroom activity,
instead “these normative understandings are continually regenerated and modified by the
students and the teacher through their ongoing interactions” (Yackel & Cobb, 1996, p.
474). Although “methodologically, both general social norms and sociomathematical
norms are inferred by identifying regularities in patterns of social interaction” (Yackel &
Cobb, 1996, p. 460), Cobb (2000b) points out that normative taken-as-shared
interpretations cannot be observed directly. Instead, conjectures about communal
mathematical activity are developed and tested through the course of analyzing what the
teacher and students say and do in the classroom (Cobb, 2000b).
It is recognized that the differences between social and sociomathematical norms are
not easily distinguished. While social norms refer to the general ways that students
participate in classroom activities, sociomathematical norms concern the normative
aspects of classroom actions and interactions that are specifically mathematical (Yackel
& Cobb, 1996). To clarify the subtle distinction between social norms and
sociomathematical norms, Yackel and Cobb (1996) explain “the understanding that when
discussing a problem students should offer solutions different from those already
contributed is a social norm, whereas the understanding of what constitutes mathematical
difference is a sociomathematical norm” (p. 461). Ultimately, Herbst (1997) recognizes
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Theoretical Framework
that social and sociomathematical norms are social constructs and the distinction between
the two is made by an observer studying classrooms, not the teacher or students.
Social and sociomathematical norms are frequently interdependent. A social norm
that is described by the expectation that students regularly offer different solution
strategies is likely related to the normative understanding of what counts as a
mathematically different solution, a sociomathematical norm. Yet a classroom governed
by such a social norm need not necessarily have constituted the related
sociomathematical norm. It is conceivable that students can describe the steps they took
to solve a problem without understanding how their solution compares and contrasts with
others already offered. Many teachers find it easy to ask for different solution strategies,
however it is a more challenging endeavor to engage students in genuine mathematical
activity (Chazan & Ball, 1999; Kazemi & Stipek, 2001)
To identify and define general classroom social norms, several researchers have
described the classroom participation structure (Lampert, 1990; Kazemi & Stipek, 2001;
McClain & Cobb, 2001). Lampert (1990), drawing from the work of Florio (1978) and
Erickson and Shultz (1981), explains that a participation structure represents the
“consensual expectations of the participants about what they are supposed to be doing
together, their relative rights and duties in accomplishing tasks, and the range of
behaviors appropriate within the event” (p. 34). When developing conjectures about
social norms, Cobb et al. (2001) focuses on regularities in joint activity rather than an
alternative approach that casts criteria for social norms in terms of the proportion of
students who act in accord with a proposed norm. Cobb et al. (2001) explain that the
latter criterion is “framed from a psychological perspective that is concerned with
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Theoretical Framework
individual students’ activity rather than from a social perspective that is concerned with
how students’ activity is constituted in the classroom” (p. 123).
In studies examining sociomathematical norms, researchers closely examine the
nature of classroom mathematical discourse and the teacher’s role in those discussions
(Kazemi & Stipek, 2001; McClain & Cobb, 2001; Pang, 2000; Yackel & Cobb, 1996).
Kazemi and Stipek (2001) used examples of classroom exchanges to suggest how
sociomathematical norms governed classroom discussions. Lampert (2001) notes that
“each word and gesture the teacher uses has the potential to support the study of
mathematics for all students” (p. 144). In analyzing the process by which
sociomathematical norms emerge, McClain and Cobb (2001) point to the importance that
the teacher’s role in symbolizing students’ offered solutions played in the development of
sociomathematical norms.
Role of Teacher in Constituting Norms
Since norms are upheld by a process of social interactions, the specific norms that
become constituted are unique to each classroom. Nevertheless, to the extent that
classroom social norms constrain and enable learning, it is possible for teachers to initiate
and guide the constitution of norms in a purposeful manner (Yackel, 2000). Blumer
(1969) posits that in any collective body “there is one group or individual who is
empowered to assess the operating situation and map out a line of action” (p. 56). The
teacher, as an institutionalized authority in the classroom, “expresses that authority in
action by initiating, guiding, and organizing the renegotiation of classroom social norms”
(Cobb, 2000b, p. 69). Lampert (1990) used her own practice to demonstrate that, as a
teacher, she could initiate patterns in the classroom to build a participation structure that
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Theoretical Framework
redefined the roles and responsibilities of both teacher and students in relation to learning
and knowing mathematics. Research by McClain and Cobb (2001) show that a teacher
can guide the development of sociomathematical norms and thus influence her students’
beliefs about what it means to know and do mathematics. For me, this is a particularly
appealing aspect of Cobb and Yackel’s (1996) emergent perspective and their work on
normative behavior in the classroom. My research topic is based on the supposition that
a practitioner can create and maintain a classroom environment that regularly makes use
of student alternate solution strategies.
Sociocultural Theory
One limitation the emergent perspective has in explicating the development of norms
is that it primarily focuses on the local classroom community as a point of reference. The
emergent perspective does not explicitly take into account that students are part of other
communities that influence how they participate in the mathematics classroom.
Sociocultural theory “proposes that teachers need to understand the mathematical
knowledge that children bring with them to school from the practices outside of school as
well as the motives, beliefs, values, norms, and goals developed as a result of those
practices” (Forman, 2003, p. 337). Cobb and Yackel (1996) found occasions when it was
essential to use a sociocultural lens to take account of broader institutional contexts in
which classroom are located. In examining a teacher’s proactive role in developing a
classroom culture it will be useful to draw from sociocultural theory to examine the way
in which teachers use their knowledge of context in their interactions with students.