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American Journal of Education.
NOVEMBER 2012 137
American Journal of Education 119 (November 2012)
!2012 by The University of Chicago. All rights reserved.
Supporting Sustainability: Teachers’
Advice Networks and Ambitious
CYNTHIA E. COBURN
University of California, Berkeley
JENNIFER L. RUSSELL
University of Pittsburgh
JULIA HEATH KAUFMAN
University of Pittsburgh and Carnegie Mellon University
MARY KAY STEIN
University of Pittsburgh
Scaling up instructional improvement remains a central challenge for school
systems. While existing research suggests that teachers’ social networks play a
crucial role, we know little about what dimensions of teachers’ social networks
matter for sustainability. Drawing from a longitudinal study of the scale-up of
mathematics reform, we use qualitative social network analysis and qualitative
comparative analysis (QCA) to investigate the relationship between teachers’
social networks and sustainability. Teachers’ social networks in the ﬁrst 2 years
of the initiative inﬂuenced their ability to sustain reform-related instructional
approaches after supports for reform were withdrawn. Social networks with
combinations of strong ties, high-depth interaction, and high expertise enabled
teachers to adjust instruction to new conditions while maintaining the core
pedagogical approach. This research contributes to our understanding of the
dynamics of sustainability and to social network theory and research.
Scaling up instructional improvement efforts remains one of the central chal-
lenges facing urban school systems. In the past decade, no longer content to
have pockets of success, school districts across the country began to develop
systems for fostering instructional improvement among teachers and schools
throughout the district (Hightower et al. 2002). Yet, while some of these ex-
Electronically published September 18, 2012
138 American Journal of Education
periments in widespread instructional reform have begun to yield success
(Elmore and Burney 1999; Marsh et al. 2005; Snipes et al. 2002), sustainability
remains a persistent challenge (Coburn 2003). School district priorities almost
inevitably change over time. Resources to support initiatives run their course.
Schools and teachers that successfully implement instructional reforms ﬁnd it
difﬁcult to sustain them in the face of changing priorities, limited resources,
and competing demands (Berends et al. 2002; Hargreaves and Fink 2000;
Mac Iver et al. 2003).
Many studies have pointed to the nature and quality of teachers’ professional
relations with one another as a key contributor to their ability to sustain
instructional reform (Cooper et al. 1998; Gersten et al. 2000; Klingner et al.
1999; McLaughlin and Mitra 2001). This research provides evidence that
teachers’ professional interactions enable them to learn from one another
(Gersten et al. 2000; Klingner et al. 1999), deepen their practice, and co-
ordinate action (McLaughlin and Mitra 2001). Yet, while we know teachers’
social relations are important, few studies measure interaction directly or in
much detail. Thus, we know little about what dimensions of teachers’ social
relations matter for sustainability.
Researchers using social network theory and analysis have developed con-
ceptual and methodological tools for investigating social relations with more
precision. They provide evidence that several dimensions of social networks,
including tie strength, level of expertise, and depth of interaction, play an im-
portant role in such relevant outcomes as diffusion of innovation, transfer of
complex information, and reform implementation (Frank et al. 2004; Hansen
is associate professor of policy, organizations, mea-
surement, and evaluation at the Graduate School of Education, University of
California, Berkeley. She studies the relationship between policy and practice
in urban schools, school district evidence use, and teachers’ social networks.
is assistant professor of learning science and policy at
the School of Education and research scientist at the Learning Research and
Development Center, both at the University of Pittsburgh. She uses the con-
ceptual tools of organizational sociology to examine the relationship between
policy shifts and the organization of public schooling. J
is a research associate at the University of Pittsburgh and a research scientist
at the Open Learning Initiative at Carnegie Mellon University. Her research
focuses on how instructional policies and programs can support high-quality
teaching and learning in schools. M
holds a joint appointment
at the University of Pittsburgh as professor of learning sciences and policy
and senior scientist at the Learning Research and Development Center. Her
research focuses on mathematics teaching and learning in classrooms and the
ways in which policy and organizational conditions shape teachers’ practice.
Coburn et al.
NOVEMBER 2012 139
1999; Obstfeld 2005; Penuel et al. 2009; Reagans and McEvily 2003). But,
because this research tends to investigate a single dimension at a time, we know
little about how, if at all, these dimensions interact. Furthermore, this research
has largely focused on adoption and implementation. To date, there is no re-
search that investigates the relationship between social networks and
Here we use the tools of qualitative social network analysis and qualitative
comparative analysis (QCA) to investigate the relationship between teachers’
social networks and sustainability. We draw on data from a study of one
district’s attempt to develop teachers’ capacities to teach mathematics in a
more student-centered, conceptually-based manner, using an innovative math-
ematics curriculum called Investigations in Number, Data, and Space. After spending
proach, including professional development, school-based mathematics coach-
ing, and time for teachers to look at data and co-plan instruction, the district
shifted its priorities and withdrew these supports. In the third year of the
initiative, teachers were faced with sustaining new instructional strategies in
the absence of formal support. Just over half of the teachers in our sample
were able to do so. We argue that the nature of teachers’ social networks in
the ﬁrst 2 years of the initiative played an important role in whether and how
teachers sustained reform-related instructional approaches in year 3. More
speciﬁcally, social networks with a combination of a high level of expertise,
strong ties, and high-depth interaction focused on substantive issues related
to mathematics and pedagogy sufﬁcient to support sustainability. Social net-
works with this combination of conditions enabled teachers to develop a
strong enough understanding and ﬂexible enough enactment of reform-
related strategies that they were able to adjust to new conditions in year 3
while maintaining the underlying pedagogical approach. After elucidating
how these different dimensions of social networks worked together to support
sustainability, we close with contributions to research on sustainability and
Social Networks and Sustainability
Sustainability is one of the central challenges for instructional improvement
efforts. Several decades of research document the challenges of maintaining
new instructional approaches in the face of changing priorities, competing
initiatives, and shifting conditions in schools and districts (Coburn 2003; Cu-
ban 1993; Datnow et al. 2002; Tyack and Cuban 1995). Yet, as important as
this issue is, existing research on sustainability is limited. Most studies inves-
tigate new initiatives at their start. Few researchers continue to collect data
140 American Journal of Education
after the initial infusion of resources and support is withdrawn (Coburn 2003;
Gersten et al. 2000; Hargreaves and Goodson 2006). Thus, they are not able
to capture sustainability, deﬁned here as the degree to which teachers use
reform-related practices in high-quality ways after support for these practices
has dissipated. Other studies of sustainability initiate investigation after support
for implementation has ended. While these studies are well suited for capturing
the degree to which an instructional approach is sustained, they typically rely
on retrospective, self-reported data to understand implementation. Because
retrospective data are often very general and can be smoothed by the passage
of time, this approach can provide only limited insight into what happens
during implementation that fosters sustainability.
implementation and sustainability in real time. These studies have mainly
focused on identifying a list of factors that appear to contribute to teachers’
abilities to maintain instructional reform over time, including effective and
stable school leadership (Berends et al. 2002; Bryk et al. 2010; Datnow et al.
2002; Gersten et al. 2000; Klingner et al. 1999), teachers’ social relations (Bryk
et al. 2010; Datnow et al. 2002; Hargreaves and Goodson 2006; Klingner et
al. 1999; McLaughlin and Mitra 2001), supportive or at least not unsupportive
district climate (Berends et al. 2002; Coburn 2003; McLaughlin and Mitra
2001), sufﬁcient resources (Klingner et al. 1999; McLaughlin and Mitra 2001),
and strong understanding of the reform on the part of teachers (Coburn 2003;
Klingner et al. 1999; McLaughlin and Mitra 2001). But few studies investigate
any given factor related to sustainability in any detail. Thus, while these studies
play an important role in identifying key factors that may be involved, it is
time for more in-depth studies of when, why, and how speciﬁc factors con-
tribute to sustainability. Here, we contribute to the nascent literature on sus-
tainability by conducting an in-depth investigation of the relationship between
one factor identiﬁed in the literature—teachers’ social relations—and teachers’
ability to sustain instructional reform over time.
Te a c h e r s ’ S o c i a l R e l a t i o n s
Existing research that attends to the role of teachers’ social relations in sus-
tainability ﬁnds that teachers’ interactions with others are important because
they provide access to knowledge, feedback, and social support that enables
teachers to deepen their understanding and enactment of new approaches
(Cooper et al. 1998; Gersten et al. 2000; McLaughlin and Mitra 2001). In
turn, teachers with at least a moderate level of implementation or deep un-
derstanding are more likely to sustain those reforms (Gersten et al. 2000;
Klingner et al. 1999; McLaughlin and Mitra 2001), likely because deep un-
Coburn et al.
NOVEMBER 2012 141
derstanding enables teachers to make principled, practical adjustments to new
students, conditions, and subject matter in ways that maintain the underlying
pedagogical approach (Coburn 2003; McLaughlin and Mitra 2001). Strong
relationships further support sustainability by fostering common approaches
and reinforcing norms that coordinate action by helping teachers move in the
same direction (Hargreaves and Goodson 2006; McLaughlin and Mitra 2001).
In the absence of such social support, teachers feel isolated (Klingner et al.
1999; McLaughlin and Mitra 2001) and have difﬁculty navigating changing
demands (Hargreaves and Goodson 2006), threatening sustainability.
However, existing research on teachers’ social relations and sustainability
suffers from several limitations. Most studies fail to measure teachers’ social
relations with much precision. Some studies discuss teachers’ social resources
quite generally, using many different terms (e.g., “professional communities,”
“peer support,” or “communities of practice,” to name a few) and providing
little information about how they measured teachers’ social relations at all
(e.g., Gersten et al. 2000; Hargreaves and Goodson 2006). Others employ
careful measurement strategies but do not measure the nature of teachers’
interaction directly. Instead these studies rely on teachers’ perceptions of the
nature of collaboration or level of supportiveness of colleagues in the school
as a whole (e.g., Bryk et al. 2010; Klingner et al. 1999; Louis and Marks
1998). This approach does not attend to who teachers are actually interacting
with and the nature and quality of that interaction. Furthermore, existing
studies tend to assume in advance the locus of professional community, focusing
on the school as the unit of analysis or on formal organizational structures
like grade-level groups or departments (e.g., Bryk et al. 2010; Louis and Marks
1998; McLaughlin and Mitra 2001). Yet we know that teachers are often
embedded in a network of relations that span multiple subgroups and include
individuals both inside and outside of school boundaries (Bidwell and Yasu-
moto 1999; Coburn 2001; Coburn et al. 2012; Penuel et al. 2009; Yasumoto
et al. 2001). The nature of teachers’ social relations can also vary greatly
within a school or even a grade level (Coburn et al. 2012; Penuel et al. 2009).
Consequently, there is still much to learn about just what it is about teachers’
social relations that matters for their ability to sustain new instructional strat-
egies over time.
Social Networks Theory and Analysis
To ad d r e ss t he se i ss ue s , we d ra w th e or et i ca ll y an d m et h od ol og i ca ll y fr o m s o c ia l
network theory. Social network theory is an approach for understanding how
an individual’s or organization’s location in a web of social relations enables
and constrains a range of organizational and individual processes and out-
142 American Journal of Education
comes. Rather than explaining social phenomenon by investigating attributes
of individuals or organizations, social network theory shifts the angle of vision
to the system of social relations within which action is embedded. It then
investigates how the conﬁguration of these relations matters for a range of
important outcomes (Borgatti and Ofem 2010; Kilduff and Brass 2010). Social
network analysis is a methodological approach developed to investigate the
nature and conﬁguration of these social relations. It involves systematically
mapping patterns of relationships by identifying the ties between individuals
(or organizations), the features of those ties, and how ties between dyads
interconnect to form a system of relations as a whole (Valente 1995).
To date, we know of no studies that use social network theory and analysis
to investigate the relationship between social relations and sustainability. How-
ever, there are studies on the relationship between social networks and reform
implementation from which we can draw guidance for building our conceptual
framework. Researchers have identiﬁed a number of dimensions of social
networks that inﬂuence diffusion of innovation, knowledge transfer, and im-
plementation. Here, we focus on three: tie strength, access to expertise, and
depth of interaction.
Tie strength.—There is evidence that strong ties—those characterized, var-
iably, by high frequency, social closeness, or a combination of the two—are
beneﬁcial for transfer of ﬁne-grained or tacit knowledge, collaboration, or
sustained problem solving (see Obstfeld  for a review). To the extent
that sustainability requires substantial reform-related knowledge (Gersten et
al. 2000; Klingner et al. 1999; McLaughlin and Mitra 2001), it is possible
that strong ties may be important for sustainability as well.
Level of expertise.—While most social network research has focused on struc-
tural and relational aspects of networks, there is a growing interest in the
nature of the “competencies and resources” that are available at the nodes
(Adler and Kwon 2002, 26). Scholars reason that if social networks are a
means to access the resources that are available by virtue of one’s position in
attend to the nature of those resources as well. Indeed, research outside of
education has shown that social networks differ markedly in the level and
diversity of resources—information, contacts, people with power—to which
they afford access (Lin 2000). Research in education has mainly focused on
the degree to which social networks create access to expertise. For example,
a series of studies by Frank and Penuel ﬁnds that the mean level of reform-
speciﬁc expertise in one’s subgroup and the other subgroups with which one
interacts predicts implementation for reform efforts ranging from use of com-
puters in the classroom (Frank et al. 2004) to complex literacy reform (Penuel
et al. 2010). These scholars suggest that reform-speciﬁc expertise in one’s
network provides access to real-time support that assists a teacher in strength-
Coburn et al.
NOVEMBER 2012 143
ening enactment. Given research on the role of strong enactment in sustain-
ability (Gersten et al. 2000; Klingner et al. 1999), access to expertise may also
foster a teacher’s ability to sustain this enactment after supports for imple-
mentation are withdrawn.
Depth of interaction.—Although social network theory is premised on the
notion that resources “ﬂow” through interpersonal connections, few scholars
directly investigate what transpires in social network transactions. Instead they
infer the nature of interaction from the structure of the ties or the attributes
of the nodes (Borgatti and Foster 2003). This has led to a range of competing
theories about the mechanism by which social networks contribute to valued
outcomes (Borgatti and Foster 2003; Kilduff and Brass 2010). Some scholars
argue that social networks inﬂuence outcomes via peer pressure and social
control (Centola 2010; Coleman 1988). Others argue that they facilitate change
by creating conditions for individuals and groups to learn from and with one
another (Hansen 1999; Reagans and McEvily 2003; Uzzi and Lancaster 2003).
Still others point to their beneﬁts for coordinating action (Obstfeld 2005).
Absent information about what actually happens in social network transac-
tions, it is very difﬁcult to adjudicate these conﬂicting claims.
Work in education has begun to addre s s t h i s i s s u e b y developing methods
to attend to the content of interaction in social networks. This work dem-
onstrates that teachers’ social networks vary considerably in the depth of
interaction (Coburn and Russell 2008; Horn and Little 2010), from quick
exchanges about how students are doing or a pending deadline to in-depth
and substantive conversations about mathematical content or the nature of
student learning. Depth of interaction varies even in networks with similar
structure or resources (Coburn et al. 2012). Interactions at contrasting depth
have different potential to foster the valued outcomes identiﬁed by social
network theorists, making it more or less likely that individuals share valuable
information even if they have it available, engage in joint problem solving,
or learn in interaction with others.
In spite of increased interest and attention to sustainability, there is sur-
prisingly little empirical research that employs research designs suitable for
investigating the phenomenon. Studies that do highlight the role of teachers’
social relations as a key factor in promoting sustainability suffer from meth-
odological limitations. Existing research on social networks is useful because
it identiﬁes dimensions of teachers’ social relations that may matter for sus-
tainability. But because each line of research tends to focus on a single di-
mension of networks to the exclusion of others (e.g., tie strength or access to
resources), we know little about how these dimensions might interact to in-
ﬂuence change in practice. Furthermore, research on social networks both
inside and outside of education has focused on either adoption or implemen-
tation. Thus, in spite of promising preliminary research in education that
144 American Journal of Education
points to the importance of teachers’ social relations, we know little about
how the quality and conﬁguration of teachers’ social networks are related to
their ability to sustain new instructional approaches over time.
To i n v es ti ga t e th e r e la ti on sh i p bet w e en t ea ch er s ’ soc i a l ne tw or k s an d su st ai n -
ability, we draw on data from a longitudinal study that investigated how district
reform strategies interacted with human and social capital in the development
of teachers’ capacities to enact new approaches to the teaching of mathe-
Speciﬁcally, we studied teachers in four urban elementary schools in
ship between their social networks and their ability to sustain new instructional
strategies after supports for the mathematics initiative were removed.
Context and Sample
Greene School District is a mid-size urban district that was in the ﬁrst year of
its efforts to improve elementary mathematics instruction in the ﬁrst year of the
The district sought to improve teachers’ instruction by emphasizing a
more conceptually-based, student-centered mathematics approach encouraged
by the National Council of Teachers of Mathematics (1989, 2000). In order to
support teachers’ ability to enact this new approach, the district adopted the
curriculum Investigations in Data, Numbers, and Space in the 2003–4 school year
after piloting selected units the year before. Investigations embodies a conceptual
approach to mathematics learning by focusing on the development of students’
understanding of the “big ideas” in elementary mathematics rather than em-
phasizing the use of correct algorithms and memorization as mathematics
curricula traditionally do (TERC 2011). By having students engage in explo-
rations and problem solving, often in real-world contexts, the curriculum
provides opportunities for students to learn to think, reason, and make sense
of mathematics. This approach is demanding for teachers to implement well
because instead of simply demonstrating one correct way to solve each type
of mathematics problem, teachers must listen to and interpret student thinking
and then subtly steer their thinking toward the canonical mathematical ideas
that are the goal of the lesson (Stein et al. 2008). Because many teachers are
unaccustomed to teaching in this manner, the curriculum and the pedagogical
approaches it promotes typically require high levels of teacher learning to
implement well (Stein and Kim 2009). Throughout this article, we refer to
this pedagogical approach as “reform-related instructional strategies.”
Coburn et al.
NOVEMBER 2012 145
In fall 2003, the district launched an initiative to support this learning.
Among other things, they initiated school-based instructional coaches, who
were supported by the district and worked with teachers one-on-one and in
groups. The district also instituted biweekly school-based professional devel-
opment in mathematics instruction, periodic district-wide workshops, and
weekly grade-level meetings to facilitate joint planning. But, in year 3, after
a change in district leadership, the district shifted its priorities and withdrew
support for the initiative. Professional development at both the district and
school level was redirected to focus on English language learning (ELL) strat-
egies. Budget and stafﬁng decisions were rolled down to the school level, and
many principals decided to cut or reduce mathematics coaches. All four schools
we studied went from either two or three half-time coaches in years 1 and 2
to a single half-time coach in year 3. Mathematics instruction was cut from
90 to 60 minutes a day to make room for 30 minutes a day of strategies for
supporting English language learners. Because virtually all supports for the
approach to mathematics embedded in the Investigations curriculum were with-
drawn in the third year of the study, we had an ideal opportunity to investigate
issues of sustainability in real time.
Consistent with the exploratory, theory-building purpose of our study, we
used purposive sampling (Strauss and Corbin 1990) to select elementary
schools and teachers within those schools for our study. Because the overall
study was interested in how schools with contrasting organizational condi-
tions—different levels of social and human capital—implemented the new
mathematics curriculum, we sought four schools that varied along these two
dimensions. Selection was based on recommendations from the district di-
rectors of mathematics. Speciﬁcally, we asked the director of mathematics in
the district to nominate schools where the faculty had, on average, relatively
high and low levels of human and social capital, with human capital described
as mathematics instructional expertise and social capital described as inter-
action about mathematics instruction. The ﬁnal sample thus included four
contrasting organizational conditions: one school with strong professional com-
munity and strong teacher expertise, one with strong professional community
and weak teacher expertise, one with weak professional community and strong
teacher expertise, and one with weak professional community and weak
We selected four focal teachers in thre e s c h o o l s . I n t h e fourth—School H—
we were only able to select two focal teachers for logistical reasons. Teachers
at all four schools were selected to represent a range of grades and attitudes
toward the new curriculum. Two of the original 14 teachers left their schools
during the 3 years of the study. Both were new teachers in year 1 and, like
many new teachers (Johnson et al. 2005; Loeb and Reininger 2004), decided
to leave the profession after a few years of teaching. For this article, weincluded
146 American Journal of Education
only the 12 teachers for whom we have 3 years of data (see the methodological
appendix [app. D], available online, for more information on sampling, the
selected schools, and the 12 focal teachers).
To analyze the relationship between teachers’ social networks and sustain-
ability, we relied primarily on interviews, observations, and document analysis.
We conducted two interviews and two classroom observations wi t h e a c h f o c a l
teacher in year 1. We then expanded our data collection in years 2 and 3,
collecting ﬁve interviews and six classroom observations for each focal teacher.
Each year we also conducted one to two interviews with mathematics coaches,
two interviews with the school principal, and one interview with six additional
teachers (whom we called nonfocal teachers). We also observed three to ﬁve
occasions in each school where teachers interacted on matters of mathematics
instruction (professional development, grade-level meetings, coaching sessions,
Social network data.—A subset of this data collection was designed speciﬁcally
to investigate focal teachers’ social networks. We took an egocentric approach
to social network analysis. In this approach, the analyst maps networks that
are centered around an individual or social unit (the ego; Wellman and Ber-
kowitz 1988). To do this, we interviewed focal teachers individually, using
questions designed to ﬁnd out who a teacher talked with about mathematics
instruction (both inside and outside of the school) and the frequency and
content of their interaction, as well as why they talked with some people and
not others. (See the last section of the methodological appendix in the online
version of the article for social network interview questions.) We then analyzed
these data and selected an additional six teachers to interview in each school
(nonfocal teachers) who were part of focal teachers’ social networks. We in-
terviewed nonfocal teachers using the same battery of social network questions,
supplemented with questions on their use of curriculum and background in
mathematics. This approach allowed us to further investigate the qualities of
focal teachers’ networks, including the location of expertise and content of
interaction. We also used the same battery of social network questions with
coaches and principals.
We supplemented the interviews by o b s e r v i n g o c c a s i o n s w h e r e focal teachers
interacted with colleagues identiﬁed in their social network interviews and by
drawing on select measures from a teacher survey.
We used sur v e y q u e s t i o n s
designed to investigate communication channels to generate our measures for
frequency of interaction. Because the survey was not administered in year 1,
we used questions from the survey in interviews in year 1 for that information.
Coburn et al.
NOVEMBER 2012 147
Classroom practice data.—To ascertain the degree to which teachers used
reform-related instructional approaches, we observed each focal teacher’s
classroom two times in the spring of 2004 and three times each in the fall
and spring of the 2004–5 and 2005–6 school years.
Trained observers took
ethnographic ﬁeld notes and then completed a structured analysis upon
leaving the classroom. The structured analysis included a lesson summary
and answers to questions about cognitive demand, teachers’ attention to
student thinking, and the location of intellectual authority during the lesson.
Each of these dimensions is described below in more depth.
To analyze how teachers’ social networks across the 3 years of the study related
to their ability to sustain reform-related mathematics instructional strategies
in year 3, we coded multiple dimensions of teachers’ social networks and the
quality of teachers’ reform-related mathematics instruction in each year of
Analysis of social networks.—We mapped ea c h o f t h e 1 2 f o c a l t e a c h e r s ’ n e t -
works for each of the 3 years, drawing upon interview data to build egocentric
networks. We then analyzed three dimensions of the networks: tie strength,
level of expertise, and depth of interaction. To assess tie strength, we analyzed
the frequency with which a teacher interacted with others in her network. We
used questions from interviews in year 1 and surveys in years 2 and 3 that
asked: how many times per month do you talk with these individuals about
mathematics? Following convention in the literature, we then created an ag-
gregate measure of tie strength across a given network (e.g., Hansen 1999;
Reagans and McEvily 2003; Uzzi 1999). Our aggregate measure was the
average of the focal teacher’s rate of interaction with each individual in her
network. We considered a network to have strong ties if teachers reported
interacting with others in the network, on average, more than twice a month.
(See the methodological appendix online for additional information about
social network measures.)
To analyze level of expertise in a network, we assessed the degree to which
individuals in a teacher’s social network had participated in prior professional
learning opportunities related to mathematics. (See app. A for deﬁnitions of
low, medium, and high expertise.) After evaluating the expertise of each in-
dividual in a focal teacher’s network, we created an aggregate measure of the
level of expertise in a given network by calculating the percentage of individuals
in a network with moderate or high expertise. To set a cut-point for a high
level of expertise, we drew on existing studies of the prevalence of expertise
in mathematics among elementary teachers. These studies suggest that ex-
148 American Journal of Education
pertise in mathematics is rare in elementary schools, with no more than one-
third of teachers in a given study having high levels of conceptual understand-
ing (Ball 1990; Ma 1999; Post et al. 1991). Therefore, we considered teachers
to have access to high levels of expertise if more than one-third of the indi-
viduals in their network had moderate or high expertise.
To analyze depth of interaction, we identiﬁed 419 instances in our data where
the 12 teachers in our sample interacted with others in their social network.
We drew on research on teacher interaction (Coburn 2003; Little 1990) to
develop criteria for assessing the depth of the content of interaction. Interaction
was judged to be at low depth when it focused on surface structures or pro-
cedures such as sharing materials, classroom organization, pacing, or how to
use the curriculum. Interaction was judged to be at high depth when it ad-
dressed underlying pedagogical principles, the nature of the mathematics, or
how students learn (see app. A for complete deﬁnitions of depth). Because
prior studies suggest that high-depth interaction is rare (Little 1982, 1990;
Lortie 1975; Sun et al. 2011), we characterized teachers’ social networks as
high depth if at least one-third of interactions that teachers had with others
in their networks was about content judged to be of moderate or high depth
(see the methodological appendix online for information about interrater
Sustainability of reform-related instruction.—We d e ﬁ n e s u s t a i n a b i l i t y a s t e a c h e r s ’
continued use of reform-related instructional practices in high-quality ways
after resources and support for such practices have been withdrawn. This
conceptualization is more restrictive than those solely emphasizing endurance
of a practice or program over time. We adopt this deﬁnition because research
suggests that many reforms falter when support for that reform is withdrawn
(Coburn 2003; McLaughlin and Mitra 2001) and that for a reform to continue
past this point, it must be self-supported and/or institutionalized (see also
Johnson et al.  on this point).
Consistent with Bryk and his colleagues (2010) and research on teacher
learning (Smith 2000; Stein et al. 2011), we assume that there are multiple
pathways to sustainability. Teachers may develop high-quality reform-related
practices early in the initiative and maintain them over time. Or they may
gradually develop and deepen their enactment over several years such that
teachers do not achieve high-quality reform-related instruction until the point
when supports for the initiative are withdrawn. Or teachers might develop
high-quality reform-related practices early on, decline in the quality of en-
actment, and regroup and deepen practice in ways that enable them to sustain
it over time.
Key to this conceptualization of sustainability is a clear deﬁnition of the
enactment of reform-related instruction with high quality. We deﬁne high-
quality reform-related instruction in terms of three hallmarks of a concep-
Coburn et al.
NOVEMBER 2012 149
tually-based, student-centered approach to teaching and learning mathe-
matics: (1) the ability to set up and maintain high-cognitive-demand
instructional tasks, (2) the ability to attend to, elicit, and build upon student
thinking in the classroom, and (3) the ability to place the intellectual authority
for mathematical correctness on thinking and reasoning rather than the
textbook or the teacher. Our conceptualization views the Investigations cur-
riculum as a vehicle for guiding teachers toward high-quality instruction,
not as an end in itself. As such, our outcome measure is the combination
of the above features of high-quality instruction, not ﬁdelity to the curric-
To an a l yz e th e d eg re e t o wh ic h te ac he rs t a ug ht m at h em at ic s us i n g hi gh -
quality reform-related instruction, we analyzed the 163 lessons from the 12
focal teachers across 3 years of our study. We drew on the ethnographic ﬁeld
notes from the observation, structured observation analysis, and artifacts from
the lesson to code the three dimensions of instruction.
Maintenance of cognitive demand.—Cognitive demand refers to the level of think-
ing and reasoning that is required for students to successfully complete a
mathematical task (Doyle 1983; Stein et al. 1996). However, even if a math-
ematical task from the curriculum has a high level of demand, teachers may
(knowingly or unwittingly) lessen the cognitive demand of the task when they
set it up in the classroom (e.g., by inserting easier numbers into the problem
or by providing “hints” regarding what to look for) or as students work on
the task (e.g., by “taking over” and doing the thinking for the students instead
of allowing them to struggle). Lessons that maintain a high level of cognitive
demand throughout are associated with greater student learning gains than
are lessons in which the level of cognitive demand starts out at a low level or
declines during the course of the lesson (Stein et al. 1996; Stein and Lane
To analyze the degree to which teachers maintained cognitive demands of
mathematical tasks, we identiﬁed ﬁve types of instructional tasks, two with
high cognitive demands (doing mathematics and procedures with connections
to meaning) and three with low cognitive demands (procedures without con-
nections to meaning, memorization, unsystematic or nonproductive explo-
ration). (See app. B for deﬁnitions of the different types of tasks.) Teachers
received a “high” rating for cognitive demand if they maintained a high
cognitive demand as they moved from the materials to the set-up of the task
in the classroom to the enactment of the task in all but one of their lessons
in a given year.
(See app. C for more detail on coding for reform-related
Attention to student thinking.—Teachers who enact reform-related instruction
with high quality attend to student thinking. That is, they are able to identify
the mathematical learning potential of strategies students use to solve math-
150 American Journal of Education
ematical problems and identify which student responses would be important
to share with the class as a whole during the ensuing discussion (Brendefur
and Frykholm 2000; Lampert 2001; Stein et al. 2008). To assess attention to
student thinking, we assigned each lesson a score of 0 to 3 depending on the
extent to which teachers uncovered student thinking and made it available to
other students to help the class’s learning as a whole, with 0 points if the
teacher did not try to uncover student thinking and 3 points if the teacher
asked students to publicly share their thinking, purposefully selected some
student to share their work, and connected or sequenced student responses
in meaningful ways. Teachers received a “high” rating for their attention to
student thinking across a year if they received a yearlong average above 1
across all lessons.
Intellectual authority.—We analyzed the extent to which the intellectual au-
thority in the classroom was vested in mathematical reasoning rather than the
teacher or text. A learning environment embodying the norm of accountability
to the discipline regularly encourages students to account for how their ideas
make contact with those of other mathematical authorities, both inside and
outside the classroom (Cobb et al. 1997; Lampert 1990). Each lesson was
assigned a score of from 0 to 2, depending on the extent to which the majority
of the students had such opportunities, where 0 meant that judgments of
correctness derived from the teacher or the text and 2 meant that judgments
of correctness derived from mathematical reasoning. Teachers received a
“high” rating for intellectual authority if they received a yearlong average
above 1 across all their lessons.
Next, we computed an aggregate rating for instructional quality that com-
bined lessons and dimensions for a given year. We considered a teacher to be
enacting reform-related instruction with high quality in a given year if her
lessons were above the bar of high quality for cognitive demand and either
student thinking or intellectual authority. While previous research has used
the maintenance of high cognitive demand as an all-encompassing measure
of high-quality reform mathematics instruction (Henningsen and Stein 1997;
Stein and Lane 1996), Greene’s reform effort also focused strongly on teachers’
interaction with students and their ability to support students’ use of math-
ematical strategies. Thus, we reasoned that requiring high attention to student
thinking or high reasoning-based intellectual authority—in addition to main-
tenance of cognitive demand—was a demanding, but realistic, requirement
for a rating of high quality.
Finally, we considered a teacher to have sustained reform-related instruction
if she or he was able to maintain or develop high-quality reform-related
instruction in year 3 after the district withdrew all resources and other supports
for this form of instruction. Figure 1 provides a representation of how we
Coburn et al.
NOVEMBER 2012 151
.1.—Sustainabilityofhigh-quality reform-related instruction
drew on multiple analyses to assess sustainability of high-quality reform-related
Qualitative Comparative Analysis
After the social network data and the classroom data were coded, we sought
to understand the relationship between the two using a technique called qual-
itative comparative analysis (QCA). Rather than relying on probability and
statistics, as is commonplace in quantitative analysis, QCA draws on set theory
and uses Boolean algebra (formalized logic) to investigate complex social phe-
nomenon. The approach is particularly well suited to research that involves
a small number of cases and a large number of conditions that may be relevant
for the phenomenon of interest (Ragin 1987; Rihoux and Ragin 2009). It is
also well suited for exploratory analysis of the sort we pursue in this article,
as it enables the researcher to engage in systematic and disciplined comparison
across multiple, multidimensional cases, using a structured dialogue between
theory and empirical data to surface new relationships that build and extend
theory (Rihoux and Ragin 2009).
Rather than having additive or linear assumptions about the impact of
various conditions on an outcome, QCA assumes that a given condition may
work differently depending on how it interacts with other conditions, forces,
or contexts. Thus, it focuses attention on how conﬁgurations of conditions—
what Ragin calls “causal recipes”—interact with one another in speciﬁc settings
with consequences for a given outcome (Ragin 2008). QCA is also rooted in
an assumption of multiple conjunctural causation. That is, it allows for the
possibility that more than one conﬁguration of conditions can lead to the
same outcome. The goal of QCA is to identify all of the conﬁgurations of
conditions that can account for all cases with a given outcome. Thus, the
152 American Journal of Education
researcher does not try to “specify a single causal model that ﬁts the data
best,” but instead seeks to “determine the number and character of the dif-
ferent causal [recipes] that exist among comparable cases” (Ragin 1987, 167).
In QCA, the analyst begins by selecting conditions that are theoretically as-
sociated with the outcomes of interest. We deﬁned the outcome of interest as high-
quality reform-related instruction in year 3. We then drew on our conceptual
framework to select conditions that preexisting theory suggested might be asso-
ciated with this outcome: the presence or absence of high-depth interactions, high
level of expertise, and strong ties in all 3 years. Following standard protocol for
QCA, we used the software fsQCA to array the conditions and outcome in a
“truth table” that lists all logically possible combinations of conditions and the
empirical outcome associated with each. The truth table enables the analyst to
assess the degree to which a given condition or conﬁguration of conditions leads
to consistent outcomes (all negative or all positive), a metric known as set-theoretic
consistency. If a conﬁguration of conditions is not associated with consistent out-
comes across cases, then the conjecture that this set of conditions is associated
with that outcome is not supported. Generally, a model—a set of conﬁgurations
of conditions—is supported if it attains a consistency of .80 or higher (Ragin 2008).
The analyst also assesses the degree to which the conﬁgurations of conditions that
are identiﬁed, taken together, can account for all the positive cases (Rihoux and
Ragin 2009), a metric known as set-theoretic coverage. Coverage is a measure of
the strength of the model. For example, one can attain consistent results but only
be able to account for a small percentage of overall cases, indicating that the
explanation is not very strong. The goal is to identify a model that can account
for all of the cases in the sample, but coverage of at least .80 is considered
We deﬁned our outcome of interest as high-quality reform-related instruc-
tion in year 3. We then drew on our conceptual framework to select conditions
that preexisting theory suggested might be associated with this outcome: the
presence or absence of high-depth interaction, high level of expertise, and
strong ties in all 3 years. We found that this set of conditions did not yield
consistent solutions. That is, there were conﬁgurations of conditions that led
to both positive and negative outcomes. This suggests that the social network
conditions across all 3 years cannot account for teachers’ ability to sustain
reform-related instruction in year 3.
In QCA, if the conﬁgurations of conditions in the truth table do not yield
consistent outcomes or if they cannot account for a sufﬁcient portion of cases, the
analyst draws on theory or in-depth knowledge of the cases to either add or
remove a condition or combination of conditions from the model. The analyst
proceeds in step-wise fashion until he or she establishes the combination of con-
ditions that yield consistent outcomes, that makes sense theoretically, and that can
account for all of the cases in the sample (Rihoux and Ragin 2009).
Coburn et al.
NOVEMBER 2012 153
We re a s o n e d t h a t a teacher’s ab i l i t y t o s u s t a i n i n s t r u c t i o n i n t h e f a c e o f
challenging conditions may be rooted in the development of strong understand-
ing of pedagogy and student learning (Gersten et al. 2000; Klingner et al. 1999;
McLaughlin and Mitra 2001). This understanding may not be developed in the
moment but rather over the course of several years, suggesting that the nature
and conﬁguration of the social network in year 3 may be less important than
in years 1 and 2. We developed a new truth table that included social network
dimensions from the ﬁrst 2 years only. This truth table attained perfect consis-
tency and could account for all positive outcomes. This ﬁnding suggests that
the three social network conditions in years 1 and 2, taken together, could
account for the presence or absence of high-quality reform-related instruction
in year 3.
Anonymous reviewers raised questions about whether the quality of teach-
ers’ enactment in years 1 and/or 2, teachers’ expertise at the start of the study,
or school leadership might also play a role in teachers’ ability to sustain reform-
related instructional practices in year 3. In response, we generated truth tables
that added membership in a school (to capture school leadership), high-quality
reform-related instruction in year 1, high-quality instruction in year 2, and
teacher expertise alone and together to the social network conditions already
in the model. The addition of membership in a school and teachers’ expertise
did not generate set-theoretic consistency at an acceptable level. Therefore,
the analysis does not support the conjecture that social network conditions in
combination with expertise, membership in a school, or both are associated
with high-quality instruction in year 3.
However, adding the presence or absence of high-quality reform-related
instruction in year 2 to our previous model did generate conﬁgurations of
conditions that had high consistency and high coverage. Thus, while a model
with social network conditions in year 1 and 2 could account for the presence
or absence of high-quality instruction in year 3, adding high-quality instruction
in year 2 created an enhanced model that illuminated the conﬁguration of
conditions associated with high-quality reform-related instruction in year 3
with more precision.
Finally, we added the quality of reform-related instruction in year 1. This
model generated the same set of causal recipes as the quality of instruction
for year 2 along with the social network conditions. In other words, including
high-quality instruction in year 1 did not add to the explanatory power nor
did it modify the inﬂuence of the other combined conditions. This suggests
that the same conﬁgurations of conditions were associated with high-quality
instruction in year 3, whether or not a teacher had high-quality instruction
in year 1 (Ragin, personal communication, January 2012). For this reason,
we proceeded with our analysis with the following conditions in our model:
the presence or absence of high-depth interactions, high level of expertise,
154 American Journal of Education
and strong ties in years 1 and 2, and the presence or absence of high-quality
reform-related instruction in year 2.
Given the number of conditions included in the model (seven), there were
128 potential conﬁgurations of these conditions that teachers could experience.
As is typical, given the combination of the limited diversity in the natural
world and the small nof our study (Ragin 2008), our focal teachers represented
empirical instances of a subset of these possible conﬁgurations of conditions:
the 12 focal teachers represented 11 of these possible conﬁgurations of con-
ditions, described in table 1.
We then used Boolean minimization to simplify the 11 expre s s i o n s . K e y t o
Boolean minimization is the idea that if cases differ in only one condition but
share the same outcome, then the condition that is not shared between the two
is irrelevant for producing the outcome and can be removed. The fs QCA
software moves in step-wise fashion, eliminating a single condition at a time
until it identiﬁes the minimum number of combinations that can account for
all the positive cases.
Once we identiﬁed a small number of conﬁgurations of conditions associated
with high-quality reform-related instruction in year 3, we turned our attention
to understanding conﬁgurations of conditions associated with the absence of
high-quality instruction. Rather than assuming negative outcomes occur be-
cause of the absence of the conﬁguration of conditions associated with positive
outcomes (what Ragin terms “the assumption of causal symmetry”), it is im-
portant to run QCA analysis to identify necessary and sufﬁcient conditions
associated with negative outcomes as well (Rihoux and Ragin 2009; see also
Bryk et al.  for a discussion of the importance of testing both positive
and negative outcomes). Thus, we ran the analysis with the same set of con-
ditions, but with the outcome “absence of high-quality reform-related instruc-
tion in year 3.”
Finally, we returned to the data from the cases to understand more about
how the conditions identiﬁed interacted with one another in each of the
identiﬁed conﬁgurations of conditions and how that related to patterns of
instruction in year 3. This last step is crucial. The results of QCA are not the
endpoint in the analysis but must be interpreted with the use of theory and
data to ensure the interpenetration of theory and data that is at the heart of
case-oriented qualitative research (Ragin 1987, 2008).
When Greene School District withdrew support for teachers’ enactment of
reform mathematics instruction during the third year of the initiative, many
teachers in our study felt unsure about the continued direction for mathematics
Presence or Absence of Social Network Conditions and Outcome of Interest
0 1 0 1 1 1 1 1 Xandria, Denise
0 1 0 0 0 1 0 0 Tara
0 1 0 1 0 0 1 0 Larissa
0 1 1 0 0 1 1 0 Winona
0 1 1 1 0 0 1 1 Kathy
0 1 1 1 1 0 1 1 Laura
1 0 0 1 0 1 0 0 Don
.—1 ppresence of a given condition or outcome; 0 pabsence of a given condition or outcome.
156 American Journal of Education
in the district. Pressure to focus attention on literacy and instruction for English
language learners and a loss of assistance to deepen mathematics instruction
contributed to this uncertainty. One teacher in School G explained: “Some-
times you don’t get the help....Everybody’s complaining, everybody’s upset
and struggling because they just don’t know what to do [with mathematics
instruction].” Compounding the challenge, the district changes also had an
impact on teachers’ social networks (see Coburn et al.  for more details).
Nearly all focal teachers’ networks had weaker ties, provided less access to
expertise, and had fewer high-depth interactions in year 3. Yet, in spite of this
loss of support from the district and teachers’ social networks, more than half
of the teachers in our sample were able to sustain high-quality reform-related
instruction in year 3. Why were some teachers able to sustain reform-related
instructional practice in the face of such challenging circumstances and others
We found that the conﬁguration of teachers’ social network conditions in
the ﬁrst 2 years contributed to or hindered teachers’ ability to sustain reform-
related instructional approaches in high-quality ways. We identiﬁed three con-
ﬁgurations of conditions that were associated with high-quality instruction in
year 3 (see table 2). All three involved a combination of high depth, high
expertise, and strong ties. Two conﬁgurations of these conditions involved the
simultaneous presence of all three in a single year, and one involved strong
ties and high expertise in year 1 followed by high depth in year 2.
We also investigated the c o n ﬁ g u r a t i o n o f c o n d i t i o n s a s s o c i a t e d w i t h t h e
inability to sustain high-quality instruction in year 3. Table 2 shows that the
absence of strong ties in year 1 and the absence of high level of expertise in
year 2 or the absence of high-depth interactions in both years was associated
with the lack of high-quality reform-related instruction in year 3.
In the section that follows, we illustrate how these various conﬁgurations
of conditions were related to teachers’ ability to sustain reform-related in-
struction in year 3. We argue that social networks fostered teachers’ ability to
enact high-quality reform-related instruction in year 3 when they provided
supports necessary for teachers to develop strong understanding and deep
enactment of new instructional strategies. This strong understanding helped
teachers maintain reform-related strategies in the face of reduced supports
and changing conditions in year 3. In the section below, we illustrate these
claims by proﬁling each conﬁguration of conditions.
Synergy: High Depth, High Expertise, and Strong Ties in the Same Year
Most teachers who were able to sustain or improve reform-related instruction
in year 3 had 1 year when their social network simultaneously had a high
Conﬁgurations of Social Network Conditions Associated with High-Quality Instruction in Year 3
Consistency Coverage Teachers
Conﬁgurations associated with year 3 high-quality instruction:
1.0 .57 Quinn, Xandria, Nina, Denise
∗yr2 tie strength)
1.0 .14 Florence
3. (Y1 expertise ∗Y1
∗y2 tie strength)
1.0 .29 Kathy, Laura
Total 1.0 1.0
Conﬁgurations associated with year 3 absence of high-quality instruction:
4. y1 tie strength ∗y2 expertise 1.0 .6 Tara, Don, Larissa
5. y1 depth ∗y2 depth 1.0 .6 Tara, Sarah, Winona
Total 1.0 1.0
.—It is the convention in Boolean algebra to represent the presence of a condition with capital letters and the absence of
a condition with lower case letters; ∗represents the logical
. As is the convention in qualitative comparative analysis, the
expressions above represent the intermediate solution, which uses conservative assumptions in the simpliﬁcation process.
158 American Journal of Education
level of expertise, a high percentage of moderate- and high-depth interaction,
and strong ties. Quinn, Xandria, Nina, and Denise had all three social network
conditions in year 2 (conﬁguration 1); Florence had them in year 1 (conﬁg-
uration 2). Having all three social network conditions at the same time seemed
to create a synergy that fostered exceptional learning conditions, contributing
to teachers’ ability to sustain the reform in year 3.
Social networks with all three social network conditions in a single year
created multiple opportunities for teachers to learn about the new instructional
strategies. For example, Quinn, a fourth-grade teacher in School H, had all
three social network conditions in year 2. That year Quinn’s social network
included her fourth-grade colleagues, two coaches, and several administrators.
or high expertise. Quinn interacted with others in her network frequently—
an average of 2.5 times per month—an indicator of strong ties. And when
she did so she was likely to discuss substantive issues of mathematics teaching
and learning: 84% of her interactions were at moderate or high depth.
This conﬁguration of network conditions was potent. Because Quinn’s in-
teractions were likely to be at moderate or high depth, she was able to work
on issues that were at the core of the instructional approaches she was trying
to enact. For example, most of her interactions with colleagues in year 2
focused on how to ask students questions to elicit their strategies for problem
solving and then draw on and leverage student strategies during class discus-
sions to foster student learning. Because there was a high percentage of people
with expertise in her network, when interaction focused on these substantive
issues, expert others were able to infuse the conversation with knowledge that
Quinn did not herself possess. In one observation, fourth-grade teachers were
discussing student responses on an open-ended assessment. Some in the group,
including Quinn, were confused about the difference between factors and
multiples. The coach, who had a moderate level of expertise, clariﬁed the
distinction and facilitated a discussion of instructional strategies to teach the
relationship between these concepts. Quinn had the additional support of
strong ties with her network: because she met with others frequently, she was
able to discuss an issue, experiment with it in her classroom, and bring it back
to her colleagues for further conversation, thus strengthening her understand-
ing of the new instructional approaches.
While Quinn had made considerable strides in her use of the new instruc-
tional approaches in year 1, the combination of high depth, strong ties, and
high expertise in year 2 appeared to enable her to strengthen her enactment
of reform-related strategies. For example, while Quinn failed to maintain the
cognitive demand of any tasks we observed in year 1, she was able to maintain
high cognitive demand in all but one in year 2. Her instruction in year 2 was
also characterized by keen attention to student thinking and a growing atten-
Coburn et al.
NOVEMBER 2012 159
tion to intellectual authority. Thus, she not only met but exceeded our deﬁ-
nition of high-quality instruction. For example, in one lesson during year 2,
Quinn provided the following problem to her students: “A pet store has 75
ﬁsh. A new shipment arrived. Now the store has 200 ﬁsh. How many ﬁsh did
the store receive in the shipment?” Quinn had several students share their
strategies for solving the problem, facilitating a discussion where the class
compared strategies. Quinn elicited student thinking and strategically linked
students’ problem-solving strategies together to assist them in moving toward
an understanding of different multiples of 5 and 25, a key goal of the lesson.
Others took notice of Quinn’s improvement in year 2. The coach remarked:
“Quinn’s really grown. When she ﬁrst started here, she was very much a
traditional teacher. ‘Do what I say, when I say.’ . . . She has really grown to
getting kids to do the talking.”
Solid enactment of reform-related instruction seemed to enable Quinn to
sustain her reform-related instruction in year 3 when the district withdrew
support for this approach. Despite limited interaction in her social network
in year 3, limited interaction with the remaining mathematics coach, and
having to contract her lessons from 90 minutes into a 60-minute time frame,
Quinn was ﬂexible enough with her enactment that she made adjustments to
the Investigations lessons in ways that continued to maintain quality instruction.
Importantly, the presence of all three social network conditions along with
high-quality enactment in year 2 was sufﬁcient to support sustainability in
year 3 regardless of teachers’ level of enactment prior to the study. Three out
of four teachers who had this conﬁguration of conditions had no experience
with this approach to mathematics teaching prior to year 1. Furthermore,
two teachers—Xandria and Deanna—continued to have low-quality enact-
ment at the end of year 1. Yet, these teachers were nonetheless able to leverage
social networks with high depth, high expertise, and strong ties in year 2 to
develop high-quality reform-related instruction by the end of year 2 and sustain
it in year 3.
Finally, having all three network conditions concurrently in year 1 worked
somewhat differently than having them in year 2; the QCA suggests that three
conditions in year 1 alone were not sufﬁcient to support sustainability without
also having strong expertise and weak ties in year 2. Although all three network
conditions in year 1 seemed to create a synergy of learning opportunities that
enabled teachers to develop strong enactment, perhaps because it was the ﬁrst
year and the approach was still so new, teachers needed the occasional support
of an expert network in year 2 to maintain high-quality enactment into year
3. For example, Florence, an experienced kindergarten teacher in School G,
had all three social network conditions in year 1 with a network that stretched
beyond her grade-level colleagues to include two coaches and members of a
district-level mathematics task force. Florence was able to leverage her fre-
160 American Journal of Education
quent, high-depth interaction with expert others to help her develop high-
quality instruction that year. In year 2, Florence did not feel that she needed
the strong ties that frequent interaction brings. She reported: “Because we
are pretty familiar with what’s to come next, there’s probably less discussion
than there was....Ithink [I’m] being pretty self-sufﬁcient with it.” She did
continue to reach out to others in her network, 43% of whom had expertise,
in year 2, but, she did so infrequently, an average of just over one time a
month. At the same time, she did need the access to expertise to maintain
high-quality instruction. Initially, Florence’s enactment declined in year 2,
mainly because she selected only low-level tasks from the kindergarten cur-
ricula during observations in the fall. But, as she engaged with more expert
others across the year, they encouraged her to select higher-level tasks such
that by the end of the year, she was enacting high-level tasks with high quality.
She was subsequently able to sustain high-quality, reform-related instruction
in year 3 in spite of the fact that she had almost no interaction with others
about mathematics in that year. In sum, Florence, like Quinn and others,
drew on the synergistic combination of strong ties and high-depth interaction
with a relatively expert network in a single year to develop the strong un-
derstanding and enactment of the reform mathematics approach necessary to
make principled adjustments that sustained high-quality instruction in the face
of changing and challenging conditions in year 3.
Adding Depth over Time: High ExpertiseandStrongTiesinYear1Followedby
High Depth in Year 2
Tea c h er s w e re a ls o a b le t o su st ai n hi g h- qu al it y r ef o rm - r e la te d in s tr u ct io n in
year 3 when they had networks characterized by high expertise and strong
ties in year 1 followed by networks characterized by high-depth conversation
in year 2 (conﬁguration 3). Two teachers had this conﬁguration of social
network conditions: Kathy and Laura.
Social networks with the combination of strong ties and expertise in year
1 enabled Kathy and Laura to develop a basic understanding of reform
mathematics strategies. Second-grade teacher Kathy had strong ties (inter-
acting an average of 3.3 times per month per person) with three grade-level
peers and two coaches. But, despite a high level of expertise in her network,
Kathy’s discussions tended to focus on basic features of the reform mathematics
approach: discussions about logistical issues, materials, and pacing. Drawing
on frequent low-depth interactions with expert colleagues, Kathy was able to
use reform-related instructional strategies consistently but was not able to enact
them with high quality in year 1.
In year 2, targeted, high-depth interactions enabled Kathy to build on the
Coburn et al.
NOVEMBER 2012 161
foundation she laid in year 1 to achieve high-quality instruction by the end
of the year. Kathy had weaker ties with colleagues in her year 2 network,
interacting an average of 1.63 times per month (compared to 3.3 times per
month in year 1). However, when Kathy did interact with others, she tended
to do so with greater depth. Sixty-four percent of Kathy’s year 2 interactions
were at moderate or high depth and targeted core features of the reform
mathematics approach. For example, she discussed how to use well-crafted
questions to uncover students’ conceptual understanding with her colleagues.
She also observed her coach questioning students about their problem-solving
strategies in a mathematics lab and then talked with the coach and other
colleagues about the ways the coach orchestrated the student discussion.
The targeted high-depth interaction in year 2 enabled Kathy and Laura
to achieve high-quality enactment that year. In Kathy’s case, high-depth in-
teraction focused on how to question students to support their mathematical
understanding. These same ideas manifested in her instruction, speciﬁcally
her use of mathematical reasoning to judge the correctness of students’ so-
lutions by asking students targeted questions. In one lesson, students worked
in groups to ﬁnd different combinations of coins to make a given amount of
money. Kathy moved to each group asking students questions about their
strategies for solving the task. When students explained their solutions, rather
than simply verifying the accuracy of solutions Kathy prompted them to justify
their answers using counting. That is, she located intellectual authority in
mathematical reasoning, a key tenet of high-quality instruction.
Weak ties in year 2 were also a condition in Kathy and Laura’s conﬁguration
of social network conditions, suggesting that high-depth interactions coupled
with high-quality enactment of reform strategies supplanted the need for fre-
quent interaction. Both teachers interacted substantially less often in year 2
than year 1. Having built a basic understanding in year 1, substantive inter-
actions targeting reform mathematics strategies were perhaps more important
than frequent interactions in year 2. Furthermore, this conﬁguration of social
network conditions was sufﬁcient regardless of whether teachers achieved high-
quality enactment of reform-related strategies in year 1. Kathy did; Laura did
Strong ties and expertise in year 1 followed by high-depth interaction in
year 2 enabled Kathy and Laura to sustain high-quality reform-related in-
struction in year 3. This is especially noteworthy since both teachers switched
to new grade levels in year 3, requiring them to learn all new instructional
materials just as network supports largely ended. Having developed sufﬁcient
understanding of reform instruction, both teachers were able to apply the
approach with ﬂexibility to a new grade level. For example, Kathy achieved
sustainability in year 3 by continuing to enact instructional strategies acquired
through network interactions in the ﬁrst 2 years: in particular, she continued
162 American Journal of Education
to progress in her ability to employ mathematical reasoning when examining
Not all teachers had conﬁgurations of network conditions that provided suf-
ﬁcient support to develop high-quality instruction in the ﬁrst place, let alone
sustain it in year 3. Here, we describe two conﬁgurations of conditions associated
with the absence of high-quality, reform-related instruction in year 3.
Missing Pieces: Absence of Strong Ties in Year 1 and Absence of Expertise
in Year 2
Three teachers had a social network characterized by an absence of strong
ties in year 1 and an absence of expertise in year 2 (conﬁguration 4): Larissa,
Tara, and Don. While teachers in this conﬁguration did some good work,
they were missing important network conditions to develop a strong under-
standing and enactment of reform-related instruction.
An absence of strong ties made it difﬁcult for teachers with this conﬁguration
of conditions to lay the groundwork for strong enactment in year 1. In contrast
with Kathy and Laura, whose combination of strong ties and expertise in year
1 enabled them to make considerable progress, Larissa, Tara, and Don did
not interact frequently enough to make use of the expertise or depth that they
had in their network. By the end of year 1, they still had not put even the
basic elements of high-quality reform-related instruction in place. For example,
Don, a ﬁfth-grade teacher in School F, had a network with high-depth inter-
action but weak ties. He did have substantive interactions with his colleagues,
many of which focused on strategies for eliciting student thinking. However,
perhaps because he was not able to revisit this difﬁcult-to-learn instructional
approach in more frequent communication with colleagues, Don was not able
to integrate this approach into his classroom instruction. By following the
teacher script in the curriculum quite closely, he was able to maintain the
cognitive demands of tasks in year 1, but he tended to give students the answer
rather than use approaches to draw out student thinking. Thus Don, like the
other teachers with this conﬁguration, had enactments of reform-related strat-
egies that were either low quality or just on the cusp of high quality but fragile.
In year 2, the absence of expertise in their networks meant that teachers
with this conﬁguration of conditions had difﬁculty further improving their
practice to achieve consistent, high-quality instruction. For example, Larissa,
than Don. However, the limited expertise in her network in year 2 was prob-
lematic for Larissa, who struggled with knowing what to do after she had
used the new instructional strategies she was learning to elicit student thinking.
In an interview after a lesson where students used straws of different lengths
Coburn et al.
NOVEMBER 2012 163
to make triangles, Larissa pointed to a moment in the lesson when a student
came up with a surprising strategy for combining the straws, remarking that
she had no idea how to respond to the student. Instead of using this student’s
novel solution strategy to help extend students’ understandings of the rela-
tionship between angles in triangles, Larissa cut the conversation short, missing
Absent access to expertise to strengthen their understanding and enactment
of reform strategies in year 2, Larissa and Don’s instruction remained stable.
For Don, this meant that his instruction remained inconsistent but ultimately
was of low quality. For Larissa, who had made considerable strides in year 1,
this meant that her instruction remained just on the cusp of high quality.
Lacking strong enactments of reform-related strategies, the teachers with this
conﬁguration were unable to sustain or improve their instruction in year 3 in
the face of changing and difﬁcult conditions. For example, even though Larissa
had achieved high-quality reform-related instruction in year 2, she lacked the
understanding and ﬂexibility to make principled mathematically and instruc-
tionally sound adaptations to the new conditions. Faced with the need to reduce
mathematics instruction from 90 to 60 minutes and adjust to a class of children
that she felt were less prepared than in previous years, Larissa made a series of
modiﬁcations to the curriculum that lowered the cognitive demands of tasks
from the outset. Then, when students encountered difﬁculties, she tended to
respond by telling students the answer, short-circuiting their engagement with
mathematical reasoning and lowering the cognitive demands of the task. While
she maintained the cognitive demands of all but one task in the ﬁrst 2 years,
she lowered the cognitive demands in every task except one in year 3.
Although all teachers in the sample faced the kind of challenges Larissa and
Don faced in year 3 (and some teachers, like Kathy and Laura, faced thearguably
greater challenge of learning a new grade-level curriculum), Larissa and Don
were not well prepared to handle these challenges. Absent the solid understand-
ing of the reform mathematics approach that Quinn, Kathy, and others de-
veloped in the ﬁrst 2 years, Larissa and Don made choices that resulted in a
substantial decline in the quality of their instruction in year 3.
Scratching the Surface: Absence of Depth in Years 1 and 2
Finally, the three teachers who lacked depth in both year 1 and year 2 (con-
ﬁguration 5)—Tara, Sarah, and Winona—were also unable to develop or
sustain high-quality reform-related strategies in year 3. These teachers engaged
in few conversations with others about substantive issues related to reform
instruction in either year. As a consequence, they either never managed to
enact the reform mathematics approach with high quality or lacked the un-
164 American Journal of Education
derstanding and enactment necessary to sustain that approach in year 3 when
the initiative was curtailed.
Tea c h er s w i t h th is c on ﬁg u ra ti on o f so ci a l ne tw or k co n di ti o ns h ad n e tw or k
interactions characterized by brief, superﬁcial conversations in years 1 and 2.
For example, Tara, a fourth-grade teacher in School G, characterized the
nature of her interaction with the coach in the following way: “I like to go
share with [the coach], ‘Hey, you know this worked out,’ ’cause a lot of times
it’ll be something that I don’t think they’ll get, and I’ll be really nervous about
it. And, sure enough, they did get it.” Similarly, Sarah, a ﬁrst-grade teacher
in School E, described her interaction with her grade-level colleagues in year
2: “I will ask Ms. Moon or the other teachers in our group and they will say
‘Well I did it this way and I found that,’ and it’s just like, ‘Oh you did that?
That was really good.’ That’s really kind of how it works.”
Lacking more in-depth interaction about the reform mathematics approach,
Tara, Sarah, a n d Winona a l l h a d p r o b l e m s m a i n t a i n i n g t h e c o g n i t i v e d e m a n d s
of tasks that required students to solve open-ended problems where the teacher
was expected to support students’ thinking. For example, Tara spoke about
her struggles to enact a facilitative teaching approach, saying: “My tendency
is to want to push them towards that [right answer], and . . . you can’t always
do that.” Tara did choose rigorous, open-ended tasks for her lessons but then
inevitably gave students procedures and algorithms to follow rather than asking
them questions about their problem-solving work and what they might do
next. As a result, Tara’s lessons were low quality in both years 1 and 2. When
she switched grade levels in year 3 and had even less district support to enact
a reform-oriented approach, Tara continued to deliver instruction at a low
Unlike Tara, Sarah and Winona enacted lessons with high quality in year
2, but their understanding of the reform mathematics approach appeared to
be too fragile to sustain in year 3. Winona’s experience is particularly striking,
as she had the same social network conﬁguration as Kathy and Laura in year
1: strong ties with high-expertise others. However, unlike Kathy and Laura,
Winona lacked the high-depth conversations in year 2 that enabled Kathy
and Laura to strengthen their understanding of the reform mathematics ap-
proach. Winona’s interactions in year 2 were mostly about “problems” and
“activities.” She speciﬁcally noted, “Sometimes we end up talking a lot about
management of free choice time and assessment during free choice time.”
However, she had few deeper discussions about students’ mathematical work.
By year 3, when Winona and Sarah had even less interaction with others
in their social networks, both teachers delivered lessons that echoed the worst
of their instruction in years 1 and 2. Absent sufﬁcient support in those ﬁrst
Coburn et al.
NOVEMBER 2012 165
articulate their mathematical reasoning, the teachers did not sustain the high
cognitive demand of most mathematical tasks in their year 3 lessons.
Taken together, our analyses highlight several dif f e rent proﬁles for social
networks that can support or forestall teachers’ sustained enactment of reform-
oriented mathematics instruction. Teachers with networks characterized by
strong ties, expertise, and high-depth interaction (in 1 year or stretched across
ditions and few network supports in year 3. Teachers missing critical pieces
of network support either never achieved or were not able to sustain high-
quality instruction when support for these approaches was removed.
Schools and classrooms are constantly changing. Teachers face a new group
of students each year with different strengths and needs. Teachers change
grade levels, requiring them to learn all new materials and adjust their in-
structional strategies. And, as vividly illustrated by the case of Greene, school
and district policy contexts change as well; new priorities, new policy initiatives,
and shifting availability of resources create changing conditions in schools and
classrooms. The bottom line is that schools never stand still. Thus, sustaining
new instructional approaches is not simply about continuing to do the same
thing. It requires that teachers and others make continual adjustments to new
conditions and needs at the same time that they maintain the underlying
pedagogical approach. We provide evidence that teachers with a solid grasp
of reform-related instructional strategies are able make adjustments that main-
tain high-quality instruction.
Teachers’ social networks are one way to develop this capacity. In this study,
social networks in year 3 were not sufﬁcient to support teachers in sustaining
high-quality reform-related instruction that year, perhaps because most teach-
ers’ networks in year 3 were characterized by weak ties and low depth. Instead,
the quality and conﬁguration of teachers’ social networks in the ﬁrst 2 years
assisted teachers to develop a strong enough enactment to sustain reform-
related instruction in year 3.
Furthermore, we provide evidence that no single dimension of social net-
works is sufﬁcient to support sustainability. Teachers with social networks
characterized by strong ties, high levels of expertise, or depth alone in a given
year may be unable to develop high-quality instruction in the ﬁrst place or
to develop instructional approaches with enough ﬂexibility to adjust to chang-
ing conditions. Strong ties only go so far toward supporting implementation
if teachers only talk with others about superﬁcial or logistical issues or if the
people they seek out lack expertise in reform mathematics. Similarly, having
166 American Journal of Education
interact only occasionally with expert others or focus their conversation on
logistical or informational matters.
We show that the combination of expertise and tie stre n g t h , a l o n g w i t h
high-depth interaction either concurrently or in the subsequent year, can
support the development of strong enactments of reform-related instruction
that enables teachers to sustain over time. Having all three social network
conditions in a single year may create a great deal of synergy as teachers are
able to discuss matters at the heart of the instructional approach with others
with expertise with enough frequency that they can experiment with new
instructional approaches, discuss them with colleagues, and push their practice
further. In this study, teachers who experienced all three social network con-
ditions concurrently in either year 1 or year 2 were able to achieve high-
quality instruction that year; however, all three network conditions in year 1
were not sufﬁcient to support sustainability unless teachers also had access to
expertise and weak ties in year 2.
Tea c h er s w er e al s o ab le t o su s ta in r e f or m-
related practices in year 3 when they had high expertise and strong ties in
year 1 followed by depth of interaction in year 2. Strong ties and high expertise
in year 1 enabled teachers to lay a foundation, which they then were able to
deepen the following year via targeted high-depth interaction.
When teachers developed a strong enactment of the new mathematics in-
structional strategies with support from their networks, they appeared to be
able to adjust their practice in ways that maintained high quality when faced
with changing conditions and limited support in year 3. Those teachers who
lacked these social network conditions in the ﬁrst 2 years were unable to make
such adjustment. The quality of their instruction declined in year 3, and they
were unable to sustain the reform.
This study is not without limitations. First, we measured tie strength by focusing
on frequency of interaction. While many scholars use frequency as a measure
of tie strength (Rivera et al. 2010; Uzzi 1996, 1997, 1999), it is important to
acknowledge that it is a proxy measure since it does not capture the affective
dimension of the relation. Importantly, however, frequency does predict af-
fective dimensions of social relations such as trust and social closeness (Coburn
and Russell 2008; Rivera et al. 2010). Frequency and closeness are often highly
correlated, suggesting that they are measuring the same underlying construct
(Hansen 1999). And, like other ways of measuring tie strength, frequency has
been shown to predict outcomes such as altruism, joint problem solving, and
information sharing (DiMaggio and Louch 1998; Uzzi 1996, 1997, 1999).
Coburn et al.
NOVEMBER 2012 167
However, given that frequency is not a direct measure of the underlying
construct, readers should view our ﬁndings related to tie strength with caution.
Second, following convention in the literature, we relied on a summary
measure of each dimension of the teachers’ social network that we were
interested in by averaging across all of an individual teacher’s ties. For example,
to create our measure of expertise, we calculated the percentage of those in
a teacher’s social network with moderate or high expertise. There are strengths
and limitations of using aggregate measures of social network dimensions in
this way. Using aggregate measures creates a summary measure of the network
as a whole, providing a concise way to compare different dimensions of a
given focal teacher’s networks or compare a given dimension across networks.
However, any summary measure necessarily simpliﬁes complexity and masks
nuance. In this case, the aggregate measures mask the ways that an individual
might, for example, have strong ties with some in their networks and weak
ties with others. It also masks the interaction between dimensions of the social
network at the tie level. For example, do strong ties with individuals with low
expertise work differently than strong ties with individuals with high expertise?
By using aggregate measures, our analysis provides a necessary ﬁrst look at
how dimensions of social networks interact. Future research can extend this
analysis by investigating these interactions at the tie level rather than the
Third, we took an egocentric approach to social network analysis. The
strength of the egocentric approach is that we did not make assumptions about
the conﬁguration of teachers’ social networks but rather took identiﬁcation of
the networks as a ﬁrst step for investigation. Because the analyst maps networks
from the ground up using nominations solicited from the interviewees, the
egocentric approach does not assume the locus of professional community is
in formal structures such as grade-level groups or even exists within preexisting
boundaries such as the school (Carrasco et al. 2006; Reagans and McEvily
2003). In fact, all teachers except for one had networks that spanned beyond
the boundaries of their school, which we were able to capture using the
egocentric approach. One potential limitation of the egocentric approach is
that the researcher typically does not have access to the alters whom the ego
nominates, and thus there is limited opportunity to validate the existence of
a tie. However, in this study, we addressed this limitation to some degree by
administrating social network protocols to nonfocal teachers in focal teachers’
networks, who provided further information about the nature of the tie, the
expertise at the node, and the content of interaction between the alter and
the ego. Another potential limitation of the egocentric approach is that, be-
cause we did not do social network analysis with all teachers in the school (as
one would with sociocentric approaches to network analysis), we are not able
to map network structure for the entire school. As a result, we are unable to
168 American Journal of Education
analyze dimensions of the structure of networks as centrality or the structure
of cliques or subgroups. Future research that uses sociocentric approaches to
network analysis is necessary to shed light on the relationship between these
additional dimensions of social networks and sustainability of instructional
Implications for Research on Sustainability
In spite of these limitations, this study contributes to research on sustainability
by extending our understanding of when and under what conditions teachers’
social relations contribute to teachers’ ability to sustain ambitious instructional
reform. Prior research on sustainability has highlighted the role of teachers’
social relations in reform sustainability (Bryk et al. 2010; Datnow et al. 2002;
Hargreaves and Goodson 2006; Klingner et al 1999; McLaughlin and Mitra
2001). We extend this research by identifying the speciﬁc conﬁguration of
social network conditions that seem to matter for sustainability.
Most studies of teachers’ social relations have tended to investigate a single
dimension of teachers’ social relations at a time (e.g., Frank et al. 2004) or to
create a scale that integrates multiple dimensions into a single construct (e.g.,
Bryk et al. 2010; Louis and Kruse 1998), making it impossible to know which
dimension or combination of dimensions is actually making the difference.
By contrast, we measure multiple dimensions of teachers’ social networks
individually and use QCA to analyze how different combinations in different
years interact to support sustainability. In so doing, we show that no single
dimension of social networks appears to be sufﬁcient to support sustainability.
Rather, it is the interaction between tie strength, expertise, and depth of
interaction that seems to matter for sustainability. Depth of interaction enables
teachers to move beyond information exchange or logistical issues to inves-
tigate substantive issues that are central to a teacher’s ability to learn a new
instructional approach. Interaction with expert others brings insight and
knowledge to bear on these issues, extending teachers’ grasp of the mathe-
matics and new and novel instructional approaches. And, strong ties enable
teachers to work iteratively with colleagues to reﬁne their instructional strat-
egies and deepen their enactment over time. We also show that these three
network dimensions could be present in at least three conﬁgurations: in the
ﬁrst year of the initiative if supplemented with access to expertise in the second
year, in the second year of the initiative, or stretched across 2 years with
expertise and tie strength in year 1 and depth and weak ties in year 2. By
identifying key qualities of teachers’ social networks that support sustainability
and elucidating the ways in which these qualities interact, we begin to uncover
Coburn et al.
NOVEMBER 2012 169
when and under what conditions teachers’ social relations serves to support
sustainability and when they do not.
This study also provides insight into the possible inﬂuence of the length of
time that teachers experience these network conditions. In this study, when
the district eliminated the structures and activities that it put in place to support
teachers’ learning, the nature and conﬁguration of teachers’ social networks
shifted as well (for further discussion on the factors that affected the nature
of social networks, see Coburn et al. ). Only two out of the 12 teachers
had social networks that were high in more than one dimension in year 3,
and most were low in all three. Yet teachers were not only able to develop
high-quality enactment of a difﬁcult-to-learn curriculum, they were able to
sustain that enactment with 1 or 2 years of favorable network conditions. This
suggests that teachers may not need powerful networks for extended periods
of time to support sustainability as long as their networks contain a combi-
nation of all three network conditions.
It is important to note that this study investigates the relationship between
social networks and sustainability of instructional reform in the context of a
mathematics initiative that promoted ambitious approaches to mathematics
reform. The instructional strategies promoted by the district initiative and the
Investigations curriculum were challenging for teachers to learn and implement
(Stein and Kim 2009). They also departed from teachers’ prior approaches
to mathematics instruction, at times substantially. This raises these questions:
Do teachers need a similar set of social network conditions to support sus-
tainability if the instructional reform is less challenging to learn and enact?
Or if it departs less substantially from teachers’ existing practice? Is depth of
interaction as important if an instructional reform requires less learning on
the part of teachers? How about expertise? This study extends research on
sustainability by identifying key dimensions of social networks and several
patterns of interaction over time that support sustainability of ambitious in-
structional reform. Future research could build on this work by investigating
the relationship between these dimensions of social networks and sustainability
for reforms that place different kinds and levels of learning demands for
Implications for Research on Social Networks
This study contributes to research on social networks by extending it to the
issue of sustainability. To date, an extensive body of research investigates the
relationship between social networks and diffusion of innovation (e.g., Davis
1991; Valente 1995). This research focuses mainly on the adoption of reform.
Yet we know that it is much easier to adopt an innovation than to implement
170 American Journal of Education
it well (Spillane and Jennings 1997; Spillane and Zeuli 1999). A smaller body
of research investigates the relationship between social networks and imple-
mentation, studying how features of social networks—most often tie strength,
but in educational settings, expertise—inﬂuence how people use new ap-
proaches in their day-to-day work (Frank et al. 2004; Hansen 1999; Obstfeld
2005; Penuel et al. 2009, 2010; Reagans and McEvily 2003). However, few
have studied social networks and implementation over time, and no one has
investigated what happens once initial energy, resources, and conditions dis-
sipate. Given how few innovations are actually sustained over time (Cuban
1988), it is important to understand not only how social networks support
adoption and implementation but how they foster sustainability as well.
This study also contributes to research on social networks by investigating
the nature of social network transactions directly. Few studies investigate what
actually happens in social network interactions (Borgatti and Foster 2003;
Kilduff and Brass 2010). Instead, they draw inferences about the process from
the structure of the network (e.g., nature of ties or the conﬁguration of the
network as a whole) or resources available at network nodes (e.g., expertise).
But prior research has shown that the content of interaction varies even in
networks with similar structure or resources (Coburn et al. 2012). We build
on this work to provide evidence that the content of interaction—measured
here in terms of depth of interaction—may play a crucial role in teachers’
ability to implement and sustain complex instructional approaches. Thus, we
uncover a new dimension of social networks that may have important con-
sequences for the degree to which social networks promote a range of valued
outcomes, including implementation and sustainability.
Gaining insight into the content of social network transactions is also im-
portant because it helps uncover the mechanism by which social networks
support these outcomes. There are currently multiple theories in this regard.
For example, in the case of tie strength, researchers argue that weak ties with
multiple people provide social reinforcement for adoption of new practices
(Centola 2010). This argument locates the power of social networks in dy-
namics of peer pressure and social control (see Frank et al.  for a similar
argument regarding expertise). Others argue that strong ties enable the transfer
of complex or tacit information (Hansen 1999; Reagans and McEvily 2003)
or joint problem solving (Uzzi and Lancaster 2003). In this account, social
networks provide their beneﬁts by creating conditions for individuals and
groups to learn from and with one another. Finally, still others argue that
strong ties facilitate increased coordination and communication (Obstfeld
2005), emphasizing the way social networks facilitatecollectiveaction.
While our study cannot resolve the debate about the underlying mechanism,
by measuring social network interaction directly we provide support for the
learning explanation. We show that when teachers have more substantive,
Coburn et al.
NOVEMBER 2012 171
more focused, more targeted talk where they grapple with teaching problems,
puzzle over the meaning of mathematical concepts, and brainstorm ways to
link instructional strategies to children’s current thinking, they are able to
move toward deeper enactment and a greater ability to sustain reform. This
suggests that the mechanism involved is one of learning rather than social
pressure or collective action. If it were social pressure, simply having other
people in one’s network actively implementing would pressure teachers to
make change in practice. But all the teachers in our study had others in their
network who were using curricular materials to some degree. And nearly all
teachers in our study were themselves trying to make the change. In this study,
social networks appeared to foster sustainability when they provided substan-
tive support to help teachers ﬁgure out how to make those changes, in the
form of talk and expertise and the frequent interaction that gave them access
to both of these things. Thus, attention to the content of interaction begins
to provide a way to open up the black box of social network transactions,
something that is crucial if we are to truly understand how social networks
actually promote such well-documented outcomes as joint problem solving,
diffusion of innovation, implementation and, now, sustainability as well.
Deﬁnitions of Codes for Dimensions of Social Networks
High High expertise is deﬁned as (1) four or more intensive professional development experiences or
(2) a math major in undergraduate or specialization in mathematics education in graduate
work accompanied by two or more intensive professional development experiences accompa-
nied by at least some opportunity to learn about pedagogical approaches consistent with the
Moderate Moderate expertise is deﬁned as (1) two or three intensive professional development experiences
or (2) mathematics major as an undergraduate or specialization in mathematics in graduate
school accompanied by at least some opportunity to learn about pedagogical approaches con-
sistent with the Investigations curriculum.
Low Low expertise is deﬁned as (1) one or fewer intensive professional development experiences and
(2) no formal mathematics training in undergraduate or graduate school, or a mathematics
major or specialization in mathematics absent at least some opportunity to learn about peda-
gogical approaches consistent with the Investigations curriculum.
Low Talk related to one or more of the following: how to use materials; how to coordinate between
the text, standards, assessments, and pacing guides; how to organize the classroom; sharing
materials or activities; general discussions of how a lesson went or whether or not students
were “getting it.”
Medium Talk related to one or more of the following: speciﬁc discussions of how lessons went that in-
cludes a discussion of why; planning for speciﬁc lessons that is detailed and includes a discus-
sion of why; speciﬁc and detailed discussion ofwhetherstudentswerelearning(butnothow
students learn); discussion of instructional strategies in the context of observations; doing math-
ematics problems together with discussion.
High Talk related to one or more of the following: pedagogical principles underlying instructional ap-
proaches; how students learn or the nature of students’ mathematical thinking; mathematical
principles or concepts.
Types of Instructional Tasks
Type of Task Description
High level tasks:
Doing mathematics (DM) Open-ended problems with limited guidance for students on how to solve them, thus requiring
complex, nonroutine thinking and reasoning such as making and testing conjectures, framing
problems, representing relationships and looking for patterns
Procedures with connections
to meaning (PWC)
Tasks designed to illuminate concepts, meaning or understanding by focusing students’ attention
on the use of procedures for the purpose of deepening students’ understanding of mathemati-
Procedures without connec-
tions to meaning (PWOC)
Tasks that focus students’ attentions on algorithms and routine procedures without any attempt
to foster conceptual understanding
Memorization (M) Tasks designed to facilitate memorization without any attempt to foster conceptual understanding
Unsystematic or nonproduc-
tive exploration (UNE)
Tasks that, as enacted, devolve into exploration that is neither systematic nor productive in de-
veloping students’ mathematical reasoning or understanding
Unrated tasks: no mathematical
Tasks that contain no work related to mathematics, including tasks designed to set up later math-
ematical activities by eliciting information/interest from students and/or providing background
Deﬁnitions of Codes for Reform-Related Instructional Practices
Maintenance of cognitive
Low In more than one lesson throughout the year, the teacher transformed the lesson’s major mathe-
matical task to a low level of cognitive demand either from materials to set up or from set up
High The teacher transformed the lesson’s major mathematical task to a low level of cognitive de-
mand either from materials to set up or set up to enactment in only one lesson or no lessons
throughout the year. For the remaining lessons, the teacher either:
(1) Maintained the same high level of cognitive demand from materials to set up and set up to
enactment without transforming a task into another type of high-level demand or to a lower
level of cognitive demand; or
(2) Transformed the task from “doing mathematics to procedures with connections” or from
“procedures with connections to doing mathematics” from materials to set up or from set up
Attention to student thinking
and intellectual authority:
Low In more than one lesson throughout the year, the teacher received a “0” rating for “attention to
student thinking”—meaning the teacher did no work to uncover student thinking—and a “0”
rating for “intellectual authority”—meaning the teacher made judgments about correctness
based on the teacher or the text only rather than students’ mathematical reasoning in the
High The teacher delivered at least one lesson where she or he either: (1) attended to student thinking
in a sophisticated way by purposefully selecting particular students to share their work and/or
sequencing students’ responses in a meaningful way or (2) vested intellectual authority in stu-
dents’ mathematical reasoning throughout a lesson and did not base judgments of correctness
on the teacher or text. Additionally, the teacher delivered no lessons or only one lesson where
she or he did not attend to student thinking at all and vested intellectual authority in the
teacher or text only.
Coburn et al.
NOVEMBER 2012 177
This work was supported from a grant from the National Science Foundation (IERI
grant REC-0228343). The content and opinions expressed herein do not necessarily
reﬂect the views of the National Science Foundation or any other agency of the US
Government. We wish to thank Marc Chun, Teresa McCaffrey, Rebecca McGraw,
Chris Nelson, Laurie Rubel, Marcia Seeley, Jaime Smith, Sarah Spencer, Stephanie
Sutherland, Mikyung Wolf, and Bahadir Yanik for help with data collection. We would
also like to thank Kristine Acosta, Linda Choi, Amy Hillen, LuAnn Malik, Willow
Mata, Andrea Miller, Wanda Nieters, Darlene Poluan, Marsha Seeley, Jaime Smith,
and Stephanie Sutherland for help with data analysis and Deanna Weber Prine for
assistance preparing the document. Thanks to Bill Penuel and Min Sun for sharing
their analysis of their measures of depth and expertise. We are also grateful to Charles
Ragin for his advice on interpreting the ﬁndings from our QCA analysis. Finally, we
extend our gratitude to all the participants in this study for permitting us into their
schools and ofﬁces and allowing us to interview and observe them engaged in their
instructional improvement efforts.
1. Because of space limitations, a methodological appendix (app. D) accompanies
the online version of this article. The appendix provides additional detail on sampling,
measures, and interrater reliability.
2. The name of the district, schools, and teachers are pseudonyms.
3. Our research collaborators administered a survey to all teachers in case study
schools in spring 2005 and spring 2006, years 2 and 3 of our study. However, the
survey, intended to serve another branch of the research project, was designed tosupport
school-level analysis rather than analysis of focal teachers’ social networks. Thus, it did
not include a full social network protocol or attention to most of the dimensions of
interest in the present study. Therefore, we only used survey data to validate and extend
our interview and observational data.
4. While we collected data in the fall and spring of years 2 and 3, we only collected
data in spring of year 1. Given that year 1 was the ﬁrst year that 11 out of 12 focal
teachers were using the Investigations materials, it is likely that scores for instructional
quality were lower in the fall and higher in the spring as teachers grew more accustomed
to this new approach to mathematics instruction. Thus, by only collecting observational
data in the spring of year 1, it is possible that our year 1 measures of instructional
quality are biased upward. However, it is not likely that this potential bias affected our
overall ﬁndings because we found no relationship between year 1 quality of instruction
and sustainability in year 3. If we found no relationship with what are possibly inﬂated
scores in year 1, it seems unlikely that we would ﬁnd a relationship if the scores were
lower given that one would expect that higher scores in year 1 would be associated
with sustainability in year 3, not lower scores.
5. For two tasks, teachers chose lessons from the text that contained “no mathe-
matical activity.” For example, in one case, a teacher had students collecting data in
response to a question that they would represent graphically in subsequent lessons. For
such tasks, there was no cognitive demand to be maintained, and we therefore did not
include a cognitive demand score from that lesson in our calculations.
6. It is true, as one anonymous reviewer pointed out, that there were no teachers
178 American Journal of Education
in our sample who had all three network conditions in year 1 who did not have access
to a high level of expertise in year 2. However, fsQCA has a procedure for incorporating
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conﬁguration of causal conditions than available in the sample. The analysis essentially
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of conditions as it engages in Boolean minimization, thus taking into account a broader
range of possible conﬁgurations of conditions. It is important to note that in doing this
analysis, fsQCA permits the analyst to enter a range of assumptions to guide the
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