AN EXPLORATORY ANALYSIS OF A VIRTUAL NETWORK OF MATHEMATICS
Anthony V. Matranga
This study is part of a larger research design project that aims to cultivate an online community
of mathematics educators. The purpose of this smaller study is to suggest interventions that will
further support community cultivation efforts. Social network analytical methods are used to
study project participants’ interactions in virtual spaces. Investigation focused upon the
connectivity of network structure, the most prominent members of the network, the presence of a
core-periphery structure and the relationship between participant types and prominence.
Analysis suggests that supporting interactions between newly involved and more long-standing
participants will enhance community cultivation efforts. And, four participants emerged as most
influential in the network, therefore being most effective in controlling and spreading novel
information. These participants are suggested to receive additional professional development.
Keywords: Communities of practice, mathematics teacher education, social network analysis
Teaching is historically conceptualized as a private practice, with emphasis on individual
teacher development through the use of didactic, lecture-based strategies, instead of teachers’
communal engagement (Newmann, 1996). To meet the demands of implementing higher quality
instruction, there recently have been calls for the development of communities focused on
enhancing educators’ instructional practices and improving student learning (Daly, 2010). In
order for these communities to be successful, it is imperative to engage in a critical examination
of instructional practices (Vescio, Ross, & Adams, 2008). One possible area of investigation is
the role of informal communication networks of mathematics educators in supporting
instructional practices (Moolenaar & Sleegers, 2010). However, there is a lack of research
concerning how to foster emergence of a community of educators, particularly those that take
place in online mediums.
This study is part of a larger research project (EnCoMPASS) designed to cultivate an online
community of mathematics educators. Due to extensive interactional data available from the
larger study, and a particular focus on structural features of the network, we used social network
analysis to investigate the patterns of interactions within the network. Social network analysis is
a quantitative analytical method derived from graph theory used to study the structure of social
networks (Scott & Carrington, 2011). Taken together, social network analysis can provide data
about relationships in social networks that can be informative for future interventions that would
not have been evident by other investigatory approaches. From an educational researcher’s
perspective, structural information about teacher networks is crucial for the implementation of
interventions designed to foster community cultivation. For example, if teachers are identified as
being sparsely connected, an intervention can be designed to support interaction between these
teachers in order to improve the flow and access of information in the network as a whole.
Interventions that serve the purpose of the one just mentioned may be critical in the process of
cultivating a community of educators. Part of the purpose of this research is to make such
suggestions for interventions.
The analytical methods briefly introduced here (and further elaborated in the methods
section) serve as the backdrop against which research questions are formulated to focus
investigatory efforts in this study. Consequently, this study seeks to answer the following
1. What are some of the structural properties of the EnCoMPASS Network?
2. Who are the most central members of the EnCoMPASS network?
3. To what extent is a core-peripheral structure present in the network?
4. What is the relationship between participant type and centrality?
Literature Review and Conceptual Framework
This study is informed by literature on sociocultural learning theories (i.e. communities of
practice and situated learning) and social network theory. These theories provide the conceptual
lens to understand the structural relationships in the community of mathematics educators, and
provide insights for interventions aimed at improving dissemination of instructional strategies
and learning in the EnCoMPASS network.
Communities of practice
Communities of practice refers to both a characterization of a social group!in terms of their
practices, beliefs and goals!and to a perspective of learning that hinges learning on individuals’
enculturation into a community through increased activity within that social group. Three
specific characteristics have been suggested for making the distinction between a group of
individuals and a community of practice, namely mutual engagement, joint enterprise and shared
repertoire (Wenger, 1998). Mutual engagement refers to the relationships and interactions in
which community members engage in practice, which serve to define a community, as
interactional patterns emerge, and impact individuals’ identities and purposes for engagement in
the community. While joint enterprise and shared repertoire rely on more qualitative research,
social network analysis is useful for initial investigation of mutual engagement (Wenger,
Trayner, & de Laat, 2011) to identify patterns of interactions.
Legitimate peripheral participation
Although mutual engagement is critical to understanding social interaction, the actual
engagement is also shaped by the context in which learning takes place. The concept, legitimate
peripheral participation, is a lens in which one can analyze learning as a byproduct of practice
(Lave & Wenger, 1991). In other words, from this theoretical perspective learning is indicative
of one’s increased participation in a community of practice. In analyzing one’s participation in a
community, participants are split into two groups, newcomers and old timers (Lave & Wenger,
1991). Newcomers are individuals that display less developed skills and practices through
minimal participation in the community. Old timers are individuals who are central members of
the community displayed through higher levels of knowledge, skills and participation. In the
transition from newcomer to old-timer, “knowledgeable skill is encompassed in the process of
assuming an identity as a practitioner, of becoming a full participant, an old-timer” (Lave &
Wenger, 1991, p. 68).
Social network analysis and communities of practice
The discussion above underscores a perspective of learning that takes group norms and
practices as the unit of analysis. What is missing, however, is the extent to which the very group
norms and practices are a function of interpersonal processes. Specifically, researchers argue that
communities of practice are comprised of social networks (Schenkel, Teigland, & Borgatti,
2001). Social network theory focuses upon the relational patterns embedded within a community
and groups. In order for group norms to emerge and common beliefs to be present amongst the
community, it is important for the social network structure to afford information flow. Social
network theory!and related analytical methods!provide critical constructs for understanding
whether the network structure is conducive to the emergence of a community of practice.
Multiple studies have used social network analysis as a tool to explore the extent to which
communities have features that are similar to those that are (theoretically) present in a
community of practice. Cowan and Menchaca (2014 ) investigated network conditions of an
online graduate program and determined that when instructors and program directors were
removed from the network overall cohesion was reduced; indicating that removing central
members reduces network cohesion. In a 2001 study, Schenkel et al. sought to operationalize the
concept of mutual engagement!one of the three foundational aspects of a community of
practice. In doing so, Schenkel et al. (2001) suggest that methods such as core/periphery analysis
highlight presence and non-presence of relationships within networks that afford and constrain
productive interactions. In a recent study of teacher learning in an online space, Dean and
Silverman (2011) sought to document collective development through the use of social network
analysis and qualitative methods to identify “local communities.” Dean and Silverman (2011)
conclude that both summative and formative results emerge from network analysis studies and
can be leveraged to support teacher learning and communal engagement.
Social network analytical methods are used to investigate the EnCoMPASS network
structure. In the following section these analytical methods are explained along with descriptions
of the research setting, participants, data collection and processing, and data analysis plan.
Study Site and Participants
Data for this study is drawn from the EnCoMPASS project, which began in 2012. This
project is designed to bring together and cultivate a virtual community of mathematics educators
across the nation. A primary goal of the EnCoMPASS project is to foster mathematics educators’
collective engagement in learning of mathematics and pedagogy. Accordingly, participants have
been recruited in groups throughout the project as incremental efforts are being made to cultivate
community. Over the past two years, multiple mediums of communication have been used as
leverage points to support teacher interactions (i.e. face-to-face workshops, online classes and
workshops, twitter discussion threads and email list serves) around different content, such as
mathematics, instruction and assessment. Therefore, the research setting is in virtual spaces that
foster online communication, such as twitter, online courses/workshops, and email listservs.
There are 92 participants (mathematics educators) involved in this study. Taking into
consideration whether (and how) they participated in the two face-to-face summer institutes (at
the end of Year 1 and Year 2 of the project) that focused on community and instructional
development, we can distinguish between 6 types of participants:
However, given our interest in participants’ interactions in the virtual spaces, we focused on
the 67 who participated in those spaces.
Data Collection and Processing:
Social network analysis is based on the premise that social life is created primarily by
relations and the patterns formed by these relations, and social network analysts study the
patterns of interactions between individuals in a network (Scott, 2013). In order to conduct the
social network analysis, we examined the records of interactions between participants in each
virtual space (twitter, online classes and workshops, email listservs) between May 2013 and
September 2014. Participant (i) is said to have “talked to” Participant (j) if the latter initiated a
communication tie to the former. In Twitter, a tie exists if Participant (i) included (j’s) hashtag in
the message (Ediger et al., 2010). On email and Blackboard learning systems, a tie is defined if
Participant (j) responded to an initial or any other posts/emails from another participant (i). From
this information, we created one communication matrix that records the interaction between
study participants. Each cell (Xij) of the matrix takes on a value indicating how frequently
(number of times) participant (i) directs or initiates a tie to participant (j). To satisfy some of the
data requirements of the social network measures and for ease of interpretation, we created a
binary matrix where the (i,j)th cell of the original valued communication matrix becomes ‘1’
when there is any communication tie between (i) and (j), and ‘0’ otherwise. The resulting 67 x 67
binary asymmetric matrix became the basis/input for all of the structural analysis presented in
this paper. The social network analysis software, UCINET version 6 (Borgatti, Everett, &
Freeman, 2002) was used in all data preparation and analysis. Analysis results are reported as
normalized indices. A normalized measurement is one in which the numerical value that results
from analysis is standardized so one could compare the particular analytical result to other
networks with different sizes.
Data Analysis Plan
This study uses four social network analysis measures (i.e. density, centralization, centrality
and core-periphery) (Borgatti & Everett, 2013; Kadushin, 2011) to address the research questions
and describe the communication structure of the EnCoMPASS network. First, density and
network centralization measures allow us to address the first research question: what are some of
the structural properties of the EnCoMPASS Network? Network density describes the general
level of cohesion in a graph (Scott, 2000). The centralization score describes the extent to which
the flow of information is organized around particular focal points. Second, centrality analysis is
employed to answer the second research question: who are the most central members of the
EnCoMPASS network? Centrality is used in social network analysis to identify important nodes
or those that occupy influential positions in a network. Three variants of centrality measures are
used in this study: in-degree, out-degree and betweenness centrality (Freeman, 1979). In-degree
centrality measures the ties an actor receives from others, while the out-degree centrality
measures the ties an actor directs to others. Betweenness centrality is a measure that indexes the
number of times a participant falls on the shortest path between two participants in the network.
Third, we employ a core-periphery model to examine the extent to which core-peripheral
structures are present in the EnCoMPASS network (Question 3). This method bifurcates the
network into a discrete model consisting of two classes. Fourth, distinguishing levels of
centrality for participant types is also of interest (Question 4). We used specialized Analysis of
Variance (ANOVA) tests for social network data to examine the differences in centrality by
participant type (e.g. whether 2013 fellows are more central then 13 & 14 fellows).
Research Question 1: What are some of the structural properties of the EnCoMPASS
network? To address this question, density and centralization measurements were used to give
information about network connectivity. The EnCoMPASS network resulted in a density of
8.3%. Out of the 5,852 possible directed edges in the network, 488 of these edges are present.
The network centralization analysis resulted in a measure of 0.222. This indicates that the
EnCoMPASS network is more representative of a network that has centrality evenly distributed
throughout the network than one that is controlled by a few central members. Density and
centralization give an overall representation of the structure of the network. And in this case
illustrate that over the 15 months in which this data was collected, close to 500 connections were
made and certain central members are potentially emerging as a centralization score of .222
indicates there is some heterogeneity among centrality in the network.
Research Question 2: Who are the most central members of the EnCoMPASS network? As
will be recalled, in-degree and out-degree centrality scores index the level of activity of
participants in the communication network. On average, the top 10 participants have normalized
out-degree scores of about 1.082, which is about 10 times larger than the average out-degree
scores for the middle ten participants (0.111), and ninety times larger than the scores for the
bottom ten participants (0.0117) respectively. On an individual level, participants 11018, 11006
and 11003 are the most active participants: actors 11006 and 11018 for instance have an
outdegree score that is about 35% higher than the next highest out degree in the network. The
indegree analysis yield similar results to those of outdegree: actors 11006, 11018 and 11003 have
the highest normalized indegree centrality scores. Betweenness centrality analysis also yields
actors 11018, 11006 and 11003 falling in the top ten, however actor 11016 also yielded a high
betweenness centrality score. While actor 11016 did not have as high a valued in/out-degree
score, her dichotomized indices where among the highest. Given the importance of betweenness
centrality for indicating potential for leadership qualities and the control of information, actor
11016 is considered as one of the most central members.
The above analysis suggests that actors 11018, 11016, 11006 and 11003 are the most central
members in the network. Each of the identified actors are 2013 and 2014 fellows, meaning they
have been a part of the project since it began in 2013, a year longer than other members that
began participation in 2014. Over a year of the project, the identified actors will have had many
more chances to participate than those that began in 2014, therefore being a possible reason for
their increased centrality. However, based on centrality alone, and drawing from previous
research, this analysis suggests these are the most central members and should receive additional
professional development opportunities.
In addition, in this analysis staff members are omitted in identification of the most central
actors. For example, staff members 1 and 7 are present in the top ten of each centrality measure,
however they are left out of this analysis. Part of the purpose of this research is to provide
additional professional development opportunities for participants that are identified through
social network analysis as most influential in the network. Therefore, identifying staff members
that are most central does not support research goals; although, staff members are included in
analysis as part of the 67 individuals because many interactions occur between staff (who act as
role models) and participants, thereby disseminating best practices in mathematics pedagogy.
Research Question 3: To what extent is a core-peripheral structure present in the network?
To address this question, we fitted a categorical core-periphery model to the EnCoMPASS data.
The core-periphery analysis provided sub-group densities, which can be used to examine the
connectivity, between/within, the core and periphery groups. The density profiles also help
evaluate the extent to which the derived groups approach an ideal core-periphery structure where
100% of ties are expected between core members, 0% between peripheral members, and most of
the connections expected between the core and periphery members. The core-to-core density of
the EnCoMPASS network is 59%, the core to periphery is 16.7%, the periphery to core is 9.6%
and the periphery to periphery is 3.2%. Although this network does not completely approach an
ideal core-periphery structure (and few are expected to be), the analysis indicates the presence of
a core periphery structure in the network (see Figure 1).
Of the 15 participants in the core of the network, 9 are 2013 & 2014 fellows, 2 are 2013
fellows, 3 are staff, and 1 is an “other” participant that was not part of either institute but was
rather active in online classes. The high concentration of 13 & 14 fellows in the core of the
network is expected, due to increased opportunities to participate throughout the life of the
project. Thus, participation in the summer institute seems to be related to communication activity
in the network. The staff members in the core of the network are also reasonable results due to
their frequent activity in twitter and email. However, a rather interesting result is participant
13010 falling into the core, as she was not part of either institute. Further research will explore
the position of participants such as 13010.
Research Question 4: What is the relationship between participant type and centrality? The
analysis above suggests that there are variations in centrality of participants in EnCoMPASS.
One determinant of this communication activity is participant type (whether they are 2013
Fellows, 2013-2014 Fellows, etc.). Table 1 displays the results of an ANOVA permutation
model examining the differences in mean (betweenness, outdegree and indegree) centrality of
participants by participant type. The ANOVA analysis shows that 2013 & 2014 fellows have the
highest centrality means in each category. Their average betweenness, outdegree and indegree
centralities are about 3 times larger than the next largest mean. For example, the mean out degree
centrality for 2013 fellows is 0.2452, while that of 2013 & 2014 fellows is 0.7148, which is 2.9
times larger then the 13 fellows. Second, 2014 fellows have substantially lower in/out-degree
(valued and dichotomized) than 13 & 14 fellows. For example, the in/out degree of the 2014
fellows is 14.8% and 10.6% of the 13 & 14 fellows, respectively. These findings illustrates a
clear divide in the amount of participation between the two groups: with the 2013 & 2014
fellows much more active than 2014 fellows.
The above analysis clearly shows the presence of a core and periphery in this network. While
the core-periphery analysis will “force” members into the two categories regardless of the nature
of the connections in the network, the density measures suggest a rather densely connected core
(59% of the potential connections are present) and a rather sparsely connected periphery (3.2%
of the potential connections are present). In the ideal core-periphery structure the core-to-core
density is 100% and the periphery-to-periphery density is 0% (Borgatti & Everett, 2000). While
the results shown here are not quite representative of the ideal core-periphery network, the divide
suggested by Borgatti and Everett is a theoretical model rarely (if ever) observed in practice.
Lave and Wenger (1991) suggest a core-periphery structure is critical to sustaining
productive interactions within a community while maintaining the structure of a community. In
particular, they note that constant interaction between core and peripheral members is critical for
knowledge creation and distribution in a community. In consideration of this guiding theory,
results of this study are used to identify connections that should be fostered between core and
peripheral members to ensure the suggested interactional patterns. Through intervention,
members that have been most recently introduced to the EnCoMPASS network (e.g. 2014
fellows) and are in the periphery, will be connected with members in the core that have been part
of the project for a longer period of time (e.g. 2013 and 2014 fellows). Furthermore, as the
periphery begins to decrease in size due to peripheral members beginning to evolve into more
centrally located members in the community; new participants should be gathered in order to
maintain the core-periphery structure.
In consideration of the broader goals of the project (i.e. documenting the emergence of a
community of practice), this study is significant in providing data that will be part of a larger
study that looks across multiple analyses similar to the one presented here. In particular,
retrospective analyses will illustrate the dynamicity of the network, where emergent patterns in
network change may provide insight into the way in which the network evolves over time in
relationship to the interventions that have been implemented. Analysis of this, and other similar
studies will document the emergence and maintenance of a community of mathematics
educators, which will fill a gap in the literature.
This study illustrates a quantitatively driven approach to enhancing professional development
efforts. Social network analytical methods were used to identify opportunities for intervention
that may have otherwise gone unnoticed. In particular, this investigation has evidenced the
presence of a core-periphery structure and has identified 4 individuals that are most central in the
EnCoMPASS network. These results provide justification for (1) providing professional
development to specific individuals in the network, and (2) fostering relationships between
specific groups of individuals in the network.
Table 4: Mean centrality measures and ANOVA tests of mean differences for each
13 & 14
Figure 1: Communication interactions by participant type
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Light blue=13&14 fellows