A persistent concern in higher education is the instructional practices used in introductory STEM courses (Koch, 2017; NASEM, 2016; NRC, 1999; PCAST, 2012; Rocard, 2007). Many have therefore investigated the factors impacting instructors’ decision-making in relation to instruction (e.g., Gess-Newsome et al., 2003; Henderson & Dancy, 2007; Landrum et al., 2017; Lund & Stains, 2015). There are many factors which appear to be at play, and in this work, we focus on one such aspect—leadership in the context of department communities. Using social network analysis, we explore overall instructional leadership structure as well as individual leaders (formal and informal) at five mathematics departments with successful calculus programs.

There is widespread consensus that good leadership enables communities and organizations to function effectively and efficiently in the pursuit of their goals—though what qualifies as “good leadership” varies significantly across contexts, as do goals. Leadership is closely and reflexively linked to an organization’s culture, with leaders able to influence the values and norms of a group, and the group’s values and norms impacting whether or not members will grant leadership to any particular individual (Carter et al., 2015; DeRue & Ashford, 2010; Schein, 2010). Universities are complex systems with multiple (sometimes competing) goals related to research, service, finance, and instruction (Gaubatz & Ensminger, 2017). Here, we embark on an investigation of instructional leadership at the departmental level, to better understand the organizational and cultural context in which instructors are making decisions about instruction.

Departments, and the universities they make up, are a form of organization. However, their goals are broader and less well-defined than formal business organizations and their cultures are often characterized by a collegial atmosphere rather than strict structural hierarchies (van Ameijde et al., 2009). This adds an additional layer to questions about instructional leadership within departments—is it characterized by central individual leaders, or distributed across peers? If there are departmental leaders when it comes to instruction, who might they be? Experience and research literature suggest three (not exclusive) answers to this question in the context of undergraduate mathematics. The first is the department chair or head, who is formally responsible in a university’s chain of command. The second are course coordinators, who formally manage particular courses and are nominally responsible for the courses they coordinate. Third, informal opinion leaders within a department who—regardless of formal title—are sources of advice about instruction which others seek and take up. Regardless of what their titles might be, it is the connections and relationships between people that characterize leaders and their influence on practice.

We use social network analysis to investigate the teaching-focused relationships among members of five university mathematics departments (details on these five are provided in the “Background studies and data sources” section) with strong calculus programs to answer the following two questions:

  1. 1.

    How vertical or distributed are the departments’ leadership structures in regard to instruction?

  2. 2.

    Who are the departmental leaders with regard to instruction, and do they correspond to individuals with formal titles?

Leadership

We focus on instructional leaders, who we conceptualize as those who influence the practice of instructors in the classroom through their relationships. To identify these leaders and understand their influence, we draw on literature related to leadership and communities.

Leadership in organizational communities

Leadership is a concept with many definitions, though all involve some form of influence. A review of definitions over the last century emphasizes that it is relational, situated, and can be formal and/or informal (Carter et al., 2015). That it is relational means that leadership involves relationships between people, and it is claimed and granted through social interactions within a community (DeRue & Ashford, 2010). By situated, we mean that a person who leads in some situations may follow in others. A key element of the situational nature of leadership is that different characteristics and expertise have different values in different contexts—a person who is considered a leader in one domain (e.g., pedagogy) may not be seen as a leader in another domain (e.g., research). Finally, leadership involves formal and/or informal influence. That is, leadership can originate from positional power (e.g., manager, chairperson), or from personal power (e.g., respect, trust), and the two are not exclusive. These features of leadership suggest the use of social network analysis, a technique which focuses on the relations between people in a particular situational setting which can detect informal leaders without losing track of formal ones (Carter et al., 2015).

Leadership has been conceived of as an organizational characteristic, something that is a feature of a group rather than a fixed characteristic of particular individuals (Ogawa & Bossert, 1995). The construct of distributed leadership (Spillane, 2006) is one that has been extensively discussed in organizational science and has been leveraged in education studies. This approach holds that leadership is a shared process among members of a group and is something that arises from interactions between group members (van Ameijde et al., 2009). We leverage the idea of distributed leadership as a mutual influence characterized by the emergence of formal and informal leaders (Devos et al., 2014). Again, this emphasizes the relationship between organizational culture and leadership which impacts individual and collective outcomes (Schein, 2010). Distributed leadership is particularly appropriate for investigating faculty groups, as higher education settings tend to have a collegial atmosphere rather than strong vertical hierarchies. Of course, vertical (formal, centralized) leadership is also at play, and can complement or even support distributed leadership (van Ameijde et al., 2009). We recognize that there is value in appreciating multiple dimensions and investigate the extent to which we have evidence of both vertical and distributed leadership in mathematics departments through a study of social networks.

Formal leaders in mathematics departments

In university mathematics departments, two common named roles inherently suggest a leadership position in regard to undergraduate instruction: chair and course coordinator. When conversations turn to challenges related to gateway courses, such as calculus, responsibility is often ascribed to the department chairs. However, department chair is a complex “middle leadership” role, with the chair formally heading up discipline-based units, perhaps managing subunits, and reporting to higher-ups in the pipeline—all with competing goals related to research, instruction, service, and finance (Gaubatz & Ensminger, 2017). Compounding the challenge of such a role, the characteristics associated with successful middle leadership are not the same as traits needed to succeed in academia, and neither of these are particularly associated with pedagogical expertise (Bryman, 2007; Thornton et al., 2018). As formal leaders, department chairs must juggle competing priorities and engage in different behaviors at different times to respond to different challenges (Gaubatz & Ensminger, 2017; Thornton et al., 2018; Yukl, 2012). The extent to which chairs engage with instructional leadership is an open question, though education is an explicit part of universities’ missions.

In the USA, many university mathematics departments offering high-enrollment introductory courses have implemented course coordination systems with the aim of ensuring coherence across instructors, sections, and terms (Apkarian & Kirin, 2017; Rasmussen et al., 2019). Robust coordination systems consist of uniform course elements (e.g., common text, exams) and regular instructor meetings, often resulting in a de facto community of practice (Rasmussen & Ellis, 2015). These systems may be led by one or more coordinators, or sometimes a committee, thereby introducing more leadership positions. In this study, we compare those with informal leader status to those in these two types of named position.

Community-driven instructional practice

Relevant to course coordination is the potential for forming a de facto community of practice where course instructors mutually engage in preparing and delivering content and assessments and support each other in refining their teaching practices. Communities of Practice (Wenger, 1998), therefore, provides a useful framework for understanding how a department can collectively impact the instructional practice of individual members. In particular, norms and practices related to instruction are part of a continuously developing shared repertoire which develops through mutual engagement surrounding the joint enterprise of teaching. There is a strong research precedent for leveraging communities of practice to understand and support instructional practice and change to that practice (Brown & Duguid, 1991; Forman, 2003; Ma et al., 2019). Central members of a community are those perceived as the most expert, and exert stronger influence over the group than newer or more peripheral members. Thus, we seek to identify the central members (i.e., leaders) of a community engaged in undergraduate instruction, by investigating patterns of mutual engagement.

Social capital is a complementary perspective which holds as a major tenet that who you know determines what you know, and is frequently used with social network analysis (Borgatti et al., 2013; Daly, 2010; Wasserman & Faust, 1994). When individuals are viewed as embedded in a complex system and engaged in social relationships and interactions, their relationships can be seen as conduits through which influence and advice are accessed and leveraged (Adler & Kwon, 2002; Bridwell-Mitchell & Cooc, 2016; Cole & Weinbaum, 2010; Daly, 2010; Lin, 2002). Social capital refers to the sharing and diffusion of expressive resources such as trust, friendship, attitude, or emotional support as well as instrumental resources, be they cognitive, material, or interactional. In the context of undergraduate mathematics instruction, examples might include transference of artifacts such as sample exams or syllabi, or discussing strategies for dealing with student difficulties, all of which can be facilitated by course coordination and a de facto community of practice. These interactions influence practice explicitly as well as implicitly, such that members of an interconnected community often draw on a common set of shared resources which result in commonalities in practice as well as beliefs (Daly, 2010; McPherson et al., 2001). Beliefs indirectly influence on practice, and instructors’ individual beliefs about teaching and learning have been shown to influence their instructional decisions (Aragón et al., 2018; Gibbons et al., 2018). Thus, we look for hubs of social capital (i.e., leaders) as they have the most explicit and implicit influence on practice.

Social network analysis and instructional practice

Social network analysis (SNA) refers to a suite of techniques for investigating social structures and interaction patterns through the use of graph theory. SNA is perhaps most closely associated with the field of sociology, but it has been grown in usage across fields including anthropology, economics, information science, criminology, and education (Borgatti et al., 2013; Freeman, 2004; Otte & Rousseau, 2002). SNA has been leveraged somewhat extensively in K-12 education settings to better understand influences on teachers’ social networks and how to navigate improvement initiatives (Daly, 2010; Penuel et al., 2012). In postsecondary contexts, faculty social networks often rely on research connections, such as co-authorship or citation networks (Kezar, 2014).

Relatively few studies have investigated faculty’s interactions in relation to instruction (Kezar, 2014). One such study concluded that faculty (members of the American Sociological Association) primarily teach in isolation, disconnected from activities related to teaching and learning and without access to social capital for instructional improvement (Spalter-Roth et al., 2010). Others have investigated networks of STEM faculty across multiple departments at the same institution, suggesting that networks related to instruction can transcend departments (Andrews et al., 2016) and that departmental affiliation alone is not a proxy for instructional networks (Quardokus Fisher & Henderson, 2015). SNA has been used to better understand specially designed faculty groups such as constructed communities of practice (Ma et al., 2018, 2019). A study centered on individuals suggested that faculty who use learner-centered instruction have stronger and more extensive networks than their peers (Middleton et al., 2015). Most connected to our work is that of Knaub et al. (2018), which suggested that using a single measure to identify leaders is insufficient and may produce lists which are biased in some way. Our study stands apart from all these in that it applies SNA to multiple types of instructional networks among the faculty of five mathematics departments without specialized interventions, to better understand pre-existing leadership structures that influence current practice.

Here, we look at four types of instructional influence relationships via social networks: seeking advice, seeking instructional materials, discussing instructional matters, and explicit instructional influence. By considering the overall patterns of these relationships, we can classify instructional leadership as more or less distributed in a particular community. These leadership structures are situated in the context of undergraduate instruction within particular mathematics departments at postsecondary institutions. These communities differ in various observable dimensions (e.g., size, research intensity) as well as in terms of institutional and departmental culture, all of which affect what and who are seen as valuable (Schein, 2010). This study assigns informal leader status to those who are central in multiple instructional influence relationships, and compares that to those with formal leadership roles and titles.

Background studies and data sources

The study reported here is a somewhat opportunistic follow-up to two large, national research projects, Characteristics of Successful Programs in College Calculus (CSPCC) and Progress through Calculus (PtC), both of which were conducted under the auspices of the Mathematical Association of America. It is opportunistic in the sense that we took advantage of the access provided by these two studies to explore the nature of leadership, rather than selecting departments specifically for this purpose. The CSPCC project surveyed a stratified random national sample of calculus 1 faculty and students to identify programs that were relatively more successful, where success was defined in terms of affective changes in confidence, interest, and enjoyment of mathematics, passing rates, and persistence rates. Once 18 of these more successful programs (covering the range of 2-year colleges to research universities) were identified, explanatory case studies were conducted to understand what it is about these programs that contribute to their success. Case studies consisted of 3-day onsite visits and included extensive interviews with faculty and students as well as classroom observations. Common characteristics of the calculus programs at the five mathematics PhD-granting departments were identified—many of which were also present across institutional types (Bressoud & Rasmussen, 2015; Rasmussen et al., 2014). Relevant to this study, coordination was one of these common features. Bressoud et al. (2015) provide more detail about the CSPCC study and its findings.

The PtC project follows up on the CSPCC study with a national census survey of university mathematics departments. The census survey examined the status and practices surrounding the common features of successful calculus 1 programs identified by the CSPCC study, but extended the course focus to include preparation for calculus courses and all of single-variable calculus (Apkarian & Kirin, 2017; Rasmussen et al., 2019). All nine departments with graduate programs that participated in CSPCC study and completed the PtC census survey were considered for inclusion in a follow-up survey which included social network questions. Of these nine, five institutions participated based on availability and willingness. To summarize, the five university mathematics departments investigated in this report come from sites that were selected in the CSPCC study as having a relatively more successful calculus 1 program, participated in the PtC census survey, and participated in an additional follow-up social network survey. The primary data set for the analysis reported here comes from the social network survey completed by members of the mathematics department at these five institutions, though we draw on data collected during CSPCC and PtC to contextualize the departments and suggest how various roles are enacted.

Methodology

Data collection

Social network surveys are used to ascertain the ties between people by asking participants to identify others with whom they have a particular interaction and/or relationship (Daly, 2010; Kadushin, 2011; Scott, 2012). Our survey asked about five relationships: seeking advice about teaching, seeking instructional materials, discussing instructional matters, friendship, and instructional influence (Coburn & Russell, 2008; Daly, 2010; Scott, 2012). These five relationships were selected in order to build a robust picture of relationships and social capital surrounding instruction. Seeking advice and seeking materials are related to leadership in that one turns to someone for specific resources (e.g., tomorrow’s exam, a particular problem) searching for guidance about specific things (e.g., tomorrow’s exam, a particular problem) while discussion and influence are more general. Seeking advice and reporting influence explicitly identify someone else with more expertise or importance, while seeking materials and discussion do not. These relationships are core to community-driven instructional practice. Friendship, which is a complex construct, can be used to ascertain how much of perceived leadership is due to popularity as a friend (i.e., do people go to Jane for advice because she is an expert or because she is personable) as well as to identify more substantive ties (i.e., discussion with a friend may be deeper than discussion with an acquaintance). In this paper, we do not delve into the friendship network, having used it only to check whether it is possible that outliers in terms of instructional influence also have outlying friendship status. In none of the departments under investigation was this the case, so we refrain from reporting those details. Focusing on the four instruction-specific relationships (seeking advice about teaching, seeking instructional materials, discussing instructional matters, and instructional influence) allows us to identify central actors who have ties with many others and who has access to different types of social capital.

Survey participants were presented with a list of names (also the participant pool) and asked to select all with whom they share a particular relationship or have particular types of interaction. These lists included mathematics instructors, department faculty members, administrators, and staff in position to oversee elements of the introductory mathematics courses, and graduate students with teaching assignments. At two of the five departments, both of which were larger in size in comparison with the other three (see Table 2), people without formal positions and who had not taught lower division mathematics courses in at least 2 years were omitted to reduce cognitive demand. The choice to approach this work via whole network analysis of multiple relations with a bounded roster and the decisions used in setting those boundaries are aligned with common and recommended practices in higher education social network research (Skvoretz et al., 2019).

Network analysis

We used graph theoretic techniques common to social network analysis to identify those with influence over instructional matters (Borgatti et al., 2013; Daly, 2010; Kadushin, 2011; Scott, 2012). A key construct in these analyses is an actor’s network centrality using measures of degree. Degree refers to the number of ties that an actor in a network has. An actor’s in-degree refers to the number of nominations they receive, while their out-degree refers to the number of nominations they provide. If an actor has degree zero, meaning that both their in-degree and out-degree are zero, they are considered an isolate. Note that non-responders are not always isolates, since respondents can nominate anyone from the list, but by definition they have out-degree zero.

Table 1 explains the interpretations of participant A selecting participant B for each of the four instruction-specific networks, which contribute to our interpretation of actors’ relative position within each network at each site. Note that of the four instruction-specific networks, only the discussion network is considered symmetric. The symmetric nature of that relation suggests the use of total degree (in-degree plus out-degree) as the appropriate degree centrality metric, but in this study, we use in-degree for all four networks in order to mitigate non-response bias (Quardokus Fisher & Apkarian, 2018). High and low degree values are determined based on the distribution of degrees across non-isolated actors in a particular network, rather than in absolute terms. This approach not only takes into account the local context and cultural norms regarding interaction but also mitigates concerns about missing data from non-respondents. Comparing inclusivity (the proportion of the pool with non-zero degree) across sites allows us to compare overall involvement in the networks and this approach to the distribution of ties allows us to understand what that involvement looks like for those participating.

Table 1 Theoretical interpretation of instruction-specific social network relationships. While discussion should in theory be treated as a symmetric relation, we consider it asymmetrically (see discussion)
Full size table

Identifying and describing leaders

Pulling from the literature about leadership and communities, we identify instructional leaders as those with extreme influence in multiple instruction-related networks, or multiplex ties among peers that imply robust influence on matters of undergraduate instruction. We put more emphasis on the advice and influence networks when interpreting overall influence, as these two are more explicitly aligned with influence and involve the designation of experts. In contrast, the sharing of materials and discussion of instructional materials likely have more implicit effects and one can imagine these interactions being quite superficial or even negative. This degree data is not normally distributed and is positively skewed, and so we use median and interquartile range of non-isolate in-degree to identify outliers in each network. We identify extreme using median and interquartile range (IQR). A value of 1.5 IQR above the median score is generally accepted as criteria for identifying outliers in data. Clean breaks in data are also sometimes used, when 1.5 IQR does not appear adequate for separating clusters of data, in particular with heavily skewed data or data with very small spread. Given the variation in contexts across sites, each network at each site was checked to see if the standard cutoff of 1.5 IQR above the median seemed appropriate, as well as if there was enough spread to declare anyone an outlier. We also considered actual difference in nominations that corresponds to the deviation in terms of IQR to avoid assigning too much importance to a small increase in the number of nominations. This is reported and discussed in context throughout the “Results” section.

The set of networks at each site was examined to identify any actors who fit our criteria for instructional leader, without consideration of those actors’ names and roles within the department. Once the instructional leaders for each department were identified, the roles of those actors were retrieved for comparison with formal hierarchies. The positions of any formally titled actors who were not identified in the initial analysis of a site’s social networks were further investigated to see their position relative to the rest of the department. A vertical leadership structure is evidenced by a small number of actors with central positions in the network, while more evenly distributed in-degree across actors’ hints at a structure of distributed leadership, and these are not mutually exclusive (Penuel et al., 2010; Spillane, 2006; van Ameijde et al., 2009). When the central actors in a network (i.e., those with highest in-degree) correspond to those with positional authority in the department hierarchy (i.e., those with formal titles), the informal and formal leadership structures are said to be in alignment; the existence of central actors in the networks who do not also have positional authority is evidence of misalignment and has the potential to subvert the official chain of command (Daly et al., 2014; Penuel et al., 2010).

The social network data was used to determine who in these departments has influence on their colleagues’ instructional practice. This information was then triangulated with the CSPCC case study interviews to understand where positional authority overlapped with instructional leadership, where it did not, and (in some cases) why. As part of the CSPCC study, all interview data was transcribed and coded via a thematic analysis (Braun & Clarke, 2006). One of the codes was “coordination” and hence, we were able to easily identify all interview excerpts from that work that related to coordination. More specifically, we took the social network interpretations and examined these excerpts for confirming and disconfirming evidence (Mathison, 1988). For the doctoral degree–granting institutions, we were also able to draw on previously analyzed interview data (Rasmussen & Ellis, 2015). Actors with positional authority were identified through their titles, either from official records or CSPCC interview data. The formal setup of coordination systems (see Table 2) was completed using the PtC census survey data. Interview data from CSPCC case studies also was used to reveal nuances about the coordination system, including characterizations of coordinators, attitudes towards the coordination system, and the extent to which cooperation with the coordination systems is enforced.

Table 2 Institution, departmental, and course coordination information from CSPCC, PtC, and publicly available data. PrTI does not offer a preparation for calculus course
Full size table

Participants and department settings

A first step in this project was to identify basic facts and features of each institution, department, and P2C2 coordination system. Table 2 gives a brief overview of each site, including institutional descriptors, the individuals identified as having positional power related to undergraduate instruction in addition to department chairs, and selected information about coordination system for P2C2 courses. This table shows the range of universities represented in our dataset, as well as the range of coordination systems. The selection criteria used in the CSPCC study focused on various measures of student success in calculus 1. It so happened that the application of these criteria resulted in a range of sizes and types of institutions, which adds robustness to the results of our research questions.

Results

In this section, we present the leadership findings at each of the five departments, which highlight the roles of important actors and those with formal titles. The appendix to this manuscript includes comprehensive tables of outliers at each site as well as additional network metrics.

Large public university

The first institution we discuss is a large, public, research university which we identify only as large public university (LPU). This institution’s mathematics department has three course coordinators (Co1, Co2, Co3) who rotate through pre-calculus, calculus 1, and calculus 2. Though their specific assignment changes every 1–2 years, these three are permanent coordinators and work together as a unit to manage the three courses. The department chair is selected from the faculty for a multi-year term, and generally serves for a single term.

The participant pool at LPU consisted of 61 people: faculty members and graduate students who taught lower division undergraduate courses in the year of this study, the department chair, and front-desk staff. Of these, 39 participated in the survey for a satisfactory response rate of 64%. At LPU, inclusivity ranges from a low of 0.53 in the influence network to a high of 0.85 in the discussion network; inclusivity is 0.59 in the materials network and 0.62 in the advice network. Of the 61 members of the participant pool, 5 were isolates in all four instruction-related networks. Together, these indicate active networks of interaction related to instructional matters. From those included in the networks, the median (interquartile range) of in-degree is as follows: advice 1 (0–1.75); materials 1 (0–1); discussion 1 (1–2); influence 1 (0–1.25). At LPU, all four networks are inclusive and spread enough for interpretation. Figure 1 is a plot of individuals’ in-degree in terms of IQR distance from the median.

Fig. 1
figure 1

Chart of LPU participants’ in-degree, scaled by IQR from the median, for all four networks

Full size image

Three actors (the coordinators, Co1, Co2, Co3) clearly emerge as instructional leaders. They are extreme outliers in all four networks, and while two other actors meet outlier criteria for one or two networks, they do not approach the level of leadership evidenced by the three coordinators. This concentration of ties indicates vertical leadership structure. The department chair is not an outlier in any of the networks.

In practice, this means that these three coordinators are the most likely to be asked for advice, to be sought out for instructional materials, to be involved in discussions about instruction, and to be considered influential on their colleagues teaching—at least in relation to lower division undergraduate mathematics courses. Not only do these three actors have formal responsibility and control of the P2C2 courses at LPU, they are also the concentrated hosts of explicit, implicit, targeted, and general influence over instruction. No other actors appeared to exert significant influence over instructional matters, including the department chair. In other words, the formal authority of all three coordinators is supplemented with real instructional influence, indicating that at LPU the coordination system is functioning as intended and the course coordinators are exactly the instructional leaders of the department.

These network-based findings are consistent with the original analyses of the CSPCC case study data (Rasmussen & Ellis, 2015). These coordinators are appreciated by other instructors as reliable resources for advice about content as well as teaching. They maintain websites to host instructor resources and sample materials, which are widely used by others. While these three were viewed by their colleagues as leaders, it was noted that they did not insist on every detail of course delivery. This allows instructors’ pedagogical autonomy, a key element of coordinated systems described by Rasmussen and Ellis (2015). This point, which was echoed by the coordinators and other instructors, is explicit in the following interview excerpt from a calculus instructor who had worked with two different coordinators:

Neither of [the coordinators] felt controlling. So it felt very much like there’s a team and the three of us were making decisions together. And in both cases they said ‘okay, here is what I typically do,’ but we can change that around. So it never felt like some control was being imposed on me as a teacher.

These three course coordinators all hold the title of “teaching faculty” at LPU, which is a faculty role with a tenure track. This role has an increased emphasis on teaching and educational scholarship, with parallel but distinct standards for review, tenure, and promotion as compared with research faculty. That these three were hired explicitly to be course coordinators may have an impact on how they see their role in the department, how others in the department view them, and how that role is realized. We do not have enough evidence to make strong conjectures about the nature of that impact, but it may be relevant for conceptualizing the extreme nature of their instructional leadership positions within the LPU mathematics department.

Public technical university

The second school in our sample is a public technical university (PTU). It is a smaller research university than LPU, but has a comparably sized mathematics department. This institution has three course coordinators and a lab coordinator, each a multi-year, semi-permanent position. One coordinator manages calculus 1 (Co1), two oversee calculus 2 (Co2, Co3) instruction, and the lab coordinator (LC) works across courses. Co2 and Co3 are newer to their role of coordinator. All calculus courses at PTU have a technology-based laboratory component, which the lab coordinator, a senior lecturer, manages.

The participant pool here was 56 people, selected using the same criteria at LPU, and 35 of these responded to the survey for a satisfactory response rate of 62.5%. At PTU, inclusivity ranges from a low of 0.61 in the materials network to a high of 0.95 in the discussion network; inclusivity is 0.63 in the influence network and 0.71 in the advice network. Of the 56 participants at PTU, only 1 was an isolate in all four networks. As at LPU, these indicate active networks of interaction related to instructional matters. From those included in the networks, the median (interquartile range) of in-degree is as follows: advice 2 (1–6); materials 1 (0–1); discussion 3 (1–4); influence 1 (0–4). As with LPU, at PTU, all four networks are inclusive and spread enough for interpretation. Figure 2 is a plot of individuals’ in-degree in terms of IQR distance from the median.

Fig. 2
figure 2

Chart of PTU participants’ in-degree, scaled by IQR from the median, for all four networks

Full size image

In the PTU networks, two actors (the lab coordinator, LCo, and calculus 1 coordinator, Co1) emerge as clear instructional leaders. They are the most extreme outliers in all four networks, and while a few actors broach the cutoff criteria in the materials and discussion network, they do not approach the pronounced and systematic leadership evidenced by LCo and Co1. The other coordinators (Co2, Co3) are not outliers in any of the four networks. The department chair appears as an outlier only in the influence network, where he has in-degree just below that of LCo and Co1. As with LPU, the concentration of ties among a few actors indicates an overall vertical leadership structure.

In practice, this means that these two coordinators are the heaviest influences on explicit elements of practice (via advice and material sharing) and are involved in many conversations about instruction (a more general, implicit influence), at least in regard to lower division undergraduate mathematics courses. Considering the alignment of formal and informal leadership at PTU, we see that LCo and Co1 have positional power backed up with real influence. The department chair is influential but only in a diffuse, generalized way. The other course coordinators have formal leadership assignments, but the network analysis indicates that this influence is not particularly deep. This is evidence of some misalignment of formal and informal leadership structures, but the absence of interlopers (i.e., no other department members are really leading without formal authority) implies that while the system may not be functioning to its full potential, it is not malfunctioning.

The original CSPCC interview analyses provide some context and explanation for the result that only one of the three course coordinators has real influence over instruction. The influential coordinator frequently takes the helm and provides default options to those working under her; the other coordinators project less authority and consider themselves to be “organizers” rather than supervisors of the coordinated courses. For example, one of Co2/Co3 put it this way:

We [referring to themselves and Co3/Co2] both try to be just at the same level as the instructors who teach the course and we’re both willing to listen to what they have to say … I’m happy coordinating this, but I’m also happy learning from her [an instructor’s] experiences. So she really brings some valuable things to those meetings and it’s good to talk to her.

Their decision to operate “at the same level” as the other instructors may explain why Co2 and Co3 do not exert significant influence over instructional practice in the department. The network approach we leveraged reveals the influence of the lab coordinator, who at the time of the CSPCC study had only just begun his role developing computer-based laboratory activities.

Public master’s university 1

The university identified as public master’s university (PU1) is a public university whose most advanced degree in mathematics is a master’s degree. It is a smaller university, with a smaller mathematics department than at the comparably sized PTU. The coordination system at PU1 is looser than that seen at some other sites, where only topics covered and one-third of the final exam items are uniform across classes. Course decisions at PU1 are made by committee, and that committee has a chairperson who we refer to as a coordinator (Co). The actor identified here as chair has a long-term position as department head, and has been in that role for over two decades.

At PU1, the participant pool included exactly the 24 department members; 15 of these responded to the survey for a 63% response rate. At this site, inclusivity ranges from a low of 0.33 in the materials network to a high of 0.83 in the discussion network; inclusivity is 0.75 in both the advice and influence networks. Of the 24 participants at PTU, 3 actors were isolates in all four networks. From those included in the networks, the median (interquartile range) of in-degree is as follows: advice 1 (1–2); materials 1 (0–1.25); discussion 2 (2–3); influence 1 (0–1). Unlike LPU and PTU, not all the networks are appropriate for outlier analysis in relation to leadership.

Only a third of the participants at PU1 (8 actors) are involved in the exchange of instructional materials, and there are only 7 ties between them. This network is also flat, as no actor has in-degree greater than 2. Due to the extreme sparseness of this network, and a lack of spread, we ignore this network for the purposes of identifying instructional leaders at PU1. This phenomenon may be related to the structure of PU1’s coordination system, where only a subset of items are common across exams and there is rarely more than one person teaching each course in a particular term. Figure 3 is a plot of individuals’ in-degree in terms of IQR distance from the median for the other three networks.

Fig. 3
figure 3

Chart of PU1 participants’ in-degree, scaled by IQR from median, for the advice, discussion, and influence networks

Full size image

At PU1, the evidence of individual instructional leadership is more scant than at LPU and PTU. The department chair and calculus committee chairperson (Co) are sought out for advice more than their colleagues, and Co is also an outlier in the discussion network. While the influence network includes 75% of the participant pool, it is flat like the materials network with a maximum in-degree of 2. All actors at PU1 are within 1 IQR (1 nomination) of the median in this network. While a flat and sparse materials network can be in part explained by the structure of a coordination system, there is not a comparably simple explanation for a flat influence network. The leadership structure at PUI is thus more distributed than at LPU or PTU, though central actors are detectable and so there is some verticality.

In practice, this indicates that the coordinator is the member of the PU1 mathematics department with the most influence over their colleagues’ instruction, and this influence is manifested through providing advice and being an active discussion partner. The department chair is also a source of advice, though less involved in other networks. In general, instructional materials are not being shared and no actors are being explicitly identified as influencing instruction.

The small size, freedom, and instructors’ beliefs that they are all good teachers may contribute to this. The CSPCC case study data underscores the importance that instructors at PU1 place on pedagogical autonomy and their belief in their colleagues’ ability to teach well. As one of the long-time faculty members put it, “anything having to do with pedagogical technique is really the concern of the individual instructors.” Another long-time instructor summed it up as “we trust each other very much.” These attitudes may in part explain why at PU1 no actor stands out in terms of instructional influence despite identifying sources for advice.

Despite sparse networks, the coordinator and chair’s advice is sought after more than anyone else’s, and no other department member is in a position to undermine that authority. This leadership structure is clearly distributed, though the coordinator and chair’s relative importance hint that there is a vertical aspect as well.

Public master’s university 2

Our study included another public university offering a master’s degree in mathematics, which we refer to at public master’s university 2 (PU2). A committee makes decisions about P2C2 courses, and for this site, we refer to the head of that committee as a coordinator (Co). Uniform course components at PU2 are restricted to topics covered and the calculus 1 final exam, with regular meetings for the instructors of pre-calculus courses but not those of single-variable calculus. At the time of the original CSPCC study, the department chair had been in that role for 12 years, and had overseen the hiring of roughly two-thirds of the department. The department chair during the time of this study had been chair for about 2 years, but had been part of the department for over a decade and had pushed for further course coordination in calculus 1 prior to her appointment as chair.

The participant pool at PU2 consisted of all 23 mathematics department members, and 14 (61%) responded. At this site, inclusivity ranged from a low of 0.52 in the materials network to a high of 0.83 in the advice network; inclusivity is 0.74 in the influence network and 0.78 in the discussion network. Overall, four actors are isolates in all four networks. From those included in the networks, the median (interquartile range) of in-degree is as follows: advice 3 (2–4); materials 1 (1–1.25); discussion 2 (2–4); influence 1 (1–2). As with PU1, the materials network is not appropriate for a similar analysis of individual leadership, and Fig. 4 is a plot of individuals’ in-degree in terms of IQR distance from the median for the other three networks.

Fig. 4
figure 4

Chart of PU2 participants’ in-degree, scaled by IQR from median, for the advice, discussion, and influence networks

Full size image

Participation in the instructional materials network at PU1 is noticeably lower than the other networks, though over half the participant pool are involved. Again, this is likely related to the structure of their coordination system, in which there are not many uniform course elements. The materials network is also problematic in this analysis because the IQR, 0.25, is less than the minimum stepwise increase in nominations. That is, a single nomination above the median corresponds to four IQR above the median, which inflates the extremity of outliers in a confounding manner (i.e., 3 total nominations is actually 8 IQR above the median in-degree of 1). Information gleaned from the materials networks is not interpreted in the same way as networks with more spread.

The coordinator (Co) is an outlier in all four networks. As noted, the criteria for outliers in the materials network is somewhat suspect, but Co has the highest in-degree in that network. The department chair is an outlier in the discussion and influence networks, and has the second-highest in-degree in the advice network (after Co) though it is not quite high enough to be an outlier. No other actors are outliers in more than one network. There is evidence of a vertical leadership structure, as there is a concentration of ties. As with PU1, one network is too spread for analysis and the outliers are not as extreme, suggesting that the leadership structure is more distributed than at LPU and PTU, though there is some verticality.

These results indicate that the coordinator at PU2 is the most influential member of the department, which is reported explicitly by their colleagues. They are also influential through the provision of advice about instruction and through frequent discussion about instructional matters with others in the department. They are also the most likely to be asked for instructional materials. The department chair is also an influential figure on instruction, but as they are not as pronounced in the advice and materials networks, the exact nature of this influence is uncertain, not dissimilar from the scenario at PTU.

Thus, at PU2, those with formal leadership (chair and calculus committee coordinator) have accompanying informal power as evidenced by their position in the advice, discussion, and influence networks. There is no one else with heavy influence in the department, so that the formal and informal authority rests with the same actors.

Similar to PU1, belief in one’s colleagues’ teaching abilities was a common theme in the CSPCC interview data. Sentiments such as “I have my department trust that I’m going to do a good job teaching calculus” and “part of the philosophy here is they try to follow people they trust to do the job and let them do the job” speak to how pedagogical autonomy manifests itself in this department. Nonetheless, interactions at PU2 result in clear indications of instructional influence stemming from certain actors.

Private technical institute

The private technical institute (PrTI) is the final institution covered in this study, and offers a PhD in mathematics. This school is roughly the size of PTU, but with a smaller mathematics department. At PrTI, there is a designated course coordinator (Co1) and a calculus committee to oversee introductory courses, and the calculus committee is managed by a chairperson (Co2).

The participant pool of 26 at PrTI included all recent instructors and a few key administrators. Of these, 16 responded for a response rate of 61.5%. At this site, inclusivity ranged from a low of 0.69 in the materials network to a high of 0.85 in the discussion network; inclusivity was 0.81 in the influence network and 0.77 in the advice network. No actors were isolated in all four networks. From those included in the networks, the median (interquartile range) of in-degree is as follows: advice 3 (1–7); materials 1 (0–3); discussion 3.5 (1–5); influence 2 (1–5). Figure 5 is a plot of individuals’ in-degree in terms of IQR distance from the median for all four networks.

Fig. 5
figure 5

Chart of PrTI participants’ in-degree, scaled by IQR from the median, for all four networks

Full size image

This department’s networks have higher median in-degree values than the previous sites, and larger IQR. This, especially in light of the smaller size of the department, describes a department in which leadership as measured by the selected social relationships is distributed. The compressed nature of the PrTI networks and the distribution of leadership indicators make it impossible to make claims about the existence of individual instructional leaders as described at the other sites.

Instead, we focus on the relative position of those with formal titles. There are no clearly emergent instructional leaders as operationalized in this study, as suggested by the median and IQR of these networks. However, the two coordinators, Co1 and Co2, are the most influential in terms of advice and discussion networks and rank highly in the explicit influence network. The department chair has low involvement in all the instructional networks.

There are some department members whose instructional materials are sought after, but they are clustered and these actors do not appear prominently in other networks. The fact that instructional materials are not coming from a unique source is made less surprising when we note that this department does not have many uniform course elements—homework and non-final exams are at the discretion of the instructor. In practice, this indicates that instructional materials are not wholly centralized and the positional authority vested in the coordinator and chair of the calculus committee is reinforced by personal power and influence. This is another instance of a good match between title and role, though it is not as emphatic as seen at other sites. The department chair’s positional authority does not appear to extend to influence over instructional practice.

This is further supported by the CSPCC interview data, which indicated that the coordinator organizes weekly meetings and provides feedback to instructors, but does not insist on the usage of pre-specified elements. Instead, as indicated by the following interview excerpt from one of the instructors, these weekly meetings provide an opportunity for faculty to productively collaborate and learn from each other without imposing a centralized instructional system:

We have this new model where we have a weekly instructor’s meeting for Calculus 1 and Calculus 2. And so we all talk about the different things that we are doing, the different approaches, things that are working, things that might not be working. And I can feel that we are using each other’s ideas.

Summary discussion

Sociocultural theories of learning and doing suggest that norms for practice are negotiated through person-to-person interaction including conversations, advice, and observations (Lin, 2002; Schein, 2010; Wenger, 1998). These principles suggest that postsecondary teaching practices are shaped by intra-departmental interactions among instructors. Understanding or changing instruction, then, can be supported by an understanding of interaction patterns and the leaders they delineate. Though some research suggests that faculty generally teach in isolation, we note that at all of the departments included in this study, at least 75% of the population pool engage in discussions about instructional matters (i.e., inclusivity ≥ 0.75). Due to non-response bias, proportion may actually be higher. This implies that at these sites, discussions of teaching are frequent and inclusive of many instructors and faculty, even those who are not currently teaching lower division undergraduate courses. Ergo, at these sites, there is a readily identifiable network through which norms are being negotiated in relation to instructional matters.

All five of these departments were selected for their apparent support of student success in calculus 1 according to measures including grades, persistence, and attitude. As such, we cannot make direct claims about the relationship between our results and student success in mathematics. However, the patterns we have observed support conjectures about mediating factors that relate departmental contextual factors and leadership structure in ways that may contribute to the observed successful functioning. We now turn to answering our research questions.

Variation in leadership structure

Our first research question asked the extent to which instructional leadership is vertical or distributed in these five mathematics departments. In fact, there is variation in the overall leadership structure. The network analysis at these five sites suggests the presence of vertical leadership at LPU and PTU, distributed leadership at PrTI, and a mix of vertical and distributed leadership at PU1 and PU2 in regard to undergraduate instruction. One contributing factor, from a methodological standpoint, is the varying usability of the materials network data across sites. This variation in usability is likely highly related to the course coordination models in use at the different sites; LPU and PTU have more uniform course elements than the other sites. More lax course coordination may reduce the importance of central figures, as there are fewer things which require consultation among instructors. The strongest evidence of vertical leadership with centralized actors also occurs at the two largest departments, the only other feature from Table 2 which seems to coincide with this pattern. Our sample is too limited for particularly conclusive statements, but these data hint at a relationship between department size, the quantity of uniform course elements, and the existence of vertical leadership structures.

The three smaller departments have much more distributed leadership structures. At PrTI, influence as measured in this study is very evenly distributed, and quite sparse, among department members. No department members, including those in coordinator-like roles and the department chair, stand out in any significant way in any of the measured networks. At PU1 and PU2, there is evidence of instructional leaders, coincident with those holding formal titles, but it is not as strong as in LPU and PTU. PrTI stands apart from the others in the extent to which network ties, and so leadership, are distributed among actors. However, the advice, discussion, and influence networks there are active and inclusive. While the formal coordination system (see Table 2) is not as robust as some others, we conjecture that the high communication activity, small department size, and general feelings of trust between faculty may result in informal coordination. That is to say, we expect the instructor was correct that as a group, they “are using each other’s ideas.” This suggests an alternative, organic, form of course coordination. Again, our sample limits the generalizability of our results, and so we have no evidence that this is possible in a larger department nor whether it has the same benefits for students as a more formalized system.

Another structural factor at play may be the permanence of the role of coordinator, as posited in previous research (Rasmussen & Ellis, 2015). The three coordinators at LPU are permanent course coordinators, though their exact assignments rotate through the P2C2 sequence. All three have pronounced influence in the networks, with the longest serving coordinator the most extreme. At PTU, the longer serving course coordinator and lab coordinator have the most pronounced influence over instruction, while two other coordinators blend into the general population. Interview analysis indicates this is in part because of distinct approaches to the role, and it may be that longer time in the position shifts both their perception of the role and others’ perception of them in that role. The three sites with more distributed leadership structures also have more committee-style course coordination, and in small departments, that committee might represent a large proportion of the population. Rotating committee assignments and intimate context may relate to the observed diffusion of influence, and without negative consequence.

(In)Formal instructional leaders

We used SNA to identify actors with informal influence, which we then compared with the formal leadership hierarchies in each department. This SNA approach is situated in departments, not only contextually but in our decision to identify central actors based on their influence relative to local norms for interaction rather than global benchmarks or cutoffs. We attended to multiple relations, so that those we identified through SNA are hubs of multiple kinds of social capital. All these decisions are aligned with the conceptualization of central actors in communities of practice. When we began this work, it was unclear the extent to any leaders would emerge who satisfied the criteria developed out of these theories of community-based leadership.

In four of the five departments, individuals were identified through SNA as instructional leaders who are positioned to influence teaching among their departmental colleagues. The most influential leaders were not department chairs, but individuals with course leadership roles. At LPU, the chair is essentially absent from the instructional influence networks; at PTU and PU2, the chair appears as an outlier only in the generic “influential on teaching” network; at PU1, the chair is prominent only in the “advice about instruction” network. One interpretation of this finding is that these coordinators, by taking up course leadership activities, allow chairs to focus on other aspects of departmental leadership. As these calculus programs are successful, it seems that this delegation structure can be effective.

The instructional leaders as identified through SNA all also had formal titles or positions which relate to instructional practice. That is to say, there are no “interlopers,” or people who exert major influence but are not part of the official department or instructional oversight team. We conjecture that the absence of interlopers is a feature of stable departments, as this kind of alignment is beneficial for organizational function and change efforts (Ma et al., 2019; Penuel et al., 2010; Schein, 2010). The existence of informal leaders who are not part of an official course leadership team might indicate factions or unrest, warranting further attention.

Not all those with formal course leadership titles realize the same level of influence within their respective departments, so there is no perfect overlap of formal and informal leadership status. At PTU, two of the three course coordinators are not identified by the SNA; at PrTI, neither of those in course leadership positions (nor anyone else) are identified. The observation that all coordinators do not have the same level of influence over instruction (even within a single department), combined with the range of network positions held by department chairs, indicates that titles are indeed insufficient for identifying instructional leaders in mathematics departments.

Future directions and implications

From this study of five more successful university mathematics programs, new conjectures have emerged, and work testing these conjectures could serve to build recommendations for developing productive instructional leadership. We conjecture that department size, the nature of the coordination system, and how an individual takes on the role of coordinator all have implications for instructional leadership. Regarding department size, we expect that distributed leadership is more possible in smaller departments where instructors can communicate with all their colleagues, while a central hub and more vertical leadership is helpful in large departments. We expect that the scope of responsibilities and the purview of the coordinators will impact the instructional influence networks. For example, when coordination does not explicitly involve shared materials (e.g., common exams), we imagine that network will be sparse. Finally, we conjecture that the ways in which individuals perceive the role of coordinator, and the ways in which coordinators take on that role, impact the extent to which they emerge as prominent instructional leaders. From the evidence collected at PTU, we see that the coordinator-leader acts more as a supervisor while the other coordinators act more as organizers; we expect that there are many more factors at play that further investigation might reveal. Targeted work to compare individuals who function as leaders with those who do not, within and across contexts, should shed additional light on what departmental and individual characteristics affect those positions. In particular, how and why certain individuals in certain contexts have influence, and the actualized impact of that influence on instructional decisions or change. Certainly, our findings support the idea that leadership and the distribution of leadership are highly contextual, and should not be overly simplified into categorizations of good or bad, strong or weak.

Other questions emerge from this work as well, which can only be answered through expansion of this approach to other contexts. For example, we noted active instructional networks and the absence of “interlopers” in these five departments, each of which are relatively more successful with their calculus programs. In departments which are less successful, are the networks sparser? Are there interlopers? Work in other contexts may allow for more causal, rather than correlational, connections between departmental function and leadership structures. Additionally, this work was done in mathematics departments with some level of course coordination. Perhaps in other disciplines, or departments which do not have any official coordination structure, leadership structures will differ—and perhaps other individuals will be more prominent. Department chairs may be more involved, or there may be informal leaders who are neither formally responsible for courses nor interlopers. These kinds of questions, and the potential implications of the answers, suggest utility in expanding this work into additional contexts.

We see two implications of this research for those seeking to understand, or change, a department’s instructional practice. First, we provide empirical evidence that in academia, department chairs do not always lead with regard to the day-to-day matters of instructional practice. We surmise that course coordination practices can take on that role, and relieve some of the burden chairs carry. This is compatible with work on the roles of department heads in general, who must promote institutional priorities, advocate for their faculty within the institution, and so on (Bryman, 2007; Thornton et al., 2018). Second, a given course coordinator may or may not have a major influence over instruction. This insight suggests that those in leadership positions, and those seeking to change practice, should carefully consider existing nuances and potential networks in their work, as well as characteristics of coordinators in regard to their desire to and expertise in promoting a community of practice.