The Duality of Courses and Students

Abstract

Significant attention has been given to how students become grouped or “tracked” through the courses they share in common. However, this work has yet to be connected to a targeted analysis of the way in which courses are grouped with other courses through the students they co-enroll. Drawing on insights from field theory, the author examines this duality with special attention to the social organization of courses and the curricular discourses they contain. Multidimensional scaling and multiple correspondence analysis are used to analyze course-taking data for a cohort of students (2005-2009, n = 494) at a comprehensive high school in the midwestern United States. The results illustrate a “field” of courses that are distributed vertically according to a principle of status that opposes different forms of curricular discourse and horizontally according to oppositions between symbolic and material forms, artistic and technical skills, and the “inner” work of the household to the “outer” work of certain occupations. While the vertical dimension of courses is associated with a racial and social class hierarchy of students, the horizontal dimension passes through a division of the sexes.

Keywords

tracking curriculum social structure courses duality field theory

Courses serve as an important medium through which students become affiliated at specific places and times, thinking, acting, and speaking in certain ways. Put simply, students and courses constitute a social structure. Similar to any game that subjects players to an existing set of rules, this social structure is a patterning of relations into which students and courses are projected. When students enter secondary school, for example, they do not encounter a randomized curriculum or enroll in courses willy-nilly. Rather, students “bump” into the residual tracings of socio-curricular configurations that have preceded them through historical struggles and compromises. In this sense, the social structure of students and courses is capable of shaping thought and action across individual schools, albeit with often disparate and unexpected results. This is not to say that this social structure determines student actions or outcomes or that students do not “push back” in meaningful ways. Rather, to persist as a social structure, courses must be continuously integrated and segregated through the assiduous work of students.

For most contemporary researchers in this area of the literature, the “shape” of this social structure is formed by vertically ordered course affiliations between students. Thus, students who share courses in common will occupy similar positions or “tracks” that form a vertical hierarchy. Researchers have developed sophisticated techniques for constructing these positions (for reviews, see Gamoran 1989, 2010), which have then been used to investigate a number of important questions. These questions have focused on the factors that predict students’ position or track placement (e.g., Kelly 2004; Lucas and Gamoran 1993), as well as the implications this position has for their standardized test scores and postsecondary trajectories (e.g., Gamoran 1987; Sorensen and Hallinan 1986).

While researchers have made significant progress in this productive area of the literature (for reviews, see Gamoran 2010; Oakes, Gamoran, and Page 1992), there remains a gap in our understanding of the social structure of students and courses. Although much has been learned about how the organization of courses has meaningful consequences for students, very little attention has been paid to how the organization of students has meaningful consequences for courses and the content contained within and between them. That is, just as students are differentiated through the courses they share in common, courses, too, become integrated and segregated through the students they co-enroll. To speak of social structure in this context, then, is to specify a duality that is constituted by affiliations within and between a space of students and a space of courses. Yet—as a distinct unit of analysis—the space of courses has received limited attention within this area of the literature.

The following article attempts to address this aspect of the literature by illustrating that a more targeted analysis of the entire space of courses can contribute to a deeper sociological understanding of the social organization of a high school’s curriculum. In particular, this article proceeds to make two distinct contributions. First, I will demonstrate how a secondary school’s curriculum becomes organized in practice through a duality of student and course affiliations that take shape over a four-year period. The social organization of secondary school curriculum is thus conceptualized as courses—and the curricular discourses they contain—occupying positions in a multidimensional space. In addition, beyond considering only the vertical hierarchy of secondary school courses (e.g., advanced/general/remedial courses), I introduce a “horizontal” form of differentiation that serves as an important principle of organization within the curriculum.

Second, I examine the space of courses in relation to student attributes and postsecondary trajectories and argue that the different regions of this space represent relational strategies associated with students’ position within this local social structure. That is, in addition to examining how the curriculum is organized through the social practice of course-taking, I also investigate “who” is organizing “which” particular segments of the curriculum. This contribution involves thinking theoretically about the duality of students and courses as a curricular “field,” and to proceed with this work I draw from field theory and curriculum theory. Further, this theoretical work is undertaken in conversation with longitudinal course-taking, demographic, and postsecondary trajectory data collected from a large comprehensive secondary school in the midwestern United States. To model the duality of students and courses at this secondary school I use multidimensional scaling and multiple correspondence analysis. These techniques contain an inherent logic that is consistent with the overall theoretical logic of the analysis, a point that speaks to a subtext underlying the entire work.

The results reported in this article may have suggestive implications for sociology of education and curriculum theory. Whereas sociologists and curriculum theorists are likely to view the social organization of the curriculum as either a mere reflection of exogenous social tensions or as an influential medium through which students’ performance on standardized test scores may vary, the analysis herein suggests that these “vertical” variations are further complicated by important “horizontal” differences. An analysis of the entire space of courses thus opens the possibility of multiple curricular configurations within the same vertical positions that can meaningfully shape academic identities and experiences. The work described in the following begins to explore these possibilities in relation to racial, class, and gender dynamics.

Curricular Tracks, Positions, and Spaces

Conventional research focusing on the social organization of secondary school curriculum has typically focused on discrete categories or “tracks” (e.g., academic/nonacademic, advanced placement/general/remedial, etc.). These categories have formed into a pervasive discourse, making it difficult to speak or think of the curriculum without them—even in the absence of an explicitly rigid tracking system. To be sure, many researchers have responded to the absence of an explicitly rigid system of tracking by developing a range of strategies for uncovering the more implicit system observed in secondary schools today. Among the more sophisticated strategies include the curriculum index developed by Hotchkiss and Dorsten (1987) and the course-based indicator conceived of by Lucas (1990; see also Lucas and Gamoran 1993).

Another insightful strategy has been to construct “social positions” using techniques from social network analysis. For example, Friedkin and Thomas (1997; see also Heck, Price, and Thomas 2004) utilize a blockmodeling approach to analyze an affiliation matrix consisting of students and courses from the High School and Beyond data set. In network terms, an affiliation matrix is a two-mode data set consisting of a set of actors (e.g., students) and a set of events (e.g., courses). Friedkin and Thomas use an iterative algorithm to categorize students into mutually exclusive positions based upon the similarity of their course-taking profiles. Their algorithm derived eight “socio-curricular” positions, which were then found to be associated with students’ status attributes, attitudes, and senior year achievement.

While the previously cited works may vary in the way they derive students’ positions within the socio-curricular order, they each similarly give primary attention to students and the way they are organized through the courses they share in common. Meanwhile, courses are only important insofar as they provide a medium through which students are grouped (or “tracked”) and subsequently provided access to specific peer relations and learning opportunities. This emphasis on students is certainly reasonable given the sociological context of the work, and as noted earlier, this approach has yielded a number of important insights. However, the result of this student-centric focus has been a theoretical understanding that foregrounds students while drowning out courses as a distinct unit of analysis. As a result, there have been few attempts to conceptualize—let alone analyze—students and courses as components comprising a distinct “whole.”

Some sociologists of education have begun to think about course-taking in this way. Field et al. (2006) and Frank, Muller, and Schiller (2008) use a similar network approach as Friedkin and Thomas, but instead utilize information from student and course affiliations when algorithmically deriving social positions of course-taking. That is, while each student has a course profile specifying the courses they share in common with other students, each course has a student profile indicating the students that course shares in common with other courses. To derive positions utilizing information from both modes of data, Field et al. (2006:103-06) draw upon the work of Skvoretz and Faust (1999) to extend Frank’s (1995) clustering algorithm for identifying cohesive subgroups for one-mode data (e.g., a symmetric student-by-student matrix) to two-mode data (e.g., a student-by-course affiliation matrix). Frank et al. (2008) then applied this algorithm to data from the Adolescent Health and Academic Achievement Study in order to examine how these “local positions” influence high school mathematics course-taking decisions.

A key insight from this work is its recognition of the inherent “duality” between students and courses. That is, on the one hand, students who share similar courses in common will occupy spatially proximal (or “structurally equivalent” in network terms) positions. Courses, on the other hand, that co-enroll similar students will also occupy spatially proximal positions with other courses. This “duality of persons and events” (Breiger 1974) means that the organization of interpersonal (e.g., student) affiliations and inter-event (e.g., course) affiliations are simultaneously distinguishable and mutually constitutive: The positions in one space constitute the relational units of analysis in the other (and vice versa). Speaking more generally, Mohr and Duquenne (1997:311) note that this duality has a number of important implications. First, the two structures of the duality—students and courses in this case—are conceptually equal. Second, these structures have an autonomous patterning and thus do not exist in a direct one-to-one correspondence. Finally, there is a “higher order” mapping of the two structures onto one another that fully captures the duality.

Unlike previous innovations in this area of the literature, the focus on social positions and the duality of students and courses is not only a methodological fix to cope with the breakdown of a rigid system of tracking. In addition, embedded in this approach is a theoretical point about the underlying social processes at work in the practice of course-taking. This point can best be described through the terminology of field theory. At the most general level, field theory explains social action by way of induced conceptions of conduct appropriate to particular positions within social environments (Martin 2011:312). Each position in a social field is associated with a finite range (or “vector”) of possible meanings and strategies for action. Thus, rather than viewing course-taking practices as a function of individual attributes, a field theorist argues that these practices arise out of action imperatives that are differentially distributed to—and experienced by—students occupying specific curricular positions within high schools.

Sociologists of education are likely to be familiar with field theory through the work of Pierre Bourdieu (e.g., 1993b, 1996). Bourdieu understood social fields as organized spaces of positions in which social actors strive for specific forms of material and symbolic resources (i.e., forms of capital). Each position is associated with a range of stances, styles, meanings, and/or strategies, which Bourdieu referred to as “position-takings.” For example, in reference to the scientific field, Bourdieu (2004:59) notes that “[o]n the basis of each scientist’s position within it [scientific field], the structure assigns to him his scientific strategies and position-takings and their objective chances of success.” In this sense, fields contain social objects with a causal “texture” that suggests to actors how they might proceed from one action to the next (Martin 2011:315-16), whether it be in adopting certain scientific strategies in a scientific field or enrolling in particular courses in an educational field.

Bourdieu (1996) put this field-theoretic approach to work in educational contexts most notably through his analysis of the French education system. In this analysis Bourdieu illustrates the positions within the field of grandes écoles, along with the array of tastes, styles, and choices associated with specific academic trajectories and credentials. Rawlings and Bourgeois (2004) utilize a similar approach to examine the driving forces underlying the differentiation of institutional categories within the field of agriculture in the U.S. higher education system. These latter analyses differ from the previously mentioned network approaches, however. Rather than relying solely on a blockmodeling algorithm to fit students and courses (or credentials) into mutually exclusive groups (i.e., positions), these field-theoretic approaches have also utilized spatial analytic techniques to analyze the latent organizing principles underlying the duality. These approaches are certainly not at odds with a focus on deriving a finite set of discrete positions. Rather, each approach enables researchers to ask different yet complementary sets of questions (e.g., see Rawlings and Bourgeois 2004).

To explore the latent principles underlying the organization of students and courses I utilize a pair of methodological tools (described in the following) that attempt to simultaneously represent the (dis)similarities between students and courses as distances in a low-dimensional space. I then draw upon field theory and curriculum theory to describe the principles that underlie these distances. We will see that the duality of students and courses is arranged vertically into a racial and class division of academic labor that opposes courses that emphasize “integrative” and “specialized” curricular discourses to those courses that have a basis in “common sense” practices (Bernstein 2000). These distances are also arranged horizontally into a sexual division that opposes courses focusing on symbolic, artistic, and intrinsic knowledge to those emphasizing material, technical, and extrinsic knowledge. While Bourdieu (e.g., 1993a) found this “symbolic/material” division at work in numerous social fields of practice (i.e., a “homology between fields”), in the present context this simple dichotomy alone does not fully capture the way in which the horizontal distribution of courses is contingent upon a particular vertical orientation.

Mapping Duality

Ronald Breiger (1974; see also Rummel 1965) provides a foundation for mathematically representing data when this type of duality exists. The initial affiliation matrix (A) for this analysis consists of binary course enrollment data, for which the present case includes data for each student across four years of course-taking (precise details are described in the following section). Thus, each row of data represents a student, and each column represents a unique course. In this matrix a 1 indicates that student i enrolled in course k, and a 0 indicates the student did not enroll in that course. By transposing matrix A, the data can also be represented such that the courses occupy the rows and students occupy the columns (A^T). Within this representation a 1 indicates that course k had student i enrolled, and a 0 indicates it did not have that student enrolled.

Using the two matrices (A, A^T) each two-mode data set can be transformed into separate one-mode data sets through matrix multiplication. If matrix A is postmultiplied by its transpose (A^T), for example, the result is a student-by-student matrix in which each cell indicates the number of courses students i and j share in common. Similarly, we can also postmultiply matrix A^T by A, resulting in a course-by-course matrix where each cell indicates the number of students courses k and l share in common. Thus, in matrix AA^T the students constitute the objects and courses are the relation, whereas in matrix A^TA courses are the objects and students are the relation.

In the analysis that follows, I use two related spatial techniques: multidimensional scaling (MDS) and multiple correspondence analysis (MCA). Multidimensional scaling is used to analyze the space of students (AA^T) and the space of courses (A^TA), whereas MCA is used to represent the duality of students and courses and to explore the underlying dimensions upon which this duality is structured. Multidimensional scaling and MCA are robust techniques for graphically representing proximities between objects as distances in a low dimensional space. In MDS, proximities (δ_ij) refer to any measure of similarity or dissimilarity between objects, whereas MCA relies on χ² distance as a dissimilarity measure. Objects in the present context refer to students, courses, or attributes (e.g., race, gender, postsecondary plans, etc.).

Multidimensional scaling is typically used to analyze a single mode of a duality (e.g., matrix AA^T) as an autonomous space. The first step in MDS is to select an appropriate proximity measure. Since the data in matrices AA^T and A^TA represent co-occurrences (e.g., students or courses shared in common) these data represent similarities. There are many similarity coefficients for binary data (see Gower 1985). One such coefficient is the Jaccard similarity measure, which is the proportion of events in which two courses (or students) co-occur when at least one of the courses (or students) occur. This conceptualization of similarity places courses that co-enroll the same students closer to one another in space than those that do not co-enroll the same students—regardless of the overall frequency (or magnitude) of enrollment.

The final outcome of this process is two square-symmetric proximity matrices of Jaccard similarity—one between students and another between courses. With an appropriate proximity measure in place, the next step is to analyze the proximity matrices using an MDS algorithm. Recall that the general purpose of MDS is to represent proximities (δ_ij) as distances (d_ij) in a low-dimensional space. In the present application of MDS, distance is conceptualized as Euclidean distance, which is simply the geodesic between two points. In a two-dimensional solution (X), the Euclidean distance (d) between points i and j can be expressed as:

d_{ij} (X) = \sqrt{{(x_{i 1} - x_{j 1})}^{2} + {(x_{i 2} - x_{j 2})}^{2}} .

Euclidean distance can be rewritten more generally to apply to m-dimensions:

d_{ij} (X) = {[\sum_{a = 1}^{m} {(x_{ia} - x_{ja})}^{2}]}^{1 / 2} .

An optimal MDS solution is one in which the distances closely approximate the proximities. Borg and Groenen (2005) refer to this as a representation function (f ):

f : {δ_{ij}}_{\to} d_{ij},

in which f indicates the type of MDS model. There are numerous types of MDS models, but by far the most common in social science research is ordinal (or “nonmetric”) MDS. In order to assess the fit of the MDS solution, analysts report the “stress” value, which is a nonstatistical measure of badness-of-fit. While Kruskal and Wish (1978) place an arbitrary cutoff point for acceptable stress at between 0.0 and 0.2 (for ordinal MDS), Borg and Groenen (2005) suggest that in practice researchers should factor in the ratio of dimensions (m) to objects (n), error and number of ties in the data, and the point at which increasing dimensionality no longer substantially improves fit. Borg and Groenen’s point is important in the context of this study in which the number of objects is extremely large for an MDS analysis (up to n = 494).

Whereas MDS is generally used to analyze a single mode of data, MCA is typically used to analyze two modes of data. In the following analysis, I use MCA to represent the duality between courses (Mode 1) and student attributes and postsecondary trajectories (Mode 2). Instead of a matrix of courses-by-students (e.g., matrix A^T), then, we begin with a matrix of courses-by-student attributes and postsecondary trajectories. Each course has a row profile (or “vector”) that shows how that course is distributed (in raw frequencies) across the different student attributes and postsecondary trajectories. Similarly, each attribute has a column profile indicating the distribution of that attribute across the different courses. The analysis uses the elements in each vector as coordinates to locate the vectors in a multidimensional space (see Greenacre and Blasius 1994). The average row or column profile (also called the centroid) constitutes the weighted average of each individual course or attribute. This weighted average distinguishes MCA from the particular application of MDS used herein.

The goal in MCA is to preserve the greatest possible proportion of the original distance in the matrix recovered by each dimension in the space. In MCA, distance refers to the χ² distance, which for two points (x, y) is simply a form of weighted Euclidean distance:

d_{x, y} = \sqrt{\sum_{j = 1}^{p} w_{j} {(x_{j} - y_{j})}^{2}},

where w_j refers to the specific weight of the jth dimension. Finally, the inertia refers to the dispersion (i.e., variance) between points in each row profile. Each row profile contributes to the total inertia (ϕ²), which is a measure used to describe the dispersion of the profiles from the centroid (i.e., average profile). In sharing the MCA results I will report the total inertia and the proportion of that inertia individually recovered by each of the first three dimensions (the output map will display the first two of these dimensions).

Cascade High School

The primary source of data for this analysis comes in the form of longitudinal course enrollment and transcript data files for a comprehensive secondary school located in a suburban midwestern city with a population of approximately 57,000 (according to a 2009 census estimate). This suburb is adjacent to an urban city of approximately 200,000 residents and resides within a metropolitan area with a population of approximately 560,000. The median household income in this city is approximately $61,000 according to a 2009 census estimate, with 91.9 percent of the total population being white, 2.5 percent African American, 3.9 percent Asian, and 4.8 percent Latino/a (of any race).

The school—which I will refer to as Cascade High School (CHS)—has an enrollment of approximately 1,900 students and 125 certified staff members (of which approximately 75 percent hold at least a master’s degree). The school offers a wide range of courses in art, business, computer science, language arts and reading, family and consumer sciences, mathematics, music, science, social studies, technology education, world languages, physical education, and a miscellaneous group of courses. There is also an extensive special education program available to those deemed to have special needs. Finally, a handful of students also take courses at an elite International Baccalaureate World School situated in the adjacent urban school district.

Students must have a minimum of 48 credits to graduate. The required distribution of credits includes 8 credits in language arts and reading, 1 credit in art, 6 credits in mathematics (including 2 credits in algebra), 4 credits in physical education, and 6 credits in both science and social studies. That is a total of 31 credits (61.4 percent of the total minimum to graduate). However, the actual number of required courses is a different matter. The only specific courses that are required are Language and Literature (2 credits, typically taken in grade nine), Speech Communication (1 credit), Fundamentals of Writing (1 credit), General Science (2 credits, typically taken in grade nine), U.S. History (2 credits) or American Heritage (2 credits), Economics (1 credit), and Government (1 credit).¹ This means that only 10 of the 48 credits (20.8 percent) required to graduate are prespecified, and all others are “chosen.” Thus, on paper, there is a considerable amount of flexibility in the range of courses that students are able to enroll.

The sample for the longitudinal course-taking analysis consists of n = 494 students who attended Cascade High School between the school years of 2005/2006 through 2008/2009. This means that all of those students who moved into or out of the district during this time frame were excluded from the analysis.² The sample used in this study is similar to the overall racial distribution of the community as a whole. That is, the vast majority (88.5 percent) of the 2005 to 2009 students from Cascade High School are white and primarily from the middle and upper middle class.³ Nearly 3 (2.8) percent of the longitudinal sample are African American, 6 (5.9) percent are Asian, and 3 (2.8) percent are Latino/a. In addition, according to postsecondary intent data collected by the school from 12th graders in 2009, two-thirds (66.4 percent) reported that they planned to attend a four-year university the following year (19.6 percent private, 46.8 percent public). An additional quarter (26.3 percent) reported that they would be attending a community college, and small percentages said they would enlist in the military (0.8 percent), seek immediate employment (2.8 percent), or pursue “other postsecondary” plans (1.6 percent).

To add greater detail to the quantitative data, the analysis that follows also draws upon a detailed (179-page) curriculum guide with expanded course descriptions and information concerning requirements, prerequisites, and sequences. Prior to the analysis I also met with the faculty chair of each department at Cascade High School in order to gain a better understanding of the course content, prerequisites (formal and informal), and curricular sequencing schemes used or encouraged by the school staff. While these data are not analyzed formally, they provide crucial background information related to the curricular content within and between each course.

The analysis will proceed in three steps. First, I present the MDS results for matrix AA^T (i.e., the space of students). This map will illustrate how students are relationally positioned through the courses they share in common across four years of course-taking. Using grayscale shading and other graphical techniques, the map will also illustrate the hierarchical structure of this space in relation to students’ postsecondary trajectories, racial attributes, and free lunch status. Next, I report on the MDS results for matrix A^TA (i.e., the space of courses), which will illustrate how courses are relationally positioned through the students they co-enroll. Finally, the MCA results model the duality of courses and student attributes and postsecondary trajectories in a single representation and enable an examination of the principles that underlie course-taking practices in relation to racial, class, and gender categories.

The Field of Course-Taking at Cascade High School

The Space of Students

As noted earlier, previous research suggests that— even in the absence of formal tracking—a typical contemporary high school in the United States will unequally distribute students into curricular positions that are associated with a stratified set of postsecondary trajectories. In addition, these positions (and corresponding postsecondary trajectories) are often segregated along racial and class categories. Thus, it should come as little surprise to find that the space of students at Cascade High School is consistent with previous findings in its overall hierarchical structure in relation to the curriculum. However, this “space” takes on a significantly more literal interpretation within a field-theoretic framework that draws upon multidimensional scaling as an analytical tool. In particular, the results demonstrate an important amount of “horizontal” differentiation that occurs within postsecondary trajectories.

Figure 1 illustrates the nonmetric MDS solution for the student-by-student Jaccard similarity matrix (AA^T).⁴ This MDS solution depicts the theoretical space of students. Thus, each point in the space represents a student at Cascade High School, and their position in this space is determined relationally vis-à-vis their normalized (i.e., Jaccard) course similarity to all other students. While the stress measure (0.205) for this solution is at the upper limit of Kruskal and Wish’s (1978) range of acceptability, applying the additional criterion outlined by Borg and Groenen (2005) suggests that this solution is an adequate (though certainly not inspiring) fit for the present purposes. First, the ratio of objects to dimensions (494:2) is dramatically larger than what is typically found in MDS applications. Second, the “payoff” for extending the solution to three or four dimensions (stress = 0.166 and 0.135, respectively) is minimal in comparison to the increase in complexity and subsequent interpretation.⁵ The primary intent here is to illustrate that a single vertical dimension is not sufficient to capture the distance between students.

Figure 1.

The multidimensional scaling (MDS) space of students at Cascade High School, 2005-2009 (stress = 0.205)

The relatively dense cluster surrounded by apparent “outliers” within the space of students in Figure 1 is the result of a limited number of students who took many of their courses at the International Baccalaureate (IB) World School (points situated at the top of the map in Figure 1) and those whose enrollment consisted primarily of special education courses (points situated at the bottom). Intuitively, then, the MDS algorithm positions students enrolled in IB courses very far from those students enrolled in special education courses. This distance between positions is the result of the fact that these students do not share any courses in common (i.e., their co-occurrence has a Jaccard similarity value at or close to 0.00). As a result, however, the MDS algorithm separates the IB and special education students by great distance while creating the illusion that the vast majority of the remaining students are very similar. A level of similarity does exist, of course, but it must be understood relative to the great dissimilarity between these two sets of unique students.

The distances observed in this space of students represent the degree of similarity in course affiliations across four years of course-taking. However, these course-based distances also point to physical distances experienced in time. While this latter dimension is not directly measured in the analysis, the relationship can be inferred from the distances in the MDS space. That is, if students do not share courses in common, then they will occupy distinctly different physical locations throughout the school (or the city, in the case of the IB students). In fact, those students who occupy great distances in the MDS space likely share very little physical space and time as it relates to activities involving the curriculum. This includes those students who are separated horizontally as well as vertically in Figure 1, a point to which we will return in the next section. Precisely how, if at all, this matters to students and courses is a matter for a future empirical investigation.

Beyond merely showing the relative positions of students, Figure 1 is also gray-scale shaded to illustrate that these positions are structured into a hierarchy associated with postsecondary trajectories. The structure of the hierarchy opposes the four-year private and public university students to those who plan to attend community college or enter directly into the workforce. The space as a whole is organized as a gradational hierarchy in which students on the four-year private and public university trajectory share a limited number of course affiliations with those bound for community college and even fewer with those pursuing active military service, employment, or other postsecondary plans. In addition, the line in Figure 1 shows that the space is segregated across racial and “free lunch”⁶ categories as well. For instance, the four-year university regions are the near exclusive domains of white and “Asian” students, while Latino/a and black students and those with free lunch status almost exclusively occupy the community college, military, and employment regions of the space.

The distances highlighted thus far speak to the vertical distribution of students and postsecondary trajectories. However, there are also important horizontal distances observed within the space, and they are not associated with differences in postsecondary trajectory. That is, many students occupy identical positions along the vertical axis and share the same postsecondary plans but are nevertheless positioned relatively far apart on the horizontal axis. It is within this horizontal form of distance that a significant portion of our understanding about the social structure of students and courses is in need of advancement. To begin to better understand this additional form of distance we now turn our attention to the space of courses.

The Space of Courses

Whereas in the space of students it is courses that serve as the medium through which students become affiliated, in the space of courses it is students that provide the medium where different courses—and their associated curricular discourses—are brought together. In this section I use MDS to illustrate the course co-occurrences that take shape across the four-year period (2005-2009) and then use the MCA results to describe the space of courses in relation to the “other side” of the duality in the form of student attributes and postsecondary trajectories. In the process I will examine the principles of differentiation that underlie the organization of the entire “field” of course-taking at Cascade High School.

To anticipate, the space of courses at Cascade High School is organized according to vertical and horizontal principles of differentiation. The vertical dimension is likely the most familiar to sociologists of education, as it represents a principle of status and prestige associated with the courses and their related curricular discourses (Bernstein 2000). In some cases, these courses are further organized into clusters of vertically ordered course sequences (e.g., Latin 1-4). The horizontal dimension of this space is, perhaps, less familiar. This dimension is structured according to a principle of differentiation that opposes courses aimed at developing technical knowledge and skills (e.g., computer science) to those whose content focuses primarily on the production and reproduction of meanings (e.g., literature). Yet this simple dichotomy does not tell the entire story. As we shall see, rather, the substance of the horizontal differentiation is contingent upon a specific orientation along the vertical (i.e., status) dimension. These divisions oppose symbolic to material forms, artistic to technical skills, and the “inner” work of the household to the “outer” work of the occupational structure.

The duality of courses and students illustrates that the different regions within the space of courses are raced, classed, and gendered both in relation to specific postsecondary trajectories and to the “vector” of courses linked with those trajectories. The different course vectors in this context reflect the range of strategies that students (and their families) invoke as they navigate their way toward different postsecondary worlds. Some students, for example, focus their course-work on advanced literature and composition while others emphasize advanced computing and technical skills. These strategies generally include different forms of linguistic resources too (e.g., taking Latin or French vs. Japanese). Other students, meanwhile, may emphasize vocational skills in business or technology as opposed to those emphasizing artistic production or interpersonal relations and household work.

The different course vectors do not only reflect positional strategies invoked in the quest for (or rejection of) curricular resources, however. They also point to different domains of the curriculum that are integrated and establish boundaries for who has access to these knowledge classifications (Bernstein 1977). That is, these course vectors are associated with multiple distinct configurations of courses and students that shape the realm of the thinkable and unthinkable. In this sense, the different regions within the space of courses have moral implications because they represent relations through which students construct meanings that have important consequences for what they are able to do and to be (Nussbaum 2011). In addition, it illustrates where associations between postsecondary trajectories, attributes, and the various elements of the curriculum are reinforced (and potentially formed) in direct relation to students speaking, thinking, and acting in distinct locations and times.

The Vertical Dimension of Course-taking: Status Hierarchy and Curricular Discourses

Many of the courses in Figure 2 (stress = 0.189) are distributed along the vertical axis according to a principle of status and relative prestige that in some cases also represents a progression through a sequence of courses. This status is not inherent to any course, but rather emerges through the students that tend to enroll (and dually, the students’ academic status emerges through the courses they take).⁷ Science courses, for example, are organized in a vertical hierarchy that includes AP Physics, AP Biology, and AP Chemistry at the top of the axis and Introductory Physics and Introductory Chemistry at the bottom. These latter introductory courses focus on “consumer chemistry topics” and the “chemistry or physics of everyday life” (CHS Curriculum Guide 2010). That is, the concepts discussed in these courses stem from a discourse that is segmented and arises out of common problems encountered in everyday life. Importantly, the credits from these courses do not meet requirements for college entrance. The same pattern is observed within mathematics where AP Calculus BC, Pre-Calculus C, AP Calculus AB, and Algebra II/Trigonometry C are situated at the top of the hierarchy and Developmental Algebras 1-4 and Math topics 1-2 are situated at the bottom. As with Introductory Physics and Chemistry, these latter math courses are rooted in “everyday” contexts (e.g., balancing a checkbook) and do not meet NCAA clearinghouse requirements.

Figure 2.

The multidimensional scaling (MDS) space of courses at Cascade High School, 2005-2009 (stress = 0.189)

Courses in social studies, language arts and reading, and technology education also follow a vertical structure. For example, APs European and U.S. History and American Political Culture are located at the top of the vertical axis. Lower down on the axis is Current Issues, a one-semester elective course that focuses on national and international events with special attention to “vocabulary, personalities, and geography” (CHS Curriculum Guide 2010). American Heritage is an alternative to meet the U.S. History requirement and focuses on teaching U.S. history and literature concomitantly in a four-credit course. Figure 2 shows the technology education courses ranging hierarchically from Computer Aided Design (CAD), Engineering, and Architectural Design to Automotive Repairs, Metals, Construction Technology, and Welding. Finally, language arts and reading courses are distributed vertically (and somewhat horizontally). These courses range from the AP, literature, and creative writing courses to Young Adult Literature and Skills for Academic Success. These latter two courses are designed to bring student reading and comprehension skills up to grade level.

Associated with this vertical hierarchy is an important conceptual distinction between three distinct structures of the curricular discourses contained within the courses. That is, AP Chemistry (or even General Chemistry) is not only different from Introductory Chemistry in terms of difficulty or status, or because they draw upon different curricular discourses, but also because of the way the discourses are organized. The latter course emphasizes “commonsense” knowledge to the extent that the curricular discourses can (at least potentially) be accessed by all and arise out of basic problems involved in living and dying (Bernstein 2000:157). These discourses are segmented (i.e., are typically not hierarchically integrated to other such discourses) and have a direct relation to practical problems encountered in daily life. In AP or General Chemistry, on the other hand, the discourses take on a hierarchical shape(s) in which the aim is to generalize from regularities underlying a range of phenomena (Bernstein 2000:161-62). In contrast to the commonsense discourses, access to the integrative discourses of “scientific” chemistry is formally and informally regulated through a variety of requirements within the education system.

While courses in social studies, language arts and reading, and others are also distributed vertically, many of them contain curricular discourses that are conceptually distinct from the hierarchical shape of (say) AP Chemistry. Rather than focusing on hierarchical discourses used to integrate knowledge, courses such as Shakespeare and Modern American Literature contain curricular discourses organized into specialized languages and methods of interrogating texts (Bernstein 2000:161-65). The specificity or “strength” of these specialized languages varies, such that it will take a student more time to acquire the specialized languages in AP Language and Literature than Young Adult Literature, for example. There are, then, three broad types of curricular discourses at work within the space of courses. Each type of discourse (or “knowledge structure” in Bernstein’s terms) has its own form of initiation that sets the parameters for styles of acquisition and contestation, as well as “who” can access the discourse. More importantly, along the vertical axis we can observe that at Cascade High School the courses with (at least some) commonsense discourses are situated far away from those courses emphasizing knowledge integration and the acquisition of specialized languages and modes of interrogation. In other words, these courses tend not to co-enroll the same students.

Finally, some of the courses are arranged into vertical clusters that represent progressions through course sequences. These progressions include sequences that move up the vertical axis, such as Latin 1-4 and Math Topics 1-2. In other cases the sequences progress down the vertical axis, such as Developmental Algebra 1-4 and Young Adult Lit 1-2. The sequence clusters located in the northern region (e.g., Latin 1-4 and CAD 1-4) tend to progress up the vertical axis, suggesting that those who progress through the entire sequence also enroll in the more advanced courses (although CAD 3 appears to be the exit point for the advanced students in that sequence). The sequence clusters in the southern region move in both directions. For instance, the Math Topics sequence moves up the vertical axis toward the center of the space (i.e., courses with greater status). Yet progression through a sequence does not always correspond to “progress.” Those who move through the entire Developmental Algebra, Keyboarding, or Foods sequences, for example, also tend to enroll in the lower status courses. In these instances, progression through the sequence is associated with distance away from the higher status courses.

The Horizontal Dimension of Course-taking

Many of the courses are not distributed along the vertical axis to the extent of those mentioned earlier. Courses offered in the subjects of computer science, world languages, business, family and consumer science, art, and music are either distributed along the horizontal axis or tend to be contained within a limited region along the vertical axis. For example, world language courses in Japanese, Spanish, Latin, German, and French stretch across the horizontal axis. However, it is notable that they are all located in the northern region within the vertical dimension. Nearly the entire range of music courses is located in the northwestern region, whereas the computer science programming courses are exclusively in the northeastern region. On the other hand, courses in business, art, and family and consumer science are located in the central and southern regions of the space (with art and business located above family and consumer science) but are not distributed vertically per se. In addition, some of the vertically distributed courses—such as technology education—are largely contained in either the western or eastern region of the space.

These oppositions achieve greater conceptual clarity when we examine them within the context of the duality of courses and students. In practice, it is difficult to illustrate the full duality of students and courses using multiple correspondence analysis since there are over 700 objects (students + courses + attributes + trajectories).⁸ However, in the present context MCA is quite helpful since the primary goal is to examine how courses are spatially organized in relation to other courses and the categorical attributes and postsecondary trajectories of students. Figure 3 illustrates the first two dimensions of this duality, with the first three dimensions accounting for 24.0 percent, 15.3 percent, and 10.6 percent of the inertia, respectively. With some exceptions (most notably the reversed positions of the technology education and family and consumer science courses), the relative positions of the attributes/trajectories and vertical and horizontal course oppositions are quite similar to those in Figures 1 and 2 (respectively). The similarity between solutions suggests robust results.⁹

Figure 3.

The multiple correspondence analysis (MCA) space of courses and student attributes/postsecondary trajectories at Cascade High School, 2005-2009 (φ² = 1.148)

At the most general level, the horizontal dimension of course-taking is associated with a sexual division that opposes courses focused on symbolic, artistic, and intrinsic curricular discourses and skills to those focusing on material, technical, and extrinsic curricular discourses and skills. The latter regions of the space—more strongly associated with male students—contain courses primarily focused on the transmission of technical knowledge, business preparation, and construction skills, such as those found within the subjects of computer science, business, and technology education (respectively). In contrast, the symbolic, artistic, and intrinsic regions—more strongly associated with female students—contain courses focused on the production of meaning, art, and household skills, such as those found in the subjects of language arts, music, art, and family and consumer science.

To gain a more complete understanding of the courses along the horizontal dimension, we must situate them along the vertical axis as well, including the student attributes and postsecondary trajectories with which these courses are associated. Consider, for example, the northwest region of the space, which is closely associated with the four-year private and public university trajectories corresponding with the entire northern region. The northwest region contains nearly the entire range of music courses, in addition to the high-status literature and composition courses such as AP Language and Composition, AP Language and Literature, Newspaper, and Advanced Publications. This region is also associated with Anatomy, AP Art, and French 4, although AP French has a more north-central location. Other courses that are in close proximity (but more toward the center) include AP European History, American Political Culture, Advanced Contemporary Literature, and Yearbook.

Moving down the vertical axis (i.e., west-central), the courses shift from the high-status music, language, and literature courses to those associated with the production of communication, art, and style. These courses include Theatrics, Journalism, Darkroom, Interior Design, Shakespeare, Painting, Fashion, Jewelry, Ceramics, and Drawing. This region of the space also includes French 1-3, Sociology, and American Heritage. This latter interdisciplinary course “integrates history, literature, and writing in an exploration of important literacy trends and themes in early and modern American literature, especially as it relates to cultural and historical events from the 17th century forward” (CHS Curriculum Guide 2010). This course is equivalent (in credit requirements) to a year (two credits) of U.S. History, one Modern American Literature credit, and one Intermediate Writing credit. Anchoring this region from the center is the set of “general” mathematics, science, and social studies courses, as well as additional courses in language arts and reading.

In the northeast region of the space are the advanced computer science courses (AP Computer Science, Advanced BASIC, and Advanced Programming Languages), along with Physics and AP Physics. This region also includes Latin and Jazz, though each of these is noticeably more toward the center than the latter courses. The courses associated with this region are also anchored by the more centrally located high-status mathematics (e.g., AP Calculus AB and AP Calculus BC), science (APs Chemistry and Biology), and social studies (e.g., AP U.S. History) courses. Just as with the symbolic region, these courses are strongly associated with the four-year private and public university trajectories (as well as the “Asian” category).

Further down in the east-central region is the realm of business and advanced technology education courses. It is here that we find the “job skills” courses such as Business Applications, Advanced Computer Applications, Business Law, Sports and Entertainment Marketing, Accounting, Entrepreneurship and Business Management, and Financial Management. In addition, this region includes a dual Marketing and Sales course and job placement combination. The course focuses on “marketing principles including market research, market segmentation and targeting, buyer behavior, pricing, product development, intermediary functions and promotion” (CHS Curriculum Guide 2010). The job placement focuses on the job training discussed in class, and students must work an average of 10 hours per week in a marketing-related job to apply these skills. Also in this region are the technical courses such as Engineering, Computer Aided Design, Architecture, and Webpage Design. The courses in this region are anchored by the centrally positioned general mathematics (e.g., Algebra 2/Trig B, Algebra B), science (e.g., Biology, Chemistry), social studies (e.g., U.S. History), and world language courses such as German, Spanish, and Japanese.

A similar pattern is observed in the southern territory of the space where the community college, military, employment, and “not graduating” postsecondary trajectories are located. While there are certainly fewer courses in this part of the space, there remains a division in the organization of courses. This region is also where students of color (i.e., Latino/a and African American) and those receiving free lunch are disproportionately positioned. The courses in this southern region are anchored by the lower status mathematics (e.g., Geometry A, Algebra A, and Developmental Algebras), science (e.g., Intro Chemistry, Intro Physics, Geology), and social studies courses (e.g., Current Issues). Oriented toward the southeastern region are a number of lower status technology education and “general skills” courses, such as Auto Mechanics, Basic Woods, Welding, Home Repairs, Construction Technology, and Skills for Living. In contrast, in the southwestern region are a number of family and consumer science courses, such as Personal and Family Living (focuses on decision making, relationship building, and interpersonal communication skills), Child Development (focuses on raising a child from birth to two years), Culinary Arts, Foods 2, Health, and Sewing.

Discussion

Toward a Field Theory for Sociology of Curriculum

At the outset of this article, I argued that a field-theoretic analysis of the duality of students and courses encourages us to consider the organization of students and courses concomitantly and as a distinct “whole.” Thus, both students and courses occupy relational positions in a field of course-taking. By examining the dimensions underlying the space of courses on its own and in relation to student attributes and postsecondary trajectories, we were able to see that these positions have a conceptual coherence (or logic) that suggests a certain “causal texture” to course-taking at Cascade High School. This causal texture contains “information on how to construct meaningful trajectories of appropriate action” (Martin 2011:316). To apply Martin’s (2011) metaphor to the present case, we might think of course-taking as similar to a scavenger hunt where each course or set of courses suggests to students which courses they should take next. At Cascade High School, this social organization of courses is not only differentiated by vertical positions of status, prestige, discursive forms, and course sequences, but also horizontal positions that oppose symbolic, artistic, and intrinsic knowledge to material, technical, and extrinsic knowledge.

We also saw that at Cascade High School the organization of course-taking corresponds to divisions of academic labor across racial, class, and sexual categories. Each student is positioned within these divisions in a way that makes certain course and student affiliations probable and improbable. At Cascade High School, this means for those intending to pursue the military, employment, or even the community college postsecondary trajectory, for example, their position does not include opportunities to think through many of the integrative and specialized discourses associated with music, language arts, mathematics, science, and world languages. In this sense, the social relations that constitute the field of course-taking differentially constrain and afford certain conceptual connections made available through specific courses and the forms of curricular discourse they offer. Note that there are no magical “mechanisms” at work here; it is the patterns of course-taking that afford some conceptual connections while making others more difficult. A student who enrolls in AP Physics and Computer Science will be afforded very different potential conceptual linkages than a student who enrolls in AP Art and Anatomy.

The field of course-taking does not only differentiate the possible conceptual connections available to particular students. So often we limit our thought to the perspective of students and how the differential access to knowledge content and forms shapes their opportunities and identities. However, as a social structure that is constituted by a duality, it is also the case that the patterns of relations constituting this “field” differentiate the students available to particular courses. To take this aspect of the duality seriously means that we must be willing to think from the perspective of courses and the curricular discourses they contain and how the differential access to (raced, classed, and gendered) students—and their postsecondary trajectories—shapes curricular organization. Heretofore, the role of courses within this duality has been relegated to the passive medium through which students are grouped and provided disparate learning opportunities. Yet the curricular discourses made available in courses are also (at least potentially) grouped and subsequently integrated and segregated through the students they co-enroll. These curricular discourses are not only influential, then, but they are also influenced (again, at least potentially). That is, whether or not particular students enroll in (say) Advanced Contemporary Literature or Sports Marketing matters to those students as well as to the curricular discourses contained within these courses and those with which they share students in common.

Does taking this aspect of the duality seriously require that we assign a form of quasi-agency to courses and the curriculum? This must be left to a future discussion. However, in closing I will note that students do act as an important medium through which curricular discourses are integrated and segregated, making the construction of certain meanings possible (and others impossible). In this case, the production and organization of the curriculum is contingent upon those who are able to think and integrate its categories of thought and perception. “Socialization into knowledge,” Bernstein (1977:97-8) says, “is socialization into order, the existing order, into the experience that the world’s educational knowledge is impermeable.” Yet we must also acknowledge the other side of the duality: that knowledge, too, is brought into order, the existing social order(s). The field of course-taking, then, is not simply an ordering and positioning of students through the organization of courses or “tracks.” Rather, it consists of the mutual constitution of these two spaces (i.e., a “field”) that together allow us to examine what can be thought, who can think it, and—with further investigation—when, where, and how such thought takes place. Thus, the field of course-taking does not explain the pattern of relations between students and courses; it is the pattern of relations (Martin 2011).

Additional work needs to be done to link the phenomenological experiences of students directly to the positions in the field of course-taking. An important step toward this end is to learn how students draw upon their cultural resources to construct educational meanings and strategies from the curricular discourses affiliated with their positions. There is already evidence that many high school students are well aware of the variety of curricular trajectories and course configurations at their school. This fact has long been suggested in the literature (e.g., see Oakes [1985] 2005), and more recent evidence indicates that students cultivate academic strategies in relation to local course-taking patterns even in the absence of formal tracking contexts (see O’Connor et al. 2011). In addition, the underlying structure of course-taking at Cascade High School suggests that students are piecing together configurations of courses with a conceptual coherence that goes beyond course sequences and requirements. Exactly how—and to what effect—students weave together curricular discourses from these course configurations remains a pressing question for sociologists of curriculum.

Another important consideration is whether the patterning of this field of course-taking derives its logic entirely from its two constituent spaces.¹⁰ Recall that the logic within the space of courses is not a mere reflection of the logic organizing the space of students. Yet the properties of this duality are such that the organization of the space of courses must pass through the organization of the space of students (and dually). Since the field of course-taking is but one of many social fields and institutional layers that students and their families simultaneously traverse, some of the information they retrieve from those social environments is conceivably at work in the field of courses. Perhaps this is why we can observe a substantively familiar social hierarchy in the space of students while simultaneously observing substantively unique principles of organization in the space of courses that are, nonetheless, familiar in form.

Conclusion

The primary purpose of this article has been to illustrate the insights that can be gained from the “course side” of the duality of students and courses. Among these insights is that in course-taking environments with some level of flexibility there is important “horizontal” differentiation in addition to the vertical hierarchies so familiar to sociologists of education. The substance of this horizontal differentiation is contingent upon a specific location along the vertical (i.e., status) dimension. At various levels this division opposes symbolic and material forms, artistic and technical skills, and the “inner” work of the household to the “outer” work of industry or construction. The case of Cascade High School thus reveals a field of curricular trajectories in which a wide array of course affiliations are possible, but certainly not randomly assembled or independent of exogenous institutional and social hierarchies.

I also highlighted the potential utility of a field-theoretic approach to studying course-taking. There is, however, a great deal of work to be done if this is to become a productive area of research. The most important next step is to connect to the insights of students in order to understand how curricular discourses between courses are integrated, shaped, transformed, and acted upon. While the present work provides a portrait of the field of course-taking, it does not tell us anything about how this field is produced and experienced by students. As this dimension is explored, we can gain a deeper understanding into how the integration of secondary school curriculum is capable of generating new forms of action, meanings, and identities, as well as the complex and creative work that goes into this process. At the same time, we can also investigate how the segregation of particular courses and their discourses prohibits the emergence of these forms and identify ways for students and teachers to build discursive “bridges” across disparate areas of the curriculum.

Footnotes

Acknowledgements

Many people have provided critical feedback and support for the work that appears in this article. I would especially like to thank the anonymous reviewers for their sincere and critical engagement with this manuscript. In addition, the following people provided detailed comments and criticisms: Michael Olneck, Adam Slez, Brian Lagotte, Quentin Wheeler-Bell, and Ross Collin. Sarah Bell contributed many hours of her data visualization expertise in the construction of the figures. I would also like to thank Michael Apple, Beth Graue, Simone Schweber, and Linn Posey-Maddox for their support of this work. Finally, this work would not have been possible without the help of district and school staff at “Cascade High School.”

Notes

Bio

Joseph J. Ferrare, Ph.D., is Associate Researcher, Wisconsin Center for Education Research, College of Education, University of Wisconsin-Madison. He is currently conducting his work from Bellingham, Washington while designing and teaching courses in sociology and education policy at Fairhaven College of Interdisciplinary Studies, Western Washington University. His research interests include sociology of education, curriculum theory, and educational policy analysis at the secondary and postsecondary levels.

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