Cultivating Quantitative Literacy in the Introductory Course: A Mathematics Education Collaboration to Teach the Gini Coefficient

Quantitative Literacy, Sociology, and the Introductory Course

Quantitative literacy has been recognized as a critical challenge for higher education in the United States for at least a quarter century (Steen 1997). The Mathematical Association of America has defined QL as “the ability to adequately use elementary mathematical tools to interpret and manipulate quantitative data and ideas that arise in individuals’ private, civic, and work lives” (Special Interest Group of the Mathematical Association of America, Quantitative Literacy 2004; for a more recent complementary definition, see Fischer 2019). That trio of applications (private, civic, and work lives) has been consistently identified as critical contributions of QL (e.g., Steen 1997, 2001; Tunstall, Karaali, and Piercey 2019), whereas more recent research has added social justice as an outcome (e.g., Hamman, Piercey, and Tunstall 2019). After decades focused on defining and legitimating QL, it has established a “stable core” in higher education (Tunstall et al. 2019). One of the critical formative questions that has emerged more recently is whether the QL movement should promote any specific kind of pedagogy (Wallace 2019).

Although QL emerged from mathematics, its underlying logic has inevitably taken it beyond the discipline. One reason is that contextual embedding of materials and topics is a definitional element of QL (see Madison 2004). Contextualization—that is, “the teaching of basic skills [reading, writing, mathematics] in the context of disciplinary topic areas” (Perin 2011:1)—has been demonstrated to benefit student learning, in part because it increases intrinsic motivation (Parker, Traver, and Cornick 2018; Perin 2011). In higher education, that embedding happens best within students’ chosen majors (or across broader general education sequences) rather than in mathematics courses. Another reason for the diffusion of QL beyond mathematics is the recognition that QL is a “cumulatively acquired intellectual skill” (Zerr 2019) that requires multicourse sequencing rather than a single QL requirement. All of that has led to QL integration “across the curriculum” (Elrod 2014). As Ganter (2006:13) writes: “QL must be everywhere in the curriculum, in all disciplines and all courses,” with the corollary that “QL is a shared responsibility.” That can take the form of QL-specific courses offered outside of mathematics or efforts to integrate QL at the level of modules, projects, or assignments in non-QL-specific courses (e.g., Friedrich and Strawn 2019; Steen 2004:91–101; Tunstall et al. 2019:47–124). The latter are particularly important because QL advocates have emphasized that it must be promoted through repeated application of progressively developed skills. As Madison (2004:11) notes: “QL is a habit of mind, and habits are developed from repeated practice.” (See Hughes-Hallett [2003] for a fully developed “habits of mind” argument.) The result is that effective QL integration demands that component skills must be sequenced across the curriculum and scaffolded within courses.

Sociology has long been a leader in promoting QL integration (e.g., Howery and Rodriguez 2006; Sweet and Strand 2006; Wilder 2009). One of the most notable early efforts in the discipline was the Integrating Data Analysis (IDA) initiative, which encouraged “introducing students to data analysis early, frequently, and sequentially throughout the curriculum” (Howery and Rodriguez 2006:23). The animating concern for IDA was the QL gap—that is, the gap between the set of quantitative tools and skills that students generally bring to classes and what they need to succeed in the major. Those gaps must be bridged across the major—starting with the introductory course and infused at all levels. Sociology has made significant progress in those efforts, identifying a range of practices for integrating QL through specific skills and applications—from reading tables (Wills and Atkinson 2007) to regression (Linder 2012) to analysis of census data (Burdette and McLoughlin 2010).

QL is particularly important in sociology due to the centrality and range of quantitative practices in our disciplinary research (Linneman 2021). Our primary pedagogical concern in that area is preparing students to succeed in statistics courses—a task complicated by anxieties they may bring to those courses (Condron, Becker, and Bzhetaj 2018; DeCesare 2007). To address that, underscoring the emphasis on sequencing, Linder (2012:51) notes: “One way to decrease students’ ‘statistics anxiety’ while improving their quantitative literacy is by introducing basic skills of data interpretation at a much earlier point in their college career and reinforcing the lessons and degree of complexity throughout their course sequence.”

Integrating QL “early and often” means that it must begin with the introductory course. Previous research has illustrated productive strategies for doing so, primarily by introducing research modules and methods into course content (Atkinson, Czaja, and Brewster 2006; Howery and Rodriguez 2006; Markham 1991). Notwithstanding its importance, QL integration presents significant challenges. Enrollments in introductory classes are often driven by general education requirements, meaning that students come from a wide range of majors with substantial differences in their preparation for and interest in quantitative material. Although exposure to QL practice is valuable for all students, we cannot risk leaving less prepared students behind (whether majors or nonmajors). Consequently, efforts to cultivate QL in introductory courses can benefit from collaboration with mathematics educators because they have specific research-based expertise in the teaching and learning of quantitative material, representing a strategic example of cross-disciplinary scholarship of teaching and learning (SoTL).

Cross-Disciplinary SoTL and Mathematics Education

SoTL advocates in higher education recognize the value of interdisciplinary collaboration and exchange (Huber and Morreale 2002; McKinney 2007; Yakura and Bennett 2003). Huber and Morreale (2002) argued that SoTL benefits from a “trading zone” across disciplines:

In this zone, one finds scholars of teaching and learning seeking advice, collaborations, references, methods, and colleagues to fill in whatever their own disciplinary communities cannot or will not provide. Their goals are to do better by their students, and they are willing (within limits) to enter the trading zone and buy, beg, borrow, or steal the tools they need to do the job. (Huber and Morreale 2002:19)

Sociology has long been a disciplinary leader in SoTL (Howery 2002; McKinney 2018; Pike 2011), but interdisciplinary SoTL has been much less common. In an analysis of trends within Teaching Sociology, Paino et al. (2012:103) found “little evidence of interdisciplinary SoTL in publications during the past decade.” Continuing disciplinary leadership, they argued, was incumbent on “join[ing] in this wider conversation” and could “help frame the next wave of the SoTL movement” (Paino et al. 2012:104).

QL represents a natural and productive “trading zone” between sociology and mathematics—or more specifically, mathematics education. Mathematics education emerged from mathematics but asks distinct questions focused on learning processes (among other topics). According to Schoenfeld (2000:641), the aims of mathematics education research are “to understand the nature of mathematical thinking, teaching, and learning” and “to use such understandings to improve mathematics instruction.” Mathematics education asks not only pragmatic and applied questions but theoretical questions as well (see Banchoff and Salem 2002) and is motivated by epistemological, ontological, and methodological considerations (Ernest 1998). Although early mathematics education research was influenced by traditional psychology and hence often employed similar quantitative research methods, contemporary research is more typically influenced by disciplines such as sociology, anthropology, and cultural and social psychology (Stinson and Walshaw 2017). The specific questions, motivations, influences, and methods in the discipline have evolved together. Early studies in math education “assumed simple causal relations between teaching methods and student achievement, that is, relied on a transmission and acquisition conception of teaching and learning” (Sfard and Cobb 2022:467). Those studies failed to answer questions about why individuals fall short of understanding certain mathematical ideas, which, in part, led to the shift away from quantitative experimental research and toward more qualitative research focused on understanding students’ conceptions of mathematical ideas and the processes by which students learn those ideas. The key point here is that the particular research methods one chooses must be closely related to one’s questions—an argument very familiar to sociologists.

Here, we briefly identify several insights that shaped our collaboration, drawing most directly from the contemporary (qualitative) body of research focused on student understanding. ² For our research, these insights represent the figurative goods exchanged in the trading zone of our collaboration.

A foundational distinction in mathematics education is that between procedural and conceptual knowledge/understanding that characterizes student learning in mathematics (Hiebert and Lefevre 1986). Procedural knowledge is “the ability to execute action sequences to solve problems,” and conceptual knowledge is “implicit or explicit understanding of the principles that govern a domain and of the interrelations between units of knowledge in a domain” (Rittle-Johnson, Siegler, and Alibali 2001:346). The teaching of mathematics has traditionally been dominated by a procedural emphasis that can lead to shallow understanding and limited transfer (i.e., the application of learning from one topic to novel problems), which grate against the aims of QL. ³ Research in mathematics education consequently emphasizes the critical importance of conceptual knowledge as a counter to the traditional emphasis on procedural knowledge—but both are critical to student learning (noting that those with conceptual understanding may have the tools to reinvent procedures in novel tasks, whereas procedural understanding is limited to familiar tasks). As Rittle-Johnson et al. (2001:346) suggest, “conceptual and procedural knowledge develop in an iterative fashion”—less distinct categories of learning than two ends of a spectrum. The import for our purpose is that the presentation of QL materials must be attentive to the proper balance between them.

Student misconceptions are another long-standing focus of mathematics education research. The term misconception is used “to designate a student conception that produces a systematic pattern of errors” (Smith, diSessa, and Roschelle 1993:119). In traditional mathematics instruction, misconceptions are generally seen as mistakes that should be prevented or avoided. Mathematics educators have reframed misconceptions, addressing them not as “wrong thinking” but as “a concept in embryo” and frequently as “a natural stage of conceptual development” (Swan 2001:154). In a seminal article that prompted a shift toward understanding (rather than simply cataloguing) student misconceptions, Smith et al. (1993:147) suggest that misconceptions often have their roots in productive knowledge and represent “novices’ efforts to extend their existing useful conceptions to instructional contexts in which they turn out to be inadequate.” By recognizing patterns of errors, we can help to build on existing foundations (seeking to “refine” rather than to “replace” understandings) and identify areas where instructor explanations have been insufficient or even misleading. That perspective on misconceptions has critical implications for cultivating QL. For example, rather than avoiding misconceptions, they can be leveraged as a source of cognitive conflict through explicit discussion, which can aid in resolving misconceptions (Swan 2005). That direct confrontation approach has been applied productively in sociology SoTL (Herda 2017).

A third insight that guided the collaboration is the importance of student noticing—that is, a student “selecting, interpreting, and working with particular mathematical features or regularities when multiple sources of information compete for one’s attention” (Lobato, Hohensee, and Rhodehamel 2013:809). The origin of the insight is the following: “Mathematical situations often present an overabundance of information, visual cues, and possible patterns, making it impossible to process everything at once” (Lobato, Hohensee, and Rhodehamel 2013:809). The result is selectivity in attention. Taking account of that inevitable selectivity can inform teaching to ensure that critical issues are effectively highlighted, hence more likely to be noticed by students. That is particularly important because selectivity can be channeled by misconceptions in ways that make them still more difficult to confront. By paying attention to student noticing, we can better understand where students’ selective attention is being drawn and how that shapes their understandings of materials and concepts presented.

Those insights by no means exhaust the benefits of applying a mathematics education lens to QL in sociology. We focus on them because they emerged as critical in our collaboration and illustrate analytical and pedagogical emphases useful for cultivating QL more effectively.

Teaching the Gini Coefficient and the Lorenz Curve

The first author has presented the Gini coefficient in his introductory sociology classes for over a decade. Our university, California State University, Channel Islands, is the newest campus (and one of the smallest) in the California State University system. It is a Hispanic Serving Institution, and a large majority of our students fit into one or more category for nontraditional students. In fall 2023, 60.9 percent of students were Latino, 60.0 percent were first-generation college students, and 48.1 percent were Pell-eligible. ⁴ As with many universities, we see high rates of nonpassing grades in quantitatively oriented coursework, both within and outside of the major, so cultivating QL is a critical challenge across the board.

Initially, the presentation of the Gini coefficient was designed simply to help students understand the role of systematic measurement in empirical research, as a way to improve general scientific literacy in the introductory course. The unit was presented in a relatively conventional manner: Students were assigned readings to introduce the Gini coefficient; classroom presentation and discussion progressed from shares of total aggregate income by quintiles to the Lorenz curve as a graphical representation of those data and culminated with an explanation of the Gini coefficient as a “ratio of areas” (explained more fully in the following). It focused solely on conceptual understanding. Over time, increased emphasis was placed on explaining how it is calculated as an opportunity to cultivate QL. However, exam responses indicated that students’ understandings were frequently underdeveloped. Curious patterns of mistakes appeared, leading the first author to engage in conversations with colleagues. The most productive of those conversations was with a mathematics educator (the second author), who was able to offer potential insights about what might be shaping (or misshaping) student comprehension. Initial conversations led to a formal interdisciplinary collaboration to assess and redesign the unit, which continued across three subsequent iterations.

For those unfamiliar with the Gini coefficient, it is a summary measure of income inequality that can be used in any society. Lamb (2012) presents a concise description for a general audience that is useful for sharing with students. The Gini coefficient is derived from the Lorenz curve, which graphs the cumulative share of aggregate income (y-axis) across the population at each percentile (x-axis, arranged from poorest to richest). The logic requires that the curve always start at the origin (0 percent of households earn 0 percent of aggregate income) through 100/100 (all households account for the total aggregate income). The level of inequality in a society determines the shape of the Lorenz curve. More unequal societies will have a greater bow because the poorest households (on the left of the x-axis) account for very little of the aggregate income (shallower slope) and the richest households (on the right of the x-axis) account for far more than their share of the aggregate income (steeper slope). If one imagines a perfectly equal society (in which each household earns the same income), the curve would be a diagonal line from the origin to 100/100; that is called the “line of perfect equality” (LPE). Therefore, for more unequal societies, the Lorenz curve will deviate more significantly from the LPE, resulting in a greater area between the LPE and the Lorenz curve. The Gini coefficient effectively summarizes that variation by finding the area between the LPE and the Lorenz curve and calculating the ratio between that area and the total area below the LPE. It varies from 0 (in the case of absolute equality, where the LPE is identical to the Lorenz curve) to 1 (in the case of absolute inequality, where the curve lies at y = 0 across the x-axis until it rises to 100/100 at the final household).

To calculate areas associated with the Lorenz curve using only basic mathematics, we use the common simplifying practice of drawing the curve aggregated by quintiles. As a result, the area below the Lorenz curve becomes a series of right triangles and rectangles whose areas can be determined via a series of relatively simple calculations—and the area between the LPE and the Lorenz curve can be found via subtraction. The quintiles-based procedure produces an estimate of the Gini coefficient that marginally underestimates the formal measure (calculated using unaggregated individual-level data). However, it avoids the need for calculus, which would make it impractical for most QL applications. ⁵ Additionally, the quintile-based process has the advantage of creating categorical data that lend themselves to a variety of graphical representations, which presents additional QL opportunities for sociology courses.

The Gini coefficient unit in the introductory course discussed here begins with a graph that tracks the share of total aggregate income earned by each quintile in the United States over the past half-century. Students see that the richest quintile has increased its share while each of the other quintiles has lost share. A graph tracking the Gini coefficient over the same span is presented next, illustrating how it tracks the diverging shares in a single measure of (increasing) inequality. Then, a bar graph with the shares of total aggregate income for each quintile is presented using the most recent data available. Subsequently, that is transformed into a cumulative bar graph to underscore that all shares must sum to 100 percent. ⁶ Finally, the Lorenz curve as a graphical representation of the cumulative shares of total aggregate income by quintile is presented. Figure 1 presents the Lorenz curve for income data from the United States in 2020.

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Figure 1.

Lorenz curve for U.S. income distribution aggregated by quintile, 2020.

Once the Lorenz curve is constructed, the class engages in a discussion based on a guided discovery model to help students understand the “ratio of areas” logic that underlies the Gini coefficient. Subsequently, the class conducts the series of calculations necessary to determine the Gini coefficient. Figure 2 presents the graphical format used to guide students and the calculations associated with each component area based on income data from the United States in 2020 (as with Figure 1).

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Figure 2.

Illustration of Gini coefficient calculation from Lorenz curve (United States, 2020).

The Gini coefficient offers a strategic focus for cultivating QL in sociology courses. First, no step in the calculation requires more than elementary mathematical knowledge, but the full process requires students to understand a logic of sequences and relationships that is at the core of QL. As Madison (2004:10) wrote of QL, although “much of the mathematics involved is relatively elementary,” their application “in multiple and unpredictable contexts requires both an understanding of mathematical concepts and practice at retrieving and applying them.” Anecdotally, students have expressed a significant sense of accomplishment in completing the process to arrive at the Gini coefficient, which certainly increases their confidence in quantitative materials. Beyond its quantitative characteristics, the Gini coefficient, as the most common summary measure for social inequality, engages a central animating concern in our discipline, which makes it relevant and essential to sociology students. That relevance, or contextual embedding, also makes it particularly effective for cultivating QL within sociology coursework.

In sum, the characteristics associated with the Gini coefficient—multiple sequenced elements applying basic mathematics, highly contextualized content, and multiple opportunities for graphical representation—make it ideal for QL development in the introductory sociology course. It also invites plentiful opportunities to discuss the benefits and limitations of quantitative and qualitative research (another critical aspect of scientific literacy), including the implications of reducing a complex issue such as inequality to a single number.

Process of Assessment and Redesign

Although the authors did not formally team teach any of the courses or units, the second author observed and assisted the classes in which the Gini coefficient unit was presented. The core of our collaboration was our collective effort to identify specific comprehension gaps and sources of misconceptions in classroom and teaching practices that presented obstacles to student learning. To do so, we began by reviewing and discussing student exam responses to prompts focused on the Lorenz curve and Gini coefficient and thinking about how they might be related not only to comprehension gaps that need to be bridged but also to misconceptions that have their roots in specific aspects of the topic presentation. We engaged in discussions after each iteration. The initial discursive exploration laid the foundation for our formal assessment framework that targeted critical elements of student comprehension and allowed us to track shifts across iterations. In developing that framework, we took a student noticing lens toward misconceptions, working under the assumption that patterns in student errors (as opposed to isolated errors) are likely have a source in the learning experience—which we then sought to identify. Although we ultimately coded (and quantified) our assessments, they were conducted primarily to understand why student learning was unsuccessful (mirroring the dominant question in contemporary mathematics education) rather than to simply measure the level of comprehension. Coding was conducted by the authors collaboratively because it was integral to diagnostic and redesign processes. Likewise, although the redesigns implemented teaching practices in an effort to increase QL gains (described in the following), our primary goal was not to isolate and measure their effects, in large part because we implemented practices whose effectiveness has already been demonstrated by research. Our goal was to focus assessment at a granular level to understand where student comprehension was failing, design remedies to target those failures, and underscore the value of our collaboration in doing so. Those goals are reflected in our research design and processes. All research was submitted to and approved by our university Institutional Review Board.

Our formal assessments focused on student responses to two exam prompts over four semesters with an average of 94 students per semester (88, 85, 97, and 104 students per respective semester). The first prompt asked students to draw a Lorenz curve (and label axes) for a hypothetical distribution in which the poorest quintile earns 5 percent of the total aggregate income and the richest quintile earns half of the total aggregate income. As explained previously, the Lorenz curve is particularly useful for gauging comprehension because an accurate representation requires integration of multiple related understandings. Responses that violate essential properties indicate comprehension gaps or misconceptions, identifying productive foci for revised teaching strategies. The second prompt asked students to identify the range of the Gini coefficient and explain the meaning of that range for actual inequalities and provide a conceptual explanation of the derivation of the Gini coefficient from the Lorenz curve. In this report, we focus on a subset of those elements that best illustrates both general comprehension gains and recurring misconceptions. For readers interested in full analyses (over 30 variables linked to nearly 400 students), the full data set is available on request.

Our collaboration comprised three iterations of assessment, redesign, and implementation following the initial/baseline course. The baseline presentation (Semester 1) followed a fairly traditional format, described previously. Each subsequent iteration identified aspects of the teaching presentation to revise to address comprehension gaps and misconceptions and to direct student noticing toward critical characteristics of the material. We integrated specific practices to facilitate that. The first redesign (Semester 2) sought to integrate additional focus on the logic and sequence of calculations to more effectively cultivate QL. The increased presentational density associated with the QL emphasis and the patterns identified in the first round of student response analyses indicated a need for more and repeated exposure to the material. We sought to do so by redesigning the unit in a “flipped” format—that is, we presented foundational materials prior to class while using the class meeting to build and deepen conceptual understandings. That redesign drew on previous research suggesting that flipping course content in introductory sociology courses can increase learning, especially when paired with active-learning practices described in the following (Luna and Winters 2017; Ward, Antoine, and Cadge 2021). We created a series of three instructional videos (less than nine minutes each) to be accessed prior to class. The series, using current and historical data from the United States, focused on (1) the logic of an income distribution, (2) the graphing of the Lorenz curve, and 3) the calculation of the Gini coefficient. All videos are publicly available (Downey, Ernest, and McGarry 2017) and currently being updated. The videos incorporated enhanced graphics in an attempt to direct student noticing toward critical concepts and procedures. Videos have the added benefit that students are able to access them before and after class, if necessary, which was particularly helpful for students needing or wanting additional engagement with the materials.

For the second redesign (Semester 3), we restructured the class meeting to incorporate an active-learning exercise, which has been shown to be particularly useful in science, technology, engineering, and mathematics (STEM fields; Freeman et al. 2014). The exercise was based on a fictional income distribution of 25 individual incomes. Student volunteers were asked to represent each income and to order themselves lowest to highest. After that “embodiment” of the distribution, we proceeded through the sequential steps to draw the Lorenz curve and calculate the Gini coefficient for the distribution. The fictional distribution also gave students an opportunity to calculate the Gini coefficient using data different from the instructional videos, which could deepen understandings. Our third redesign (Semester 4) was a more subtle redesign that expanded the active-learning component to include more group work, mostly to extend practice with graphing. The expansion of group work drew from research indicating that peer discussions can be particularly effective in addressing misconceptions (Swan 2001). We concluded our collaboration after the fourth iteration—because further progress would require the creation of a new assessment framework to identify new/emerging patterns and because our introductory course was moved online.

Analyses and Interpretations

We now turn to analyses and interpretations of patterns in student responses. We organize our presentation to illustrate the value of our analytical framework focusing on critical components of comprehension and how specific insights from mathematics education guided our interpretations and interventions. We first present sources of comprehension gains from our interventions and subsequently focus on persistent misconceptions linked to specific pedagogical challenges.

Patterns and Sources of Comprehension Gains

The most significant gains in comprehension were visible following the first redesign, with diminishing marginal returns in subsequent iterations. ⁷ Gains were most evident in responses to the prompt asking students to draw a Lorenz curve for the data specified (i.e., in which the poorest quintile earns 5 percent of the total aggregate income and the richest quintile earns half of the total aggregate income). At the most basic level, we saw a significant decline in graphs evidencing minimal comprehension—for example, a diagrammatic representation of a curve without any connection to axes or labels. An example of such a response is provided in Figure 3. Such responses decreased from nearly a third of students (30.7 percent) in the initial class to below 10 percent in the first redesign and then down to under 2 percent in subsequent classes. Those gains are critical because they suggest we were able to direct student noticing toward key ideas among those who were previously least likely to do so.

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Figure 3.

Student response indicating no conceptual understanding.

We also saw significant increases in strategic points of comprehension—most notably, whether graphs ran from the origin to 100/100. Those data points were not identified in the prompt but represent fundamental logical/quantitative properties of the Lorenz curve, for any society regardless of the level of inequality. Thus, responses integrating those points suggest some understanding of the logical quantitative properties. In the first iteration, slightly over half of responses (52.3 percent) had the graph starting at the origin; by the final iteration, that rose to 85 percent. At the other end, less than one in five (19.3 percent) ended the curve at 100/100 in the first iteration; that more than doubled by the final iteration to 40.6 percent. The sequence of response patterns is presented in Table 1, indicating nearly perfect monotonic increases.

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Table 1.

Critical Properties of Lorenz Curve in Student Responses.

Response Property	Semester
	1(N = 88)	2(N = 85)	3(N = 97)	4(N = 104)
Curve starts at origin	52.3%	63.5%	68.00%	84.9%
Curve ends at 100/100	19.3%	24.7%	41.20%	40.6%

In addition to those critical comprehension gains, we saw vacillation in other areas over subsequent iterations such that gains in one iteration would be followed by declines in the next. In many ways, those more complicated patterns and our efforts to diagnose them best illustrate the value of our analytical framework and collaborative process. Specifically, those patterns were only noticeable because we had an iterative and reflective process and because we created a framework designed to assess specific granular response patterns. Moreover, identifying the plausible sources of those patterns and designing interventions to address them relied fundamentally on insights from research in mathematics education. In the following sections, we address those more complicated patterns, highlighting their implications for effective QL pedagogy.

Calibrating Procedural/Conceptual Balance

One initially discouraging pattern concerned responses to the second exam prompt, which asked for a conceptual explanation of the derivation of the Gini coefficient from the Lorenz curve, presented in Table 2. Although we observed significant improvement from Semester 1 to Semester 2 (with responses indicating full comprehension increasing from 6 percent to 25 percent), that was followed by partial reverses in Semester 3 (declining to 9 percent). Responses describing the Gini coefficient as the area between the LPE and the Lorenz curve (which represents only the numerator in the ratio of areas definition) increased significantly and then vacillated. Responses indicating no comprehension remained generally level after significant declines from Semester 1 to Semester 2, whereas responses that evidenced incomplete comprehension (stable from Semester 1 to Semester 2) doubled and tripled over the subsequent semesters. Finally, responses regarding the range of the Gini coefficient and the meaning of that range saw significant vacillation across iterations beyond the initial increases.

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Table 2.

Response Patterns: Derivation and Meaning of the Gini Coefficient.

Response Component: Coding Category	Semester
	1(N = 88)	2(N = 85)	3(N = 97)	4(N = 104)
Describe (/derivation)
Correct	5.7%	24.7%	9.3%	11.3%
Area between Lorenz curve and line of perfect equality	14.8%	21.2%	14.4%	19.8%
Evidence of incomplete comprehension	11.4%	11.8%	25.8%	34.9%
Evidence of no comprehension	60.2%	38.8%	39.2%	32.1%
No response	8.0%	3.5%	4.1%	1.9%
Range: correct	52.3%	71.8%	51.5%	68.9%
Meaning of range: correct	42.0%	57.6%	37.1%	59.4%

In a closer examination of the declines and vacillations, we observed a general pattern in which students were drawing on critical information but struggling to make it gel into full comprehension. After renewed discussion of those patterns and reexamination of responses, we observed that students had integrated much more quantitative content in responses in Semester 2 and beyond (a pattern that we did not track formally because the insight emerged after our framework had been established and implemented). Further examination of recordings of classroom presentations and instructional videos made clear that in our effort to increase the contribution of the unit to QL gains, we significantly increased time spent emphasizing procedures for calculating the Gini coefficient from the Lorenz curve while reducing in relative terms the time spent making sense of those calculations. That goes to the heart of procedural-conceptual balance. Counterintuitively, that dynamic was exacerbated with the introduction of the active-learning exercise. Because we ran into several areas where students did not have the quantitative foundation to move through materials as quickly as expected, the final portion of the presentation (which was to be where we concluded the discussion and tied together all of the details and meaning of the calculations) was cut short—as often happens when applying new pedagogical strategies. As a result, students were more deeply engaged in calculations but came away with a less solid conceptual foundation. Essentially, we had moved too far toward a procedural emphasis and away from a conceptual understanding of the meanings of those procedures. This was a critical lesson for QL pedagogy. Collaborative assessment and discussion were essential to subsequent efforts to recalibrate the procedural-conceptual balance, producing improvements in the final iteration (e.g., an uptick in correct responses and partially correct responses). In part, that was because we devoted an additional class meeting to the unit. Moving forward, we are also focusing more on “big picture” comprehension in current video revisions.

A Stubborn Misconception and a Comprehension Cul-de-Sac

Among the most complicated response patterns were those associated with students’ labeling of the x-axis for the Lorenz curve. The x-axis is critical because it represents the fundamental conceptual foundation of the income distribution—that is, the arrangement of poorest to richest across the full population. Labels for the x-axis in students’ responses are presented in Table 3. We can see that the percentage of students with fully accurate labeling of the x-axis (percentage of households, requiring both components of that label) remained stable from the first to the second iteration and then declined from 17 percent down to 6 percent (although if we consider correct and partially correct responses—e.g., only “households” or only “percentage”—the decline is less dramatic). That pattern of vacillations mirrored what we discussed previously.

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Table 3.

Patterns of x-Axis Labeling in Lorenz Curve Responses.

x-Axis Label	Semester
	1(N = 88)	2(N = 85)	3(N = 97)	4(N = 104)
No label	28.4%	16.5%	12.40%	9.4%
Correct	17.0%	16.5%	6.20%	20.8%
Correct without %	2.3%	5.9%	9.30%	4.7%
%	5.7%	5.9%	7.20%	20.8%
Quintiles	21.6%	36.5%	47.40%	21.7%
Aggregate income	13.6%	7.1%	7.20%	10.4%
Time or years	2.3%	—	1.00%	.9%
Other incorrect	9.1%	11.8%	9.30%	11.3%

The most common error made by students was to label the x-axis as “quintiles”—an error that rose from 22 percent to 37 percent and then to nearly half of all responses (47 percent) in the third iteration. Students who did so were clearly drawing on the understanding that we had aggregated income shares by quintiles to draw the Lorenz curve and as such, exemplifies a conceptually rich misconception. Students making that error focused their attention on a critical aspect of the presentation and clearly engaged with the material. However, they conceptually transposed the categorical x-axis associated with bar graphs onto the continuous x-axis of the Lorenz curve—and in the process, neglected the critical cumulative nature of the curve. We refer to that pattern as a “comprehension cul-de-sac” because the further they followed the categorical logic of quintiles, the more difficult it was to understand the cumulative and continuous logic necessary to graph the Lorenz curve properly—and inevitably led to more comprehensive errors in the drawing of the Lorenz curve. To explain how the former led to the latter, we can use the example presented in the prompt.

The prompt specified that the richest quintile earns 50 percent of the aggregate income—meaning that all other quintiles together must earn the other 50 percent. As such, the Lorenz curve must pass through the 50 percent mark on the y-axis at the 80 percent mark on the x-axis, with the final 50 percent of aggregate income (y-axis) covered by the final/richest 20 percent of households (x-axis). That is a logical requisite of the cumulative nature of the curve across both axes and the fact that it must conclude at 100/100. However, responses in which the x-axis was labeled as quintiles would have that final coordinate on the x-axis aligned with 50 percent of aggregate income on the y-axis. That would make sense if we were charting the percentage earned by each quintile rather than the cumulative income of all quintiles. Responses making that mistake had to complete the graph in a way that violates fundamental logical properties. Some students simply ended the curve at 100 percent on the x-axis and 50 percent on the y-axis. Others completed the graph by continuing the curve from 100/50 vertically to 100 on the y-axis or beyond 100 percent on the x-axis to reach 100 on the y-axis. An example of that final response is reproduced as Figure 4. Patterns in those types of errors are presented in Table 4.

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Figure 4.

Student response indicating lack of understanding of cumulative properties.

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Table 4.

Specific Cumulative and Categorical Confusions in Lorenz Curve Responses.

Cumulative/Categorical Confusions	Semester
	1(N = 88)	2(N = 85)	3(N = 97)	4(N = 104)
Ends at 100% on x and 50% on y	10.2%	9.4%	6.2%	16.0%
5th quintile at 50%, curve continues	11.4%	24.7%	20.6%	13.2%

Recognizing the increase in those errors, we emphasized the cumulative and continuous nature of the variables graphed in our classroom presentation. However, the mistake remained prominent even as other aspects of comprehension were increasing (as noted previously)—hence the “stubbornness” of the misconception. In part, that is because we were not able to revise the instructional videos to direct student noticing to the crucial distinctions—so while we revised classroom presentations, those misconceptions were already planted in students’ understandings. The critical point to underscore here is that our ability to address the specific misconception was only made possible by the extensive and granular assessment and discussion of student responses and by applying the insights from mathematics education associated with misconceptions and student noticing. In other words, it allowed us to chart a path out of the comprehension cul-de-sac (that we are now taking with revisions to the instructional videos).

Discussion and Conclusion

Our primary intended contribution is to offer a model of interdisciplinary collaboration between sociology and mathematics education to more effectively cultivate QL in sociology courses. Secondarily, we present the Gini coefficient as a strategic topic for sociologists interested in cultivating QL in our major. We begin our summary by focusing on lessons for the latter and then return to the former.

The Gini coefficient represents a strategic tool for QL in sociology because of both its thematic relevance to the discipline and the associated elements that lend themselves to cultivating QL. We have identified a range of practices and materials for presenting the Gini coefficient in the introductory course. Although our research was not designed to formally measure the relative gains associated with each specific practice, our analyses provide insight into how they generally contributed to learning. The use of instructional videos worked well to provide students with a basic understanding of the Gini coefficient prior to class (and were accessible repeatedly after class, which is an important additional benefit), leaving class time for higher level learning—as is the design goal of flipped instruction. That is critical for the QL applications where we need to provide significant essential foundation to students. We recommend the videos to those interested in integrating the Gini coefficient into courses. Revised videos will apply the insights that emerged later in our collaboration—most notably, directing student noticing to how the categorical data presented in bar charts translate to the continuous/cumulative x-axis of the Lorenz curve.

We also found the active-learning and group exercises to have made important contributions to student learning. We are revising those exercises to more fully integrate questions and associated discussions designed to generate cognitive conflict among students so that misconceptions can be identified and addressed among peers, as encouraged by Swan (2001).

Finally, we recommend setting aside at least a full week for the unit to effectively calibrate the appropriate procedural-conceptual balance. Although the Gini coefficient might be effectively introduced in a single class, cultivating QL gains requires additional time and attention. The unit could also be productively expanded beyond that, based on the goals of the course—integrating more examples of the Gini coefficient in research, additional graphing exercises, multiple data sets, and lengthier structured discussions. ⁸ In addition, it could be integrated productively into a range of courses in the major (from Social Inequalities to Globalization), and it can be used to emphasize a variety of different QL elements (Catalano, Leise, and Pfaff 2009; Warner and Lim 2019). One might even purposefully and productively integrate the Gini coefficient at multiple levels of the curriculum, emphasizing different skills and requiring different levels of independent work. It is a rich pedagogical tool for QL in the major.

Our primary intended contribution speaks more broadly to QL and to SoTL addressing QL within our discipline. From the sociologist’s perspective, the first author found the insights of mathematics education (which go beyond what we were able to present here) to offer invaluable guidance for more effectively cultivating QL in sociology courses. For sociologists interested in doing so, we suggest that greater engagement with that field will help to realize and extend interdisciplinary SoTL in our own discipline (Paino et al. 2012).

We also sought to provide a systematic and exploratory model for pedagogical assessment and intervention. Specifically, our assessment of specific elements and patterns of student comprehension and misconceptions allowed us to identify gaps in understanding that need to be addressed and identify elements embedded in teaching that inadvertently contributed to misconceptions. Much of SoTL seeks to identify and encourage meso-level practices that will lead to learning gains, which we strongly support. Our focus in this project, however, is on more granular analyses of comprehension following the emphases of contemporary mathematics education. In many senses, our process more closely mirrors the approach and assumptions of the qualitative tradition in SoTL (McKinney 2007:67–82). That deep exploration of student comprehension (gains, gaps, barriers) is particularly critical in the realm of QL, in which student learning must be scaffolded and sequenced and where specific misconceptions can lead to a cascade of misunderstandings. Mathematics educators are uniquely equipped to connect student misconceptions and student noticing to sources in the learning experience and identify ways to address them.

As we integrate QL across the major, we encourage sociologists to seek out the insights of mathematics educators to tap into their specialized expertise. Extensive quantitative skills do not necessarily produce insights into how to cultivate those competencies in students—as is evident in the disciplinary distinction between mathematics and mathematics education. Most universities have mathematics educators—either within the Mathematics Department, within the School of Education, or in some instances, in their own dedicated department. The benefits to our students of making those connections—from formal collaborations to informal discussions to engagement with the research—can be significant.