
District DecisionMakers’ Considerations of Equity and Equality Related to Students’ Opportunities to Learn Algebraby Beth A. HerbelEisenmann, Lindsay Keazer & Anne Traynor  2018 Background/Context: In this article we explore equity issues related to school district decisionmaking about students’ opportunities to learn algebra. We chose algebra because of the important role it plays in the U.S. as a gatekeeper to future academic success. Current research has not yet explored issues of equity in districtlevel decisionmaking. Purpose/Objective/Research Question/Focus of Study: We examine the extent to which district decisionmakers for mathematics attend to aspects of equity when they make decisions about resources related to the teaching and learning of algebra. The research questions guiding this study were: How do district decisionmakers for mathematics report considering issues of equity when making decisions about students’ opportunities to learn Algebra I? How do district characteristics, particularly students’ racial and ethnic diversity, affect the extent of equity considerations by mathematics decisionmakers? Research Design: We surveyed a national probability sample of 993 district decisionmakers for mathematics about criteria that they consider when they select and distribute resources and structure learning opportunities in algebra for students and teachers in their districts. These survey items were our attempt to identify districtlevel practices in relationship to an equity framework. In this study, we examine national patterns in criteria for decisionmaking about algebra resources and examine the relation of these criteria to district features using a structural equation model. Findings: Our findings suggest that fewer decisionmakers considered equityrelated criteria in their decisionmaking about algebra, while many tended to endorse equalityrelated items addressing considerations for all students, such as giving all students the same resources or attending to preparation for standardized testing. The vast majority of decisionmakers reported considering real life contexts for algebra when making decisions about professional development (PD) and curriculum, while fewer considered the students’ culture or culturally relevant teaching. Decisionmakers in only about half of the districts reported considering structural aspects, such as tracking. Modeling of the survey responses indicates that decisionmakers in the most racially or ethnically and linguistically diverse districts have the greatest tendency to consider equity criteria in structuring students’ opportunities to learn algebra. Conclusions/Recommendations: The extent to which district decisionmakers for mathematics attend to aspects of equity is noteworthy because their decisions inform the selection and distribution of educational resources for learning algebra across districts. These findings raise important concerns with respect to how district decisionmaking mobilizes and shapes the resources available to teachers and students. Recommendations include supporting district decisionmakers in a) expanding their conceptions of real life contexts to include students’ culture, b) considering different framings of the problem of participation gaps, c) reconsidering ability grouping and understanding the negative consequences of tracking, and d) carefully examining the kinds of stated and unarticulated rules, rewards, and sanctions that get put into place to uncover how inequitable practices get perpetuated. Currently in mathematics education few issues are receiving more attention from the policy and professional practice community than increasing the mathematical attainment of every student. Contributing to work in this area, this article focuses on equity of opportunities to learn in mathematics education. In particular, we report on results of a largescale survey that seeks to understand what district decisionmakers consider when they make decisions about resources related to the teaching and learning of algebra. Spillane and colleagues have convincingly shown the importance of studying districts because they are where policy actually happens and can illustrate variation in how policy gets enacted (e.g., Spillane, 1996). The capacity of districts to enact policy well entails “human capital (knowledge, skills, and dispositions of leaders within the district), social capital (social links within and outside of the district, together with the norms and trust to support open communication via these links), and financial resources (as allocated to staffing, time, and materials)” (Spillane & Thompson, 1997, p. 199). Because districts’ financial resources are limited, decisions about what resources to provide, to whom, and for what purposes may affect the equity of students’ opportunities to learn. District administrators are likely players in mobilizing support for the effective implementation of policy (e.g., Elmore & McLaughlin, 1988). Rorrer, Skrla, and Scheurich (2008), in fact, have argued: As institutional actors, districts have an “organized interest” in (Wong & Jain, 1999) and unique ability to be the “carriers and creators of institutional logics” (Scott, Ruef, Mendel, & Caronna, 2000, p. 20). That is, their role in improving achievement and advancing equity, in this instance, is connected to their collective identity and their ability to create change by altering institutional scripts that tacitly and explicitly govern behavior of organizational members. (p. 332) Although scholars in educational leadership have made central the district context in which schools are embedded and the influence of district and site leadership on district decisionmaking (e.g., Bowers, 2008; Honig, 2003^{1}), researchers in mathematics education have largely overlooked the role of the district in enacting policy. Exceptions to this claim include, for example, Spillane’s (2000) examination of district responses to the reforms suggested by the National Council of Teachers of Mathematics and Cobb, McClain, Lamberg, and Dean’s (2003) description of an analytic approach for situating teachers’ work in the context of the school and district. Little of the work outside of mathematics education and none of the mathematics education work we could find, however, empirically addressed issues of equity in this districtrelated work. Here we seek to contribute to work focused on districts, equity, and resource allocation by reporting findings from a quantitative study that takes into consideration four factors that shape the availability, distribution, and nature of opportunities to learn in relationship to districtlevel mathematics decisionmaking about algebra. We chose algebra because of the important role it plays in the U.S. as a gatekeeper to future academic success (e.g., Moses & Cobb, 2001; National Governors Association [NGA], Council of Chief State School Officers [CCSSO], & Achieve, Inc., 2008). In this study, we surveyed nearly 1,000 district decisionmakers for mathematics in order to examine how they interpret demands associated with offering algebra for all students, the strategies they have developed to respond to these demands, and how their strategies shape students’ opportunities to learn algebra (Steele et al., 2016). In particular, we focus on findings from analyzing a subset of survey items we developed that drew on an equity framework articulated by Gutiérrez (2012). In the sections that follow, we first explain issues associated with algebra for all, address opportunities to learn, explain Gutiérrez’s equity framework, and connect opportunities to learn to her equity framework. BACKGROUND The issue of providing access to algebra to all students lies at the nexus of several powerful forces shaping education in the country at this time: • global competitiveness reforms resting on a recognition of the mediocre performance of American students on international mathematics assessments when compared with students in most other economically advanced nations, and a commitment to make students in this country more competitive; • public and professional calls for higher educational standards and expectations, which have led many states to adopt more ambitious and rigorous high school graduation requirements that include several years of mathematics coursework; and • equitydriven reforms responding to longstanding differences in participation in mathematics and science fields among demographic subgroups of students based on race, income, and neighborhood, and a commitment to reduce or eliminate these differences. In response to these forces, there has been a call for activities aimed at improving the mathematical attainment of all students. Achieve, Inc. (2004), for example, recommended that states increase their mathematics graduation requirements to align with a “college and careerready curriculum,” or 4 years of challenging mathematics. In response to this and similar recommendations, nearly half of U.S. states have implemented mathematics proficiency examinations that students must pass in order to graduate from high school; other states are likely to join this movement (Holme, Richards, Jimerson, & Cohen, 2010). More recently, the Common Core State Standards (National Governors Association Center for Best Practices [NGACBP] & CCSSO, 2010) were developed for mathematics in order to respond to the challenge that “the mathematics curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country” (p. 3). The area of mathematics that is often at the center of efforts to improve mathematics teaching and learning is algebra. Nomi (2012) claimed that reports focusing on algebra as a gateway course to future achievement (i.e., National Mathematics Advisory Panel, 2008) have resulted in policy makers pressuring schools to offer Algebra I to all students at or before Grade 9. Researchers have suggested offering algebra in middle school (Gamoran & Hannigan, 2000; Raudenbush, Fotiu, & Cheong, 1998) and mandating or creating stronger links between arithmetic and algebra in K–8 classrooms (Carpenter, Franke, & Levi; 2003; Paul, 2005). Those advancing equitybased arguments have pointed to early access to algebra as a key factor in opening pathways for students to pursue a range of career and postsecondary educational opportunities. Raudenbush et al. (1998), for example, showed that social background, as determined by parent’s education level, is significantly related to the probability of attending a school that offers algebra in eighth grade (rather than in a later grade). Analyses of mathematics achievement and mathematics coursetaking in the U.S. consistently show that proficiency is strongly correlated with the number and level of high school courses completed, with algebra coursetaking identified as a particularly strong determinant of later mathematics achievement (Gamoran & Hannigan, 2000). Researchers have emphasized the importance of providing more students with access to algebra at an earlier age and have highlighted contexts in which policies were implemented to require all students to enroll in algebra in eighth or ninth grade. Initial reports at the local and the state level, however, indicate that students and teachers have had a great deal of difficulty meeting these often highstakes expectations, with some states reporting failure rates in algebra courses in excess of 50% (Clotfelter, Ladd, & Vigdor, 2015). Similarly, findings from a study involving 160,000 Chicago high school students who were required to take algebra in ninth grade beginning in 1997 showed that the new policy has not resulted in test score gains or the taking of additional higherlevel mathematics courses and that the percentages of students failing mathematics in ninth grade rose after the new policy took effect (Allensworth, Nomi, Montgomery, & Lee, 2009). Two notable findings in that study were that, on average, students who traditionally would have succeeded under the former algebra policy failed at higher rates under the algebraforall policy; and there was little impact on later outcomes. These and other wellintentioned efforts of policymakers, practitioners, developers, and funders are currently plagued by a dearth of solid information about how the policy elements are interacting and impacting teaching and learning in mathematics classrooms. In short, organizations and other actors who want to intervene now have too little basis on which to decide how and where their interventions would likely be most beneficial to students’ learning of algebra. The purpose of the broader study from which we draw for this article was to investigate school districts’ policies and practices, and the perceptions of districtlevel mathematics decisions makers related to Algebra I. Differences in opportunities to learn mathematics for students from dominant versus nondominant racial, ethnic, linguistic and socioeconomic backgrounds have resulted in visible “participation gaps” in mathematics classrooms (Hand, 2003) and in STEM fields (National Research Council, 2011). The extent to which district decisionmakers for mathematics attend to aspects of equity in structuring students’ opportunities to learn is important because their decisions could potentially reduce participation gaps in learning Algebra I across school districts. The research questions guiding this study were: “How do district decisionmakers for mathematics report considering issues of equity when making decisions about students’ opportunities to learn Algebra I?” and “How do district characteristics, particularly students’ racial and ethnic diversity, affect the extent of equity considerations by mathematics decisionmakers?” In the following sections, we explain what we mean by opportunities to learn and elaborate the framework for equity that informed these particular survey items we designed to collect information about the criteria decisionmakers use in selecting resources to foster algebra learning, and distributing these resources across their district. OPPORTUNITIES TO LEARN The phrase “opportunities to learn” has had a growing presence in mathematics education literature. This term was used originally to refer only to the amount of time devoted to a particular topic in the classroom (Floden, 2002). Hiebert (2003) then offered a perspective that accounted for the nature and quality of the time devoted to particular topics: “Providing opportunities to learn means setting up the conditions for learning” (p. 10). At the classroom level, Hiebert argued that the conditions for learning needed to include the nature of the tasks students encounter as well as how students’ engagement with those tasks would be structured and supported. Although much of the work related to opportunities to learn has focused on classroom experience, we extend this work by considering how school districts put in place structures and resources to direct and facilitate classroom algebra instruction. At the school district or system level, taking an opportunity to learn view considers how resources are deployed and the nature of support built into the system to facilitate their quality deployment toward a particular goal (Cohen, Raudenbush, & Ball, 2003). The key dimensions of opportunities to learn at the district level consist of the kinds of structures and tools available to districts to shape learning in classrooms. At a minimum, four important structures and tools that shape the availability, distribution, and nature of opportunities to learn are a) the curriculum resources used, b) the development and use of professional expertise (human resources), c) the organization and structure of learning, and d) how learning is assessed. Curriculum resources refers to print and electronic curriculum materials and programs used and how they are employed. Studies on curriculum materials that preceded national standards in mathematics and science education suggested, for example, that textbooks can impact both what and how teachers teach, as well as what and how students learn (e.g., Alexander & Kulikowich, 1994; Remillard, 2005; Stein, Remillard, & Smith, 2007; Tobin, 1987; Usiskin, 1985). Professional expertise refers to human resources, including the use of external expertise, the development of internal capacity through professional development, and the emphasis and balancing of the two within an individual district. The teacher has become recognized as an important player in influencing students’ opportunities to learn, and consensus has been reached that professional development of teachers is critical to improving the quality of education (Sowder, 2007). The organization and structure of learning refers to how the district and schools structure and make available algebra instruction, including scheduling and sequencing. For instance, the curricular tracks that are created by school structures, separating students into mathematics course sequences with varying levels of rigor (e.g., Oakes & Guiton, 1995), can affect students’ mathematics achievement gains (Gamoran, Porter, Smithson, & White, 1997). Finally, assessment refers to how algebra learning is assessed by the district. That is, we wanted to understand how students were expected to demonstrate knowledge and the types of understanding or knowledge that were prioritized by these assessments. We probed decisionmakers’ criteria for selecting standardized, formative, and summative assessments. Formative assessment, in particular, requires adapting instruction to students’ needs and providing feedback that is useful to students’ progress (Wiliam & Thompson, 2007). As noted earlier, differences in opportunities to learn mathematics have resulted in visible participation gaps. As society becomes more and more technologically driven, Moses (Moses, 1995; Moses & Cobb, 2001) has argued that participation gaps in mathematics are a presentday civil rights issue. Some scholars have referred to this as an “educational debt” in order to: begin to think about the incredible debt that we as a nation have accumulated. So rather than focusing on telling people to catch up, we have to think about how we, all of us, will begin to pay down this mountain of debt that we have amassed at the expense of entire groups of people and their subsequent generations. (LadsonBillings, 2007; cf. LadsonBillings, 2006) Addressing equity in mathematics education as a function of opportunities to learn can shift the focus of mathematics education reform away from remediation following the occurrence of unequal educational outcomes among particular student groups, and toward ensuring powerful mathematics learning for these groups within classrooms, schools, and districts (Gutiérrez, 2002; Martin, 2009a; Moschkovich, 2010). Such powerful learning requires that the system itself operate differently. This kind of systemic equity, Scott (2001) suggests, transforms [the] ways in which systems and individuals habitually operate to ensure that every learner—in whatever learning environment that learner is found—has the greatest opportunity to learn, enhanced by the resources and supports necessary to achieve competence, excellence, independence, responsibility, and selfsufficiency for school and for life. (p. 6) Thus, when one takes an equity approach to distributing structures and tools that support students’ opportunities to learn, one needs to consider what each student needs to be successful, bearing in mind the demands of social justice, and not presume that every student should get equal resources (Espinoza, 2007; Secada, 1989). AN EQUITY FRAMEWORK TO CONSIDER OPPORTUNITIES TO LEARN In order to better understand students’ opportunities to learn algebra, we were convinced by Gutiérrez’s (2012) argument that we needed to include access and achievement (which she calls the “dominant axis” of equity work) as well as issues of identity and power (which she labels the “critical axis”). In her framework, she articulated these aspects of equity (see pp. 18–21) in the following ways: • Access relates to the resources available that students can use to participate in mathematics, such as highquality teachers, a rigorous curriculum, classroom supplies, technology, and a classroom environment that promotes positive participation and offers additional support when needed. • Achievement is linked to students’ participation in class, course grades, standardized test scores, participation in advanced mathematics courses, and participation in mathematicsbased careers. • A focus on identity necessitates that we consider how participation in mathematics offers students a "mirror" to see themselves in the curriculum, as well as a "window" to understand how mathematics is a part of the broader world. It also relates to questions about whether students find mathematics personally meaningful to their lives and whether they are able to draw on their cultural and linguistic resources when they do mathematics. • Power relates to issues of social transformation. It includes students' options to influence what happens in their mathematics classrooms and whether they are equipped to engage in the world through the use of mathematics as a critical analytical tool. It recognizes that mathematics is a humanistic enterprise and includes the consideration of alternative epistemologies. Using Gutiérrez’s framework as a lens to carefully consider the four main structures and tools that shape the availability, distribution, and nature of opportunities to learn described earlier, one can consider how decisions about these might be equitable. For example, questions might be asked about the ways in which districtlevel decisionmakers select curriculum resources or the choices they make when they put resources into developing human expertise. That is, do districtlevel decisionmakers consider adopting curriculum materials that allow students to see themselves in the curriculum? Do they provide opportunities for teachers to learn about the cultures and communities of their students in the professional development they offer? Do they incorporate historically accurate images of mathematics, as developed by Asian and African communities? Do they think it is important for students to use mathematics as an analytic tool to critique society or to make sense of the politics involved in their lives? Taking account of district decisionmakers’ considerations of both the dominant axis (i.e., access and achievement) and the critical axis (i.e., identity and power) as they make decisions about factors known to support students’ opportunities to learn is important because they have the capacity to influence the selection and distribution of educational resources for learning. This is particularly important in the context of Algebra I, given its status as a gatekeeper to higher education and other life and employment opportunities. METHODS SURVEY DESIGN We investigated the landscape of how districts structure students’ opportunities to learn Algebra I through a national survey of district decisionmakers. When sampling public school districts across the U.S., we targeted district mathematics decisionmakers as the most suitable respondents to answer dataoriented and philosophical questions regarding Algebra I policy and practice because they are often the ones making decisions about how to allocate resources. Drawing on the mathematics education and policy literatures, interviews with 12 district mathematics leaders, and the principles of total survey design (Fowler, 2002), we devised a set of survey items to address the broader study’s research questions. A subset of these items was written to address the particular research questions investigated here: (1) “How do district decisionmakers report considering issues of equity in making decisions about students’ opportunities to learn Algebra I?” and (2) “How do district characteristics, particularly student diversity, affect the extent of equity consideration by mathematics decisionmakers?” The items were reviewed by a panel of mathematics education and policy experts, revised, and then piloted by 38 district mathematics decisionmakers. Before administering the final survey nationwide, we conducted cognitive interviews with 10 decisionmakers from the pilot sample to explore response meanings, and finalized the survey items. In this paper, we focus on district decisionmakers’ responses to 23 survey items about decision criteria they considered that might affect students’ opportunities to learn in Algebra I. These particular survey items were motivated by Gutiérrez’s (2012) equity framework, which accounts for both the dominant axis (i.e., access and achievement) and critical axis (i.e., identity and power). In the survey, for each of the four important structures and tools that shape the availability, distribution, and nature of opportunities to learn (e.g., curriculum resources), respondents were given a list of criteria and asked to select the criteria that influenced decisions about that particular opportunity to learn in the district. Eighteen of the listed criteria (items) were intended to measure districts’ consideration of dominantaxis criteria; for instance, “We considered or will consider ways to ensure that our best teachers work with our most struggling students” indicated consideration of ideas related to access. To measure consideration of criticalaxis criteria, we devised five items, including, for example, “We considered or will consider whether the curriculum materials represent the culture of the students who are in our district.” (See Appendices for further examples of the items we analyzed here.) We found these criticalaxis items particularly difficult to write in such a way that respondents would interpret them as intended. We did not see evidence, however, that the district decisionmakers who piloted the items had difficulty interpreting them or interpreted them differently than we intended. Yet, we recognize that the difficulties we encountered writing these items relate to the fact that aspects of power and identity tend to function below the radar screen of school personnel and often go unquestioned (Cobb & Hodge, 2002)—they relate to aspects of the hidden curriculum (Jackson, 1968) or hegemonic practices (Darder, Baltodano, and Torres, 2003) in schools. PARTICIPANTS AND DATA COLLECTION A power analysis corresponding to the broader study’s research questions indicated a target sample size of 1,000 district decisionmakers. To ensure our sample of decisionmakers was as representative as possible with respect to key features of school districts, we used a complex sampling design (Kish, 1995), with districts stratified by size, urbanicity, and state algebra graduation policy type, to select survey respondents. We obtained contact and demographic information for all public school districts in the United States from the Common Core of Data (National Center for Education Statistics, 2011). Because the largest 19% of districts enroll a disproportionate number of the nation’s students, to improve the precision of any findings regarding large districts, we oversampled decisionmakers from these large districts to comprise 40% of our sample, while decisionmakers from smaller districts comprised 60%. We sampled proportionately from three urbanicity strata (urban, suburban/town, and rural) and four state algebra policy type strata (Algebra I required for high school graduation by 2011, Algebra I to be required for graduation by 2015, indefinite plan to increase the rigor of state graduation requirements, and no plans to increase the rigor of state graduation requirements [Achieve, Inc., 2009]). We sought to administer the survey to the individual within a school district most clearly responsible for decisions in mathematics education, who would be best able to answer both dataoriented and philosophical questions regarding Algebra I policy and practice. We identified the most likely districtlevel decisionmaker by searching school district and state education agency websites. In larger districts, the decisionmaker tended to be a mathematics coordinator or an assistant superintendent for curriculum and instruction. In smaller districts, this person tended to be a superintendent, principal, or teacher. In each initial invitation to participate, we asked respondents to forward the email to whoever was responsible for districtlevel decisionmaking in mathematics, if it was not them. We sent the survey to identified decisionmakers in 1,400 districts selected by our stratified random sampling procedure in Spring 2012. Based on the initial nonresponse rate, we contacted an additional 932 districts selected using the same sampling method in Fall 2012, for a total of 2,332 surveyed school districts. Data cleaning to remove substantially incomplete or duplicate surveys yielded 993 respondents, for a total participation rate of 43%. The distribution of the districts with respect to state policy grouping [χ^{2} (3) = .721] and urbanicity [χ^{2} (2) = 3.994] was statistically representative of our intended sample, showing no evidence of response bias. The realized sample contained a greater proportion of large districts than intended by the sampling plan [χ^{2} (1) = 26.439, p < .001], but a nonresponse adjustment was applied to the sampling weights so that weighted estimates from the sample would reflect the national distribution of district characteristics. Within each of the 24 cells of the stratification table, the initial weights for districts that responded were scaled so that their sum was equal to the known population proportion of districts in that cell, producing final, nonresponseadjusted sampling weights. The final analytic sample included decisionmakers representing districts in all states except Hawaii and Nevada. ANALYSIS Our analysis consisted of three main phases: examination of descriptive statistics for the survey items about decisionmaking criteria, exploratory and confirmatory factor analysis in a subset of the sample to determine the best measurement model for the items, and structural equation modeling (SEM) using the remainder of the sample to investigate how particular school district features were related to the two latent decisionmaking factors suggested by the exploratory modeling. All analyses were conducted using Mplus software (Muthén & Muthén, 2012), which implements estimation methods described in Rao and Thomas (2003). So that our results would approximate those expected from a simple random sample of district decisionmakers, all analyses accounted for the stratification and unequal probabilities of selection (i.e., sampling weights) that described the sampling design. We first split the sample into two subsamples, one for exploratory analysis and one for hypothesis testing. Results of a recent simulation study (Wolf, Harrington, Clark, & Miller, 2013, p. 922) suggested 200 cases would be adequate for exploratory analysis with factor analytic methods, even if the indicator items had an average factor loading of only .50. We reserved the remainder of the sample, 793 cases, for hypothesis testing using SEM. To address Research Question 1, we estimated the proportion of district decisionmakers affirming consideration of each listed criterion item, and identified general patterns of criteria use. Because the measurement properties of the newly developed survey had not previously been investigated, we used exploratory, followed by confirmatory, factor analysis to determine a reasonable measurement model for the survey data. We initially hypothesized a twofactor model for the item set based on Gutiérrez’s (2012) framework, with items written to represent the dominant axis loading highly on one latent factor, and criticalaxis items loading on another factor. To determine whether the hypothesized model for the collected responses was plausible, we first conducted an exploratory factor analysis for categorical data using an oblique geomin rotation that allowed any underlying factors to be correlated. Information from decisionmakers with missing data on some items was included in the analysis by use of robust maximum likelihood estimation. To determine the preferred factor solution, we considered substantive interpretability, model fit, and parsimony (e.g., Bollen, 2014). Five decisionmaking criteria that did not load highly on any factor in solutions of up to three factors, and had meanings that were noticeably different from those of the remaining 18 criteria (e.g., “Materials will be easy for our teachers to implement”; “Feedback from colleges and universities about the rigor of our math offerings”) were removed, one by one, from the analysis. We then used samplesize adjusted Bayesian information criterion (BIC) values for model fit comparison. Complete results of these models are available from the authors. A BIC difference greater than 10 points is typically interpreted as support for the model with the lower BIC value (e.g., Raftery, 1995). BIC values suggested the twofactor solution (BIC = 3,375.9) was a more likely population model than the onefactor solution (BIC = 3,413.1). The threefactor solution (BIC = 3,359.9) fit better than the twofactor solution, but had a solution that was not readily interpretable, and seemed unlikely to be reproducible in a more constrained independentcluster confirmatory model due to substantial crossloadings on many of the items (Marsh et al., 2009; van Prooijen & van der Kloot, 2001). The factor loading patterns in the exploratory factor analysis results suggested that districts’ decisionmaking criteria were related to two unobserved latent factors, but these factors did not appear to represent the dominant/critical axis distinction as we originally thought. To determine whether the loading pattern in the twofactor model had a meaningful substantive interpretation, we examined the text of the survey items that loaded most heavily on each factor. An observable theme among the 7 items related to Factor 1 was that all the items were written to consider the needs of identified student subgroups in the district, including “underserved populations,” “struggling students,” students from different demographic backgrounds, and those representing nondominant cultures. We interpreted the Factor 1 items as measuring an equity orientation to decisionmaking because of the emphasis on considering the needs of particular subgroups in order to support their success. Examining the 11 items that loaded most highly on Factor 2, we observed that the criteria referred, more generically, to “students” or “all students.” Noticeably absent from these criteria was a consideration of students’ cultural backgrounds or other demographic identifiers. Overall, the exploratory factor analysis results indicated two separate facets of decisionmaking about Algebra I structures and resources: attention to recognizing that some subgroups may need different resources in order to be successful, and attention to making decisions that would give all students the same resources. This difference in language choices reflects discussions in the field about focusing on equity versus equality. Secada (1989), for example, adamantly argued that these two terms (i.e., equity and equality) do not represent the same perspective and thus should not be treated interchangeably. Gutiérrez (2012) agreed that “equity means fairness, not sameness” (p. 18). A focus on giving all students the same amounts and types of resources thus may be characterized as an equality view, while considering the possible needs of student subgroups may be evidence of taking an equity view toward distribution of resources. Drawing on these distinctions, we labeled these two factors equity view and equality view. The two factors represented distinct unobserved variables underlying district mathematics decisionmakers’ responses to the survey items. Based on the exploratory results, we devised a competing confirmatory factor analysis model for comparison with the critical/dominant axes model we had originally proposed. To judge which model was a more likely representation of the latent processes underlying decisionmaking about algebra resources, we then estimated the parameters of the original and competing models using confirmatory factor analysis. The BIC values for the original and competing models were 3,172 and 3,140, respectively, implying that the competing model should be preferred on the basis of model fit. The estimated correlation between the two latent factors underlying the competing model, which we had labeled the equity view and equality view, was .74, suggesting that the survey items were capturing two distinct, although interrelated, aspects of district decisionmaking. It is possible that the magnitude of the correlation may capture some degree of affirmation bias (i.e., tendency to affirm more items than warranted) among certain survey respondents. Factor loading estimates from the original and competing CFA models are presented in Appendix Tables A1 and A2. Based on the model fit results and our reconsideration of the items’ wording, we determined that the competing model, which represented district decisionmakers’ consideration of equity versus equality in decisionmaking, was the better measurement model for survey item responses in our population of district decisionmakers. The binaryitem omega reliability coefficients for the final equity and equality sum scores in this population, accounting for features of the sampling design (Dimitrov, 2003; Raykov & Traynor, 2016), were .76 and .71, respectively, which would generally be considered acceptable levels of score precision for lowstakes use. Addressing Research Question 2, we used structural equation models to regress the “equity view” factor on a set of observed variables representing districts’ student diversity, and other district characteristics likely to be associated with both diversity and mathematics decisionmakers’ consideration of equity. To create an indicator for the proportion of racial or ethnic minority (nonwhite) students, we summed the proportions of American Indian/Alaskan Native, Asian/Pacific Islander, Black, Hispanic, and multiracial students reported by each district, drawn from the Common Core of Data (National Center for Education Statistics, 2011). We similarly constructed variables containing the proportions of limited English proficient students, and students eligible for free or reducedprice lunch in each district, as well as indicators representing different district sizes and locales. We hypothesized that district mathematics decisionmakers’ consideration of students’ culture or communities might increase with the racial and ethnic diversity of each district’s student composition. That is, we hypothesized that equity consideration would increase with the proportion of minority students in the district while the proportion was low or moderate, and then decrease as the proportion neared 1. Therefore, we represented the relationship between equity and the proportion of minority students as parabolic by including both the minority student proportion and its square as predictors in the model. Consistent with the relations among predictors in observed variable regression models, all predictors were allowed to be correlated with one another. FINDINGS To address the first research question regarding the extent to which district decisionmakers report considering issues of equity when making decisions about resource use for algebra learning, we examined descriptive statistics for the survey items that listed criteria respondents might consider when making decisions for their districts. The weighted proportion of respondents considering each specific criterion is reported in Table 1. We focus primarily on those criteria that were affirmed by more than 80% of district decisionmakers, in order to identify the most widelyused decision criteria. Table 1. Weighted Sample Descriptive Statistics for Survey Items About Decisions Structuring Algebra Opportunity to Learn and School District Features (n = 793)
When district decisionmakers select assessments, more than 80% reported considering whether the assessments help to identify information students do not know and whether they help to predict future student performance. In making decisions about professional development (PD), district decisionmakers reported that they consider whether the PD focuses on applications or real life contexts for algebra and on how to teach algebra to all students. For the selection of curriculum materials, more than 80% reported that they consider whether the materials will help students do well on achievement tests, are relevant to real life situations, are accessible to students, and will prepare students for university mathematics courses. Notably, two of the three structural items were endorsed by fewer than 60% of respondents, indicating that relatively many of the district decisionmakers do not consider changing structures, such as scheduling, course sequences or grouping within classrooms, to be important (or, possibly, within their purview) in fostering opportunities to learn. The two criteria selected by the fewest decisionmakers, about half, included whether PD focused on teaching that was culturally relevant and whether curriculum materials reflected cultures of their students, criteria which loaded most highly on the “equity view” factor in our exploratory factor analysis. Our second research question was: How do district characteristics, particularly student diversity (i.e., racial demographics, as reported by the district), affect the extent of equity consideration by mathematics decisionmakers? Figure 1 presents SEM results relating school district features to decisionmakers’ latent tendency to consider equityrelated criteria when they are selecting and distributing resources for algebra learning. Correlations between all possible pairs of structural model variables depicted on the left side of the figure, with the exception of the correlation between large district size and the proportion of free or reducedprice lunch eligible students, were all significantly different from zero, but are not printed in the diagram for simplicity. (They are available from the authors, however.) As shown in Figure 1, controlling for other district features, we found that district mathematics decisionmakers’ consideration of equity was significantly related to the squared proportion of minority students in the district. The negative coefficient indicated that the relationship between equity consideration and the proportion of minority students was concave down; that is, the results suggest decisionmakers’ consideration of equity increases with minority student enrollment until the average proportion is nearly reached, and then decreases with increasing minority student enrollment in districts with higher proportions of minority students. Further, equity consideration was associated with rural district locale. Compared to mathematics decisionmakers in urban or suburban districts, those in rural districts gave less consideration to equity when making decisions—controlling for other variables, their equity scores were .16 of a standard deviation lower, on average. Equity consideration scores also tend to be higher in large than in small districts, and to increase with the proportion of limited English proficient students enrolled in a district, although we note that these predictors are only marginally statistically significant. Figure 1. Structural Equation Model Predicting District Mathematics Decisionmakers’ Latent Equity Consideration in Decisions About Algebra Resources (n = 793)
Although we observed that, generally, larger proportions of district mathematics decisionmakers reported using equalityrelated rather than equityrelated criteria to make decisions about Algebra I structure and resources, there was little reason to believe, a priori, that the extent to which they rely on equality considerations to guide their decisionmaking would be influenced by any of the district characteristics represented in our model. As a robustness check, we estimated a SEM model with the same observed district features as predictors for the latent “equality” factor, which was measured by its 11 indicator criteria. Unlike in the model for latent equity consideration in decisionmaking, there was no linear or nonlinear relation between decisionmakers’ equality scores and the predictors that described districts’ racial or ethnic diversity. Only large district size was a marginally statistically significant predictor (p = 0.079) of equality consideration. (Results from this model are available from the authors.) DISCUSSION, IMPLICATIONS, AND SOME RECOMMENDATIONS This article highlights an issue of equity in decisionmaking at the district level that has not been closely considered in mathematics education. We recognize that, on any selfreport survey, some district decisionmakers may have responded in ways they thought others would find socially desirable, or that there could have been other intentional or unintentional response biases of which we are unaware. We also recognize that we gathered reported perceptions and practices, which do not imply causal relationship between district characteristics and considerations in decisionmaking about algebra instruction. Honig (2008) has pointed out that little research has focused on the actual practices of central office administrators. The findings here indicate some areas of decisionmaking and associated practices that might be important to target in future research. Our findings also have significant implications for the education and professional development of administrators. LOWER FOCUS ON EQUITY WHILE ENDORSING A FOCUS ON REAL LIFE AND STANDARDIZED TESTING Relatively few decisionmakers tended to consider the equityrelated items in their decisionmaking, while many tended to endorse items related to preparation for standardized testing and real life situations. The equityrelated items are framed in ways that attend to particular demographic groups and those who are underserved in schools, as well as draw attention to considerations of students’ cultures. Yet, these items were ranked as important by only 45–70% of district decisionmakers (with most occurring at about 50%). The highest percentages of items (ranging from 87 to 95%) reported by district decisionmakers focus on a) identifying information students do not understand in order to teach that content better, b) knowing and having available curriculum that relates to real life contexts, and c) doing well on achievement tests. First, we note the inherent contradiction in the fact that most district decisionmakers think “real life” is important to consider but only about half of them report that students’ cultures are considered in their decisions about professional development or curriculum material selection. In mathematics education research, there has been an enduring and thoughtful consideration of applications and real life contexts as sites for learning for students (e.g., Boaler, 1993; Chazan, 2000; Frankenstein, 2009; Moses & Cobb, 2001). District decisionmakers’ recognition of the importance of real life in students’ learning experiences is noteworthy as it potentially shows that they have moved beyond a decontextualized, exclusively symbolic view of algebra (c.f., Chazan, 2000; Usiskin, 1988). In line with district leaders’ focus on real life contexts, national policy documents providing standards and recommendations for mathematics teaching (e.g., National Council for Teachers of Mathematics [NCTM], 2000; NGACBP & CCSSO, 2010) have consistently emphasized the importance of incorporating “realworld” or “real life” contexts into the mathematics curriculum. In contrast to this finding, however, is the fact that two items selected by the lowest proportions of decisionmakers relate to the importance of providing resources in the form of professional development and curriculum materials that might help teachers connect to students’ cultures. This suggests a disconnect between district decisionmakers’ consideration of real life contexts and the contexts that are real in the lives of their students. Not surprisingly, there is a similar disconnect in national policy documents, which also neglect discussing the local and cultural nature of real life contexts, or how the cultural nature of the mathematics curriculum can influence students’ opportunities to learn. If district decisionmakers’ conception of real life involves only problems about amusement parks or shopping, for example, then it is possible that their notions of real life need to be expanded to include aspects of students’ culture. In literacy, for instance, Au (2001) has argued that culturally responsive instruction should be one strategy used to improve the learning of students from diverse backgrounds. This includes “teaching approaches [that] build upon the strengths that students bring from their home cultures, instead of ignoring these strengths or requiring that students learn through approaches that conflict with their cultural values” (p. 4). This particular kind of work is more recent in mathematics than in literacy, but the existing work has shown promise for supporting students who have been historically marginalized in schools (see, for example, Aguirre & Zavala, 2013; Averill et al., 2009; Barton, Fairhall, & Trinick, 1998; Borden, 2011; Civil, 2002; 2007; Gutstein, 2003, 2006; Gutstein, Lipman, Hernandez, & de los Reyes, 1997; Hand & Taylor, 2008; LadsonBillings, 1995b; Leonard, Brooks, BarnesJohnson, & Berry, 2010; Meaney, Trinick, & Fairhall, 2013). One possible explanation for the perceived disconnect between real life and students’ culture could be that district decisionmakers buy into the false assumption that mathematics is culturefree. Mathematics education researchers have argued that culture is important (e.g., Bishop, 1988, Nasir, 2007), especially if what constitutes mathematics is considered (e.g., Joseph, 1987) as well as quality pedagogy. Although the disconnect is probably greater in mathematics education than in some other content domains because people tend to believe mathematics is culturefree, it is important to recognize that these issues also tend to be further amplified in relationship to racial and ethnic minority students and students from low socioeconomic backgrounds. Another possible explanation could be that many of the administrators who make mathrelated decisions do not have a mathematics background and, as Lochmiller and colleagues (Lochmiller, 2016; Lochmiller & AckerHocevar, 2016) have shown, focus more on pedagogical aspects of teaching mathematics rather than contentrelated aspects grounded in philosophical or epistemological concerns. This finding points toward a need for more research on what district decisionmakers see as “real life” and/or how those views might relate to their content backgrounds and the policies they are exposed to and are trying to enact. In order to help district leaders see the connection between real life contexts and the culture of their students, policy recommendations could be more explicit about what it means to select contexts that are both real life and locally relevant. Additionally, programs that prepare teachers and administrators could include critical examination of such ideas. The arguments for the need to emphasize social justice in teacher education (e.g., Nieto, 2000) and educational leadership (e.g., Brown, 2004) programs relate to this finding. Additionally, there are tools now available that can make culturally relevant mathematics teaching an explicit focus (see, for example, Aguirre & Zavala, 2013) and could be useful for teachers and administrators. Mathematics education researchers and teacher educators have also suggested many ways to integrate the professional development of mathematics teachers and important issues of equity (e.g., Battey & Franke, 2015; Foote, 2010). The remarkably strong attention to assessment and achievement tests was not necessarily surprising. This finding may reflect the federal Elementary and Secondary Education Act requirement for schoollevel reporting of average achievement test scores by demographic group (e.g., Linn, Baker, & Betebenner, 2002). Some researchers have argued that a focus on achievement without a consideration of different framings of the problem, such as thinking about opportunity gaps, participation gaps, or an education debt, could lead to deficit framing of students (Gutiérrez, 2008; LadsonBillings, 2006). Such framing is, unfortunately, common in many schools and has been described as an “equity trap” or patterns of thinking and behavior that trap the possibilities for creating equitable schools for children of color. In other words, they trap equity; they stop or hinder our ability to move toward equity in schooling. (McKenzie & Scheurich, 2004, p. 603) McKenzie and Scheurich propose that improving equity in relationship to deficit thinking requires getting to know students, their families, and their communities: school staffs need to get to know their students and their students’ families and community on a personal level, they need to learn to dignify the culture of their students, and they need to actively solicit and incorporate the community into the decision making of the school (pp. 609–610). Yet, in order to do so, they would need to know more about students’ cultures, something that about half of decisionmakers reported as not affecting their selection of curriculum materials or professional development. Strategies like doing home visits or neighborhood walks, gathering oral histories from community members, and facilitating conferences that involve students, parents, and teachers/administrators in goal setting and reflection help administration and staff to learn information about students and their families that run counter to this equity trap (McKenzie & Scheurich, 2004). These recommendations, along with a call to prepare educational leaders who can learn to experience emotional safety and risks, echo those made by others (e.g., Capper, Theoharis, Sebastian, 2006). LITTLE ATTENTION TO DETRACKING The relatively lower attention to structural aspects in district decisionmaking sheds light on the pervasive hegemony of tracking and paucity of attention toward making changes to better support diverse representation in collegebound mathematics courses. A wide array of research has studied the harmful effects of tracking (i.e., grouping of students based on perceptions of ability) on students’ mathematical achievement (see Boaler, 2011; Cogan, Schmidt, & Wiley, 2001; Oakes, 2005 for a summary). Given the overwhelming evidence, dating back to the 1980s, about the negative effects of tracking on students (e.g., Chunn, 1988; Gamoran, 1992; Harklau, 1994; Lucas, 2001; Welner & Oakes, 1996), it is surprising that only a little more than half of the district decisionmakers report that they consider enrolling students of multiple ability levels together in a single class. Yet, as some researchers have shown, the politics of a district can influence how evidence is used (e.g., Daly, Finnigan, Jordan, Moolenaar, & Che, 2014). A reconsideration of ability grouping requires educators (including district decisionmakers) to move beyond a personal perspective, yet research has shown that white people (which most teachers and administrators are) tend to approach issues of inequality from a personal perspective rather than approaching them as societal, systemic, and institutional manifestations (McIntosh, 1989; McIntyre, 1997). As the recent position statement released by the National Council of Supervisors of Mathematics and TODOS: Mathematics for All (NCSM & TODOS, 2016) has pointed out, the detrimental effects of tracking start early in elementary school with readiness labels and ability grouping structures that provide vastly different mathematical experiences. In practice, children placed in “low” groups experience mathematics as an isolating act consisting of factdriven low cognitive demand tasks and an absence of mathematics discourse opportunities. This is because of a pervasive misguided belief that students must “master the basics” (e.g., know the times tables or “basic facts”) prior to engaging with complex problems solving. (p. 2) Without support for district decisionmakers to make such shifts in thinking, however, students will continue to fall victim to structures like tracking, which have enduring negative effects on educational, work, and income trajectories (NCSM & TODOS, 2016). This finding suggests further research and action to be done by researchers and teacher educators in order to provide more compelling arguments for attention to this structural aspect of students’ mathematics experience. Current research and evidence about what happens when schools detrack (e.g., Boaler & Staples, 2008; Burris, Heubert, & Levin, 2004, 2006; Burris & Welner, 2005) is extremely compelling in terms of, for example, increased enrollment in mathematics courses, students’ development of positive conceptions and attitudes toward mathematics, the performance of all students, the narrowing of the achievement gap, and the increased interactions and development of friends between students across race and ethnicity (e.g., Boaler, 2006). Such findings must be communicated in professional and public spheres so that better informed decisions can be made in practice. CONSIDERATION OF EQUITY IN DISTRICT DECISIONMAKING ABOUT ALGEBRA AND STUDENT POPULATION HETEROGENEITY Our results suggested that consideration of equity by mathematics decisionmakers is highest in districts with the most diverse student compositions—districts with moderate proportions of ethnic or racial minority students (i.e., not nearly allwhite or nearly allminority), in urban or suburban rather than rural areas, or with relatively high proportions of limited English proficient students. When searching for literature to try to explain why the scores on the equity factor tended to be highest in districts that were the most racially and ethnically diverse, we considered how district administrators’ awareness of social identityrelated differences (e.g., differences in race) might change with observed differences in their environments. For example, social psychology research has suggested that people notice visually the racial and gender demographics of spaces and that this impacts how they react to/in the space (e.g., Murphy, Steele, & Gross, 2007; PurdieVaughns, Steele, Davies, Ditlmann, & Crosby, 2008). As the racial diversity of a district increases, district leaders might be more likely to attend to this diversity in equitable ways. Additionally, because women and minorities make up a disproportionally small percentage of administrators (Marshall, 2004; Riehl, 2000), and some research has shown that “colorblindness” is a common belief of teachers (and is not addressed well in teacher education or administrator education) (e.g., CochranSmith, 1995; 2000; Marshall, 2004; Milner & Self, 2012; Watson, 2012), we imagined a possibility that district leaders working in districts with little or no racial diversity may be less likely to notice and respond to issues of equity, possibly due to colorblind ideologies (e.g., McKenzie & Scheurich, 2004). Many scholars have argued that paying careful attention to race and ethnicity “allows one to perceive and acknowledge the genuine and important influence of racial and ethnic stigma in students’ educational lives” (Zirkel, 2005; as cited in Zirkel, 2008, p. 1168). Previous research that has focused on principals or other administrators has suggested that those who approach their work with a social justice orientation tend to burn out due to resistance and the effort it takes to go against the status quo and the system (e.g., Armstrong, 2010). A case study by Trujillo (2013), however, suggested that the equityoriented but controversial policy decisions made by central administration in one urban district were repeatedly nullified by principals and teachers. Thus, educators across the district could be culpable in resisting equitable practices. The findings reported here provide broader evidence that more work needs to be done in central administration to identify and emphasize issues of equity, so central administrators can learn practices that allow them to support principals with this stance rather than undermine them, force them into compliance with inequitable practices, or allow them to burn out. Daly and Finnigan (2012) suggest, however, that work should be done to create environments in which principals feel supported by central office administrators, even when highstakes equity work is not the focus. Carefully examining the stated and unarticulated rules, rewards, and sanctions that get put into place would be an important practice for central administration to uncover how inequitable practices are perpetuated. Ways to make issues of colorblindness a focus of discussion and reflection include, for example, book clubs and study groups focused on relevant texts about race and racism in educational experiences (see McKenzie & Scheurich, 2004 for some examples, as well as http://www.bustle.com/articles/15339010booksiwishmywhiteteachershadread). Although not focused specifically on issues of equity, Honig and Rainey (2014) have shown that professional learning communities can be important to central office staff’s effective mentoring of principals. They also suggest, however, that more research is needed to understand how central office administrators might productively facilitate this process. Another strategy districts could employ is to have regular equity audits (Skrla, Scheurich, Garcia, & Nolly, 2001), which “are a simple way to start a discussion of inequities within a school or district. Equity audits are simply ‘auditing’ the school’s or district’s data for inequities by race” (McKenzie & Scheurich, 2004, p. 617). Examining data like AP course enrollment or other coursetaking patterns can help bring attention to inequities in the district. Additionally, surveys and other information could be gathered from students and parents to better understand their perspectives on school climate and the support students receive for their mathematics learning. As Welton (2013) has argued, leaders must be willing to engage in discussions but also then take action that leads toward enacting systematic change. CONCLUSION Spillane (1996) has pointed out that district administrators have the power to “mobilize and shape a number of key instructional guidance instruments—staff development, curriculum guidelines, curricular materials, teacher supervision, and student assessment—to support particular notions of teaching and learning” (p. 84) but that creating change in districts goes beyond only attending to administrators’ beliefs and knowledge. He also contends that the local context plays a role in how administrators take up various policies. If district decisionmakers continue to see real life as separate from students’ culture, allow statusquo tracking practices to continue unquestioned, concentrate more often on equality, and attend to aspects of equity only in schools that are racially and ethnically diverse, then their decisions about mobilizing and shaping resources teachers can draw on, as well as those made available to students, will continue to be problematic. Because most research in mathematics education has focused on teachers and teacher education, further study of administrators who make decisions about mathematics learning resources is needed. Although focused on how administrators enabled new small autonomous schools to be implemented, Honig (2009) suggested that attending to central office administrators’ bridging and buffering activities might be a fruitful way to conceptualize how institutions manage nontraditional demands. That is, further related research might consider how administrators both bridge—“bring organizations in greater contact with their environments to garnering information and other resources they might use to deepen or change organizational activities and ultimately achieve organizational goals” (bridging activities)—and buffer—make “attempt(s) to reduce the extent to which [an organization] is externally inspected, scrutinized, or evaluated by partially detaching or decoupling its technical activities from external contact (Oliver, 1991, p. 155; Pfeffer & Salancik, 1978)” (Honig, 2009, p. 392). Conceptualizing administrators’ work in this way would run counter to the prevalent view of administrators as managers and bureaucrats (Marshall & Ward, 2004). Additionally, Theoharis (2010) has shown that administrators must be able to disrupt four kinds of injustice in order to advance social justice: school structures that marginalize, segregate, and impede achievement, a deprofessionalized teaching state, a school climate that is not welcoming to disenfranchised families or connected to the community, and disparate and low student achievement. Research on such practices might find additional bridging and buffering activities beyond those reported in Honig (2009). We see it as imperative for future work in mathematics education to better understand what these stakeholders do and why—to understand the ways in which administrators acknowledge that the mathematics education system is “unjust and grounded in a legacy of institutional discrimination based on race, ethnicity, class, and gender” (NCSM & TODOS, 2016, p. 4), take action “to create and sustain institutional structures, policies, and practices that lead to just and equitable learning opportunities, experiences and outcomes for children” (p. 4), and put in place systems of accountability (p. 5) for the actions they take—especially because of the privileged nature of mathematics achievement in influencing access to higher education and other life outcomes. Acknowledgments The work reported in this article was supported by the National Science Foundation (Grant Nos. 1108828 and 1108833). Any opinions, findings, conclusions, or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation. We would like to acknowledge the contributions of additional members of the LANDSCAPE research team (Janine Remillard, Michael D. Steele, John Baker, Nina Hoe, Paul McCormick, and Josh Taton) and our advisory board members (Edward Silver, Diane Briars, Jim Lewis, Jon Star, Mary Kay Stein, and Bob Floden). Finally, we thank Terry Flennaugh, Niral Shah, Tonya Bartell, and Amy Parks, the three anonymous reviewers and the editorial team of Teachers College Record for their helpful feedback. Notes 1. See also, for example, many of the references in the Discussion, Implications, and Some Recommendations section of this article. References Achieve, Inc. (2004). The expectations gap: A 50state review of high school graduation requirements. Washington, DC: Author. 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APPENDIX Table A1. Confirmatory Factor Analysis of Items About Decisions Structuring Algebra Opportunity to Learn: Hypothesized Model (n = 200)
Table A2. Confirmatory Factor Analysis of Items About Decisions Structuring Algebra Opportunity to Learn: Alternative Model (n = 200)


