Child Development, September/October 2014, Volume 85, Number 5, Pages 1948–1964

The Link Between Middle School Mathematics Course Placement and Achievement Thurston Domina University of California, Irvine

The proportion of eighth graders in United States public schools enrolled in algebra or a more advanced mathematics course doubled between 1990 and 2011. This article uses Early Childhood Longitudinal Study’s Kindergarten Cohort data to consider the selection process into advanced middle school mathematics courses and estimate the effects of advanced courses on students’ mathematics achievement (n = 6,425; mean age at eighth grade = 13.7). Eighth-grade algebra and geometry course placements are academically selective, but considerable between-school variation exists in students’ odds of taking these advanced courses. While analyses indicate that advanced middle school mathematics courses boost student achievement, these effects are most pronounced in content areas closely related to class content and may be contingent on student academic readiness.

Driven by concerns about mediocre performance on international mathematics assessments, the United States is in the midst of a decades-long push to intensify middle and high school mathematics curricula. While educational commentators raise concerns about the performance of American schools on a wide array of indicators, American students particularly trail their peers in other highly developed countries in mathematics achievement, and this achievement gap widens as students progress through their formal schooling (Kelly et al., 2013). American students study algebraic concepts 1–2 years later than their peers in other economically developed nations, and low-achieving U.S. students lag even further behind in exposure to advanced mathematics instruction (Gonzalez et al., 2008; Organisation for Economic Co-operation and Development, 2009). Based on the assumption that students are more likely to learn that which they are taught, educators and policy makers across the United States have begun to accelerate mathematics instruction for a wide range of students. This movement has particularly focused on boosting the proportion of middle school students enrolled in The author thanks Peg Burchinal, Greg Duncan, Andrew McEachin, Andrew Penner, Morgan Polikoff, and Deborah Vandell for helpful advice on this manuscript. In addition, various colleagues at the 2013 annual meeting of American Educational Research Association and a visiting lecture at the University of Southern California’s Rossier School of Education provided valuable feedback on this manuscript. Correspondence concerning this article should be addressed to Thurston Domina, School of Education, University of California, Irvine, Irvine, CA 92697-5500. Electronic mail may be sent to [email protected].

algebra courses (Stein, Kaufman, Sherman, & Hillen, 2011). Between 1990 and 2011, the proportion of eighth graders in U.S. public schools enrolled in algebra or a more advanced mathematics course more than doubled. Data from the National Assessment of Educational Progress (NAEP) indicate that 35% of public school students take algebra in the eighth grade and an additional 9% take geometry or a more advanced mathematics course (Loveless, 2013). Efforts to enroll more U.S. students in early algebra assume that accelerated mathematics course placements positively influence students’ achievement. However, the developmental theory of person–environment fit raises questions about that assumption, raising the prospect that placing students into contexts for which they are not suited may have negative developmental consequences (cf. Eccles et al., 1993). This article uses panel data from the nationally representative Early Childhood Longitudinal Study’s Kindergarten Cohort (ECLS–K) to: (a) investigate the processes through which students are placed in middle school mathematics courses and (b) estimate the consequences of these mathematics course placement decisions for student academic achievement. In doing so, it provides a unique test of person–environment fit theory to a diverse, nationally representative sample and suggests new insights into the relevance of © 2014 The Author Child Development © 2014 Society for Research in Child Development, Inc. All rights reserved. 0009-3920/2014/8505-0016 DOI: 10.1111/cdev.12255

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developmental theory for the design and sequencing of middle school mathematics curricula. Opportunity to Learn and Person–Environment Fit Middle school mathematics is an important nexus of educational inequality. Over the last several decades, U.S. high schools have dramatically intensified their curricula in an effort to meet rising student demands and prepare more students for higher education (Domina, Conley, & Farkas, 2011; Domina & Saldana, 2012). In the same period, gender inequalities in high school mathematics course completion closed, whereas ethnic and class-based inequalities narrowed (Domina & Saldana, 2012). While this curricular intensification trend has far outpaced sluggish growth in student academic achievement (Domina & Saldana, 2012; Loveless, 2013), access to rigorous middle school mathematics courses remains highly stratified along ethnic and class lines (Walston & McCarroll, 2010). Nearly half of White eighth graders and 67% of Asian eighth graders are enrolled in algebra or a more advanced course. By contrast, algebra enrollment rates for Black and Hispanic students lag considerably (16% and 38%, respectively). Furthermore, students from nonpoor families are twice as likely as those from poor families to take algebra or a more advanced mathematics course in eighth grade (Walston & McCarroll, 2010). To address these inequalities, policy makers (e.g., Duncan, 2009), scholars (e.g., National Mathematics Advisory Panel, 2008), educators (e.g., Moses & Cobb, 2001), and business leaders (e.g., Business-Higher Education Forum, 2005) argue for universal access to accelerated middle school mathematics curricula. Policies implemented in California, North Carolina, Washington, DC, and elsewhere aim to enroll all eighth graders in algebra (Loveless, 2013; Stein et al., 2011). These policies clearly influence middle school mathematics course placement practices, dramatically increasing the odds that relatively low-achieving students take advanced middle school mathematics courses (Clotfelter, Ladd, & Vigdor, 2012; Domina, McEachin, Penner, & Penner, 2014; Schiller & Muller, 2003; Stein et al., 2011). While the implementation of the Common Core State Standards in mathematics— which recommend prealgebra in the eighth grade and algebra in the ninth grade—may slow this trend, the acceleration of middle school mathematics curricula remains a major force in the U.S. education.

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Accelerated algebra efforts aim to expand opportunities to learn in U.S. schools. Defined as “the opportunities which schools provide students to learn what is expected of them” (Herman, Klein, & Abedi, 2000, p. 16), the phrase “opportunity to learn” is predicated on the intuitive notion that students are more likely to learn that which they are taught (Porter, 2002). This proposition enjoys considerable empirical support. Research focusing on national curricula, state standards, student course trajectories, and teacher practices indicates that students who are exposed to rigorous mathematics curriculum and instruction experience greater achievement gains on average than those who are not (e.g., Attewell & Domina, 2008; Gamoran & Hannigan, 2000; Long, Conger, & Iatarola, 2012; Schmidt et al., 2001). However, a broad range of theory and research in developmental psychology suggests that not all students are equally likely to benefit from broader opportunities to learn. Csikszentmihalyi and Larson (1986, p. 266) provide a particularly elegant statement of this developmental perspective when they invoke the “balanced tension between challenge and skills” that is central to learning and development. More formally articulated as a theory of person– environment fit, this perspective suggests that placing students in academic environments for which they are academically and developmentally unprepared may have unintended negative consequences for student self-efficacy, motivation, and achievement (Duncan & Vandell, 2012; Eccles et al., 1993). Issues of person–environment fit are particularly relevant to middle school mathematics. Several studies indicate students’ academic motivation and interest in school decline during early adolescence, particularly in mathematics and particularly during transition to middle or junior high school (Anderman & Maehr, 1994; Eccles et al., 1993; Gottfried, Fleming, & Gottfried, 2001; Rumberger, 1995; Wigfield, 1994; Wigfield & Eccles, 1992). Person– environment fit theory attributes motivational declines in the middle school years to a mismatch between the adolescents’ developmental needs and middle school educational environments. This mismatch may be particularly pronounced in accelerated middle school mathematics courses, where instruction may be more teacher driven and less social than in lower level and elementary school classrooms (Eccles et al., 1993; Larson, 2000) and where teachers may assume mathematical proficiencies that students lack (Allensworth, Nomi, Montgomery, & Lee 2009; Nomi, 2012; Stein et al., 2011).

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Empirical Evidence Regarding Accelerated Mathematics Students who enroll in early algebra clearly enjoy higher mathematics test scores than their peers who enroll in less advanced mathematics courses (Gamoran & Hannigan, 2000). At first glance, this skills gap seems to suggest that accelerated algebra boosts student achievement. However, most U.S. middle schools selectively place students into mathematics courses, placing students with relatively high levels of mathematics skills, interests, and motivation into accelerated algebra and less high-achieving students into grade-level courses. This selection makes it difficult to make strong statements about the causal effects of eighth-grade algebra based on observational data, even after controlling a wide range of student characteristics (including demographics and prior achievement). Experimental and quasi-experimental evaluations of accelerated algebra efforts return mixed results. In one small-scale experimental study, researchers offered online algebra courses to high-achieving eighth graders in 68 randomly selected rural middle schools that did not already offer eighth-grade algebra. In this case, access to online eighth-grade algebra significantly boosted mathematics performance (Heppen et al., 2012; effect size = 0.39). However, these results may not generalize to settings in which a broader range of students have access to accelerated algebra. Quasi-experimental evidence from North Carolina suggests that eighthgrade algebra has negative achievement effects for the average student. Clotfelter et al. (2012) take advantage of policy-driven efforts to enroll nearly all eighth graders in Charlotte–Mecklenberg and a handful of other North Carolina school districts in algebra to generate instrumental variable estimates of the effects of early algebra. Their analyses indicate the course has particularly pronounced negative consequences for the mathematics achievement of eighth graders who enter the course with relatively low levels of prior achievement (Clotfelter et al., 2012). Stein et al. (2011) present a broader and more comprehensive review of the available evidence regarding accelerated algebra. Consistent with person–environment fit theory, Stein et al. challenge the assumption that accelerating algebra for a wide array of students necessarily leads to improved mathematics achievement. They find that some efforts to accelerate algebra return positive results, particularly when these efforts allocate extra time and focused instruction to allow struggling students to master challenging algebra content. However,

Stein et al. (2011, p. 486) caution that “relying on universal access alone to produce increased achievement and to guarantee advanced mathematics course taking and enhanced careers is incomplete.” They hypothesize that underprepared students may particularly suffer in accelerated middle school mathematics in the absence of these supports. Person–Environment Fit in Middle School Mathematics This study uses nationally representative data from the ECLS–K to investigate the relation between accelerated algebra and student mathematics achievement. It addresses the following two central issues regarding the relation between middle school mathematics placement and student achievement growth. First, by modeling the factors that predict placement into middle school mathematics courses, the study investigates the extent to which placement into advanced mathematic courses are demographically biased in contemporary U.S. middle schools. In light of the recent expansion in eighth-grade algebra placement, the study investigates the extent to which middle school mathematics course placement rates vary with students’ ethnicity, gender, family background, as well as test scores and teacher perceptions (measured both in kindergarten and in fifth grade). Second, the study provides estimates of the relation between eighth-grade algebra placement and student achievement. Taking advantage of the rich panel data available in the ECLS–K, the analyses that follow trace mathematics test score growth for U.S. public school students across the elementary and middle school years, noting the ways in which achievement growth patterns vary with students’ eighth-grade mathematics courses. These analyses bridge conceptual gaps between correlational studies in the “opportunity to learn” tradition and evaluations of accelerated algebra programs. Like Gamoran and Hannigan (2000) and other correlational studies in the “opportunity to learn” tradition, the analyses reported here are based on broadly representative data that measure advanced courses as they are routinely administered in U.S. schools. However, this study’s developmental approach addresses the problems associated with nonrandom selection into advanced courses that have plagued earlier studies using nationally representative data. By contextualizing middle school course enrollments in students’ elementary to middle school mathematics achievement trajectories, the study is able to identify the consequences of

Middle School Mathematics and Achievement

advanced courses, controlling for all characteristics of students that do not vary over time. Doing so makes it possible to develop a richer understanding of the factors that influence exposure to accelerated middle school mathematics courses and to estimate the consequences of these course placements for student mathematics achievement. These analyses provide a new empirical test of person–environment fit theory using longitudinal data from a large, diverse, and broadly representative group of eighth graders. In doing so, the study aims to provide new insights into the ways in which schools can contribute to—or undermine— student development in the domain of mathematics achievement.

Method Participants The ECLS–K is sponsored by the U.S. Department of Education’s National Center for Education Statistics. The study drew a nationally representative stratified sample of 21,260 kindergarteners from 1,277 schools in 100 U.S. counties in the fall of 1998 (Tourangeau, Nord, Le, Sorongon, & Najarian, 2009). In addition to collecting comprehensive baseline data from students and their teachers, school administrators, and parents, the study longitudinally tracked students, collecting data in spring 1999, spring 2000 (when most respondents were in the first grade), spring 2002 (when most were in the third grade), spring 2004 (when most were in the fifth grade), and spring 2007 (when most were in the eighth grade). This article’s analytic sample includes all ECLS–K respondents who participated in multiple ECLS–K survey rounds and provide both test score and course enrollment data in the final wave of data collection. Approximately half of the initial sample is missing from these analyses due to attrition. All analyses utilize the study’s full panel weight to address this attrition. The analyses also exclude students who enrolled in private schools in eighth grade (approximately 10% of the initial sample), students who were not in eighth grade in the final survey round (approximately 5% of the initial sample), and students who enrolled in integrated mathematics or algebra II courses in eighth grade (< 8% of the initial sample). The final sample (N = 6,425) is thus representative of students who enrolled in U.S. kindergarten in 1998 and were enrolled as eighth graders in traditional middle school mathematics course sequences at U.S. public

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schools in the 2006–2007 school year. The sample is split evenly between males and females; the mean age for sample participants is 5.7 years and 13.7 years at the end of the panel. Key Measures These data provide unique opportunities to investigate the predictors of middle school mathematics course placement and its effects on achievement trajectories. Course Enrollment The analyses that follow use eighth-grade mathematics teacher reports to characterize students’ middle school mathematics course enrollment. The ECLS–K provides no information on students’ mathematics course enrollment prior to this final survey round. Mathematics Achievement The basic mathematics skills test that the ECLS– K administers to each student at each round of data collection is at the center of this article’s analyses. This assessment is administered adaptively in order to tailor questions to students’ demonstrated skill level and capture the entire range of student mathematics achievement. Based on the Mathematics Framework for the 1996 and 2005 NAEP, which were in turn derived from the National Council of Teachers of Mathematics’ curricular standards, the ECLS–K mathematics test captures mathematics achievement in five areas: number sense (including symbolic representation of numbers; the ability to deal with proportion, probability, and rates; and irrational numbers such as square roots and pi), measurement, geometry and spatial sense, statistics and probability, and algebraic concepts such as function and variable. This article utilizes mathematics test scores that are vertically integrated using item response theory (IRT) to make it possible to track student achievement growth over time. To create this score, ECLS– K statisticians drew upon the set of 49 test questions that are available across multiple test administrations to generate IRT weights for test questions. These weights aim to reflect the empirical difficulty of test questions to differentiate student achievement across the distribution. IRT “a” parameters indicate that these measures form a highly cohesive scale of mathematics achievement growth across the ECLS–K panel (Najarian, Pollack, Sorongon, &

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Hausken, 2009). This IRT-scaled test score makes it possible to track student progress from kindergarten through eighth grade on an integrated mathematics achievement metric. The distribution of ECLS–K mathematics IRT scores is highly truncated for relatively high-achieving eighth graders enrolled in algebra and geometry courses, but not for students enrolled in lower level mathematics courses. (see online Appendix S1 for kernel density plots). There is no evidence of right censoring on the ECLS–K mathematics IRT score (the highest score IRT score observed in the data is 172.2, while the test has a maximum score of 174). However, these plots indicate that students who are placed into advanced middle school mathematics courses have less room to grow on this measure of mathematics skills than their peers who are placed into less advanced mathematics courses. As a result, these mathematics test scores may return negatively biased estimates of the effects of advanced mathematics course enrollment on student achievement. It is therefore possible that the IRT test score fails to capture positive effects of accelerated middle school mathematics course enrollment. Several sensitivity analyses described below address this concern. Controls In addition, all analyses take advantage of the rich array of control variables available for ECLS–K respondents. These controls include time-invariant student demographics measures such as gender, ethnicity, and age at kindergarten entry, as well as time-varying measures of parental education, family income, and free-reduced lunch program participation. The analyses also utilize teacher assessments of students’ approaches to learning, measured in kindergarten and fifth grade. This highly reliable measure (alpha = .91) is composed of six items rating students’ attentiveness, task persistence, eagerness to learn, learning independence, flexibility, and organization (Tourangeau et al., 2009).

Analytic Approach Modeling Eighth-Grade Mathematics Course Placement The analyses begin with a consideration of the factors that influence eighth-grade mathematics course placement. Using the student as the unit of analyses, these models compare the characteristics of students placed into accelerated eighth-grade

prealgebra, algebra, and geometry courses with students placed into eighth-grade general mathematics courses. The first of these multinomial logistic regression models considers the association between student demographics and eighth-grade mathematics placement. The second model considers the relation between students’ fifth-grade test scores and eighth-grade mathematics placement, controlling for demographics. Comparing across the first and second models makes it possible to assess the extent to which student skills and teacher perceptions mediate demographic inequalities in opportunities to learn in eighth-grade mathematics. The third model investigates eighth-grade course enrollments controlling for a full set of kindergarten and fifth-grade test scores, as well as kindergarten and fifth-grade measures of teacher perceptions of students’ approaches to learning and demographic controls. This third model provides suggestive evidence regarding the extent to which opportunities to learn in early elementary school influence students’ later opportunities to learn. Placement and Mathematics Achievement The second set of analyses investigates the relation between eighth-grade mathematics course placement and mathematics achievement growth. Several of these analyses treat the ECLS–K data as a panel, in which the student observation is the unit of analysis. Students with no missing data are observed five times in these analyses—at kindergarten, first grade, third grade, fifth grade, and eighth grade. The first of these models takes the following general form: X Yit ¼ poi þ p1 ðYrÞit X þ p2 ðG8 CourseÞit þ p3 ACHitðn1Þ

ð1Þ

þp4 Controlit þ eit : In this model, Yit is the IRT P mathematics test score for student i at time t. p1 ðYrÞit is a matrix of dummy variables capturing variation in student achievement across waves (with the fifth-grade wave withheld as the reference category). P p2 ðG8 CourseÞit is a matrix of dummy variables measuring the extent to which achievement growth between fifth and eighth grade varies with student eighth-grade mathematics course placement. ACHit(n1) is a lagged measure of student achievement, measured using students IRT-scaled mathematics and English language arts (ELA) test

Middle School Mathematics and Achievement

scores as well as teacher “approaches to learning” ratings from the prior wave. The model also includes a student-level random effects term to account for the model’s repeated observation design. This model captures the association between middle school mathematics course taking and fifth- to eighth-grade student mathematics achievement growth for students with similar measured skill levels and teacher ratings in the fifth grade. However, if student exposure to opportunities to learn in early elementary school and other unmeasured differences between students who enroll in different middle school mathematics courses correlate with differences in student achievement, this model may return biased estimates of the effects of middle school mathematics coursetaking. The second model attempts to address these potential confounds in order to generate a more internally valid estimate of the effect of middle school mathematics course taking on student achievement. X X Yit ¼ poi þ p1i ðYrÞ þ p2i ðG8 CourseÞ X þ p3i ðMath TrackÞ þ p4i ACHitðn1Þ

ð2Þ

þ p5i Control þ ei : P This model adds p3i ðMath TrackÞ, a set of timeinvariant controls for student mathematics course track, to Model 1. This model makes it possible to compare divergences in the rate of achievement growth for students in different middle school mathematics courses. The vectorPof eighth-grade mathematics course dummies, p2i ðG8 CourseÞ, indicates the extent to which students on the prealgebra, algebra, or geometry tracks experience unique perturbances in their fifth- and eighth-grade mathematics achievement growth, compared to students with similar demographic and skills backgrounds who are enrolled in eighth-grade general mathematics. In this model, divergences in test score trajectories for students placed in these four middle school mathematics groups that occur during their middle school years provide evidence of causal effects of middle school mathematics course placement. Meanwhile, divergences in achievement growth that occur during respondents’ elementary school years indicate that U.S. schools differentiate mathematics learning opportunities prior to placing students into middle school mathematics tracks (Raudenbush, 2001; Tach & Farkas, 2006).

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A third model adds school fixed effects to the above second model to assess the extent to which variation between school districts and schools confound estimates of the consequences of middle school mathematics course taking. X X Yit ¼ poi þ p1i ðYrÞ þ p2i ðG8 CourseÞ X þ p3i ðMath TrackÞ þ p4i ACHitðn1Þ X þ p5i Control þ p6i ðK schoolÞ þ ei :

ð3Þ

As a result of the ECLS–K’s stratified sampling scheme, students in the data are clustered in schools at kindergarten. This school clustering does not hold throughout the panel, since students move from one school to another over time. Nonetheless, the addition of school fixed effects in Model (3) makes it possible to measure the consequences of middle school mathematics course enrollment among students who start at the same schools. As a robustness check, a fourth model takes a student fixed-effects approach to estimating the effects of middle school mathematics courses on student achievement. This model takes the following basic form: X X Yit ¼ poi þ p1i ðYrÞ þ p2i ðG8 CourseÞ X þ p3i Z þ p4i Control þ ei :

ð4Þ

In place of the time-invariant mathematics course track variable above as well as the time-varying measures of student prior achievement, this P model includes a matrix of student fixed effects, p3i Z, which accounts for all time-invariant student characteristics. This model therefore focuses on withinstudent comparisons, investigating the extent to which eighth-grade mathematics course placements boost or slow student achievement growth rates. It arguably provides the most internally valid available estimate of the achievement effects of accelerated eighth-grade mathematics courses since it controls for all time-invariant student characteristics. If the ECLS–K mathematics IRT score fails to adequately capture relatively advanced mathematics skills and the variation that exists among highachieving middle school mathematics students, these models may understate the effects of middle school mathematics courses on student mathematics achievement. To address this concern, all models estimating the achievement effects of middle school mathematics courses include controls for students’

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lagged test scores as well as lagged test scores squared. In addition, all models include two interaction terms, to account for the possibility that achievement may be particularly slow for highachieving students at the end of these panel data. The first of these variables, labeled Eighth Grade 9 Math IRT, is the product of the eighthgrade indicator variable and students’ IRT mathematics score. The second, labeled Eighth Grade 9 Math IRT2, is the product of the eighthgrade indicator variable and the squared term of students’ mathematics IRT score. Assuming the bias in the eighth-grade ECLS–K mathematics test is entirely a function of student fifth-grade achievement, these controls should yield equally accurate mathematics achievement measures for students in all eighth-grade mathematics courses. An additional set of analyses take advantage of the IRT-based subscale scores that measure the predicted probability that students have mastered the following advanced mathematics concepts: the use of rate and measurement to solve simple word problems, the concept of fractional parts, and solving word problems that involve calculating area and volume, including the use of relevant units of measurement. Possible values on these scores are bounded at 0 and 1 and student scores typically cluster near the bounds. Many students performing at lower levels receive no questions in these advanced areas and thus score a 0 on the subscale. On the other hand, many students who correctly answer one question in these areas answer all of them correctly, and thus score 1 on the subscale. Accordingly, these subscales are not amenable to full panel data analyses. Rather lagged cross-sectional models are estimated to capture the extent to which growth in students’ odds of mastering advanced topics between fifth and eighth grade varies with their middle school mathematics course placement.

Results What Factors Predict Placement in Eighth-Grade Algebra? Table 1 provides a descriptive portrait of eighthgrade mathematics course placements among public school students in the ECLS–K sample. Consistent with other studies demonstrating curricular intensification in U.S. secondary schools, it indicates that 40% of students enroll in algebra or a more advanced course in the eighth grade and an additional 38% enroll in prealgebra.

As Table 1 makes clear, middle school mathematics course placements are stratified by student ethnicity, gender, and family background. Black students are greatly overrepresented in eighth-grade general mathematics and underrepresented in eighth-grade algebra and geometry. The reverse holds for Asian students, who enroll in eighth-grade geometry at more than twice the national norm. (The ethnicity labels utilized in this manuscript are consistent with the ECLS–K survey items from which the ethnicity measures are derived.) Boys are slightly overrepresented in eighth-grade general mathematics but also substantially overrepresented in eighth-grade geometry. In addition, eighth-grade mathematics course placements vary with socioeconomic status. Students in eighth-grade general mathematics classes come from considerably less affluent and less highly educated families than their peers in eighth-grade algebra and geometry. Table 1 also indicates that eighth-grade mathematics course placements are also highly stratified by students’ prior test scores and teacher assessments of students’ approaches to learning in mathematics. The relation between mathematics scores and eighth-grade mathematics course placement is clear, monotonic, and apparent in each wave of the ECLS–K study. In the spring of eighth grade, the mean mathematics achievement gap between students who took eighth-grade geometry and students who took eighth-grade general mathematics is approximately equal to a full standard deviation. General mathematics students score approximately half a standard deviation lower on this posttest measure than prealgebra students. Prealgebra students, meanwhile, score approximately one third of a standard deviation lower than algebra students, and algebra students score approximately 0.15 SD lower than geometry students. However, much of this relation predates middle school mathematics course placement. This is particularly apparent in Figure 1, which provides a graphical representation of ECLS–K student mean test score trajectories. As early as the spring of kindergarten, students who are destined to enroll in eighth-grade algebra score more than two thirds of a standard deviation higher than their peers who are destined to enroll in eighth-grade general mathematics. These gaps widen as students progress through elementary school, and in the spring of fifth grade, students bound for eighth-grade algebra score approximately 0.85 SD higher than students bound for eighth-grade general mathematics. These data clearly indicate that students’ prior

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Table 1 Background Characteristics for Spring K to Spring Eighth-Grade ECLS–K Panel Respondents by Eighth-Grade Mathematics Course Enrollment; Public School Students Only

Race/ethnicity Black (%) Hispanic (%) Asian (%) Other (%) White (%) Gender Male (%) Female (%) Region Midwest (%) South (%) West (%) Northeast (%) Parental education Less than HS (%) HS (%) Some/AA (%) BA (%) MA+ (%) Age, fall K ( x, months) HH income, fall K ( x, $1000) Sp 5th approaches to learning ( x) Sp K reading scores ( x, IRT) Sp 5 reading scores (x, IRT) Sp 8 reading scores (x, IRT) Sp K math scores ( x, IRT) Sp 5 math scores ( x, IRT) Sp 8 math scores ( x, IRT) % of sample Unweighted N

General math

Prealgebra

Algebra

Geometry or higher

Total

31.7 22.7 10.7 23.2 19.4

37.7 35.7 24.7 32.5 39.0

27.5 38.3 54.7 40.0 37.0

3.1 3.3 10.6 4.3 4.6

11.1 17.5 5.1 5.3 60.9

22.4 21.8

38.5 36.6

34.3 38.0

4.7 2.7

50.4 49.6

22.5 27.9 13.0 18.9

39.8 40.2 33.4 33.7

33.8 27.6 49.7 42.6

3.6 4.3 3.9 4.9

27.6 33.8 20.4 18.2

27.6 30.2 23.6 14.5 11.6 68.5 34.9 2.8 0.43 0.63 0.70 0.49 0.66 0.76 22.1 1,293

34.0 38.0 39.9 40.0 29.2 68.6 44.5 2.9 0.13 0.12 0.24 0.12 0.12 0.21 37.6 2,344

36.4 28.5 33.9 39.9 49.7 68.5 57.2 3.2 0.15 0.18 0.11 0.19 0.23 0.13 36.1 2,486

2.0 3.3 2.7 5.6 9.6 68.8 56.3 3.2 0.58 0.36 0.25 0.44 0.44 0.28 4.2 302

7.7 21.1 35.6 21.9 13.6 68.5 47.8 3.0 0.06 0.10 0.19 0.07 0.09 0.19 6,425

Note. Weight = c2_7fc0. K = kindergarten; ECLS-K = Early Childhood Longitudinal Study, Kindergarten Cohort; HS = high school; AA = associate’s degree; BA = bachelor’s degree; MA+ = master’s degree or higher; SP = spring; IRT = item response theory.

mathematics achievement plays an important role in predicting middle school mathematics course placements. Furthermore, the data on student reading scores and teacher assessments of students’ approaches to learning reported in Table 1 suggest that student skills in subjects other than mathematics as well as subjective teacher evaluations contribute to the process of selection into middle school mathematics courses. Indeed, the association between middle school mathematics course enrollments and students’ reading test scores is nearly as pronounced as the association between mathematics course enrollments and mathematics test scores. The multinomial regression analyses presented in Table 2 investigate these predictors of eighth-grade mathematics course placement in a multivariate fashion. The coefficients in this table represent each

predictor’s contributions (on a log odds scale) to students’ likelihood of enrolling in prealgebra, algebra, and geometry rather than the base category of general mathematics. This table’s first model focuses on the demographic predictors of eighthgrade mathematics course placement. This model indicates that student odds of eighth-grade algebra and geometry placement rise with family income and parental education. After controlling for these indicators of students’ socioeconomic status, Asian students are significantly more likely to enroll in eighth-grade algebra and geometry than their White peers, while Black students are significantly less likely to enroll in eighth-grade algebra. However, Hispanic students are significantly more likely to enroll in eighth-grade algebra than White students, controlling for family background. However,

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Figure 1. Kindergarten to eighth-grade mathematics test score growth by mathematics track. Sp = spring.

perhaps the most striking aspect of this model is its low R2, which indicates that student age, ethnicity, gender, parental education, family income, age, family income, and school poverty together explain just 3% of the variance in eighth-grade mathematics course placements. The second and third models in Table 2 add students’ fifth-grade test scores to the analysis, to explore the extent to which student background influences placement over and above their prior achievement. The results indicate that middle school mathematics placement selection mechanisms take account of students’ measured achievement as well as teacher evaluations of students’ approaches to learning. Model 2 indicates that students’ odds of enrolling in eighth-grade prealgebra, algebra, and geometry rise with their fifthgrade mathematics test scores. The associations between fifth-grade mathematics test scores and student odds of enrolling in eighth-grade algebra and geometry are particularly large and statistically significant at 0.63 and 0.76, respectively. This model further indicates that student academic skills in reading influence middle school mathematics course placements. Fifth grade reading test scores positively predict eighth-grade prealgebra, algebra, and geometry placement. These odds ratios are approximately half the size of the odds ratios describing the relation between fifth-grade mathematics test scores and middle school mathematics course placement. However, the significant independent relation between fifth-grade reading scores suggests that mathematics course placement

is not purely a function of measured mathematics skills. These measures of fifth-grade mathematics and reading achievement largely explain ethnic, gender, and class-based inequalities in middle school mathematics course placement. After accounting for these controls, Black students are no longer significantly less likely than White student to enroll in eighth-grade algebra. Furthermore, controlling for fifth-grade test scores explains nearly all of the association between parental education at eighthgrade mathematics course placement and approximately half of the association between family income and eighth-grade mathematics course placement. Furthermore, Model 3 indicates that kindergarten mathematics and reading test scores influence course placement, even after controlling for the more proximal fifth-grade measure of students’ skills. Two likely explanations for the positive relation between kindergarten skills and eighth-grade mathematics course placement bear consideration: First, since fifth-grade test scores are measured with error, kindergarten test scores may provide additional information about students’ latent ability. Second, it is possible that the students who demonstrate high levels of proficiency in early elementary school receive higher levels of mathematics instruction throughout their elementary school careers. This second possibility is intriguing. U.S. elementary schools rarely sort students into hierarchically differentiated classes for mathematics instruction and most research on within-class differentiation in

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Table 2 Predictors of Eighth-Grade Mathematics Course Placement (Multinomial Logistic Regression Coefficients) Model 1

Black Hispanic Asian Other Male Less than HS parent Some/AA parent BA parent MA+ parent Family income (ln) School % free lunch Age, fall K (months) Math IRT, fifth grade (std) Reading IRT, fifth grade (std) Learning approach, fifth grade (std) Math IRT, K (std) Reading IRT, K (std) Learning approach, K (std) Constant N Pseudo R2 BIC

Model 2

Model 3

Prealgebra

Algebra

Geometry

Prealgebra

Algebra

Geometry

Prealgebra

Algebra

Geometry

0.19 0.02 0.29 0.09 0.00 0.10 0.17 0.46*** 0.30 0.20*** 0.01** 0.031 —

0.29** 0.25* 1.09*** 0.28 0.14* 0.46*** 0.27** 0.62*** 0.92*** 0.39*** 0.00* 0.03 —

0.09 0.24 1.63*** 0.34 0.24 0.21 0.18 0.66** 1.22*** 0.43*** 0.01** 0.03 —

0.03 0.09 0.30 0.01 0.02 0.15 0.04 0.23 0.01 0.09 0.00* 0.06 0.31***

0.13 0.37*** 1.06*** 0.39* 0.22** 0.60*** 0.03 0.21 0.37* 0.19*** 0.00 0.08 0.63***

0.38 0.37 1.57*** 0.44 0.12 0.06 0.44* 0.21 0.62* 0.21 0.01* 0.09 0.76***

0.05 0.06 0.26 0.01 0.05 0.13 0.03 0.22 0.05 0.08 0.00* 0.11* 0.22***

0.15 0.30** 0.93** 0.48** 0.01 0.56*** 0.03 0.14 0.25 0.16** 0.00 0.19*** 0.45***

0.29 0.522 1.15** 0.64 0.34* 0.04 0.42* 0.16 0.37 0.16 0.01 0.27** 0.56***

—

—

—

0.22***

0.33***

0.19***

0.22***

0.07

—

—

—

—

—

—

0.00

0.25***

0.31***

— — —

— — —

— — —

— — —

— — —

— — —

0.04 0.17* 0.02

0.22*** 0.34*** 0.12

0.38*** 0.48*** 0.39***

1.47** 6,339 0.0326 15,091

3.73***

6.35***

0.56 6,339 0.0669 14,655

0.61

1.24

0.30**

0.23 6,339 0.0832 14,636

0.58

2.76*

Note. Early Childhood Longitudinal Study, Kindergarten Cohort panel respondents; public school students only (weight = c2_7fc0). Missing values on mathematics and reading IRT scores, as well as teacher assessments of student approaches to learning are mean substituted, and the model includes dummy variables flagging students with mean-substituted values. HS = high school; AA = associate’s degree; BA = bachelor’s degree; MA+ = master’s degree or higher; K = kindergarten; IRT = item response theory; std = standaridzed; BIC = Bayesian information criterion. *p < .05. **p < .01. ***p < .001.

elementary school is focused on reading instruction. However, the persistent association between kindergarten skills and eighth-grade mathematics course placement may suggest that instruction in elementary mathematics is more differentiated than commonly assumed. It should be noted, however, that the pseudo R2 in this model is just 0.08, suggesting even after controlling for this extensive set of kindergarten and fifth-grade test scores and teacher evaluations, only a fraction of the between-student variation in eighth-grade mathematics course taking has been explained. Supplementary models that add fixed effects for the school in which students are enrolled as kindergarteners increase the pseudo R2 to 0.45 without substantively altering the associations between students’ background characteristics

and measured skills and their eighth-grade mathematics course enrollment. Together, these findings indicate that middle school mathematics course placements clearly respond to various measures of student academic readiness. However, even among students with similar prior skills, school-level factors, as well as unmeasured characteristics of students and their families play a substantial role in determining students’ middle school mathematics course trajectories. Does Accelerated Algebra Placement Influence Students’ Mathematics Skills? Having investigated the factors that predict middle school mathematics course placement, the remainder of this article is concerned with the con-

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sequences of these placements for student mathematics achievement. The data reported in Table 1 and Figure 1 provide an initial descriptive treatment of the relation between middle school mathematics placement and achievement. While these data clearly indicate that access to advanced middle school mathematics courses is in part a function of student achievement, it is less clear from these data that access to eighth-grade algebra and geometry boosts student achievement. Indeed, data reported in Table 1 and Figure 1 suggest that students placed into advanced eighth-grade mathematics courses actually experience slower mathematics test score growth in this period than their peers in general mathematics courses. As discussed earlier, the curvilinear pattern of IRT mathematics test score growth may indicate that ECLS–K’s mathematics test score may not differentiate student mathematics achievement at the top of the skills distribution as well as it does lower in the distribution. Nonetheless, these descriptive findings suggest that some of the association between middle school mathematics placement and student achievement is attributable to time-invariant characteristics of students and schools, as well as the processes through which students are selected into middle school math. If the fit between students and courses influences student achievement growth —rather than the courses themselves—these data may point to an important limitation to the notion that expanding opportunities to learn improves achievement. The analyses that follow provide a more detailed investigation of the factors that influence selection into advanced middle school mathematics courses and the consequences of this selection process for student achievement. Table 3 reports the results of four different panel data models estimating the effects of middle school mathematics course enrollments on student mathematics achievement. Each of these models treats the student observation as the unit of analysis, tracking student achievement scores kindergarten through first, third, fifth, and eighth grades. The first is a lagged student random effects model, regressing student mathematics IRT scores on student middle school mathematics course enrollment, controlling for wave and a rich set of prior achievement and demographic measures. This model indicates students who enroll in eighth-grade prealgebra courses score 0.08 SD higher on the ECLS–K’s spring eighth-grade mathematics IRT test compared to students with similar fifth-grade test scores and demographics who enroll in eighthgrade general mathematics courses. Students who

enroll in eighth-grade algebra and geometry experience larger achievement gains. Students who enroll in eighth-grade algebra score 0.17 SD higher on spring eighth-grade mathematics tests than their peers in eighth-grade general mathematics, controlling for demographics and prior achievement. Students who enroll in eighth-grade geometry, meanwhile, score 0.25 SD higher than eighth-grade general mathematics students, after accounting for controls. This model indicates that challenging middle school mathematics courses have positive effects on students’ mathematics achievement. However, omitted variable bias presents a substantial threat to validity for this model. If, for example, students who enroll in advanced middle school mathematics courses tend to have more academic goal orientations than their peers in less rigorous middle school mathematics courses, the association between mathematics course enrollment and mathematics achievement reported in Model 1 may be spurious. The remaining models in Table 3 represent different strategies to address omitted variable bias. Model 2 adds a set of time-invariant controls for students’ mathematics course track. This model estimates separate mathematics test score trajectories across the entire kindergarten to eighth-grade panel for students who are bound for eighth-grade general mathematics, prealgebra, algebra, and geometry. The prealgebra track coefficient in this model indicates that by the spring of eighth grade, students who are bound for eighth-grade prealgebra score 0.06 SD higher on mathematics tests than their peers who are bound for eighth-grade general mathematics. These pre-enrollment gaps are even more pronounced for students who are bound for eighth-grade algebra and geometry. This model thus indicates that at least some of the achievement gaps associated with eighth-grade mathematics courses in Model 1 are actually due to unmeasured student characteristics. This model indicates that eighth-grade geometry has no effect on mathematics achievement after controlling for these unmeasured student characteristics. Nonetheless, this model indicates that enrolling in eighth-grade prealgebra and algebra significantly improves student mathematics achievement. Students who enroll in eighth-grade prealgebra score an average of 0.06 SD higher, and students who enroll in eighth-grade algebra score an average of 0.08 SD higher than they would have had they enrolled in eighth-grade general mathematics. The results reported in Model 3 add school fixed effects to this analysis to address the possibility that the measured relation between eighth-grade mathematics course

Middle School Mathematics and Achievement

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Table 3 Panel Data Predictors of Mathematics Test Scores, Kindergarten Through Eighth Grade (1)

Spring K First grade Third grade Fifth grade Eighth grade Eighth-grade prealgebra Eighth-grade algebra Eighth-grade geometry Prealgebra track Algebra track Geometry track Math score (Yn1, std) Reading score (Yn1, std) Math score (Yn1, std)2 Reading score (Yn1, std)2 Learning approach (Yn1, std) Black Hispanic Asian Other Male Midwest South West < HS parent Some/AA parent BA parent MA+ parent Family income (ln) K entry age Eighth Grade 9 Math IRT Eighth Grade 9 Math IRT2 School FE Student FE Constant N (observations) N (students)

(2)

(3)

(4)

Coef

SE

Coef

SE

Coef

SE

Coef

SE

0.32*** 0.38*** 0.43*** — 0.13*** 0.08*** 0.17*** 0.25*** — — — 0.44*** 0.33*** 0.01* 0.04*** 0.09*** 0.20*** 0.01 0.02 0.03 0.19*** 0.02 0.03** 0.01 0.02 0.06*** 0.13*** 0.14*** 0.06*** 0.01*** 0.04*** 0.07*** — — 0.63*** 29,017 7,133

0.07 0.08 0.09 — 0.02 0.02 0.02 0.04 — — — 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.01 0.01 — — 0.05

0.36*** 0.42*** 0.48*** — 0.09*** 0.06** 0.08*** 0.09 0.03* 0.11*** 0.19* 0.44*** 0.32*** 0.01*** 0.04*** 0.09*** 0.20*** 0.01 0.02 0.03* 0.19*** 0.01 0.03*** 0.01 0.02 0.06*** 0.12*** 0.14*** 0.05*** 0.01*** 0.02** 0.07*** — — 0.76*** 29,017 7,133

0.07 0.08 0.09 — 0.02 0.02 0.02 0.04 0.01 0.01 0.02 0.01 0.01 0.00 0.00 0.00 0.01 0.01 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.01 0.00 — — 0.06

0.55*** 0.63*** 0.72*** — 0.09** 0.06 0.08* 0.12 0.03 0.15*** 0.18*** 0.38*** 0.33*** 0.00 0.04*** 0.10*** .19*** 0.04 0.01 0.04 0.19*** 0.34 0.90 0.14 0.03 0.07*** 0.11*** 0.13*** 0.05*** 0.01*** 0.02 0.07*** Yes — 0.87 29,017 7,133

0.14 0.15 0.17 — 0.03 0.04 0.03 0.08 0.02 0.02 0.05 0.01 0.01 0.00 0.00 0.00 0.03 0.03 0.05 0.04 0.01 2.85 2.85 2.53 0.03 0.02 0.02 0.03 0.01 0.00 0.02 0.01 — — 0.19

0.04*** 0.05*** 0.06*** — 0.17*** 0.09*** 0.10*** 0.01 — — — — — — — — — — — — — — — — 0.00 0.03 0.04 0.12** 0.02* — — — — Yes 0.19** 31,179 7,157

0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.03 — — — — — — — — — — — — — — — — 0.02 0.02 0.02 0.04 0.01 — — —

0.06

Note. Early Childhood Longitudinal Study, Kindergarten Cohort panel respondents; public school students only (weight = c2_7fc0). Dependent variables are z-score standardized so that coefficients represent effect sizes. K = kindergarten; HS = high school; AA = associate’s degree; BA = bachelor’s degree; MA+ = master’s degree or higher; IRT = item response theory; FE = fixed effects. *p < .05. **p < .01. ***p < .001.

and achievement actually reflects unmeasured school effects. The results of this model are quite similar to the results of Model 2, suggesting that the effects of eighth-grade mathematics course enrollment on student mathematics achievement are largely independent of broader school effects. Model 4 estimates the effects of eighth-grade algebra on student achievement using a student fixedeffects approach. This model controls for all time-

invariable student characteristics essentially by estimating a separate test score growth trajectory for each student. Model 4 indicates that enrolling in eighth-grade prealgebra boosts student achievement by 0.09 SD and enrolling in eighth-grade algebra boosts student achievement by 0.10 SD compared to enrolling in eighth-grade general mathematics. This model’s results are thus largely consistent with the findings reported in Models 2 and 3.

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Overall, Table 3 indicates that enrolling in eighthgrade prealgebra and algebra courses has positive effects on student mathematics achievement. In interpreting these findings, however, it is important to consider the challenges associated with measuring mathematics achievement growth that are common across these analyses of ECLS–K mathematics IRT test scores. As discussed earlier, student growth on this measure slows considerably in the panel’s later years as large numbers of students begin to cluster at the upper end of the test score scale. This pattern of test scores suggests that the exam may not effectively capture mathematics achievement for relatively advanced upper elementary and middle school students who are moving beyond basic arithmetic to more advanced mathematics. The quadratic prior achievement and Eighth Grade 9 Prior Mathematics Achievement interaction terms in the models reported in Table 3 should partially address this issue. However, if high-level middle school mathematics courses focus their instruction on high-level mathematics skills that are not well represented in the IRT mathematics test score, the estimates of the effects of advanced middle school mathematics classes reported in Table 3 may be negatively biased. One strategy to address this issue is to replicate the models reported in Table 3 using tobit regression models that explore the extent to which truncation at the right side of the test score distribution influences estimates of the effects of advanced course enrollment in eighth grade. These analyses, available from the author by request, return similar results to those reported above, providing some evidence to suggest that these findings are not artifacts of ceiling effects on the ECLS–K measures. A second strategy to address this issue is to focus on the relation between eighth-grade mathematics course enrollment and achievement on relatively advanced mathematics skills. Table 4 reports the results of a series of lagged cross-sectional analyses that do so using ECLS–K test score subscales measuring students’ proficiency in three areas: (a) the use of rate and measurement to solve simple word problems, (b) the concept of fractional parts, and (c) solving word problems that involve calculating area and volume. To interpret the results of these analyses as causal, one must assume that the lagged achievement measures and control variables fully capture the differences between students in various middle school mathematics courses that predict later achievement. This assumption is more restrictive than the assumptions underlying the panel models reported in Table 3. However, the findings reported in the first column of Panel A of Table 4 indicate

that the lagged cross-sectional approach returns findings that are similar to—if somewhat more conservative than—the panel approach. The remaining three columns in Panel A of Table 4 provide estimates of the effects of middle school mathematics course taking that are more tightly focused on student mastery of advanced mathematics skills. These analyses indicate that eighth-grade mathematics courses have fairly narrow effects—with each course boosting students’ odds of mastering the mathematics content that is central to that course, but not influencing students’ odds of mastering other advanced mathematics content. Enrolling in eighthgrade prealgebra and algebra courses significantly boosts the probability that students have mastered the use of rate and measurement to solve simple word problems by approximately 3 percentile points. Similarly, enrolling in eighth-grade algebra has a fairly large positive effect on student fractional mastery, boosting the probability of mastery for the average student from 24% to approximately 30%. This finding is striking, given the strong conceptual overlap between fractional mastery and algebra success (Siegler et al., 2012). Finally, Panel A of Table 4 indicates that eighth-grade geometry improves the probability that students demonstrate proficiency at solving word problems involving area and volume. Notably, however, this effect does not spill over to other advanced mathematics courses. There is no relation between eighth-grade algebra enrollment and proficiency in this area. Further, students who enroll in eighth-grade prealgebra are significantly less likely than students with similar prior achievement and demographics who take eighth-grade general mathematics to demonstrate proficiency in these basic geometric skills. Panel B of Table 4 adds a series of interaction terms to investigate the extent to which these measured effects of middle school mathematics courses are contingent on students’ prior mathematics achievement. While the results reported in this panel’s first column indicate that no such interaction exists on the broad mathematics IRT test, an interesting pattern emerges on the subscale analyses. In the analyses of students’ proficiency in rates and measurement, the positive effect of eighth-grade pre-algebra varies significantly with student prior mathematics achievement. This finding indicates that enrolling in eighth-grade prealgebra has nearly no effect on the probability that a relatively low-skilled student will demonstrate proficiency in this central prealgebra concept, but that the course has a substantial positive effect on proficiency probability for a relatively high-skill student. Similarly, the analysis

Middle School Mathematics and Achievement

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Table 4 Predictors of Eighth-Grade Mathematics Subscale Probability Proficient Scores Panel A

Eighth-grade prealgebra Eighth-grade algebra Eighth-grade geometry Math score (fifth, std) Reading score (fifth, std) Math score (fifth, std)2 Reading score (fifth, std)2 Learning approach (fifth, std) Eighth-grade prealgebra 9 Math (fifth) Eighth grade Alg 9 Math (fifth) Eighth grade Geo 9 Math (fifth) K scores Demographic controls Constant

Panel B

Math IRT

Rate

Fractions

Area

Math IRT

Rate

Fractions

Area

0.03 (0.03) 0.07* (0.03) 0.07 (0.06) 0.68*** (0.02) 0.12*** (0.02) 0.01 (0.01) 0.05*** (0.01) 0.04** (0.01) —

0.03* (0.01) 0.03** (0.01) 0.01 (0.02) 0.24*** (0.01) 0.04*** (0.01) 0.01 (0.01) 0.00 (0.00) 0.01* (0.00) —

0.00 (0.01) 0.06*** (0.01) 0.03 (0.03) 0.29*** (0.01) 0.03** (0.01) 0.02** (0.01) 0.10*** (0.00) 0.01* (0.00) —

0.03*** (0.01) 0.02 (0.01) 0.07** (0.03) 0.18*** (0.01) 0.02** (0.01) 0.01* (0) 0.09*** (0.00) 0.01** (0.00) —

0.04*** (0.01) 0.01 (0.01) 0.05* (0.02) 0.19*** (0.01) 0.02** (0.01) 0.01** (0) 0.08*** (0.00) 0.01** (0.00) 0.06*** (0.01)

0.03 (0.03) 0.07* (0.03) 0.08 (0.07) 0.67*** (0.02) 0.12*** (0.02) 0.01 (0.01) 0.04*** (0.01) 0.04** (0.01) 0.02 (0.03)

0.04** (0.01) 0.04** (0.01) 0.00 (0.03) 0.22*** (0.01) 0.04*** (0.01) 0.01 (0.01) 0.01 (0.00) 0.01* (0.00) 0.03*** (0.01)

0.00 (0.01) 0.06*** (0.01) 0.04 (0.03) 0.28*** (0.01) 0.03** (0.01) 0.02** (0.01) 0.09*** (0.00) 0.01* (0.00) 0.01 (0.01)

—

—

—

—

—

—

—

—

+

+

+

+

0.01 (0.01) 0.06*** (0.02) +

0.04 (0.03) 0.00 (0.05) +

0.01 (0.01) 0.01 (0.02) +

0.03** (0.01) 0.01 (0.01) +

+

+

+

+

+

+

0.66*** (0.08) 6,882

0.24** (0.08) 6,882

0.03 (0.19) 6,882

0.01 (0.05) 6,882

0.03 (0.05) 6,882

0.02 (0.19) 6,882

+

+

0.64*** (0.08) 6,882

0.24** (0.08) 6,882

Note. Early Childhood Longitudinal Study, Kindergarten Cohort panel respondents; public school students only (weight = c2_7fc0). Dependent variables are z-score standardized so that coefficients represent effect sizes. Robust standard errors in parentheses. IRT = item response theory; std = standardized; K = kindergarten. *p < .05. **p < .01. ***p < .001.

of proficiency with fractions indicates that the positive effects of eighth-grade algebra are significantly larger for students with relatively high prior achievement than for students with relatively low prior achievement. The analysis of proficiency with the concepts of area and volume once again suggests that the positive effects of eighth-grade geometry are most pronounced for relatively high-achieving students. Intriguingly, however, this analysis also points to a negative interaction between eighth-grade prealgebra and student proficiency probabilities, indicating that eighth-grade prealgebra coursework has a particularly pronounced negative effect on achievement in this mathematical area for students with relatively high levels of prior skills.

Discussion Over the last several decades, policy makers and educators across the United States have undertaken a broad-based effort to intensify mathematics curricula in U.S. secondary schools. American middle schools enroll more eighth graders in algebra and other relatively advanced mathematics courses today than ever before. This movement is predicated on the assumption that accelerating mathematics course placements will increase students’ “opportunity to learn” and thus raise average achievement levels, and narrow educational inequalities. However, the theory of person–environment fit raises important questions about the

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link between advanced course taking and achievement. If students are not developmentally prepared for the academic challenges associated with advancement mathematics course enrollment in early adolescence, accelerated algebra efforts and other related educational policies may have unintended negative consequences for student achievement. Furthermore, research in child development indicates that students vary considerably in their capacity to benefit from opportunities to learn (Duncan & Vandell, 2012; Eccles et al., 1993). This article applies these ideas, which are central to developmental psychology, to explore the relation between middle school mathematics curricula and student achievement. Doing so provides unique opportunities to test theories of person–environment fit using nationally representative data and to apply insights from developmental theory to central questions in educational policy and curriculum development. Analyses of course placement decisions indicate that racial and class-based inequalities in access to advanced middle school mathematics courses persist in contemporary U.S. middle schools. However, these inequalities in access to eighth-grade algebra and geometry are largely explained by inequalities in student achievement that predate middle school mathematics course placement. This article’s uniquely detailed look at the factors that influence placements in middle school mathematics indicates that mathematics achievement at the end of elementary school influences middle school mathematics course placements. Furthermore, a wide range of other achievement indicators also exert an independent influence on these course placements, including reading test scores, teacher perceptions of student approaches to learning, and even student academic skills measured at kindergarten. In addition to influencing middle school mathematics course placement, these early skills clearly influence student achievement growth long before schools sort students into stratified middle school mathematics courses. The findings indicate that student opportunities to learn are path dependent, giving students who start school with advanced skills advantaged access to opportunities to learn that persist through middle school. In particular, this pattern of findings may indicate that U.S. students receive differentiated mathematics instruction within elementary school classrooms. While prior research has investigated within-classroom differentiation in U.S. elementary schools (e.g., Tach & Farkas, 2006), this work has focused primarily on reading instruction. Future research should investi-

gate formal and informal differentiation in elementary mathematics instruction. While the ECLS–K provides some evidence to suggest that student mathematics achievement growth rates slow during the middle school years, the analyses reported here indicate that enrolling students in eighth-grade prealgebra, algebra, and geometry helps to slow this slide. The evidence suggesting that eighth-grade algebra boosts student achievement is particularly robust, with effect sizes of approximately 1/10th of a standard deviation. These positive effects are smaller than the positive effects associated with advanced course enrollment identified in earlier analyses of nationally representative data (Gamoran & Hannigan, 2000). Nonetheless, the finding that enrolling students in advanced middle school mathematics courses has positive average effects on achievement provides important evidence to support the push to intensify mathematics curricula in American middle and high schools. However, these analyses also provide reason for caution. Rather than influencing student mathematics achievement broadly, eighth-grade mathematics courses seem to have relatively narrow effects on the domains of mathematics that are most closely linked to course content. Furthermore, not all students benefit equally from access to advanced courses. Instead, the achievement effects of advanced middle school mathematics course enrollment are somewhat contingent on the fit between students’ mathematics competency and their course exposure. Students who come into advanced mathematics courses with relatively strong prior mathematics achievement seem to benefit more from these classes than their peers with lower mathematics skills. These findings lend qualified support to the person–environment fit perspective, suggesting that the expanding advanced course enrollments may reach a point of diminishing returns if students are placed into courses for which they are unprepared. The findings reported here may thus help to explain the perplexing disconnect that exists between many studies that point to positive individual-level effects of advanced mathematics course taking and other, more recent, studies suggesting that the broader policy-driven curricular intensification may be ineffective. If some of the positive consequences of advanced course enrollment are contingent on student readiness, efforts to accelerate middle school mathematics curricula may see diminishing returns if they lead students to enroll in courses for which they are unprepared.

Middle School Mathematics and Achievement

Enhanced curricula and personalized learning opportunities for relatively low-achieving students and other strategies to tailor instruction to students’ skill level may help prepare students for advanced middle school mathematics courses and improve person–environment fit in accelerated middle school mathematics classrooms. Future research should consider curricular innovations to help prepare students for algebra instruction (Siegler et al., 2012), as well instructional and organizational reforms that help underprepared students succeed in advanced mathematics courses (Nomi & Allensworth, 2013; Stein et al., 2011).

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Supporting Information Additional supporting information may be found in the online version of this article at the publisher’s website: Appendix S1: Truncation of Mathematics IRT Test Score Distribution for Upper-Grade Students