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A Journal of the American Sociological Association

SOCIOLOGYOF

EDUCATIONVolume 82 January 2009 Number 1

SOC

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OF ED

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e 82, Num

ber 1, Jan

uary 2009

Rethinking the Attitude-Achievement Paradox Among Blacks

DOUGLAS B. DOWNEY, JAMES W. AINSWORTH, AND ZHENCHAO QIAN

The Black-White-Other Achievement Gap:Testing Theories of Academic Performance Among

Multiracial and Monoracial AdolescentsMELISSA R. HERMAN

The Black-White Gap in Mathematics Course TakingSEAN KELLY

How African American Is the Net Black Advantage?Differences in College Attendance

Among Immigrant Blacks, Native Blacks, and Whites PAMELA R. BENNETT AND AMY LUTZ

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SOCIOLOGICALMETHODOLOGY

SOCIOLOGYOF

EDUCATIONVolume 82 January 2009 Number 1

Contents

Rethinking the Attitude-Achievement Paradox Among BlacksDOUGLAS B. DOWNEY, JAMES W. AINSWORTH, AND ZHENCHAO QIAN 1

The Black-White-Other Achievement Gap: Testing Theories of AcademicPerformance Among Multiracial and Monoracial Adolescents

MELISSA R. HERMAN 20

The Black-White Gap in Mathematics Course TakingSEAN KELLY 47

How African American Is the Net Black Advantage? Differences in CollegeAttendance Among Immigrant Blacks, Native Blacks, and Whites

PAMELA R. BENNETT AND AMY LUTZ 70

Ainsworth, JamesAlexander, KarlAlon, SigalAndrew, MeganArchbald, DouglasArum, RichardAyalon, HannaBaker-Degler,

ThereseBarnett, Steve Bellmore, AmyBennett, KatieBennett, PamelaBills, DavidBlau, Judith Bodovski, KatarinaBoli, JohnBorman, KathrynBouffard, SuzanneBoyadjieva, PepkaBozick, RobertBrand, JennieBraxton, JohnBrown, KevinBrown, SusanBuchmann, ClaudiaByun, Soo-yongCabrera, NatashaCamburn, EricCarbonaro, William Carter, PrudenceCatsambis, SophiaCheadle, Jacob Chudgar, Amita Clark, RebeccaClotfelter, CharlesCoburn, CynthiaCole, WadeCollier, PeterConchas, Gilberto Connelly, Rachel Cooksey, Elizabeth Crosnoe, RobertCrowley, Martha D'Agostino, JeromeD'Angiulli, AmedeoDavies, ScottDavis, JamesDavis-Kean, PamDeil-Amen, ReginaDeLuca, StefanieDesimone-Porter,

LauraDiamond, JohnDiemer, Matt Dimitriadis, GregDiPrete, Thomas Dixon-Roman,

EzekielDodge, Ken Domina, Thurston

(Thad)

Dornbusch, SandyDougherty, KevinDowney, DouglasDoyle, Jamie

MihokoDreeben, RobertDronkers, JaapDrori, GiliDrummond, Todd Dumais, SusanEccles, JacqueEder, DonnaEspenshade,

ThomasFarkas, GeorgeFeliciano, CynthiaFernandes, Danielle

CirenoFine, Michelle Fisherkeller, JoEllenFlache, AndreasFletcher, JasonFox, CybelleFrank, David Frank, KennethFrankenberg, EricaFuchs, LynnGalindo, ClaudiaGamoran, AdamGangl, MarkusGibbs, Benjamin Glanville, Jennifer Glenn, NorvalGlick, JenniferGold, Steve Goldrick-Rab, SaraGoyette, KimberlyGrodsky, EricGuest, AndrewGuetzkow, JoshuaHallinan, MaureenHamilton, StephenHao, LingxinHarding, DavidHarris, AngelHarris, DougHearn, JamesHedberg, EricHerd, Pamela Hermanowicz,

JosephHess, FrederickHeuveline, PatrickHigginbotham,

ElizabethHill, LoriHirschman, CharlesHorvat, ErinHurtado, Sylvia Jencks, ChristopherJensen, LeifJepsen, Christopher

Johnson, OdisKaftan, JoannaKamberelis, GeorgeKaren, DavidKelly, SeanKim, Doo HwanKingston, PaulKinney, David Kirkpatrick-Johnson,

MonicaKlinger, Janette Konstantopoulos,

SpyrosKremer-Sadlik , TamiKubey, Robert Kubitschek, WarrenKurlaender, MichalKwiek, MarekLabaree, DavidLareau, AnnetteLarson, ReedLauen, DouglasLeahey, ErinLechner, Frank Lee, JenniferLee, Stacey Lee, Yun-sukLeicht, Kevin Lewis, KristineLimage, LeslieLong, Bridget Low , MarkLynn, FredaMa, XinMacMillan, RossMagnuson,

KatherineMaier, Kim Maier, RobertMark, NoahMarks, GaryMarks, HelenMarsh, HerbertMateju, PetrMcFarland, DanielMcGhee-Hassrick,

Elizabeth McLanahan, SaraMcLaren, PeterMcWayne, ChristineMechur Karp ,

MelindaMetz, MaryMeyer, JohnMickelson, RoslynMilesi, CarolinaMillett, CatherineMollborn, StefanieMoody, JamesMorgan, StephenMortenson, TomMortimer, Jeylan

Mulkey, Lynn Muller, ChandraMuschkin, ClaraMustard, DavidOffer, ShiraOno, HiromiPalardy, GregoryPallas, AaronPapachristos ,

AndrewPascarella, ErnestPenner, Andrew Penuel, WilliamPerreira, KristaPettit, BeckyPewewardy, CornelPfeffer, Fabian Pigott, TerriPlank, StephenPomerantz, EvaPong, Suet-lingPost, DavidPowell, Brian Powers, Jeanne Price, DerekRaley, KellyRam, UriRamirez, FranciscoRandolph, Antonia Rasinski, KennethReam, RobertReardon, SeanReay, DianeRedstone Akresh,

Ilana Reininger, MichelleRenzulli, LindaReyhner, JonRiegle-Crumb,

CatherineRoksa, JosipaRoscigno, VincentRosenbaum, JamesRosenbaum, JamesRothstein, JesseSadovnik, AlanSaporito, SalvatoreSauder, MichaelSchafer, MarkSchalliol, DavidSchiller, KathrynSchmeer, KammiSchmidt, Bill Schmidt, JenniferSchofer, EvanSchoon, IngridSchweinhart, LarrySchwille, JohnSedlacek, WilliamSharkey, Amanda Shavit, YossiShernoff, David

Shouse, RogerShu, Xiaoling Smardon, ReginaSmith, ThomasSmith, TomSomers, Patricia A.Song, ShigeSouth, ScottStanton-Salazar,

RicardoStaples, MeganStearns, ElizabethStephan, JenniferStevens, MitchelStocke, Volker Suter, LarrySwanson, ChrisTam, TonyTeachman, JayTienda, Marta Turley, RuthTyler, JohnTyson, KarolynUrzua, SergioVaquera, ElizabethVeenstra, Rene Warren, John RobertWeininger, ElliotWeiss, MichaelWillms, DouglasWilson, GeorgeWilson, Suzanne Wimberly, GeorgeWooden, Mark Yair, GadYoungs, PeterYu, Wei-hsinZeng, Zhen

SPECIAL EDITORIAL REVIEWERS

We thank the following persons who served as special reviewers duringthe past year:

The Black-White Gap in MathematicsCourse Taking

Sean KellyUniversity of Notre Dame

Using data from the National Education Longitudinal Study, this study investigated differences

in the mathematics course taking of white and black students. Because of lower levels of

achievement, prior course taking, and lower socioeconomic status, black students are much

more likely than are white students to be enrolled in low-track mathematics courses by the

10th grade. Using multilevel models for categorical outcomes, the study found that the black-

white gap in mathematics course taking is the greatest in integrated schools where black stu-

dents are in the minority and cannot be entirely accounted for by individual-level differences

in the course-taking qualifications or family backgrounds of white and black students. This

finding was obscured in prior research by the failure to model course taking adequately

between and within schools. Course placement policies and enrollment patterns should be

monitored to ensure effective schooling for all students.

Sociology of Education 2009, Vol. 82 (January): 47–69 47

Black students are found disproportion-ately in lower ability groups and acade-mic courses as early as the first grade

(Entwisle, Alexander, and Olson 1997). Kelly(2004) found that by high school, whites areabout twice as likely as are blacks to beenrolled in advanced mathematics courses.This disparity can be thought of as a form ofwithin-school segregation (Mickelson 2001a,2001b). By the time students reach sec-ondary school, within-school segregation canaccount for over half the total segregation ina district (Clotfelter, Ladd, and Vigdor 2003).In this article, I examine the determinants ofenrollment in mathematics courses amongblack 10th graders in different school set-tings. My analysis was motivated by fourresearch questions:

1. To what extent can differences in coursetaking among black and white students beattributed to differences in academic achieve-ment or other factors that are associated withindividual students, such as family back-ground?

2. To what extent can the lower levels ofacademic course taking in mathematicsamong black students be explained bycourse-enrollment patterns at the schoolsthat black students attend?

3. To what extent can lower levels ofcourse taking be attributed to a contextualeffect within integrated schools, wherebyblack students are disadvantaged in predom-inantly white schools?

4. Do inequalities in black-white coursetaking vary across school sectors?

BACKGROUND

Effects of Mathematics CourseTaking on Students’ Lives

Over the course of the school career, an indi-vidual will pass through many structural loca-tions, from within-class ability grouping in ele-mentary school to the tracked courses taken inmiddle and high school. I use the terms course

48 Kelly

taking to refer to enrollment in specific courses(e.g., geometry, Algebra II, and calculus) andtracking to refer to students’ overall pattern ofcourse taking across subjects or to overall char-acterizations of course taking (e.g., vocational,regular, and high or college prep). In highschool, academic course taking is an importantstructural predictor of students’ achievement,especially in mathematics (Gamoran 1987). AsGamoran (1987) reported, the high-track effectin mathematics—the amount that high-trackstudents gained above and beyond low-trackor vocational students, independent of pretestscores and other factors—amounted to about2.5 times the typical two-year gain of students.In another metric, the high-track effect wasabout three times the difference betweendropouts and low-track students. Analyses thathave used more rigorous methods to controlfor selection bias have found somewhat attenu-ated but still large effects (Carbonaro 2003;Gamoran and Mare 1989).

The specific courses that students take andthe overall system of tracking have a wide vari-ety of other effects as well, from their impact onaspirations (Heyns 1974) and friendship pat-terns (Kubitschek and Hallinan 1998) to moregeneral psychosocial outcomes, such as attach-ment to school (Abraham 1989; Hargreaves1967; Rosenbaum 1976; Schwartz 1981). Overthe course of schooling, these effects accumu-late, contributing greatly to a student’s finaleducational attainment (Kerckhoff 1993;Rosenbaum 1980). Mathematics course takingplays an especially important role in enteringscience, technology, engineering, and mathe-matics (STEM) fields that require a strong “cur-ricular momentum” coming out of high school(Heckel 1996). For example, Adelman (1998)reported that students in the High School andBeyond (HS&B) study who took precalculus inhigh school, but had mediocre grades, weremore likely to attain bachelor’s degrees in STEMfields than were students with high grades whohad completed only Algebra II.

Achievement, Family Background,and Course Taking

Much of the variance in course taking amongstudents in general is attributable to priorschool performance at the start of the year

(Alexander and McDill 1976; Heyns 1974).Lower track placements among black stu-dents are due, in part, to lower achievementscores. To the extent that black students havelower achievement scores, this can explaintheir lower track placements. Course taking isalso related to family background, and thecorrelation of race with family backgroundincreases the black-white course-taking gap(Kelly 2004; Lucas 1999). Prior school perfor-mance and family background account formost of the racial gap in course taking. Somestudies have shown a negative effect of beingblack on track placement when they havecompared whites and blacks of similar familybackgrounds and academic achievement(Gamoran 1992; V. E. Lee and Bryk 1988;Lucas 1999; Mickelson 2001a, 2001b), butthe import of these findings has often beenqualified by small samples or the appearanceof negative effects in only subsets of the data.In other studies, black students have seemedto have equal chances (Alexander and Cook1982; Garet and Delany 1988; Hallinan 1992;Vanfossen, Jones, and Spade 1987) or evenan advantage in track placement (Gamoranand Mare 1989; Jones, Vanfossen, andEnsminger 1995; Lucas and Gamoran 2002;Rosenbaum 1980; Stevenson, Schiller, andSchneider 1994).

Course-Enrollment Patterns atPredominantly Black Schools

The segregated nature of U.S. public schoolsmay contribute to racial differences in coursetaking. Despite historical and some continu-ing efforts to integrate U.S. schools, blackscontinue to be schooled in separate facilitiesfrom whites (Mickelson 2001a). The LewisMumford Center (2002) estimated that in1999–2000, only 28 percent of average blackstudents’ schoolmates were white. To assesspossible racial inequalities in course taking,researchers must consider disparities across aswell as within schools. Before a student cantake a course, the course has to be offered bythe school. Course-enrollment patterns arenot purely a function of the achievement dis-tribution of the students who attend theschool (Garet and Delany 1988:Tables 5 and7; Hallinan 1992). Instead, some schools offer

The Black-White Gap in Mathematics Course Taking 49

the majority of students the opportunity totake academic courses, whereas at otherschools, students of similar achievement lev-els take vocational courses or fewer academiccourses. The propensity to enroll students inacademically rigorous courses is oftenreferred to as the inclusiveness of the school(Sørensen 1970).

Prior research has suggested that school-to-school differences in inclusiveness helpexplain differences in track placement bysocioeconomic status (SES) (Kelly, 2004), butthat these differences may or may not be amajor factor in explaining the black-whitegap in course taking. Lucas and Gamoran(2002) found that students who attendedschools with a higher percentage of black stu-dents actually had a higher probability ofbeing placed in an upper track. Then again,one robust finding was that private schools,particularly Catholic schools, are more inclu-sive than are public schools (Gamoran 1996;V. E. Lee and Bryk 1988). One would expectthis pattern of greater inclusiveness to be amajor reason why some parents enroll theirchildren in private rather than public schools(Epple, Newlon, and Romano 2000). If whitestudents are more likely than black studentsto be enrolled in Catholic and other privateschools, this enrollment pattern would con-tribute to both the between-school and thetotal course-taking gap between blacks andwhites.

Black and White Students’ CourseEnrollments in Integrated Schools

The way racial inequality operates within indi-vidual schools may depend on the context ofthe schools. I hypothesized that school racialcomposition may be related to black-whiteinequality within schools, such that black stu-dents are disadvantaged, but only in integrat-ed or predominantly white settings. Theeffect of school racial composition on racialinequality within schools has mostly beenignored in prior research on tracking.Descriptive research on classroom-level segre-gation has found that segregation is the high-est in schools with a moderate percentage(30 percent–70 percent) of minority students(Clotfelter et al. 2003). This finding certainly

suggests that inequality in course taking maybe related to racial composition, although themost important predictors of track place-ment—achievement and family back-ground—were not considered. Lucas andBerends (2002) found that an importantorganizational dimension of tracking—theassociation between a student’s placement indisparate subjects, or the scope of the track-ing system in a school—is linked to theracial/ethnic and socioeconomic diversity in aschool. Tracking systems of broad scopeincrease differences in the opportunity tolearn by making placements consistent acrosssubjects. The finding that the scope of track-ing correlates with racial composition sug-gests that racial disparities in course takingmay also vary systematically across schools.

The placement process itself certainlyallows for the possibility of racial inequality.Track placements are a function not only ofgrades and test scores, but of more subjectivecriteria, such as teachers’ recommendationsand decisions by students, parents, and guid-ance counselors (Kelly 2007). Thus, just asthere is room for the well-documented social-class inequality in course taking, there is cer-tainly room for racial inequality, even if place-ment is mostly meritocratic. Researchers havesuggested that social-class inequality incourse taking is caused by differential levels ofparental involvement (Baker and Stevenson1986) or students’ expectations (Kelly 2004).What mechanisms may lead to racial inequal-ity in course taking? Numerous studies havetested the effects of race on track placements,but unfortunately few have discussed the rea-sons why race may be related to track place-ments. In this section, I discuss two potentialsources of racial inequality: discrimination byschool personnel, either intentional or statisti-cal, and the decisions of students themselves.The focus is on sources of racial inequalitythat are not mediated by known predictors oftrack placement, like social class, and thatmay vary by the racial composition of theschool.

Actions of School Personnel Intentionaldiscrimination by white school personnel,favoring white students over black students,could result in the uneven allocation of stu-

50 Kelly

dents to mathematics sequences. This type ofin-group bias, which results in categoricalinequality based on membership in a racialgroup, has been termed “opportunity hoard-ing” (Tilly 1999). If whites hoard access toupper-track classes in individual schools, thelarger black-white inequality in society atlarge would be reinforced, since course takingis a determinant of educational achievementand future attainment.

Racial bias by school personnel could alsoreflect “statistical discrimination” (Arrow1972), rather than opportunity hoarding.Placement decisions by school staff are likelyto favor the “high-status” group in a contextof incomplete or unreliable information abouta student’s skills. Because black students enterhigh school with lower average levels ofachievement than do white students, theymay suffer from biased assessments of abilityby teachers and guidance counselors whomake placement decisions. There is evidence,for example, that teachers perceive that blackstudents put forth less effort in their school-work (Ainsworth-Darnell and Downey 1998).

Statistical discrimination and opportunityhoarding both entail collective action bywhites. As forms of collective action, the like-lihood of statistical discrimination and oppor-tunity hoarding may be affected by the racialcomposition of a school. This helps us under-stand why the racial composition of a schoolmay affect the likelihood of these practices.Discriminatory actions are carried out largelyby individuals (e.g., a teacher’s failure to rec-ommend a black student for a high-trackcourse), but the fundamental process of dis-crimination is collective. School personnel donot benefit directly from discriminatory prac-tices and thus have no incentives as individu-als to carry them out. A guidance counselor,for instance, does not benefit directly fromrecommending that black students enroll inlow-track courses. As forms of collectiveaction, opportunity hoarding and statisticaldiscrimination are more likely to occur whenthere is a set of beliefs and practices to sustainthem (Tilly 1999:155). For example, for sta-tistical discrimination to be widespread,school personnel must believe it is better tohave the best possible fit, on average,between students’ skills and the courses they

take than to have equality of opportunity andsome degree of mismatch between skills andcourse taking. Otherwise, it would be difficultto ignore the fact that course-taking decisionsuniformly benefit whites.

Social learning theory (Bandura 1977) pro-vides a framework for understanding how thediscriminatory practices of individuals arelinked to group settings. Individuals learn dis-criminatory beliefs and practices through theprocesses of direct reinforcement and model-ing (Hechter 1987). Differential association—exposure to different groups with differentnorms—partly explains why some individualsadopt discriminatory behavior and some donot. In predominantly white schools, schoolstaff who make placement decisions with sub-jective criteria could act on norms that favorwhite students.

Students’ Agency An additional source ofracial inequality in course taking may bederived from students’ and parents’ roles incourse-taking decisions. For example, blackstudents who are qualified for higher-trackclasses may hold antischool norms and actu-ally choose a lower track themselves. Orwhite students and their parents may feel asense of entitlement relative to black stu-dents, desiring placement in upper-trackmathematics classes even when the students’school performance is marginal, in essenceconfusing the color of their skin with acade-mic skills. Both situations would lead to dis-proportionately more black students in thelower tracks.

It seems possible that black students’course-taking decisions may be affected bythe racial composition of a school. In a schoolwhere black students are in the majority,these students may be unlikely to accept low-track assignments because many black stu-dents are enrolled in high-track courses, pro-viding a visible testament to the possibility ofsuccess (and thus a viable individual alterna-tive to embracing antischool norms). Blackstudents are also more likely to be exposed tonorms of collective struggle in a school inwhich they constitute the majority of stu-dents, an experience that leads to positiveeducational outcomes. In a study of high-achieving inner-city black students, O’Connor

The Black-White Gap in Mathematics Course Taking 51

(1997) found that the most resilient studentswere those who were exposed to norms ofcollective struggle among blacks and that thisexposure helped them cultivate pro-schoolattitudes and behavior (see also Sanders1997).

The salience of a particular social identity,such as a racial identity, is not fixed; it is influ-enced by the social setting itself (Mullen1983). For black students who attend inte-grated schools, race may be a more salientelement of their identities, and being one ofthe few black students in predominantlywhite high-track courses may be socially iso-lating.1 Tyson, Darity, and Castellino (2005)found evidence of just such feelings of isola-tion among black students in honors/APclasses at an integrated high school. In choos-ing to take a lower-track course, a studentmay not be responding to peer pressure ordenying the importance of school success,but may be seeking the setting in which he orshe has friends and feels comfortable.

If the effect of a student’s racial identity oncourse taking varies across schools and is afunction of school racial composition, thenpast modeling strategies and data have beeninadequate for detecting racial inequality incourse taking. Consider, for example, theanalysis of ability grouping in elementaryschools by Pallas et al. (1994), which report-ed average effects across schools. In this sam-ple of 19 schools, two-thirds of the schoolshad either a 99 percent black student popu-lation or more than a 90 percent white stu-dent population. Would one expect the effectof being black to be similar in these schools?Perhaps, if it is truly a null effect.

School Sector Effects onMathematics Course Taking

Early research on Catholic schools (Coleman,Hoffer, and Kilgore 1982; Greeley 1982)asserted that the achievement gap amongstudents from different family backgroundswas smaller in Catholic schools than in publicschools. The finding that disadvantaged stu-dents did relatively better in Catholic schoolsthan in public schools was dubbed the “com-mon school effect.” Subsequent researchusing longitudinal data and other methods to

control for selection bias has concluded thatat least on some dimensions, such as achieve-ment growth in mathematics, Catholicschools do appear to be more meritocraticthan public schools (Bryk, Lee, and Holland1993; Morgan 2001; Morgan and Sørensen1999). This finding can be explained, in largepart, by the fact that Catholic schools havemore inclusive track placements (Gamoran1996), at least in mathematics, and smallerdifferences in achievement gains across tracks(Bryk et al. 1993).

Specific findings on the black-white gap incourse taking in Catholic schools have beenmore equivocal. In two analyses of race andtrack placement in Catholic and publicschools using data from the HS&B, V. E. Leeand Bryk (1988) showed that there may bepublic–private sector differences in the effectsof race on track placement, but the differ-ences were relatively small and difficult todetect. In a sample of 12 public and Catholicmiddle schools, Hallinan (1992) found thatthe negative effects of race were confined tothe public sector, but again the differencesbetween sectors were small and insignificant.It is interesting that Gamoran (1992) foundnegative effects for racial minorities in a smallsample of public and private school districtsthat claimed to have purposefully meritocrat-ic selection criteria. Racial inequality in coursetaking may be lower in Catholic than in pub-lic schools, but so far the findings have beeninconclusive.

In summary, prior research has identifiedthe segregation of black students in low-trackclassrooms as a major source of educationalinequality. However, it is unclear whetherinequality in course taking is due primarily tosegregation at the school level, where pre-dominantly black schools are less inclusive, orthe segregation of blacks and whites withinintegrated schools. Indeed, some researchhas found that predominantly black schoolsare actually more inclusive, suggesting thatinequality in course taking is primarily a with-in-school phenomenon. Actions of schoolpersonnel and students’ agency may explainwhy inequality in course taking among blacksand whites occurs primarily within integratedand predominantly white schools. Racialcomposition is a potentially important factor

52 Kelly

affecting inequality within schools. Moreover,research on inclusiveness in Catholic schoolshas suggested that inequality in course takingoccurs primarily in the public sector.

My study investigated the contextual fac-tors that influence inequality in course takingwithin and between schools. In this article, Ibegin by investigating whether the well-established course-taking differences amongblack and white students can be attributedsimply to individual differences in academicachievement or other factors. I examine amore reliable measure of course taking thanin previous studies, a five-category, transcript-coded measure of course taking. Second, Iinvestigate course-taking patterns in predom-inantly black schools, independent of individ-ual-level factors. Most research has ignoredhigh levels of black-white segregation and theeffect that school-level differences may haveon course-taking opportunities. Third, I deter-mine whether course enrollments amongblack and white students within a school varyas a function of the school’s racial composi-tion, which has not been explicitly testedbefore. Finally, I examine whether inequalitiesin black-white course taking vary acrossschool sector.

DATA AND METHODS

I used the National Education LongitudinalStudy of 1988 (NELS:88), which began col-lecting data on 8th graders during the1987–88 school year, to examine course-tak-ing patterns among white and black students.In the analyses, I included data from the 1988student, school administrator, and parent sur-veys, along with the 1990 student and schooladministrator surveys and the high schooltranscript file. I used measures of achieve-ment and other indicators in the 8th grade topredict mathematics course taking in the 9thand 10th grades in a longitudinal analysis.

Of the 17,424 students from 1988 whowere enrolled in school in both 1988 and1990, 16,489 were selected to participate inthe transcript study, of whom only 14,283, or86.6 percent, actually participated (NationalCenter for Education Statistics, NCES, 1995).2Of these cases, 13,548 had transcript data for

both the 9th and 10th grades, as well as keyachievement and socioeconomic data. I usedthese 13,548 students for Models 1–6 (Tables3 and 4), which do not include multilevelinteraction terms. Figure 1, which relies onmultilevel interaction terms from Table 4describing the effects of school racial compo-sition, uses a subset of schools attended byboth black and white students in the NELS:88sample: 5,000 students in 367 schools. Thesample loss for those three coefficients (inbold face in Table 4) occurs naturally becausea within-school effect of being black cannotbe estimated in a school in which no blackstudents are enrolled. The results on theeffects of school racial composition areintended to generalize only to schools attend-ed by both white and black students.Appendix A presents the means and standarddeviations of the variables that were used.

Dependent Variable

I used an indicator of sophomore-year coursetaking (mathematics sequence) as the depen-dent variable. Mathematics sequence is anordinal variable with five categories thatcodes the student’s sophomore-year mathe-matics sequence. This coding scheme buildson the work of Stevenson et al. (1994), whodeveloped a method of identifying a singlecourse sequence for sophomores using tran-script data from the 9th and 10th grades.Using the Classification of Secondary SchoolCourses (NCES 1982), I assigned individualcourses one of five codes describing the con-tent and complexity of the course work. Ithen assigned students to a unique mathe-matics sequence measuring the level of math-ematics course work taken by the 10th grade,on the basis of the combination of classestaken in the 9th and 10th grades. For exam-ple, a student who took Geometry in the 9thgrade and Algebra II in the 10th grade wouldbe assigned to Algebra II and Geometry.3Students can be assigned to an orderedmathematics sequence even if they were notenrolled in a mathematics course during theirsophomore year. A student who took AlgebraI as a two-year sequence would be in thesame mathematics sequence as a studentwho took Algebra I as a 9th grader but no

The Black-White Gap in Mathematics Course Taking 53

mathematics class as a 10th grader. The low-est sequence is Less than Algebra I, followedby Algebra I; Algebra II or Geometry, but notboth; Algebra II and Geometry; and, finally, thehighest category, Greater than Algebra II orGeometry. The major difference between thisscale and the one used by Stevenson et al. isthat additional mathematics sequences werecoded for the upper and lower ends of thescale (for a full description of the coding pro-cedure that was used, see Kelly 2004).

Course-based indicators like the one I usedhave several properties that make thempreferable to more subjective, student- orteacher-reported indicators. First, manyschools do not formally track students com-prehensively across all subjects (Moore andDavenport 1988). It may make more sense touse student-reported indicators on only asubset of schools with comprehensive formaltracking systems. Second, the typical student-reported indicator has only three categories(e.g., vocational, general, and academic inHS&B and NELS:88); any variation in struc-tural location beyond the three-category dis-tinction is therefore lost. Third, student-reported measures confound the within-school and between-school component oftrack placement (Lucas 1999). In otherwords, the comparison groups being evokedby the respondent are not known. For exam-ple, a student may consider trigonometry tobe a high-track course on the basis of theother courses available in his or her school,whereas in an elite school, trigonometry maybe considered a general-track course. Finally,comparisons of student-reported and course-based indicators have shown that variation inreporting is nonrandom across social groups(Lucas and Gamoran 2002). Different resultsmay be obtained from the two types of indi-cators of track placement. For example, theSpearman rank-order correlation coefficientfor the 11,560 NELS:88 cases with data onboth the mathematics sequence indicatorand the traditional student-report measure ofoverall track placement is only .396.

There are two reasons why the NELS:88data are well suited to examining mathemat-ics sequences in particular. First, school track-ing policies may be specific to individual sub-jects. Mathematics course sequences are rela-

tively easy to code without explicit knowl-edge of each school’s tracking structurebecause of relatively high levels of standard-ization in the naming and content of courses.Second, the mathematics achievement testsare the most reliable of the achievement testsin NELS:88.4

Why examine 10th-grade course taking?Course taking is important throughout highschool, but it is especially important to consid-er course taking in the first two years becausethese years set the stage for a student’s eventu-al mathematics attainment. Mathematicscourse taking in high school, especially after the9th grade, is based largely on prerequisites.Because summer school is offered primarily asremediation, rather than as a vehicle forupward mobility, it is difficult for students totake the mathematics prerequisites they needto move up the track ladder. Few studentsexperience upward mobility in mathematicsafter their sophomore year. Lucas (1999) esti-mated that only about 12 percent of studentsare upwardly mobile in mathematics betweentheir sophomore and senior years. Thus, it isimportant that students get off to a good startin mathematics, taking the most rigorouscourse work they are qualified to take. In addi-tion, attrition between the 10th and 12thgrades because of students’ transfers, dropout,and other forms of nonresponse makes analyz-ing course taking in later years more difficult.This property is particularly important when themodels require reliable estimates of within-school effects, which necessitates having ade-quate within-school sample sizes.

Independent Variables

To account for differences in students’achievement, I took measures from theeighth-grade student file, including testscores in mathematics, English, and historyand grades in mathematics. Initially, I consid-ered the full set of grades and test scores ineach subject. However, since these indicatorswere highly collinear, I eliminated theachievement variables that were insignificantor inversely related to course taking once themathematics test scores and grades wereaccounted for. I controlled for prior trackplacement by using a student-report indica-

54 Kelly

tor of the ability level of the mathematicsclass taken in the eighth grade. Since place-ment decisions are influenced by the coursesthe student has taken in the past, this is a nec-essary part of the model.

Parental background variables included a6-category variable coding for parent’s high-est educational level, which was transformedto be linear in years of education; a 15-cate-gory variable coding for family income,coded to the category midpoints in $1,000units and treated as linear; and the Duncansocioeconomic index (SEI) score for 12 occu-pational categories. However, the SEI scorewas dropped from the final models because itwas insignificant once education and incomewere considered. Dummy variables were alsoincluded for the marital status of the parent.McLanahan and Sandefur (1994) illustratedthe importance of family structure for a vari-ety of educational outcomes. Missing dataprocedures are reported in Appendix B. Ingeneral, multiple measures were adminis-tered for different constructs broadly defined,so the impact of missing data is somewhatattenuated (e.g., family background or math-ematics achievement). Cases missing all dataon a construct were dropped.

Measures of the racial composition of stu-dents in schools came from administrators’reports. Because of the within-school samplesizes in NELS:88, administrators’ reports aremore reliable than are aggregated studentdata. Missing data on these variables wereimputed from aggregate data on students—2.7 percent of students in the case of schoolpercentage black. Percentage free lunch wasincluded as a proxy for school SES, which hasbeen shown to influence the offering ofmathematics courses (Useem 1992). Schoolmean income and parental education werealso included in some models. Dummy vari-ables denote urban and rural schools relativeto suburban schools. Sector variables wereincluded comparing Catholic, private reli-gious, and private nonreligious schools withpublic schools. A single private school ofunknown religiosity with four students wasincluded in the omitted category, but did notappear in the final analysis because all fourstudents were white.

Modeling Strategy

To analyze the effects of race, academic back-ground, and school racial composition andsector on mathematics course taking, I usedmultilevel models for categorical outcomes.According to the logic of the models, stu-dents’ course enrollments are assumed to bedetermined by two factors: the school theyattend and their position within that schoolrelative to other students. Course taking inthe 9th and 10th grades is modeled as a func-tion of individual-level variables in the 8thgrade and school-level variables in the 10thgrade. Using hierarchical linear modeling(HLM6) software (Raudenbush, Bryk, andCongdon 2005), I estimated a series ofordered logistic regressions.

Ordered logit models, in this case, multi-level ordered logit models, are an extensionof simple logistic regression models. In bothcases, the models assume there is an underly-ing latent continuous outcome that mapsonto the observed categorical outcomes,where any variable in the model affects thevalues of the latent continuous outcome and,hence, the observed categorical outcome. Forordered logit models, the dependent variablehas multiple ordered outcomes (m = 1, . . . ., M). To develop a single regression model,ordered logit models estimate cumulativeprobabilities (Prob R ≤ m), for example, theprobability that the outcome is less than orequal to a given category of the dependentvariable. In addition to the traditional regres-sion parameters (β) for each independentvariable i, ordered logit models estimate M-1“threshold” parameters (θm). Thus, the finalmodels specify the cumulative log odds ofattaining category m for a given value of Xi asa logistic regression equation of the form(Raudenbush and Bryk 2002:319):

ηmi = θm + βXi.

Thus, just as in a simple logistic regression,the regression parameters (β) refer to anincrease or decrease in the latent outcome.For the ordered logit model, this can bethought of as a generic increase in the proba-bility of attaining a higher category of thedependent variable, with the precise probabil-

The Black-White Gap in Mathematics Course Taking 55

ities dependent on which threshold is beingconsidered. As in a simple logistic regressionmodel, the expected log odds of an outcomefor two cases, one where X = X1 and onewhere X = X2 is just a function of the value ofthe independent variable X and the estimatedcoefficient β:

ηm1 – ηm2 = β(X1 – X2).

The ordered logit specification is well suit-ed to modeling mathematics course taking,which is an ordered outcome. The mathe-matics sequences that I coded are not nomi-nally related as different types of vocationalcourses may be; rather, they are ordered fromremedial or low mathematics sequences toadvanced or high mathematics sequences.The ordered logit models capture this verticalordering in the dependant variable and, byestimating a series of threshold parameters,relax the assumption that the data areordered on equal intervals as in an ordinaryleast-squares (OLS) regression. In addition,ordered logit models are much more efficient(McCullagh 1980; Whitehead 1993) and par-simonious than a set of simple logistic modelsfor each transition. Preliminary model build-ing using binary logit estimates suggestedthat the track- placement process is similar foreach category of the dependent variable andsatisfied the proportional odds assumption. Ifthis were not the case, a set of separate logis-tic models would be preferred. It turns out,though, that certain students’ traits—likemathematics achievement, prior track place-ment, and family background—always oper-ate in the same direction. For example, beingin a high mathematics track in the eighthgrade has a strong positive effect on bothavoiding placement in the lower tracks andobtaining placement in the higher tracks.

Like simple logistic regression models, theordered logit coefficients and thresholds canbe used to calculate predicted probabilities toassess the impact of independent variables onthe dependent variable. In discussing effects,I use discrete change calculations (Long1997:135), cumulative probabilities calculat-ed with respect to a specific category of thedependent variable such that explicit proba-bility comparisons can be made (e.g., the

probability of taking Algebra II and Geometryor higher among blacks and whites).5 I referto the probability of being in the top twomathematics sequences because these cours-es are generally considered “elite collegepreparatory.” In the final models (Table 4:Models 6a–c), I present coefficients from sim-ple binary logit models (Models 6b and c) inaddition to the ordered logit models (Model6a). These models provide further details onthe probabilities of attaining a specific math-ematics sequence among blacks and whites,but are less efficient than the ordered logitmodels.

To produce unbiased estimates of the pop-ulation parameters of the relationship amongschool racial composition, sector, and theblack-white gap in mathematics course tak-ing, school weights were used to adjust forthe NELS88 sample design. Unfortunately,exact school weights based on the inverseprobability of selection from the universe ofsecondary schools in 1990 are not available.NELS did not sample schools as units of analy-sis in 1990; rather, the 1988 8th graders werefollowed to their schools in 1990. I construct-ed approximate school weights by aggregat-ing school weights from the base year to thefirst follow-up schools.6 Unweighted esti-mates are similar to those reported here andare available from me on request.

Unless otherwise noted, all models useuncentered achievement and family back-ground variables, such that compositionaleffects of race are estimated directly(Raudenbush and Bryk 2002). The Level 1racial variables are school mean centeredusing HLM’s group mean-centering com-mand. Thus, at Level 1, students of differentracial/ethnic backgrounds are comparedwithin the same schools. Administrator-report-ed school racial composition is included as apredictor of course taking at the school levelbecause it is more accurate than the random-ly selected sample proportions. All reportedcoefficients are unit specific.7 All student-levelcoefficients are constrained to have the sameeffects across schools (i.e., they are “fixed,” inHLM terminology), except for the coefficientfor black students in Table 4, where multilevelinteractions are estimated. An examination ofQ-Q plots from OLS models confirmed that

56 Kelly

the distributional assumptions of the modelshold; Level 2 residuals are normally distrib-uted, and there was no evidence of outliers.8

RESULTS

To what extent can differences in course tak-ing among black and white students beattributed to differences in academic achieve-ment or other factors that are associated withindividual students, such as family back-ground? Table 1 reports the baseline differ-ences in course taking between blacks andwhites. White students are almost twice aslikely to be in the top two mathematicssequences as are black students (22.1 percentversus 11.9 percent). Whites are also muchmore likely to avoid placement in the lowestmathematics sequences (35.3 percent versus56 percent). Model 1 in Table 2 shows thereduced-form estimate of the black-white gapin mathematics course taking within schools;the results are similar to those in Table 1 butare expressed as a logistic regression coeffi-cient (-.84) that captures only within-schoolinequality. Once achievement and prior trackplacement are controlled for in Model 2, theblack-white gap is greatly reduced, but thereis still some small disadvantage for blacks.Model 3 shows that after family backgroundis controlled, the difference diminishes almostto zero and is no longer statistically signifi-cant. Consistent with prior research, the

answer to Research Question 1 is that, onaverage, there is no black-white gap in math-ematics course taking after test scores,grades, prior track placement, and SES aretaken into account.

In Table 2, school-to-school differences incourse taking were set aside. Is some of thelarge baseline difference between course tak-ing among blacks and whites in Table 1attributable to course-enrollment patterns atthe schools that black students attend?Approximately 7 percent of the schools in thesample (76 of 1,087) were predominantlyblack (more than 60 percent black). Another19 percent were predominantly nonblack(less than 15 percent black), and the remain-ing 74 percent were relatively integrated (15percent–60 percent black).9 Table 3 reportsregression estimates of the effect of attendingschools of different racial compositions beforeand after school- and individual-level vari-ables were controlled.10 Model 4 in Table 3confirms that at least descriptively, fewer stu-dents are enrolled in upper-track mathemat-ics courses in predominantly black schools.Further calculations revealed that in predom-inantly nonblack schools, 21.6 percent of stu-dents are in one of the top two mathematicssequences. In predominantly black schools,only 17 percent of the students are in thesesequences.

However, the conclusion is quite differentwhen the characteristics of the students whoattend schools with different racial composi-

Table 1. Cell Frequencies of the Dependent Variable Among Students (N = 13,548; per-centages in parentheses)

Mathematics Sequence Black Students White Students All students

5 Greater than Algebra II or Geometry 46 (3.89%) 607 (6.21%) 855 (6.31%)

4 Algebra II and Geometry 95 (8.02%) 1,553 (15.9%) 2,022 (14.92%)

3 Algebra II or Geometry,but not both 380 (32.09%) 4,164 (42.63%) 5,414 (39.96%)

2 Algebra I 334 (28.21%) 2,021 (20.69%) 2,996 (22.11%)

1 Less than Algebra I 329 (27.79%) 1,423 (14.57%) 2,261 (16.69%)

The Black-White Gap in Mathematics Course Taking 57

tions are considered. After the individual-levelvariables, student achievement, prior coursetaking, and family background, as well as sev-eral school-level variables in Model 5, areadjusted for, predominantly black schoolsactually have a higher level of mathematicscourse taking (as in Lucas and Gamoran2002). A calculation using the coefficientsfrom Model 5 highlights the inclusive natureof predominantly black schools. If all the stu-dents in a school were average (and otherschool-level variables were held constant),then increasing the proportion of black stu-dents from 5 percent to 90 percent in anurban public school would more than doublethe proportion of students in the top twomathematics sequences, from 7.3 percent to24 percent.

Considering the results of Tables 2 and 3together, one begins to see why manyresearchers have found null, or even positive,effects of race on track placement in single-level models. The schools that black studentsattend have an inclusive approach to coursetaking in mathematics, with a greater numberof students than would be enrolled at a pre-dominantly white school with students ofsimilar achievement levels and backgrounds.Because many black students attend predom-inantly black schools, they benefit from inclu-sive course taking in mathematics.

In past research, confounding the effectsshown in Tables 2 and 3 did not lead to acompletely erroneous conclusion aboutinequality in course taking because the effectsare not off-setting. The effect in the student-

Table 2. The Effects of Race, Social Class, and Academic Achievement on MathematicsCourse Taking (Mathematics Sequence): Ordered Logit Regression Coefficientsa

Models

Variable (1) (2) (3)

Interceptb -1.72 (.028)*** 2.17 (.14)*** 3.36 (.16)***

Black -.84 (.087)*** -.23 (.092)* -.16 (.093)

Hispanic -.63 (.075)*** -.077 (.079) .027 (.079)

Asian .69 (.093)*** .56 (.097)*** .52 (.098)***

Other -.70 (.13)*** .39 (.14)** -.30 (.14)*

Male -.13 (.031)*** -.17 (.034)*** -.19 (.034)***

Math grades -.42 (.020)*** -.42 (.020)***

Math test .083 (.0025)*** .077 (.0025)***

English test .035 (.0032)*** .031 (.0032)***

History test .042 (.0057)*** .033 (.0058)***

Prior track placement (low) -.51 (.076)*** -.57 (.077)***

Prior track placement (high) .82 (.042)*** .82 (.043)***

Stepparent -.15 (.065)*

Intact family .11 (.046)*

Parental education .12 (.0083)***

Family income .003 (.0005)***

*p < .05. **p < .01. ***p < .001.a HLM models: Within-school (Level-1) coefficients, 13,548 students in 1,087 schools.

Racial/ethnic coefficients are school mean centered; all other coefficients are uncentered.Standard errors are in parentheses.

58 Kelly

level model is null and the effect at the schoollevel is positive, leading to a small positiveeffect when modeled as a single coefficient. Amore serious problem, though, is treatingblack and white students as if they all attend-ed a hypothetical “average” school. As washypothesized in Research Question 3, coursetaking among black and white students with-in schools may be influenced by school con-text. Specifically, black students may be at adisadvantage in predominantly white schools.

Table 4 reports a combined between- andwithin-school analysis that allows the effect ofrace on mathematics course taking to varyacross schools. Both an ordered logit modeland binary logit models of high- and low-track placement are presented. The orderedlogit model provides a good summary ofinequality in course taking across the full 5-category range of mathematics sequences.The coefficient for black students in Model 6a(-.37) reveals that black students are, in fact, ata course-taking disadvantage in predominantly

white schools.11 This disadvantage graduallydisappears as the school becomes more inte-grated and eventually predominantly black.In these data, calculations from Model 6areveal that the disadvantage of black studentsremains until the enrollment of black studentsreaches about 59 percent. That calculationrefers to a hypothetical effect where differ-ences in academic background and othervariables in Model 6a are set aside.Subsequent calculations examined the effectof school racial composition while maintain-ing individual differences in academic back-ground. Models 6b and 6c focus on specificmathematics sequences, avoiding placementin the lowest sequences, or obtaining a spotin one of the top two sequences. The per-centage black effect in Model 6b (.010) andModel 6c (.020) is consistent with theordered logit models. However, owing to theunreliability in the black slope and the lowerstatistical power of the simple logistic models,the effect in Model 6b is not statistically sig-

Table 3. The Effects of Race, Social Class, Urbanicity, and Sector on School-Level AverageMathematics Sequence: Ordered Logit Regression Coefficientsa

Model Adjusted for Model Unadjusted for Student-Level

Student-Level Effects in Effects Model 3

Variable (4) (5)

Intercept -1.69 (.026)*** 4.12 (.30)***

Percentage black -.0029 (.0009)*** .0060 (.0011)***

Percentage Hispanic .0040 (.0009)*** .0050 (.0011)***

Percentage free lunch .0037 (.0012)**

School-mean income .0022 (.001)

School-mean parental education .055 (.022)*

Urban .060 (.055)

Rural -.26 (.042)***

Catholic .57 (.070)***

Private, other religious .63 (.11)***

Private, nonreligious .29 (.11)**

*p < .05. **p < .01. ***p < .001.a HLM models: Level-2 coefficients, 3,548 students in 1,087 schools. Standard errors are in

parentheses.

The Black-White Gap in Mathematics Course Taking 59

nificant.12 The effect of attending a Catholicschool (2.09 in Model 6c) on the course tak-ing of black and white students is evenstronger than the effect of racial composition.However, the Catholic school effect is restrict-ed to course taking in the highest mathemat-ics sequences. Supplementary analysesrevealed that a similar effect does not hold forother private schools.

Figure 1 depicts white and black students’chances of being in one of the top two math-ematics sequences in public and Catholicschools with various racial compositions usingthe coefficients from Model 6a. To producethese probability estimates, a specific valuemust be entered for each independent vari-able in the model. For each group of students(white and black), the probabilities are evalu-ated assuming that they have the averageattributes of the students in their samplegroup.13 Figure 1 is a complex graph becauseit shows many findings simultaneously. First,it highlights the large black-white gap inmathematics course taking in public andCatholic schools, regardless of the racial com-position of the schools, which is due to indi-vidual factors, including academic and family

background. Figure 1 also illustrates differ-ences in inclusiveness across school sector;both blacks and whites attending publicschools have lower absolute chances of beingin an upper-track mathematics sequence thanif they attended a Catholic school. Finally, ifthe models are correctly specified, the graphalso depicts the effect of different schoolracial compositions in each sector.

In public schools, the relative advantage ofwhites decreases as the racial compositionchanges, becoming predominantly black,although it is difficult to tell in the graphbecause the effect is small relative to the largeeffect of individual-level variables. In a publicschool where 20 percent of the students areblack, an average black student has about a1.8 percent chance of being in one of thehighest two mathematics sequences, com-pared to about an 8.1 percent chance for anaverage white student. At a public schoolwhere 40 percent of the students are black,the probability of the average black studenttaking a high-track mathematics course isabout 33 percent higher, at 2.4 percent,whereas the probability for the average whitestudent increases by only about 10 percent,

Figure 1. Mathematics Sequence Placement as a Function of School Racial Composition(probabilities calculated from Model 6a).

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60 Kelly

Table 4. The Black-White Gap in Mathematics Course Taking as a Function of School RacialComposition and the Catholic School Effecta

Logit Model of Logit Model ofOrdered Logit Low-Track High-Track

Model Course Takingb Course Takingc

Model (6a) (6b) (6c)

Between-School ModelIntercept 4.02 (.30)*** -7.02 (.93)*** -8.21 (.97)***Percentage black .0062 (.0011)*** .0049 (.0034) .009 (.0046)*Percentage Hispanic .0050 (.0011)*** .0044 (.0036) .0075 (.0045)Percentage free lunch .0037 (.0012)** .0061 (.0041) .0064 (.0066)Urban .058 (.055) -.092 (.16) .18 (.21)Rural -.26 (.042)*** -.18 (.14) -.48 (.17)**Catholic .57 (.070)*** 1.32 (.30)*** -.25 (.32)Private religious .64 (.11)*** .70 (.44) .32 (.37)Private nonreligious .29 (.11)** .45 (.41) .16 (.35)

Within-School ModelBlack

Intercept -.37 (.11)** -.36 (.16)* -.65 (.33)*Percentage black effect .0084 (.0037)* .010 (.0054) .020 (.0087)*Catholic school effect .88 (.34)* .13 (.64) 2.09 (.60)***

Hispanic -.022 (.080) -.13 (.13) .15 (.16)Asian .54 (.098)*** .49 (.18)** .51 (.16)**Other -.32 (.14)* -.16 (.30) -.07 (.41)Male -.19 (.035)*** -.21 (.074)** -.10 (.083)Math grades -.41 (.020)*** -.54 (.045)*** -.21 (.069)**Math test .081 (.0026)*** .082 (.0054)*** .082 (.0070)***English test .029 (.0032)*** .034 (.0066)*** .0093 (.0076)History test .034 (.0058)*** .048 (.013)*** .014 (.016)Prior track place (low) -.59 (.078)*** -.58 (.20)** -.13 (.31)Prior track place (high) .84 (.043)*** .60 (.10)*** 1.27 (.12)***Stepparent -.006 (.066) -.060 (.12) -.15 (.17)Intact family .22 (.047)*** .26 (.089)** .037 (.12)Parental education .090 (.0092)*** .088 (.017)*** .073 (.018)***Family income .001 (.0006) .002 (.001) .000 (.001)

*p < .05. **p < .01. ***p < .001.a HLM models. Multilevel interactions estimated with 5,000 students in 367 schools.

Racial/ethnic coefficients are school mean centered; all other coefficients are uncentered,except the percentage black effect (shown in bold), which is grand mean centered in the mul-tilevel interaction. Standard errors are in parentheses.

b Model of avoiding placement in Sequence 1 or 2, Algebra I or less.c Model of placement in either Sequence 4 or 5, Algebra II and Geometry or higher.

The Black-White Gap in Mathematics Course Taking 61

to 9.1 percent.14 White students’ net advan-tage, regardless of the racial composition ofthe school, stems from higher levels ofachievement, prior course taking, and moreadvantaged family backgrounds.In Catholicschools, because black and white students’placements in mathematics sequences aremore similar to begin with, the gap decreas-es substantially as school racial compositionincreases. In Catholic schools, effects areshown until the school reaches 75 percentblack, the highest percentage of black stu-dents in the Catholic schools that were sam-pled.

DISCUSSION

I began this analysis by posing four researchquestions about differences in course takingbetween blacks and whites and the effects ofschool context on course taking. With respectto the first question, the black-white gap incourse taking in mathematics can indeed beexplained primarily by differences in academ-ic and family background upon entry to highschool. However, black students are at aremaining disadvantage, after other factorsare controlled, in specific school contexts.The second research question concerned theoverall pattern of course taking, on average,in predominantly black schools. I found that astudent’s chances of being enrolled in a high-track mathematics course are actually greaterin predominantly black schools than in non-black and integrated schools. The thirdresearch question concerned course takingamong students within the context of pre-dominantly white schools. It is within pre-dominantly white public schools that blackstudents are disadvantaged. The fourthresearch question involved sector differencesin the black-white gap in course taking. I didnot find a similar disadvantage in predomi-nantly white Catholic schools. Moreover, theaverage level of mathematics course takingamong all students was higher in Catholicschools.

It is important to remember that by thetime students reach the eighth grade, thestarting point in this analysis of mathematicscourse taking, they have already had diverse

opportunities to learn, and their levels of aca-demic achievement vary widely. In manycases, the academic achievement of disad-vantaged students and minorities has alreadybeen “deflected” by the schools and coursesthey have encountered; the initial differencesamong students have been magnified(Kerckhoff 1993). Entering high school withlower levels of academic achievement and ahistory of less rigorous course taking is detri-mental to many black students becauseachievement and prior course taking are suchimportant predictors of the courses that stu-dents will be enrolled in by their sophomoreyear. The problem is further exacerbated bythe fact that students from lower-SES familiesexperience additional course-taking disadvan-tages. Beyond these explanations, pastresearch has found no glaring evidence ofracial inequality in course taking betweenblacks and whites per se.

However, because of high levels of black-white segregation at the school level and thefailure to model between- and within-schoolcourse taking among blacks and whites prop-erly, the models used in past studies may bemisleading and obfuscate the black-whiteinequality in course taking in predominantlywhite schools. There appears to be a connec-tion between the racial composition of aschool and the chances of black and whitestudents enrolling in high-track mathematicscourses. The link between school-racial com-position and course-taking opportunitieswithin schools deserves further study. A sig-nificant weakness of this study was that theobserved black-white gap within schools wasnot measured reliably because of the relative-ly small sample of students within schools.Further analyses of databases that containlarger samples of students within schools ofvarious racial compositions are needed.

Furthermore, since the early 1980s, theaverage number of mathematics coursestaken by high school students has increased.Planty, Provasnik, and Daniel (2007) reportedthat the percentage of graduates who com-pleted a semester or more of Algebra II rosefrom 40 percent in 1982 to 67 percent in2004. It is certainly possible, then, that theincreasing focus on maximizing students’ per-formance on standardized tests has led

62 Kelly

schools to reduce the stratification of coursetaking, opening opportunities for minorityand disadvantaged students to enroll indemanding college preparatory courses. Thisanalysis may present an overly pessimisticportrait of course-taking opportunities forblack students in today’s educational system.

However, other recent research on black-white course-taking patterns has noted thatthe total segregation of blacks and whites inU.S. schools has often been underestimatedbecause of the glaring levels of within-schoolsegregation (Clotfelter et al. 2003; Mickelson2001a). Even when black students attendintegrated schools, they face resegregationwithin these schools. The analysis presentedhere suggests that this resegregation goesbeyond what would be expected purely onthe basis of academic achievement and evenbeyond what is faced by lower-SES white stu-dents, who are themselves segregated in low-track classrooms. Course taking has powerfuleffects on students’ growth in achievementand other important educational outcomes,and these effects hold even when rigorousmethods to control for selection bias are used(Carbonaro 2003; Gamoran and Mare 1989).Thus, the resegregation that occurs withinintegrated schools undermines the goal ofschool integration, which is to provide diversestudents with effective learning environ-ments. As students continue their educationand enter the workforce, the black-whiteinequality in mathematics course taking willhelp perpetuate existing inequalities in edu-cational and occupational attainment.

The finding that predominantly blackschools had higher average levels of coursetaking in mathematics than would have beenpredicted by students’ achievement levels isconsistent with other research (Lucas andGamoran 2002). How should this finding beinterpreted? Upper-track mathematics cours-es in predominantly black schools may not becomparable, in terms of curriculum andinstruction and learning outcomes, to thosein other schools. In understanding a predom-inantly black school with demanding course-taking requirements, Metz (1989) reportedthat not all the content of the coursesmatched the course titles and that studentstook courses without having fully completed

the typical prerequisites. Metz argued thatthe rigorous official curriculum that wasenforced by administrators upheld the imageof a “real school” and helped maintain a pos-itive social identity for the teachers, staff, andstudents. Further research is needed to docu-ment whether predominantly black schoolsreally do have higher average levels of coursetaking, given students’ achievement levels,and whether the nature of instruction and thegrowth in achievement in upper-track classesis different in predominantly black schools.Yet even if a “real school” phenomenonexplains the results of this study, students inpredominantly black schools may still benefitfrom an inclusive approach to enrollment inupper-track courses. In most cases, the addi-tional growth in achievement that is associat-ed with high-track courses cannot be attrib-uted purely to instructional effects (Pallas etal. 1994). Moreover, academic course takingplays a strong role in the college admissionsprocess, especially in mathematics, and stu-dents may benefit in that way as well(Adelman 1998).

Past analyses of course taking amongblacks and whites pooled data across schools,districts, and even states, but course-takingpolicies are implemented and often designedat the school level. If further research showsthat the findings of this analysis are robust,then there is an important policy implication:Educators can address racial inequalities incourse taking by designing and implement-ing course-placement procedures within inte-grated schools. Educators have been reluc-tant to abandon the practice of curriculumdifferentiation, which is at the core of thesocial organization of U.S. schools. But evenas the core practice of curriculum differentia-tion continues, changes in its implementationmay reduce educational inequality.

I have suggested two possible sources ofcourse-taking inequality among students:decisions and recommendations by teachersand school officials in the context of subjectiveplacement criteria and the agency of studentsand parents in different school contexts.15

This analysis did not investigate whether theobserved relationship was actually caused bythese mechanisms. But whether discrimina-tion, students’ choices, or some other social

The Black-White Gap in Mathematics Course Taking 63

force is at work, these mechanisms occur with-in the context of specific school policies. Atthe school level, a host of formal rules governcourse taking, from requirements for prerequi-sites and corequisites to grades and test scoresto teachers’ recommendations (Kelly 2007).The formal rules of course taking are furtheraugmented by guidance procedures that helpallocate students into course sequences.Further research is needed to investigate whatspecific placement policies at the school levelact to perpetuate or interrupt inequality inblack-white course taking.

NOTES

1. Being in a predominantly white schoolmay enhance the salience of racial identitiesfor black students, but it may also reduce thesalience of racial identities for white students,subsequently reducing biased behavior. In ameta-analysis of 137 studies on in-group bias,Mullen, Brown, and Smith (1992) found thatthe majority group was likely to express muchless in-group bias as the group increased pro-portionately in size. The relationship betweenbias and the relative size of the in-group maycounteract some of the effect of social learn-ing that I posit leads to biased behavior bywhites in predominantly white schools.

2. Nonresponse occurred primarily at theschool level; few students explicitly refused toparticipate in the transcript portion of thestudy.

3. Almost all schools in the sample offeredthe full spectrum of mathematics courses forsophomores. Data from the NELS:88 adminis-trator survey indicate that 2 percent of thestudents who were sampled attended schoolsin which Algebra II was not offered. However,there are multiple labels for similar material,including Technical Mathematics andMathematics 2 Unified. Because the adminis-trator survey asked only about the most com-mon course label, Algebra II, this indicator isunreliable, and 2 percent is likely to be anupper-bound estimate of students attendingschools that did not offer Algebra II..

4. The total reliability (across all races andethnicities) is .90 for mathematics, but only.84 for reading and .75 for science.

5. Raudenbush et al. (2005) providedannotated sample calculations of predictedprobabilities in the ordered logit frameworkin the Help module of HLM6 (see “teacherdata: ordinal model” under the “GeneralizedLinear (HGLM) examples” section).

6. This procedure produced schoolweights whose distributions have the samerange as the base-year school weights inNELS (1.54–387.3 or 1–251), which is similarto the 10th-grade school weights used in theHigh School Effectiveness Study (range of1–360). However, the variation across schoolswas slightly compressed, as would be expect-ed as multiple middle schools feed into a sin-gle high school. The constructed 10th-gradeschool weights have a mean of 33.09 and avariance of 1,463 compared to a mean of37.46 and a variance of 2,109 for the base-year school weights. Final weights were nor-malized to a mean of 1.

7. HLM6 produces both unit-specific andpopulation-average coefficients for models ofcategorical outcomes, where the unit-specificoutcome is more closely analogous to stan-dard output in continuous models. These esti-mates are used here to compare the effect onhypothetical individual students of changingschool racial compositions.

8. At Level 2, school percentage black isused in its original metric (0–100) and has askewed distribution. However, as indicated bythe Q–Q plots, use of the untransformed scaleof school racial composition does notadversely affect the model estimates and pre-serves meaningful variation in the data.Because of the relatively small number of stu-dents who were sampled within schools,every effort was made to preserve statisticalpower by maintaining variation in the inde-pendent variables.

9. I chose unbalanced cutoffs for “pre-dominantly” black and white schools becauseresearch has shown that minority concentra-tions of only 10 percent–30 percent ofteninfluence behavior, such as the choice of aresidential neighborhood (B. A. Lee andWood 1991). Thus, a school with 60 percentblack students is likely to be perceived bywhites as “substantially” or “importantly”minority.

10. The reliability of the estimates of the

64 Kelly

school-level intercepts, which are based onthe sample size for each school, is .71 onaverage.

11. This set of coefficients is grand meancentered at the average racial composition ofthe schools in the sample (14.55 percentblack). At first glance, the statistically signifi-cant negative effect may appear to contradictthe estimates in Model 3. However, manyblack students attend segregated schools,and the models in Table 4 allow the effects ofbeing black to vary across schools. If theschool racial composition were centeredaround the mean for black students, which ismuch higher at 44.8 percent black, the inter-cept would approximate that of Model 3.

12. Wald tests comparing the black coeffi-cients in adjacent binary logit modelsrevealed that the parallel regression assump-tion for the black coefficient holds acrossmathematics- sequence transitions (Prob > F= .99, .72, .94, .16). Prior analyses usingunweighted models suggested that the effectmay be somewhat stronger in the highermathematics-sequence transitions.

13. An alternative way to depict theseeffects would be to give white and black stu-dents the same values on the other indepen-dent variables in the model—namely, acade-mic achievement and test scores. But ignor-ing differences in achievement and prior trackplacement in this way would not do justice tothe glaring inequalities that students actuallyface. Black students do begin high schoolwith lower test scores, and unfortunately bythe eighth grade, there is already a large gapin mathematics course taking. To depict theeffect of sector independent of the studentswho attend public instead of Catholicschools, the average Level 1 attributes forwhite and black students, respectively, arepooled across sectors. At Level 2, this simula-tion assumes that the effect of school racial

composition and the Catholic school effectoccur independently of the other Level 2 vari-ables. Percentage free lunch and percentageHispanic are held at their overall samplemeans. The urban school category, in whichmany of the black students in the sample fell,was used.

14. Probabilities change “faster” at levelscloser to .50, and because the higher levels ofachievement among whites give them a start-ing probability closer to .50, the absolute dif-ference between whites and blacks does notchange much initially and appears toapproach parity slowly as the enrollment ofwhite students decreases.

15. Perhaps the results reported herereveal a more benign process, having more todo with the dynamics of course taking in pre-dominantly black schools. If predominantlyblack schools can find few sophomores whoare “qualified” to enroll in upper-track math-ematics courses, then some underqualifiedblack students may be enrolled simply to fill aclass. Could this queuing process generatethe findings presented in Table 4? It seemsunlikely because unlike models of individualcourse taking used in prior analyses (e.g.,Garet and DeLany 1988), the ordered logitmodeling strategy used here considers the fullrange of course taking and thus is inherentlyless sensitive to queuing effects.Supplementary models were run on the sub-set of schools that had at least 30 percentwhite students and thus were unlikely to besensitive to queuing effects because theywould not run out of white students. Thesemodels still picked up the improved marginalprobabilities of high-track enrollment amongwhites. In these models, the racial composi-tion effect from Model 6a remained about thesame size (84 percent as strong).

The Black-White Gap in Mathematics Course Taking 65

APPENDIX A

Descriptive Statistics (Unweighted)

Mean/Sample Mean/SampleProportions SDa Proportions SD

Variable Full Sample Reduced Sample

School Level N = 1,087 N = 367Percentage black 14.55 23.08 25.85 25.58Percentage Hispanic 10.84 20.40 9.48 16.87Percentage free lunch 19.76 21.42 23.00 21.56Urban .37 .48 .42 .49Rural .27 .44 .25 .44Catholic, proportion of .08 .28 .06 .24Private religious, proportion of .03 .17 .02 .13Private nonreligious,

proportion of .05 .22 .05 .22Individual Level N = 13,548 N = 5,000

Black, proportion of .09 .28 .21 .41Hispanic, proportion of .11 .31 .10 .30Asian, proportion of .06 .25 .07 .25Other, proportion of .02 .13 .01 .12Male, proportion of .50 .50 .50 .50Math grades 1.97 .94 2.0 .94Mathematics test 37.49 11.98 36.40 12.21English test 27.78 8.53 27.21 8.65History test 29.94 4.51 29.62 4.55Prior track placement (low) .06 .23 .05 .21Prior track placement (high) .34 .47 .36 .48Stepparent, proportion of .11 .31 .11 .31Intact family, proportion of .71 .45 .66 .47Parental education 14.39 2.55 10.00 20.00Family income 43.62 40.76 0.00 250.00

a Standard deviations of dummy variables (e.g., Catholic) are a function of the sample pro-portions. They increase as sample proportion approaches .5 and decrease as sample propor-tions approach (0,1).

66 Kelly

APPENDIX B

Procedures for Missing Dataa

Number of Cases Variable Procedure Used with Missing Data

Parental education Mean substitution; one case had jointly missing cases on all social- class variables and was dropped 80

Income Regression imputation using occupation and education 1,210

Intact family Dummy variable for missing data Omitted from the final analyses 154

Stepparent family 154

Mathematics grades Grade data is imputed from test data in that subject, and vice versa, using a regression analysis; 27 cases jointly missing mathematics grades, and test scores were dropped 294

Math test 473

English test 475

History test 513

Percentage black Missing racial composition data at the school level was replaced by aggregating from student data.a Three schools with low numbers of students were replaced using median substitution 365

Percent Hispanic 386

Percentage free/reduced-price lunch Missing data replaced with a regression

of school mean SES (aggregated from student data) on percentage free/reduced-price lunch 958

a Aggregations based on the unweighted, longitudinal sample (freshened students notincluded).

The Black-White Gap in Mathematics Course Taking 67

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Sean Kelly, Ph.D., is Assistant Professor, Department of Sociology and Center for Research onEducational Opportunity, University of Notre Dame. His main fields of interest are sociology of edu-cation, social stratification, social psychology, and quantitative methods. His research has focusedon several educational issues facing America’s schools, including the process of matching teachersto classrooms and the assignment of diverse students to course sequences in high school.

An earlier version of this article was presented at the August 2002 meeting of the AmericanSociological Association (ASA), and it received the 2003 David Lee Stevenson Award for an out-standing paper by a graduate student from the Sociology of Education Section of the ASA. Theauthor thanks Adam Gamoran, Eric Grodsky, and Sean Reardon for their helpful comments.Address correspondence to Sean Kelly, Center for Research on Educational Opportunity, 1015Flanner Hall, Notre Dame, IN 46556; e-mail: kelly.206@nd.edu.