Sociodemographic Diversity, Correlated Achievement, and De Facto Tracking
Author(s): Samuel R. Lucas and Mark Berends
Source: Sociology of Education, Vol. 75, No. 4 (Oct., 2002), pp. 328-348
Published by: American Sociological Association
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Sociodemographic Diversity,
Correlated Achievement, and
De Facto Tracking
Samuel R. Lucas
Universityof California-Berkeley
Mark Berends
VanderbiltUniversity
De facto tracking-the association between students' courses in disparate subjects, regardless of the decline of institutional mechanisms that organized de jure
tracking-is a contested feature of secondary schools. Some analysts imply that
de facto tracking arises simply because students who do well in one area often do
well in other areas. Other analysts contend that pronounced tracking systems
maintain racial, ethnic, and social-class segregation and thus that de facto tracking is driven, in part, by the sociodemographic composition of schools. This article investigates the school-level correlates of de facto tracking. An analysisof data
from High School and Beyond suggests that the higher the correlation between
students' achievements in different domains, the more pronounced the de facto
tracking. However, racial-ethnic and socioeconomic diversity are also positively
associated with de facto tracking, even though the achievement correlation is
controlled. These findings suggest that de facto tracking may be maintained by
both technical and demographic aspects of schools, both of which must be considered in any evaluation of tracking.
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
he basis of contemporary tracking is contested. Some analysts
have noted that tracking allows
the formation of homogeneous groups
based on students' achievement and that
such groups are easier to teach (e.g.,
Hallinan 1994a, 1994b). For these analysts, tracking is best understood as a technical pedagogical device. Others, however,
have contended that tracking inescapably
involves racial, ethnic, and class segregation (e.g., Oakes 1994a, 1994b). Forthese
D
=
analysts, the sociodemographic composition of schools is the primary basis for the
maintenance of tracking systems.
It is well known that researchon the correlates of students' track placement has
routinely revealed important associations
between placement and students' socioeconomic status and race-ethnicity (e.g.,
Gamoran and Mare 1989; Lucas and
Gamoran 2002). However, such individuallevel analyses do not speak to the debate
about systems of tracking directly.This arti-
Sociology of Education 2002, Vol. 75 (October): 328-348
328
De Facto Tracking
329
F
m
cle addresses the school-level correlates of
tracking systems, in an effort to assess
whether the contours of students' achievement, their sociodemographic characteristics,
both, or neither actually undergird the tracking systems of schools.
Researchers have documented the existence of school-level variation in tracking systems (Garet and DeLany 1988; Oakes 1985),
linked that variation to students' outcomes
(Gamoran 1992), and suggested a relationship between students' sociodemographic
characteristics and tracking systems
(Braddock 1990; Lucas 1999). However, the
Braddock and Lucas studies did not control
for the distributionof student achievement, a
competing explanation for the structure of
track systems. Thus, it remains unclear
whether the apparent association between
sociodemographic characteristics and the
degree of tracking is spurious.
Hence, despite the documented schoollevel variation in tracking systems and the
known importance of features of tracking systems for students' achievement, the factors
underlying variation in tracking systems
remain unclear. In this article, we examine the
factors underlying tracking systems in U.S.
secondary schools.
Before we pursue this task, however, we
must navigate a more fundamental dilemma,
a dilemma forged in an emerging conflict
over how to define tracking itself. We can
identify two polar perspectives on the definition of tracking. On one side are those who
regard curriculum differentiation-the division of a constructed intellectualfield into distinct courses of instruction-as tracking. Such
analysts view dividing mathematics into arithmetic, geometry, algebra, and calculus or
dividing English into teenage fiction, the
American novel, and comparative literatureas
tracking (e.g., Loveless 1999; Wheelock
1992). On the other side are those who contend that curriculum differentiation is not
tracking. For these analysts, tracking exists
when there is an association between the
level of courses that students take in two subjects or when there is an association between
the level of courses that students take in the
same subject over time (e.g., Lucas 1999).
Such analysts argue that curriculumdifferen-
tiation is a necessary condition for tracking,
but not a sufficient one. In short, for the latter analysts, a school with curriculum differentiation may or may not be a tracked school;
for the former analysts, a school with curriculum differentiationis a tracked school.
These two polar positions have implications for the research questions that analysts
will ask, as well as the data that researchers
will collect. And because the researchwe pursue herein makes more sense under one of
these two views of trackingthan it does under
the other, we discuss the definitions directly,
rather than leave our definition implicit.
Hence, in this article we describe the definitional disagreement concerning tracking.
Afterward,we explain the position we take in
this article and our reasoning as well.
Afterarticulatingour definition of tracking,
we turn to the primary focus of the article,
namely, assessing the two focal explanations
for tracking in the literature-students'
achievement, on the one hand, and students'
sociodemographic characteristics, on the
other. In this discussion, we outline the logic
underlying each explanation. Then, after
describing our methods, we present the
results of our analysis and conclude with a
discussion of the implications of our findings
for the substantive understanding of tracking.
In studying tracking systems, a foundational
issue concerns the important question of
what is tracking; hence, it is to that question
that we now turn.
DEFININGTRACKINGFOR
U.S. RESEARCH
Identifying Trackingin the
Context of TraditionalSystems
Priorto the mid-1960s, a small set of overarching programs existed at the high school
level (e.g., Cicourel and Kitsuse1963; Conant
1967; Hollingshead 1949). On entering high
school, students were assigned to one of
these mutually exclusive programs that determined their course taking for the three or four
years of high school.
Under this regime, schools seemed to
330
3 Berends
Lucas and
allow little mobility across tracks (Rosenbaum
1976). Furthermore, the institutionalization
of track assignment should have constrained
students' course taking across subjects on the
basis of their track assignments.1 In such an
environment, analysts may have come to
regard a school with curriculum differentiation as a school that tracked students.
Indeed, the tendency at the time to see
schools as tracked if they differentiated the
curriculum was not inconsistent with what
appears to have been the on-the-ground reality, in that access to courses was routinely
governed by assignment to track positions. In
that context, distinguishing between curriculum differentiation and tracking, for all its
analytic accuracy, may have made no difference.
now able to take courses of different levels in
different subjects. Indeed, this appears to be
more than an analytic possibility because
Lucas (1999) found that the association
between mathematics and English implied
that three-quartersof students take courses of
different levels. But an association between
students' courses in disparate subjects was
also evident. A nonzero association is consistent with de facto tracking. Thus, it appears
that there is more flexibility in course taking
while de facto tracking remains. In these circumstances, it is imperative to distinguish
between curriculumdifferentiationand tracking.
Identifying Tracking After the
Unremarked Revolution
This article does not address tracking priorto
the unremarked revolution, but afterward
students appear still to be tracked. However,
the mechanisms of tracking may be much
more subtle than in the past. Clearly, the
extent to which students' course taking is
constrained by institutionalized overarching
programs has greatly diminished (Carey,
Farris,and Carpenter 1994; Oakes 1981). If
analysts continue to study schools through
the lens formed prior to the dramatic transformation in school practice, they will be
searching for schools with a differentiated
curriculum.And they will find such schools in
abundance because the differentiated curriculum is still the dominant form of pedagogical organization in secondary schools in
the United States (e.g., Carey et al. 1994).
However, analysts who equate curriculum
differentiationwith tracking not only will find
tracking on a massive scale but, because of
the poor resolution of the old lens in the new
reality,will also likelymiss more subtle schoolto-school variation in the arrangement of
tracking systems. By equating all schools with
a differentiated curriculum, they will be
unable to discern the factors associated with
variation in how the differentiatedcurriculum
plays out. In short, they will miss the varying
occurrence of de facto tracking.
At the national level, analysts have found a
connection in students' course taking across
subjects (Lucas 1999). But investigation of
Yet, research now suggests that this traditional system of tracking was dramaticallytransformed in the late 1960s and early 1970s.
During that period, many urban school systems seem to have retreated from assigning
students to mutually exclusive all-determinative overarching programs. Instead, their students enrolled in courses in different subjects,
and the courses were vertically differentiated
(Moore and Davenport 1988). Lucas (1999,
p. 1) termed this transformation the unremarked revolution in school practice, in that
"its occurrence has been noted but its implications . .. have been incompletely recognized."
One unrecognized implication is that a
school with curriculum differentiation may
have neither de facto nor de jure tracking. De
jure tracking exists when schools have registration procedures that assign students to
overarching programs that determine their
course taking in academic subjects. And de
facto tracking exists when, absent such institutional procedures, students' levels of study
in disparate subjects remain associated. Thus,
after the unremarked revolution, curriculum
differentiation may or may not eventuate in
de facto tracking.
Surely, the differentiated curriculum continues to exist. But, in principle, students are
Studying Variation in Tracking
Systems
De Facto Tracking
i
the school-level correlates of these factors has
barely begun, and it is just such investigation
that is needed to assess comparatively the
competing explanations for the maintenance
of tracking systems in the United States. Such
investigation can occur only if analysts are
willing to draw an analytic distinction
between curriculumdifferentiationand tracking, a distinction that research suggests is
consistent with the lived experience of students in schools.
331
Trackingand Sociodemographic
Composition
An alternative understanding of tracking systems is based, in part, on the history of tracking. Trackingarose in concert with the expansion of educational systems, the promulgation
of compulsoryschooling, and the construction
of comprehensive schools. Documents suggest
that in many locales, advocates of compulsory
schooling aimed at forcing immigrantchildren
into state-run institutionsin which they could
be socialized or, in the word of the time,
Americanized(e.g., Kelley1903; Massachusetts
PEDAGOGICAL AND
Teacher 1851/1974). Yet, a serious debate
POLITICAL PERSPECTIVES
raged as to what to do with these new students
once they were deposited at the schoolhouse
door. What emerged was the differentiatedcurTracking and Correlated
Achievement
riculum, which prepared some students for
higher education and others for (at best) other
Students of like achievement levels are often
opportunities(e.g., Kliebard1987).
grouped for instruction. Grouping students
Recent critics of tracking have drawn on
for instruction on the basis of prior achieve- this
history to maintain that contemporary
ment may facilitate learning in many ways.
tracking, especially in public schools, is based
For example, with such grouping, it may be on an
ideology of paternalism and a brazen
possible for teachers to fine-tune instructional desire to maintain class and racial segregaapproaches to students' different levels and tion. For example, in her Sociology of
types of prior knowledge and may facilitate Education exchange with Hallinan (1994a,
the teaching task by reducing the range of
1994b) Oakes (1994a, 1994b), drawing on
students to which teachers must orient at any the historical evidence
just mentioned, suggiven time.
gested that a race-coded hierarchycontinues
For these potential advantages to be real- to reinforce
stereotypes and perpetuate disized, however, assignments must be made on advantage. She maintained that
continuing
the basis of prior achievement in the relevant
political support for tracking appears to have
subject. Ostensibly it is possible to do so, for many of the same motivations that charactercurriculum differentiation in secondary ized the
inception of tracking.
schools allows students to be sorted for math
according to their priorachievement in math,
Evidence
to be sorted for English according to their Existing
Some claim that de facto tracking may be the
prior achievement in English, and so on.
Note, however, that students' achievement inadvertent result of defensible pedagogical
in different subjects is associated. Thus, if stu- practice. Others contend that de facto trackdents enroll in levels of course work owing to ing is associated with the racial and class
their levels of achievement in each subject, it composition of the school. Research has
is possible they will find themselves in similar shown an association between sociodemolevels of courses for different subjects because graphic school composition and the tracking
their achievements in different subjects are system. Braddock(1990) found that the mix
associated. Therefore,even when subject-spe- of black and white students in the school is
cific achievement is the only determinant of associated with more pronounced tracking.
placements, the association between stu- Lucas (1999) found that schools with more
dents' prior achievement in different subjects socioeconomic diversity had a higher associacan create a de facto tracking system.
tion between students' courses in math and
332
English. These studies suggested that demographic composition matters.
Despite these findings, however, the
debate as to whether tracking systems are
based in correlated achievement or demographics remains unresolved.2 Although evidence suggests that tracking is more pronounced in racially and socioeconomically
diverse schools, it is possible that it may be so
because achievementmay be more variable in
those schools, and the correlation between
achievement in different domains may be
more pronounced in such schools as well. If
so, the true basis of researchers'observations
may be achievement variation and covariation. Hence, to assess the thesis that sociodemographic composition matters, it is necessary to control for the correlation between
students' levels of priorachievement in different domains.
Furthermore,researchershave yet to assess
both class and racial-ethnicdiversity simultaneously. The effect of racial-ethnic diversity
may actually be zero if the key factor underlying de facto tracking is class ratherthan race.
Or the effect of socioeconomic diversity may
be zero once racial-ethnic diversity is controlled. Thus, simultaneously investigating
the role of social class and racial-ethnicdiversity is essential to determining just what, if
anything, is the sociodemographic factor that
matters.
This article addresses the questions just
posed. However, note that the larger frame
encompassing our consideration of demographic correlates of de facto tracking was
forged by the evidence suggesting that political interests and potential mobilization
around tracking issues are linked to the
sociodemographic composition of schools
(e.g., Wells and Serna 1996). It is worthwhile
to keep this largerframe in mind. But we hasten to add that although these larger issues
framed our analysis, our analysis could not
assess whether political action lies behind de
facto tracking. Indeed, ethnographies and
research using in-depth interviews may be
better suited to that task. Nevertheless, the
analysis could assess whether sociodemographic composition is associated with de
facto tracking once competing explanations
are simultaneously considered; this is a task
Lucas and Berends
3
that is particularlywell suited to statistical
analysis.
Thus, although further research will be
needed to assess fully the links among
sociodemographic composition, political
mobilization, and de facto tracking, this
research remains useful, for we assessed
whether sociodemographic composition matters once competing explanations are considered. Before we search for mechanisms
through which sociodemographic composition may matter, it is imperative to assess
whether, on balance, sociodemographic composition matters at all. At this point, the literature is suggestive, but a major competing
explanation-correlated achievement-has
not been simultaneously considered. This article takes this step and thus provides important
substantive knowledge about, as well as the
groundwork for further study of, the mechanisms by which de facto tracking is maintained in the United States.
METHODSOF ANALYSIS
We used data from High School and Beyond's
(HS&B)base year (data collected 1980), first
follow-up (1982), high school transcript
(1983), and administrators' and teachers'
(1982) surveys. The unit of analysis for our
study was the school, and because students'
placements were drawn from data in the
transcripts, the study was dependent on the
975 schools that completed the transcripts
surveys. Of these 975 schools, 948 had nonmissing data on the dependent variable, and
thus these 948 schools served as the sample
for this study.
We used HS&Binstead of the more recent
National EducationLongitudinalStudy of 1988
(NELS)data, primarilybecause HS&Ballowed
us to make inferencesto and about the school.
Incontrastto HS&B,the NELShigh school sample did not constitute a nationallyrepresentative sample of high schools, nor did the students in any given high school sample constitute a probabilitysample of the students in the
high school (Ingels et al. 1998). Under these
conditions, we deemed it inappropriateto use
informationabout the student sample to make
inferencesabout the schools.
De Facto Tracking
m
Data
Independent variables, all of which were
measured in the 10th grade, are described in
the Appendix. A more detailed discussion is
needed, however, of our indicators of racialethnic diversity. The racial landscape of the
United States is changing, such that now (if
not earlier) it is advisable, when possible, to
include more than just blacks and whites in
the analytic picture. The multiplicityof identifiable and potentially salient racial-ethnic
groups only increases both the number of
ways one may measure diversityand the controversies that attend any measurement strategy.
In our study, we used two sets of indicators
of racial-ethnic diversity. To construct the
measures of diversity, we used principals'
reports of the racial composition of the
school. Principals' reports are advantageous
because a sample of only 36 or 72 students
may not include some relatively rare racialethnic groups (such as Native Americans)and
thus would lead one to underestimate the
degree of racial-ethnic diversity in some
schools. The HS&B principals' questionnaire
asked about the percentage of students in the
school who were white, black, Latino/Latina,
Asian, and Native American. These five
groups were used in calculating both sets of
measures of racial-ethnicdiversity.
One set of measures reflects the number of
different racial groups in the school. The
more groups there are, the more racially
diverse the school is, all else equal. We used a
set of dummy variables in the models to capture this dimension of diversity, which we
regard as capturing the presence of racial-ethnic diversity.
The second measure we used was the
index of racial dispersion (Brewster 1994).
This index provides a comparable measure of
racial diversity for different schools even
though some schools have more racialgroups
than do others. This index ranges from zero,
for schools with no racial diversity (schools
with only one racial-ethnic group), to 1 for
schools with several groups represented in
equal measure (see the Appendix for a formal
explication of the calculation of this index).
Essentially, schools with one numerically
333
dominant group and one or two "token"
groups obtain low scores on this index. In
contrast, schools with a more nearly equal
distribution of students across two or more
racial-ethnic groups obtain high scores. We
refer to this index as capturing the incidence
of racial-ethnicdiversity.
Both the presence and incidence of diversity are important to study because both may
be important determinants of de facto tracking. On the one hand, heightened de facto
tracking may be associated with the mere
presence of more groups in the school. On
the other hand, relative proportions of different groups may be most relevant for tracking
systems. Because we are aware of no research
on the correlates of track systems that has
gone beyond the issue of black-white diversity, we investigated both dimensions of racialethnic diversity.
The second important construct concerns
the correlation between students' achievement in pairs of domains in each school. We
used this variable to control for a potentially
confounding variable and to assess whether
de facto tracking is connected to the contours
of students' achievement. Other control variables in the analysis included the natural log
of school size, sector, urbanicity, and the
mean and variance of achievement in specific
domains.
The dependent variablewas a count of the
number of students in a school who occupied
the same level of courses for both Subject A
and Subject B in the 1 1th grade (see the next
section for an explanation of these subjects).
We used subject-specific versions of Lucas's
(1990, 1999) course-based indicators (CBIs)
of track location. These CBIswere constructed from the HS&Btranscript data and classified students into six categories: none, remedial, business or vocational, lower college
prep, regular college prep, and elite college
prep (for more details on this measure, see
Lucas 1999). The dependent variablewas the
count of students who were in the same level
of courses in Subject A and B in grade 11.3
The dependent variable was measured in
grade 11 because measures of prior achievement were obtained for grade 10. Using a
grade 11 measure as the dependent variable
means that all independent variables were
334
33
measured immediately and clearly before the
dependent variable was measured.4
Plan of Analysis
We conducted two separateanalyses;what distinguishes them is the definition of Subject A
and Subject B. In the first analysis, Subject A
was math and Subject B was English;thus, the
focus was on math and Englishonly. These are
arguablythe centralsubjects in the curriculum,
and thus considering them in particular is
important. In the second analysis, Subject A
and Subject B were any two of the following
four academic subjects: mathematics, English,
social studies, and science. Hence, six different
course combinations were investigated to
determine whether the findings from the first
analysisgeneralizeacrossthe academic curriculum.
Neither analysis included placement in foreign language study or "nonacademic" subjects, such as physical education, computer
programming, and home economics. Some
of these subjects may provide key gatekeeping functions in secondary schools (e.g.,
Alexander,Cook, and McDill 1978), but there
are no test scores for these subjects in HS&B.
Because a key competing explanation for the
existence of de facto tracking is that students
are placed in similar levels across different
subjects because students who do well in one
subject do well in other subjects, it was necessary to confine the study to subjects for
which data on test scores were available.
Finally,the analysisfocused on placement in
pairs of subjects. There is at least one other
approach one could follow. Given the four academic subjects, one could analyze the proportion of students in the same level of courses
acrossall four subjects. However,with the small
school-specific sample sizes available in the
HS&B data (the mean within-school sample
size is 12.3 students), such an analysiswould
likelyunderstatethe presence of de facto tracking and distortthe estimation of its correlates.
Models
The aim was to investigate whether the proportion of students in the same level of courses is associated with school-level factors. The
ua and
n Berends
eed
Lucas
negative binomial regression model is one of
a class of models that allow investigation of
this question under conditions of different
sample sizes within schools (Long 1997). The
negative binomial regression model for the
incidence of same-level course taking is specified as
ez
k+ Os
"+
6(1- e ?k)
where Ys represents the number of students
in school s who are in the same level for math
and English, Os represents the number of students sampled in school s in math and
English, Rk represents a coefficient capturing
the association of school-level variablexk with
the incidence of same-level course taking, xsk
represents independent variables measured
on each school for math and English, and 6
represents an overdispersion parameter
(Hardinand Hilbe 2001).
Note that this model is essentially a
Poisson regression model with an extra parameter, 6. This parameter relaxes the restrictive
assumption of Poisson regression that the
mean of the dependent variable equals the
variance of the dependent variable.The parameter allows the variance to be larger than
the mean, and one perspective views 6 as a
parameter capturing unmeasured sources of
heterogeneity. If 6 is statisticallysignificant, it
indicates that the negative binomial regression model is a more appropriate model for
the data. In such a situation, were one to use
a Poisson model instead, one would underestimate the standard errors.
Note also that Os represents the students
in the sample for whom there were valid data
on their tracks for the subjects under study
(i.e., the students who could possibly be
observed in the same level for both subjects).
In this model, Os is entered as a covariate
whose coefficient is constrained to equal 1.
Doing so transformsthe model of counts into
a model of rates (Yamaguchi 1991).
The foregoing describes the model for the
first analysis,the analysisof math and English.
In the first analysis, each school contributes
only one instance. For the second analysis,
however, each school contributes six
De Facto Tracking
i
instances, one for each course combination.
Moreover, the instances are interdependent
because each subject area is involved in three
of the six cases for each school. Under these
conditions, we could not use the usual model
without modification because the standard
errors would be underestimated. A common
approach to the problem posed by nonindependence of observations is to calculate
Huber/White(or sandwich) standard errorsto
account for the clustering of cases. Thus, in
the second analysis, we present Huber/White
standard errors (Huber 1967).
Finally, the debate about the role of
sociodemographic composition has focused
primarily on public schools. Private schools
are much more able to alter their sociodemographic composition than are public schools.
Furthermore, parents of private school students may more easily use mechanisms (e.g.,
voice, exit) to react to undesirable features of
the school (e.g., Chubb and Moe 1988).
These factors make it likelythat the cross-sectional relationship between de facto tracking
and sociodemographic factors will differ for
public and private schools. Hence, we estimated models separately for public and private schools.
335
tracking. Note also that in Model 2, the presence of racial-ethnic diversity is associated
with de facto tracking; the association
appears curvilinearin that schools with 2, 3,
and 4 racial-ethnicgroups have higher levels
of de facto tracking than do schools with 1
group or 5 groups. Once both indicators of
racial-ethnic diversity are included, the incidence of racial-ethnic diversity is no longer
associated with de facto tracking, but the
presence of diversity is.
Controlling for the association between
math and English achievement (Model 4)
does not change the relationsfound in Model
3. Indeed, the correlation between achievement and de facto tracking is not statistically
significant in Model 4. However, once urbanicity is added as a control in Model 5, the
achievement correlation is statisticallysignificant. The higher the association between students' prior achievement in math and
English,the greater the level of de facto tracking in the school.
Model 6 adds controls for the mean and
variance of achievement in Englishand math.
Adding these controls greatly reduces the
coefficient for mean socioeconomic status,
and it is no longer statistically significant.
However, the coefficient for the correlation
between math and English achievement
remains statistically significant. Thus, it
RESULTS
appears that one basis for de facto tracking is
the similarityof students' achievement levels
Math and English
in different domains.
Math and English are central subjects in the
Yet, the coefficients for socioeconomic
high school curriculum and the transition to diversity and for two, as opposed to one,
college and postcollegiate institutions. Thus, racial-ethnicgroup are essentially unchanged
the role of racial-ethnic and socioeconomic and remain statistically significant in the
diversity in structuringthe provision of math- move from Model 5 to Model 6. Thus, even
ematics and English instruction is, in itself, the model with the most extensive set of conimportant. The results from six models of de trols continues to provide evidence that one
facto tracking in math and English in public basis for de facto tracking is the diversity of
schools are presented in Table 1. Models 1, 2, students' social backgrounds. With respect to
and 3 include only school size and variables social class, the more socioeconomically
that capture the socioeconomic composition diverse the student body, the more proand racial-ethnic diversity. They differ in the nounced de facto tracking appears to be. And
with respect to race-ethnicity, schools with
specification concerning racialdiversity.
Across Models 1-3, socioeconomic diversi- two racial-ethnicgroups appear to have more
ty is associated with higher levels of de facto pronounced de facto tracking than do monotracking in math/English. When the incidence racialschools.
of racial-ethnic diversity is considered alone,
The foregoing concerned public schools.
it, too, is positively associated with de facto Table 2 reestimates the six models on private
Lucas
and Berends
Berends
Lucas and
336
336
Table 1. Correlatesof the Incidenceof Students Enrollingin the Same Levelof Math and English,
Grade 11: Public Schools (N = 841 schools; standard error below the coefficient)
Variable
Model 1
Model 2
Model 3
Model 4
Model S
Model 6
Intercept
-1.562a
0.241
-1.764a
0.249
-1.712a
0.252
-1.883a
0.265
-1.669a
0.290
-1.718a
0.289
Ln(schoolsize)
0.023
0.034
0.038
0.035
0.029
0.036
0.032
0.036
0.004
0.039
0.007
0.039
Socioeconomic diversity
0.195a
0.073
0.197a
0.073
0.193a
0.073
0.187a
0.073
0.187a
0.073
0.1 72a
0.073
Mean socioeconomic
status
0.281 a
0.064
0.242a
0.061
0.275a
0.065
0.265a
0.065
0.265a
0.067
0.100
0.083
Index of racialdiversity
A). II (W
OAa
0.134
0.089
0.135
0.089
0.121
0.090
0.174
0.091
0.207a
0.078
0.1 72a
0.082
0.175a
0.082
0.183a
0.082
0.181a
0.081
0.208a
0.081
0.171a
0.086
0.172a
0.086
0.167
0.086
0.165
0.086
0.206a
0.082
0.157
0.088
0.154
0.089
0.146
0.089
0.139
0.089
0.007
0.096
-0.045
0.102
-0.043
0.102
-0.051
0.102
-0.061
0.102
0.225
0.115
0.229a
0.114
0.259a
0.124
0.045
0.057
0.056
0.057
-0.075
0.057
-0.082
0.057
-
C).082
2 racial-ethnicgroups
(versusl)
-
3 racial-ethnicgroups
(versus1)
4 racial-ethnicgroups
(versus1)
5 racial-ethnicgroups
-
(versus)
Achievementcorrelation
~-
Urbanversussuburban
-~-
Ruralversussuburban
~
0.108a
0.042
Math achievement mean
-0.009
0.042
Englishachievement mean
0.004
0.015
Math achievement variance
-0.010
0.011
Englishachievement
variance
Dispersionparameter
Log-likelihood
0.057a
0.018
-1641.5
0.052a
0.018
-1635.7
0.048a
0.018
0.050a
0.018
-1633.8
-1631.6
0.046a
0.017
-1630.0
0.040a
0.017
-1622.4
a Coefficientis discerniblydifferentfrom zero at or below a = .05.
schools alone. The most striking feature to
notice is that in all six models, only one substantive independent variable is statistically
significant. The statistically significant independent variable is the correlation between
students' level of achievement in Englishand
math, and even this is true only in Model 6. It
is interesting that Model 6 has more controls
than any other, and as one may expect, the
standard error for the achievement correlation coefficient is the largest in that model.
Yet, the achievement correlation coefficient is
also nominally larger in Model 6 than in any
other model, making it the only factor for
De Facto Tracking
i
337
Table 2. Correlatesof the Incidenceof Students Enrollingin the Same Levelof Math and English,
Grade 11: Private Schools (N = 107 schools; standard error below the coefficient)
Variable
Model 1
Intercept
Model 2
Model 3
Model 4
Model 5
Model 6
-0.217
0.526
-0.314
0.546
-0.314
0.534
-0.812
0.587
-0.728
0.321
-0.954
0.645
Ln(school size)
-0.127
0.078
-0.118
0.079
-0.125
0.078
-0.119
0.077
-0.125
0.079
-0.102
0.081
Socioeconomic diversity
-0.100
0.215
-0.031
0.220
-0.070
0.217
-0.059
0.218
-0.063
0.218
0.031
0.230
Mean socioeconomic status 0.284
0.204
0.217
0.199
0.279
0.204
0.222
0.209
0.219
0.212
0.275
0.236
0.171
0.240
0.142
0.239
0.122
0.249
0.058
0.253
0.183
0.246
0.181
0.253
0.148
0.249
0.128
0.252
0.067
0.261
0.125
0.237
0.112
0.251
0.118
0.247
0.105
0.248
0.087
0.253
0.020
0.241
-0.027
0.258
-0.051
0.253
-0.073
0.258
-0.082
0.264
0.071
0.276
0.098
0.293
0.073
0.297
0.041
0.304
0.019
0.319
0.753
0.388
0.751
0.387
0.954a
0.441
n
Index of racialdiversity
v.
1QA
I Ut
(0.219
2 racial-ethnicgroups
(versus 1)
3 racial-ethnicgroups
(versus 1)
4 racial-ethnicgroups
(versus 1)
5 racial-ethnicgroups
(versus 1)
Achievement correlation
Urban versus suburban
-0.025
0.162
-0.036
0.164
Ruralversus suburban
-0.095
-0.203
-0.067
-0.207
Math achievement mean
-0.093
0.157
Englishachievement mean
0.026
0.156
Math achievement variance
~-
_-
Englishachievement variance
Dispersion parameter
Log-likelihood
0.232a
0.050
-306.6
-0.060
0.069
0.044
0.058
0.237a
0.051
-307.4
0.220a
0.049
-304.7
0.211a
0.047
-302.7
0.211a
0.047
-302.6
0.205a
0.046
-301.7
a The coefficient is discerniblydifferent from zero at or below a =.05
which we have evidence of an effect on de
facto tracking in private schools.
Note that the failure of socioeconomic
diversityand racial-ethnicdiversityto "reach"
statistical significance in Model 6 does not
appear to be based on the smaller number of
private schools in the sample. Even if we sub-
stituted the standard errors from the public
school analysis into the private school analysis, neither the single coefficient for socioeconomic diversitynor any of the five coefficients
for racial-ethnicdiversity would be statistically significant. These results suggest that variation in the level of de facto tracking in private
338
338
andBerends
Lucas
Lucas and
Berends
schools is not connected to the socioeco- expected, the higher the correlation between
nomic or racial-ethnicdiversityof the schools. prior achievement in two different domains,
Instead, de facto tracking appears to be relat- the more pronounced de facto tracking is in
ed to the contours of students' achievement. the school. Despite the impact of correlated
achievement, socioeconomic diversity is positively associated with de facto tracking. The
Math, English, Social Studies, and
coefficient for socioeconomic diversity is disScience
cernibly different from zero, while the joint
Table 3 presents the results of six models of test of no association with racial-ethnicdiverde facto tracking in private schools. These sity (as reflected in the incidence of diversity
models are estimated on data reflecting four and the effect for four racial-ethnic groups)
central subjects of the academic curriculum. leads to a 2 = 7.83, with 2 degrees of freeThus, they reflect a fuller picture of the dom for a p-value of .0199. Thus, racial-ethnic diversity seems to be associated with de
course-taking regime.
However, consideration of the wider set of facto tracking, and the more socioeconomic
academic subjects in private schools does lit- diversity, the more pronounced de facto
tle to alter the results obtained when we tracking is in the school, even though correfocused only on mathematics and English. lated achievement is controlled.
Model 5 adds controls for urbanicity.
Socioeconomic and racial-ethnicdiversity are
not associated with de facto tracking in pri- Including these controls leaves racial-ethnic
vate schools. Instead, variation in specific diversity, socioeconomic diversity, and corredomains of achievement is associated with de lated achievement clearly and positively assofacto tracking, although the direction of the ciated with de facto tracking.5 And once the
association varies. The factors underlying this full set of controls are included (see Model 6),
pattern of results are not immediately obvi- the story from Model 5 with respect to racialous, primarily because we controlled for ethnic diversity, correlated achievement, and
mean achievement in each domain, the cor- socioeconomic diversity does not change.
relation between achievement in different
Figure1 illustratesthe connection between
domains, and many other factors. Thus, we de facto tracking and racial diversity using
are at a loss to explain this pattern and are coefficients from Model 6. Urbanicityis held
reluctant to construct off-the-cuff specula- constant at zero, so that the figure reflectssubtions on the basis of this result. Instead, we urbanschools; school size is held constant at its
would suggest that this pattern of results may mean; variables for the distribution of math
profit from closer scrutiny in future research. and Englishachievement are held constant at
Table 4 contains the results for public their means, as is the correlation between
schools. Models 1 and 2 show that, as before, them; and all other variablesare fixed at zero.
both socioeconomic and racial-ethnicdiversi- Because in monoracial schools the index of
ty are associated with more pronounced de racialdiversitywill be zero by definition, Figure
facto tracking in public schools. In Model 3, 1 contains a point (*) for that type of school.
in which both aspects of racial-ethnicdiversi- The dashed line represents how the predicted
ty are included, none of the individualaspects values for schools with four differentracial-ethis statisticallysignificant. However, a joint test nic groups change as the index of racialdiverof whether the coefficient for the incidence of sity changes. A school with four racial-ethnic
racial diversity and the coefficient for four groups but low incidences of three of the
racial-ethnic groups are both zero is statisti- groups is predicted to have a little over sixcally significant below the .05 level (X2 = tenths of its students in the same level of
8.09, with 2 degrees of freedom, p-value = course across subjects. In contrast, a school
.0175), suggesting that racial-ethnicdiversity with four racial-ethnic groups represented
is associated with heightened de facto track- equally will have an index of racialdiversityof
1.0, and such a school is predicted to have
ing for the academic curriculumas a whole.
Model 4 adds the correlation between nearly three-fourthsof its students enrolled in
achievement in the relevant domains, and, as the same level course across subjects.
De FactoTracking
339
m
Table 3. Correlates of the Incidence of Students Enrolling in the Same Level of Math, English, Social
Studies, or Science, Grade 11: Private Schools (N = 107 schools, standard error below the coefficient)
Variable
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Intercept
-0.159
0.262
-0.305
0.253
-0.277
0.251
-0.361
0.258
-0.485
0.277
-0.498
0.296
Ln(schoolsize)
-0.137a
0.042
-0.127a
0.040
-0.135a
0.040
-0.131a
0.040
-0.122a
0.040
-0.124a
0.043
Socioeconomic diversity
-0.149
0.106
-0.148
0.101
-0.151
0.104
-0.130
0.106
-0.139
0.105
-0.130
0.112
Mean socioeconomic status
0.168
0.108
0.129
0.096
0.174
0.107
0.152
0.109
0.172
0.109
0.208
0.129
Index of racialdiversity
0.208
0.121
0.191
0.127
0.174
0.127
0.214
0.139
0.207
0.145
0.261
0.146
0.206
0.153
0.199
0.152
0.230
0.168
0.176
0.183
0.212
0.140
0.142
0.163
0.141
0.163
0.162
0.175
0.127
0.179
0.147
0.137
0.064
0.147
0.057
0.148
0.095
0.166
0.056
0.174
0.115
0.150
0.046
0.163
0.006
0.163
0.080
0.183
-0.007
0.191
0.114
0.128
0.122
0.127
0.145
0.135
Urbanversus suburban
0.035
0.083
0.021
0.093
Ruralversus suburban
0.147
0.115
0.126
0.128
2 racial-ethnicgroups
(versus 1)
3 racial-ethnicgroups
(versusl)
4 racial-ethnicGroups
(versusl)
5 racial-ethnicgroups
(versusl)
Achievement correlation
Math achievement mean
--0.027
-~-
Englishachievement mean
--0.033
~
0.046
~
0.017
0.062
--0.013
-~-
0.036
-~-~-
0.027a
0.012
~Socialstudies achievement
mean
-~-
Science achievement mean
~-
Math achievement variance
Englishachievement
variance
~~-
Social studies achievement
variance
-~-
~-
Science achievement
variance
Dispersionparameter
Log-likelihood
0.038
~
~
0.035a
0.017
~
0.038a
0.018
--0.023a
-
0.494a
0.055
-1916.8
0.493a
0.057
-1915.9
0.490a
0.057
-1914.7
0.489a
0.057
-1914.1
a The coefficient is discerniblydifferentfrom zero at or below a = .05.
0.487a
0.057
-1913.5
0.008
0.432a
0.057
-1889.7
Lucas and Berends
340
Table 4. Correlates of the Incidence of Students Enrolling in the Same Level of Math, English, Social
Studies, or Science, Grade 11: Public Schools (N = 841 schools; standard error below coefficient)
Model 2
Model 3
Model 4
Model 5
Model 6
-0.858a
0.112
-0.908a
-0.879a
0.120
-0.934a
0.124
-0.929a
-0.904a
0.139
0.142
-0.046a
-0.043a
-0.047
0.017
-0.048a
0.017
-0.048a
0.017
-0.049a
0.016
0.019
0.019
0.110a
0.041
0.113a
0.040
0.110a
0.041
0.108a
0.041
0.109 a
0.041
0.1 04a
0.041
Mean socioeconomic status
0.057
0.035
0.033
0.034
0.051
0.036
0.048
0.035
0.055
0.036
0.040
0.044
Index of racialdiversity
A r%n7ya
(J.uV/"
0.076
0.046
0.078
0.047
0.072
0.047
0.082
0.049
Variable
Model 1
Intercept
Ln(schoolsize)
Socioeconomic diversity
0.119
-
(0.041
2 racial-ethnicgroups
(versus1)
0.074
0.039
0.052
0.042
0.050
0.042
0.052
0.042
0.043
0.042
3 racial-ethnicgroups
(versus1)
0.061
0.042
0.039
0.045
0.037
0.045
0.039
0.045
0.032
0.045
4 racial-ethnicgroups
(versus1)
5 racial-ethnicgroups
(versus1)
0.094a
0.044
0.066
0.048
0.062
0.049
0.064
0.049
0.058
0.050
0.029
0.050
-0.001
0.054
-0.002
0.054
-0.001
0.054
0.000
0.055
-
0.098a
0.044
0.1 36a
0.047
Urbanversussuburban
0.021
0.031
0.028
0.031
Ruralversussuburban
0.009
0.030
0.010
0.031
Achievementcorrelation
_-
~-
~
0.097a
0.044
~
Math achievement mean
~-
Englishachievement mean
-~-
~-
~
0.012
0.013
Socialstudies achievement
mean
~-
~
-0.019
0.016
~-
~
-0.009
0.009
Science achievement mean
Math achievement variance
--0.01
-~-
Englishachievement
variance
-~-
Socialstudies achievement
variance
-~-
~~-
Science achievement
variance
Dispersionparameter
Log-likelihood
0.040a
0.014
3a
0.005
~
~
0.012a
0.004
~
0.008
0.007
-0.0111a
0.002
0.228a
0.013
-11137
0.228a
0.013
-11135
0.228a
0.013
-11133
0.227a
0.013
-11130
a The coefficient is discerniblydifferentfrom zero at or below a = .05.
0.227a
0.013
-11130
0.205a
0.013
-11044
De Facto Tracking
i
341
Consider predicted values one standard
deviation above and below the mean. A
school that is one standard deviation below
the mean on the index of racial diversity (D)
is expected to have approximately 63 percent
of its students in the same-level course. A
school that is one standard deviation above
the mean on the index of racial diversity (<)
is expected to have approximately 69 percent
of its students in the same-level course.
Although this difference is substantially less
than that estimated on the basis of the
extreme values of diversity, note also that
both predicted values are substantially larger
than the predicted value for a monoracial
school. For this reason, it would appear that
both the presence and the incidence of racial
diversity matter.
The impact of racial diversity is more than
modest. As Figure 2 reveals, socioeconomic
diversity also appears to be consequential.
Figure 2 illustrates the connection between
de facto tracking and the variance of socioeconomic status for schools with and without
racial diversity, again using coefficients from
Model 6. The dashed line reflects a monoracial school, while the solid line is for schools
-c
0)
c
LJ
1.0
I
I
with four racial-ethnicgroups. The incidence
of racial diversity is held constant at its mean
for the solid line, but obviously monoracial
schools have zero racial diversity, so for the
dashed line, the incidence of racialdiversity is
set at zero. Figure2 reveals that in monoracial
schools with no socioeconomic diversity,
more than half the students take courses of
different levels in disparate subjects. However,
in schools with four racial-ethnic groups but
no socioeconomic diversity,approximately 57
percent of the students are in courses of the
same level. In a monoracial school with the
maximum amount of socioeconomic diversity
observed in the sample, approximately threequarters of the students are in the same level
of courses across subjects. However, in a
racially diverse school with the maximum
amount of socioeconomic diversity, over 90
percent of the students are predicted to be in
the same level of courses across disparate
subjects. Note that the figures assume constant distributions of achievement, as well as
a constant correlation between achievement
in different domains. Taken together, Figures
1 and 2 indicate not only that coefficients for
racial-ethnic and socioeconomic diversity are
I
I
0.9
Io
c
cu
-
0,7
(D
0.6
a)
E 05
co
c
p
0,4
c
a) 03
-
-
D 0.2
0-
0.1
0
Q.
o
-
0.0
D
c)D.0
I
x
0.2
I
0.4
I
0.6
0.8
1.0
Indexof Racial Diversity
Figure 1. Levelof De FactoTrackingby RacialDiversity.Note: * = monoracialschool, >
X = mean IRD, <i = +1 SD, and -
= 4 racial-ethnic groups.
= -
1 SD,
-
342
Lucas and Berends
.? 10.
l
l
l
l
c
CT'
C
W 09
m
0.9
c 0.8
C0)
-
-
-
/
E03
~~~
~~~
E 0,5
a)
c 0.4
-c
0.2
0.1
'0
0.2
)>
C 0.(
300
x
0.404
14
0.808
1
1.212
1.616
2.(020
Variancein Socioeconomic Status
Figure 2. Levelof De FactoTrackingby Socioeconomic Status and RacialDiversity.Note: > = 1 SD,X = mean SESvariance,< = +1 SD,-= 4 racial-ethnicgroups, and --- = monoracialschool.
discernibly different from zero, but that the
differences they reflect across schools suggest
a substantively important relation between
diversity and de facto tracking in public
schools.
DISCUSSION
With respect to private schools, socioeconomic and racial-ethnicdiversityare not associated with de facto tracking, regardless of
whether the core subjects of math and
Englishare analyzed alone or in concert with
the additional academic subjects of science
and social studies. It may be tempting to
regard the finding of no association between
sociodemographic school diversity and de
facto tracking in private schools as evidence
of no effect of diversity on private school
practices. This temptation must be resisted.
Earlier we noted that private schools are
much more able to alter their sociodemographic composition than are public schools,
and parents of private school students may
more easily use other mechanisms, such as
exit, to react to features of the school they
regard as undesirable. We noted that these
factors make it possible that the cross-sectional relationship between de facto tracking
and sociodemographic factors is different in
public and private schools. In comparison to
public schools, private schools of the period
allowed a greater degree of sorting between
schools in the sector.
Parents'greater ability to sort between different private schools makes treating sociodemographic composition as an input to the
system to which school personnel respond
pedagogically problematic. Instead, for private schools, sociodemographic composition
may be better viewed as an output of admissions policy. Hence, the status of the modeling is less secure for private schools than for
public schools. Our finding, therefore, is simply that no association is visible in the cross
section. Although we regard our finding as
suggestive and potentially important, given
the greater latitude availableto privateschool
actors (schools and parents), researchon a set
of private schools over a series of cohorts will
be needed to ascertain whether processes of
student recruitment, selection, and retention,
as well as changing school demographics, are
De Facto Tracking
m
associated with de facto tracking in the private school sector. Should such research replicate our results, our confidence in the irrelevance of race and class for curricularorganization in private schools will greatly increase.
For both the central subjects of the public
school curriculum-math and English-and
for the curriculum considered more broadly,
it is apparent that both racial-ethnic and
socioeconomic diversity are positively associated with de facto tracking. Adding the correlation between math and English achievement and a host of other covariates reflecting
urbanicity and the contours of students'
achievements leaves intact the qualitative
assessment that socioeconomic and racialethnic diversity matter.
Our findings were obtained even though
we controlled for the correlation between
achievement in two subjects. Controlling for
the correlation between students' achievement was important in that we found an
association between the correlation of prior
achievement and de facto tracking in public
schools for all the analyses and for math and
English in private schools. Including correlated achievement, however, did not eliminate a
role for sociodemographic diversity in public
schools. Including the control for correlated
achievement was useful, but we observe that
correlated achievement itself may be an outcome of de facto or de jure tracking systems
that the students experienced earlier in their
educational careers. Thus, it is possible that
some of our models understate the role of
diversity in determining de facto tracking, for
if tracking priorto grade 11 creates an association between students' achievement in different domains and if sociodemographic
diversity partly determined prior tracking,
then controlling for the association between
students' achievement is to riskattributing to
correlated achievement effects that actually
flow from the sociodemographic composition
of schools that students experienced in earlier years.
However, even if understated, the findings
are such that it appears appropriate to conclude that de facto tracking is associated with
diversity.The sum total of these findings is consistent with Oakes's(1994a, 1994b) contention
as to the basis of trackingin public schools.
343
We did not study mechanisms, and thus
we cannot identify the processes that may
underlie the patterns we have documented.
However, the claim that sociodemographic
diversity is associated with the maintenance
of tracking was articulated by analysts who
believed that political action was likely in
diverse schools. The particular action envisioned by these analysts included exclusionary moves by whites and members of the
upper class as they endeavor to segregate
their children from others. Whether this particular story is apt or not, the motivation for
focusing intently on the politics of tracking
does seem deepened by our analysis.
Although we did not measure "politics," we
believe it would be just as unwise to interpret
our findings as documenting the role of politics as it would be to ignore the larger forces
that make diversity of both social theoretical
and social policy interest. Diversity was not
randomly chosen for attention; ongoing
debates and a long history of sociological reasoning and research suggest that diversity is
not merely a staid demographic feature of
institutions but, instead, a key determinant of
structures, individuals' experiences, and
opportunities (e.g., Kanter 1977; Simmel
1950).
Given our findings and the foregoing
observation, the kind of research needed at
this juncture is that which embeds analyses of
the technical aspects of tracking in a framework that explicitly considers the socioeconomic and racial-ethnic diversity of schools
and acknowledges a potential role for variously based political action. Such research
should adopt a definition of tracking that is
more consistent with the operation of
schools; that is, definitions that essentially
regard every school with curriculumdifferentiation as tracked likely suppress real and
important differences between tracking
regimes to which analysts should attend.
Given the thrust of our findings, it is apparent
that evaluations of tracking and analytical
strategies that ignore the complex variability
of tracking or deemphasize the linkage
between tracking and racial-ethnic and/or
socioeconomic diversity are likely to misunderstand the maintenance and complex
transformations of tracking. And because
344
344
these sociodemographic factors are themselves undeniably linked to both obvious and
subtle political interests, ignoring or deemphasizing the role of sociodemographic
Lucas
Lucas and
and Berends
Berends
diversity in the construction, development,
and maintenance of track systems is likely to
misunderstand the implications of tracking
for students, schools, and the wider society.
APPENDIX
Independent Variables
Allvariableswere recoded to the midpoint for missing cases. In the models, a control for missing on each particularvariable was used.
Socioeconomic Composition
Student-levelsocioeconomicstatusis measuredwith a linearcombinationof mother'seducation, father's education, father's occupation, family earnings, and a household items index.
The meanof this linearcombinationis usedto measurethe levelof socioeconomicstatusin the
school. The varianceof this linearcombination is used to measure the degree of socioeconomic
diversityin the school.
Racial-Ethnic Diversity
Principals'reports of the proportion of students who are white, black, Latino/Latina,Asian, or
Native American, coupled with principals' reports of the number of students in the school,
were used to construct measures of racial-ethnicdiversity.
To measure the presence of racial-ethnicdiversity,a count of the number of groups in the
school was constructed from the principals' reports. Four dichotomous variables were constructed: 2 groups in the school, 3 groups in the school, 4 groups in the school, and 5 groups
in the school. The omitted category is 1 group in the school.
To measure the incidenceof racial-ethnicdiversity,an index for each school was calculated
as follows: If k > 1, then Ds= (k(N2- >fsk2 )) / (N2(k-1));if k = 1, then D, = 0, where k is the
number of racialgroups in the school, N is the total number of students in the school, and fsk
is the number of persons of race k in school s.
Distribution of Achievement
Seven Oth-grade tests were administered to HS&Bsophomores. We constructed a measure
of mathematics achievement by adding together the scores on Math I (range 0-28) and Math
II (range 0-10), rescaling the test to range between zero and 10, and subtracting the mean.
A measure of English achievement was constructed by summing the scores for vocabulary
(range 0-21), reading (range 0-19), and writing (range 0-17) and similarly rescaling the
scores to range between zero and 10 and subtracting the mean. The science test (range 0-20)
and civics test (range 0-10), which were also centered and scaled to have a range of 10, measure science and social studies achievement.
We used the school-specific mean to indicate the level of achievement in each domain and
used the school-specific variance to indicate the degree of diversityin achievement in each
domain. And we used the school-specific correlationbetween any two areas to measure the
degree to which achievement is associated across domains.
De Facto Tracking
345
o
Sector, School Size, and Urbanicity
Schools were categorized as private (1) or public (0). The size of the school is the total enrollment of students; we used the natural log of the total enrollment. Two variables-urban (1)
or not (0) and rural(1) or not (0)-capture urbanicity.With both in the model, the comparison is to suburban schools.
TableA1. DescriptiveStatisticsfor Public(N = 841) and Private(N = 107) Schools
BothAnalyses
Ln(schoolsize)
Ln(schoolsize) missing
Socioeconomicdiversity
Meansocioeconomicstatus
Indexof racialdiversity
Racialdiversitymissing
1 racial-ethnic
group
2 racial-ethnic
groups
3 racial-ethnic
groups
4 racial-ethnic
groups
5 racial-ethnic
groups
Urban
Suburban
Rural
PrivateSchools
PublicSchools
Variable
Mean
SD
Min
Max
Mean
SD
Min
Max
6.958
0.100
0.534
-0.141
0.352
0.132
0.132
0.249
0.206
0.234
0.107
0.251
0.459
0.290
0.783
0.300
0.309
0.356
0.286
0.339
0.339
0.432
0.404
0.424
0.309
0.434
0.248
0.454
3.40
0.00
0.00
-1.15
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
8.58
1.00
2.02
1.15
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
6.188
0.047
0.675
0.309
0.350
0.056
0.093
0.206
0.262
0.308
0.093
0.234
0.654
0.112
0.833
0.211
0.501
0.441
0.280
0.230
0.291
0.404
0.440
0.462
0.291
0.423
0.226
0.316
3.04
0.00
0.00
-0.67
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
7.63
1.00
3.01
1.39
0.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.737
0.141
0.755
0.148
-3.69
0.00
-3.85
0.00
3.94
1.00
3.52
1.00
0.363
0.187
0.374
0.187
0.925
0.136
0.830
0.136
-2.38
0.00
-1.66
0.00
4.73
1.00
3.69
1.00
0.704
0.157
1.499
0.141
2.254
2.826
-4.93
0.00
-9.97
0.00
0.00
0.00
3.07
1.00
5.53
1.00
20.61
97.22
0.291
0.187
0.431
0.187
1.484
1.456
0.648
0.136
1.385
0.136
2.163
2.277
-1.16
0.00
-3.12
0.00
0.00
0.00
3.07
1.00
5.03
1.00
21.81
25.81
3.493
10.474
0.255
0.238
0.00
0.00
-0.90
0.00
96.43
163.80
1.00
1.00
1.696
5.302
0.537
0.084
2.045
7.415
0.221
0.278
0.00
0.00
-0.90
0.00
15.76
71.91
0.99
1.00
1.042
1.061
0.200
1.865
2.642
0.239
-3.69
-3.85
0.00
0.06
0.48
0.00
3.94
3.34
1.00
16.73
44.09
1.00
0.724
0.749
0.037
2.967
2.908
0.084
1.210
1.050
0.191
2.233
2.481
0.279
-2.38
-1.66
0.00
0.42
0.31
0.00
4.73
3.69
1.00
21.81
25.81
1.00
0.207
-0.70
1.00
0.654
0.199
-0.70
0.99
0.239
0.00
1.00
0.084
0.279
0.00
1.00
AnalysisOnly
All-Subject
Mathachievementmean
-0.078
Mathmean missing
0.020
-0.111
Englishachievementmean
0.022
Englishmean missing
Socialstudiesachievement
-0.084
mean
Socialstudiesmean missing 0.025
Scienceachievementmean -0.126
0.020
Sciencemean missing
Mathachievementvariance 1.807
Englishachievementvariance 1.787
Socialstudiesachievement
variance
2.180
Scienceachievementvariance 6.990
0.582
p achievement
0.060
p achievementmissing
AnalysisOnly
Math-English
Mathachievementmean
-0.155
-0.228
Englishachievementmean
mean missing 0.042
Math/English
Mathachievementvariance 3.616
Englishachievementvariance 3.557
variancemissing 0.061
Math/English
p achievement
0.703
(Mathand English)
p achievement
(Mathand English)missing 0.061
3 and Berends
Lucas
346
NOTES
1. Unfortunately,
there is no researchon
this point. Perhaps because tracking worked
in such a deterministic manner, we were
unable to find data from the period that
would allow us to assess the veracity of the
claim that tracking completely constrained
students' course taking in academic subjects.
Thus, we simply accept the logic of the previous regime as described by Hollingshead
(1949), Cicourel and Kitsuse(1963), and others and will do so unless and until research is
able to address the issue.
2. It should be noted that other possibilities for the maintenance of de facto tracking
have been implied or articulated. For example, it is possible that placement in nonacademic courses (e.g., keyboarding or band) may
affect the scheduling of academic courses. As
another example, consider that research has
suggested that some schools, especially large
urban institutions, assign students to classes
and thus courses more on the basis of logistical aims than pedagogic planning (e.g., Riehl,
Pallas, and Natriello 1999). Such processes
may drive de facto tracking in some schools.
We accounted for this possibility by including
controls for school size in all models and by
estimating some models with controls for
school urbanicity.Thus, if size and urban location are at the root of observed associations
between sociodemographic diversity and de
facto tracking, models with those controls
should show no association between diversity
and de facto tracking.
3. Recall our definition of tracking. Under
our definition, not taking a course in an academic subject is a structural category. For
example, students who take neither math nor
English occupy an important location in the
school stratificationsystem that has ramifications for their later success. For this reason,
we treated students who took neither math
nor English as being in the same level of
courses across subjects and included them in
the count variable. Under our definition of
tracking, this is appropriate. However, we
estimated all models deleting students who
took neither math nor English. The results
presented here were replicated almost completely.
4. The data can be downloaded from
http://sociology. berkeley.edu/faculty/lucas/
datasets.html
5. The joint test of the index of racialdiversity coefficient and the four racial-ethnic
groups coefficient produces a X2of 7.35 with
2 degrees of freedom for a p-value of .0253.
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Samuel R. Lucas, Ph.D., is Associate Professor,Sociology Department, Universityof CaliforniaBerkeley.His main fields of interest are social stratification,sociology of education, and research
methods and statistics. He is currentcompletinga study on discriminationand studyingchanges in
the structuraleffects of tracking.
Mark Berends,Ph.D., is AssociateProfessor,Departmentof Leadership,Policy,and Organizations,
PeabodyCollege,VanderbiltUniversity,PayneBuilding,Room207B, Nashville,TN37203. Hismain
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currentlystudying the structureand effect of trackingand investigatingthe effects of teachers'certificationon student outcomes.
348
Lucas and Berends
The authors thank Grace K. Kimfor providingthe measures of racial diversityused in this article,
CarrieS. Horsey and Michael Hout for their comments, Carl Mason for computing assistance,
GretchenStockmayerfor aid with computercode, EdBunnand CheriMintonof HarvardUniversity
for additionalhelp in computing,and EmilySperfor assistance with graphics. The researchfor this
article was supportedby grant R305F960164 from the FieldInitiatedStudies Program,Officeof
EducationalResearchand Improvement,U.S. Departmentof Education,and by grant 199700213
from the SpencerFoundation.However,the views in this articledo not necessarilyreflectthe views
7. 0, runningon
of the granting agencies, and all errorsare the responsibilityof the authors. STATA
a Sun MicrosystemsE-450 with 2 CPUsand 1.5 GBof RAM,were used in estimating the models.
All workfor this articlewas conducted with the assistance of the DemographyDepartmentof the
A versionof this articlewas presentedat the annual meeting of the
Universityof California-Berkeley.
AmericanSociologicalAssociation,New York,August 1996. Addressall correspondenceto Samuel
410 Barrows Hall #1980,
R. Lucas, Sociology Department, Universityof California-Berkeley,
e-mail:
CA
[email protected].
Berkeley, 94720-1980;