EMPIRICAL RESEARCH

Big School, Small School: (Re)Testing Assumptions about HighSchool Size, School Engagement and Mathematics Achievement

Christopher C. Weiss Æ Brian V. Carolan ÆE. Christine Baker-Smith

Received: 28 August 2008 / Accepted: 13 February 2009 / Published online: 3 March 2009

� Springer Science+Business Media, LLC 2009

Abstract In an effort to increase both adolescents’

engagement with school and academic achievement, school

districts across the United States have created small high

schools. However, despite the widespread adoption of size

reduction reforms, relatively little is known about the

relationship between size, engagement and outcomes in

high school. In response, this article employs a composite

measure of engagement that combines organizational,

sociological, and psychological theories. We use this

composite measure with the most recent nationally-repre-

sentative dataset of tenth graders, Educational Longitudinal

Study: 2002, (N = 10,946, 46% female) to better assess a

generalizable relationship among school engagement,

mathematics achievement and school size with specific

focus on cohort size. Findings confirm these measures to be

highly related to student engagement. Furthermore, results

derived from multilevel regression analysis indicate that, as

with school size, moderately sized cohorts or grade-level

groups provide the greatest engagement advantage for all

students and that there are potentially harmful changes

when cohorts grow beyond 400 students. However, it is

important to note that each group size affects different

students differently, eliminating the ability to prescribe an

ideal cohort or school size.

Keywords School size � Mathematics achievement �School engagement � High school students �High school organization

Introduction

The last two decades of school reform in the United States

have seen the emergence of a number of initiatives advo-

cating for the restructuring of secondary schools into smaller

educational units. Examples of these efforts include the

Coalition of Essential Schools and the Carnegie Founda-

tion’s joint initiative, which focuses on more personalized

teaching and learning (National Association of Secondary

School Principals 1996; Sizer 1992); the Annenberg Foun-

dation’s emphasis on reducing students’ alienation in

schools (Chicago Annenberg Challenge 1994); and the

Child Development Project’s focus on restructuring schools

to promote caring communities (Developmental Studies

Center 1998). Most prominent among recent initiatives is

the one promoted by the Bill and Melinda Gates Foundation,

which, as of 2005, had invested more than $800 million to

create 2,000 small high schools, particularly ones that focus

on underserved children of color (SRI/AIR 2002). Partly as a

result of this support, New York City opened 20 new small

schools in September, 2008, bringing the number of such

schools created in the past 5 years to more than 200

(Herszenhorn 2007).

C. C. Weiss (&)

Quantitative Methods in the Social Sciences (QMSS), Institute

for Social and Economic Research and Policy (ISERP),

Columbia University, 420 West 118th Street, Room 811,

Mail Code 3355, New York, NY 10027, USA

e-mail: cw2036@columbia.edu

B. V. Carolan

College of Staten Island, The City University of New York,

Building 3S-224, 2800 Victory Boulevard, Staten Island,

NY 10314, USA

e-mail: carolan@mail.csi.cuny.edu

E. C. Baker-Smith

Quantitative Methods in the Social Sciences (QMSS), Institute

for Social and Economic Research and Policy (ISERP),

Columbia University, 420 West 118th Street, Room 820,

Mail Code 3355, New York, NY 10027, USA

e-mail: ecb2125@columbia.edu

123

J Youth Adolescence (2010) 39:163–176

DOI 10.1007/s10964-009-9402-3

One of the underlying rationales of this set of reforms

(whether creating small schools from scratch or through

subdividing a large comprehensive school) is that the

learning settings of smaller schools facilitate greater stu-

dent engagement, which is associated with increases in

achievement, rates of graduation, and the likelihood of

post-secondary attendance (National Research Council the

Institute of Medicine 2004). That is, one of the primary

mechanisms through which smaller schools are believed to

benefit students is through enhanced student engagement;

however, initiatives to improve students’ achievement

through engagement are based more on theory and anec-

dotal evidence (e.g., Theroux 2007), while empirical

research evidence linking size to better outcomes through

student engagement is thin.

A small number of rigorous studies has linked school size

with academic performance (e.g., Lee and Smith 1997), with

many suggesting that engagement is the proximate mecha-

nism of this benefit. However, empirical work on this topic

has two key limitations. First, individual-level measures of

school engagement have been restricted to either behavioral

or psychological dimensions, neglecting to fully capture the

richness of this construct (Glanville and Wildhagen 2007).

Methodologically, studies that have measured the effects of

school size on engagement have either used small samples of

students and schools, which limit the generalizability of the

findings, or have used less current observational data from

cross-sectional designs, limiting the extent to which con-

clusions are currently applicable.

This study addresses these limitations and contributes to

the growing body of research on high school size, engage-

ment and achievement in three ways. Primarily, we define

and measure the construct of school engagement in terms of

both behavioral and psychological dimensions; we employ a

dynamic predictive variable that includes measures of both

sets of constructs. Next, by using data from one of the most

recent nationally-representative studies of a cohort of high

school students Educational Longitudinal Study (ELS:

02:02:2002), we estimate the effect of school engagement on

standardized mathematics scores using cohort size as com-

pared to the more commonly measured school size. Finally,

we examine separately the relationship of both school size

(the total number of students in a school) and cohort size (the

total number of students in a grade) on both the mathematics

score and on the predictive engagement measure.

Extant Theory on School Size and Engagement

School Size

Size as a structural characteristic of schools has received

much attention in the scholarly and policy literatures, with

the particular dimension of size varying based on the level

of schooling studied. While research on elementary school

size has generally focused on the size of classrooms (Finn

and Achilles 1999), research on high school size has

focused on the size of the aggregate unit (either total school

or school-within-school; Cotton 2002). The current

research focus on school size, to a great extent, stems from

research on the academic and social shortcomings of large,

comprehensive, ‘‘shopping mall’’ high schools (Powell

et al. 1985). Policy responses to these well-studied issues,

such as the creation of schools-within-schools, focus on

creating smaller schooling units in order to foster engage-

ment among students, as well as between teachers and

students (e.g., Fine 1991).

However, previous research has used a wide variety of

measures and values of school size, with authors employing

a number of different values as cut points in examining the

effects of the number of students. While the different

measurements represent a potential difficulty in operation-

alization, a more fundamental difficulty in examining school

size relates to what Bidwell and Kasarda (1980) identified as

the distinction between easily measurable characteristics of

students and schools, and the activities that occur within

schools. For example, although relations generally were

more positive and intimate in the smaller schools studied by

Lee et al. (2000), this situation did not always benefit all

students, particularly those who preferred the anonymity of

large schools due to the fact that their reputations or those of

their families followed them at school.

Another concern surrounding small schools is their

ability to increase achievement by creating a more com-

munal climate. If schools are successful in strengthening

the sense of community and developing a positive school

climate, but are not able to raise achievement at the same

time, it would appear that this reform may not be func-

tioning as intended (Battistich et al. 1995; Ravitch 2006).

This may simply be a function of the problems of scala-

bility and replicability. When schools are the unit of

innovation, effective change should be located in the

school itself and be specific to the school, each of which is

likely to have a unique organizational character and student

population (Stevenson 2000). Therefore, simply creating

smaller schools and transferring students into them from

larger schools may not produce the desired effect.

In addition, the research on the appropriate size of school

unit for student benefit has yielded inconsistent results.

There is little agreement about what specific size works best

for students. Garbarino (1980), echoing Barker and Gump

(1964), described the advantages for high schools with more

than 500 students, while Goodlad (1984) advocated for

schools between 500 and 600 students (see Lee 2000, for a

review of this literature). Lee and Smith (1997) concluded

that learning was greatest in middle-sized schools (i.e., 600–

164 J Youth Adolescence (2010) 39:163–176

123

900 students) compared with larger or smaller schools. They

also found that learning was more equitably distributed in

smaller schools, that school size has important effects on

learning, that many high schools should be smaller than they

currently are, and that high schools can be too small. It could

also be that the size of the school itself yields no benefits, but

appears to, given that school size is a feature of schools that

is often correlated with a number of other factors that predict

both engagement and achievement (e.g., Iatarola et al.

2008). The benefits of school size may be confounded with

features of the districts and student populations in which

small schools are found.

While some have offered specific recommendations for

size, others (e.g., Meier 1998; Raywid and Osiyama 2000)

have used qualitative criteria, such as the sense of com-

munity, to define what a ‘‘small school’’ is. Such authors

prefer instead to describe size in relation to a school’s

ability to provide collaborative opportunities for faculty

and possibility for personalization and safety for other

actors within the school. With this information, as well as

the knowledge that access to diverse and quality curricu-

lum may affect student achievement, we intend to

encompass a broader level of organizational characteristics

in our analysis to more carefully reflect the true effects of

school size (Gamoran and Hannigan 2000; Oakes 2005).

School Engagement

One key to small schools’ effectiveness, according to

theory, is that they generate greater levels of engagement,

which serves as a key link between school size and student

achievement. Previous research has established a strong

relationship between school engagement and student out-

comes (e.g., Fredericks et al. 2004; Jessor et al. 1998; Finn

and Rock 1997). Students who are better connected with

aspects of their schooling perform better academically and

have lower levels of problem behaviors (e.g., Newmann

et al. 1992; Bryk and Thum 1989; Gutman and Midgley

2000). More recently, a publication by the National

Research Council the Institute of Medicine (2004) draws

attention to how engagement with school can improve

academic achievement as well as reduce student disaffec-

tion and dropout rates.

The emerging body of work on school engagement

suggests that it is an essential student-level characteristic

and an important predictor of student success; however,

conceptual issues have inhibited the research community

from employing a consistent and robust indicator. Student

engagement is customarily defined as having both psy-

chological and behavioral dimensions. Psychological

dimensions are typically defined by enthusiasm, interest,

and intensity (Smerdon 2002; Newmann et al. 1992; Bollen

and Hoyle 1990), while behavioral dimensions are often

defined by students’ preparedness, attendance, and time

spent on academic work (Finn 1989; Lee et al. 1996).

Studies of alienation, or disengagement, examine the same

behaviors as engagement studies, only in reverse: cutting

class, tardiness, violence, and vandalism (Natriello 1984;

Newmann 1981). These various measures of school

engagement are used as both outcomes, as in the work of

Smerdon (2002) and Johnson et al. (2001), and more

commonly as predictors as in the work of Newmann et al.

(1992), Smerdon (1999) and Raudenbush (1984). Because

of the evidence that many of the measures discussed here

are related to student engagement, we combine measures

typically used as outcomes into our predictors in an attempt

to better capture all facets of student engagement.

A large number of studies has investigated the relationship

between engagement and desired school outcomes, generally

concluding that there is a consistent, positive relationship

between the two (Fredericks et al. 2004). For example, Rod-

erick’s (1993) analysis shows that students with higher levels

of school engagement are less likely to drop out of school

before completing their degrees (see also Bryk and Thum,

1989; Newmann et al. 1992; Crosnoe et al. 2002). Several

studies report a positive relationship between levels of

engagement and students’ grades and scores on standardized

tests (e.g., Lee and Smith 1995; Connell et al. 1994; Finn and

Voelkl 1993; Roeser et al. 1996). Roscigno and Ainsworth-

Darnell (1999) found that students who work hard in school

and pay attention in class have significantly higher scores on

achievement tests in high school. Finn (1989), (see also Finn

and Rock 1997) shows that more engaged students have

higher grades and fewer disciplinary problems than those who

are less engaged. Finally, Murdock et al. (2000) has a similar

finding, documenting the relationship between engagement

and a series of school discipline troubles. Overall, this body of

research strongly suggests that engagement with schoolwork

and the school community is a proximate determinant of

students’ achievement.

There has been a small, but influential number of studies

that examine the relationship between school size and

student engagement, within which a few merit mention.

For example, a study by Wehlage and Smith (1992) found

that smaller high schools were more likely than larger ones

to have the conditions that promote student engagement for

students at risk of dropping out. Similarly, Lee and Smith

(1997) found that students in smaller, more communally

organized schools had higher levels of engagement.

Efforts to increase students’ engagement with school,

however, are ultimately intended to increase students’

achievement. In particular, a recent focus of these efforts has

been the improvement of students’ mathematical ability.

There are several reasons for focusing on mathematics

achievement. Researchers agree that, in contrast to other

school subjects, mathematics learning is most likely to occur

J Youth Adolescence (2010) 39:163–176 165

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in school and be particularly sensitive to instruction.

Mathematics learning is thus more school dependent than

other subjects, such as reading or general knowledge (Bur-

kam et al. 2004; Porter 1989). Additionally, achievement in

mathematics has been shown to be associated with college

attendance. Students who score higher in mathematics on

standardized tests (Hoffer 1995) and who take more

advanced mathematics and science courses (Schneider et al.

1998) are more likely to attend competitive 4-year colleges.

Finally, the emphasis on mathematics is in response to recent

federal legislation, mathematics achievement is one of the

criteria by which students and schools are judged to make

adequate yearly progress under the No Child Left Behind

legislation (2001). In general, when it comes to shaping

students’ achievement, particularly in areas such as mathe-

matics that are sensitive to school-based instruction, the

consensus in the research literature is that that it may not be

the ‘‘smallness’’ of the school that matters most. Rather it

may be the community atmosphere that the school creates

which, in turn, enhances student engagement and ultimately

achievement.

Hypotheses

Given the well-established relationship between school

engagement and an array of desirable school outcomes, we

further investigate how its influence is shaped by students’

immediate organizational context, i.e., the size of the high

school that one attends, as well as the size of students’ grade-

level cohort. Moreover, because student engagement is a

concept that encompasses both sociological and psycho-

logical properties, we construct and employ a measure that

more completely reflects the totality of students’ schooling

experiences. Specifically, we hypothesize that this new

measure of engagement will be equally important to stu-

dents’ achievement in mathematics as previous measures, if

not an even more significant contributor to this relationship.

With regard to the relationship between school engagement

and mathematics achievement, based on school and cohort

size, we expect to find similar, if not stronger, relationships

between engagement and mathematics achievement in

cohorts than we do in school size. However, we also predict

that this relationship will vary along several key student

characteristics, meaning that the ability to prescribe an ideal

school or cohort size for all students is limited.

Methods

We test the relationship among high school size, school

engagement and achievement using the public-use data file

obtained from the Educational Longitudinal Study of 2002

(ELS: 02), conducted by the National Center for Education

Statistics (NCES). ELS: 02 is a nationally-representative

sample of over 16,000 students in 750 high schools and

provides detailed information about the nation’s high

schools and students. ELS: 02 used a two-stage sample

selection process. First, schools were selected with proba-

bility proportional to size, and school contacting resulted in

1,221 eligible public, Catholic, and other private schools

from a population of *27,000 schools containing sopho-

mores. Of the eligible schools, 752 participated in the study.

These schools were then asked to provide sophomore

enrollment lists. In the second stage of sample selection,

*26 students per school were selected from these lists.

Additional information on the base-year sample design can

be found in the base-year data file user’s manual (Ingels et al.

2005), chapter 3 and appendix J. ELS: 02 is similar to

National Educational Longitudinal Study of 1988 (NELS:

88) and contains many school and student measures,

including information related to student achievement, aca-

demics, interests, and demographic information.

Educational Longitudinal Study: 02 contains data from

multiple sources, not just from students and school records

but also from their parents, teachers, and administrators of

their high school, including the principal and library media

center director. The data collected from their teachers

provides direct information about the student as well as the

credentials and educational background information of the

teacher. This array of information provides a comprehen-

sive picture of the home, school, and community

environments and their influences on the student.

Sample

The final analytic sample includes only those students with

valid measures on school engagement and school size from

the base-year, tenth-graders in the year 2002 (n = 10,946).

For this wave, the overall weighted school participation rate

was 67.8% and the overall weighted student response rate

was 87.3%, although the response rate for certain domains

was below 85%. We focus on this tenth-grade cross-section

of students to capture a critical transition in adolescents’

secondary educational career. The first 2 years (ninth and

tenth-grades) of the high school experience have been

identified as a key turning point in the educational trajectory

of adolescents who are at risk (Natriello et al. 1990). At-risk

students are an important target population of the contem-

porary small school reform movement, particularly in urban

school districts that serve a disproportionately large number

of disadvantaged students (Fine 2005). Therefore we use this

tenth-grade cross-section as a way to examine students at a

critical juncture and to expand our understanding of what

organizational level’s size is of most importance to these

students.

166 J Youth Adolescence (2010) 39:163–176

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Measures

School Engagement

The key outcome variable of interest is students’ level of

school engagement in the tenth grade. Our measure is a

composite created from items on the base-year student,

parent and teacher questionnaires. This composite captures

students’ psychological and behavioral connections with

the values and aims of school. The social and academic

correlates of school engagement are determined from stu-

dent reports of their experiences in school. Details on the

specific measures, including the particular variables used in

creating the measures and their alpha scores require some

elaboration.

Several composite measures were created to provide

comprehensive information on specific dimensions of a

student’s experience in school that ultimately constitute our

measure of school engagement. Two facets of this step of

analysis require elaboration. First, although these measures

appear disparate, they all are related to school engagement

(see below and our forthcoming work for more detail on

engagement measure). Second, when testing the larger

composite measure of engagement with confirmatory factor

analysis, we find that excluding the weaker composites

does not significantly increase the reliability of the factor’s

eigenvalue and therefore include each of the following

seven variables in our composite measure of school

engagement.

Teacher Experience

First, teacher experience consists of two variables that

measure the teacher’s years of experience teaching math-

ematics at all grade levels and, separately, at the 7–12th

grade level. These variables are measured on a scale of

0–40 where each integer represents 1 year of teaching

experience (a = .985).

Delinquent Behavior

Second, delinquent behavior measures the student’s actions

with regard to truancy and delinquency. Variables provide

a measurement of the number of times a student performed

acts against school or legal codes. These items were

measured on a Likert scale with 1 being ‘‘never,’’ 2 =

‘‘1–2 times,’’ 3 = ‘‘3–6 times,’’ 4 = ‘‘7–9 times,’’ and 5

‘‘10 or more times.’’ The variables measured number of

times: late for school, cut/skipped classes, got in trouble,

got suspended or put on probation in school and out of

school (a = .745).

Academic Friend

Third, academic friend measures the importance of scho-

lastic grades to the student’s closest three friends. The

measurement scale for these variables was 1–3 with one

measuring no importance while three was ‘‘very impor-

tant’’ (a = .617).

Educational Motivation

Fourth, educational motivation measures the student’s

perception of school importance (a = .776). Each of the

variables compiled for this measure are based on the Likert

scale with 1 being ‘‘strongly agree’’, 2 and 3 ‘‘agree’’ and

‘‘disagree’’, and 4 meaning ‘‘strongly disagree.’’ The

variables ascertain, respectively, if classes are interesting,

if the student is satisfied with class performance, if edu-

cation is necessary for future work attainment, if the

student will learn skills directly related to future employ-

ment in school and if the student’s teachers expect success

in school.

Teachers’ Beliefs about Ability

Fifth, teachers’ beliefs about ability measures whether the

teacher believes that students can learn to be good at

mathematics or if they must have innate ability. The vari-

ables used were reversed to provide a consistent

measurement (a = .572).

School Preparedness

Sixth, school preparedness measures the extent to which a

student arrives at school prepared to learn (a = .813). The

three variables included in this composite measure how

often the student attends class without pen and paper,

without books and without homework completed on a scale

of 1–4 with 1 meaning ‘‘never’’, 2 = ‘‘seldom,’’ 3 =

‘‘often,’’ and 4 = ‘‘usually.’’

Parental Involvement

Seventh, parental involvement asks parents if they were

involved they are with their child’s school by measuring

involvement with various school-related organizations such

as parent–teacher associations and the like on a binary, yes

or no, scale (a = .696). These variables measure, the

extent of parental involvement by asking not only about

belonging to parent–teacher organizations, but if parents

participate in the organizations’ activities, if they volunteer

in the school, and if they belong to other parent associa-

tions. These items are measured on a binary scale of with

answers being yes or no scored with no = 0 and yes = 1.

J Youth Adolescence (2010) 39:163–176 167

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Mathematics Achievement

The mathematics outcome is a standardized score derived

from students’ performance on the ELS: 02 mathematics

assessment, which is based on item response theory. This

assessment maximizes the accuracy of measurement that

could be achieved in a limited amount of testing time while

minimizing floor and ceiling effects by matching sets of

test questions to initial estimates of students’ achievement.

This was accomplished by means of a two-stage test in

which all students received a short multiple-choice routing

test, scored immediately by survey administrators, who

then assigned each student to a low, middle, or high dif-

ficulty second-stage form, depending on the student’s

number of correct answers in the routing test. Test speci-

fications were adapted from frameworks used for NELS:

88. Most items were multiple choice, with about 10% of

the base-year mathematics items being open-ended. The

standardized scores are overall measures of status at a point

in time, but they are norm-referenced rather than criterion-

referenced (M = 48, SD = 9.4).

School and Cohort Size

The final variable of interest is school and cohort size. We

employ an ordinal measure of size, which is the measure

available in the public use dataset and is derived from the

administrator’s questionnaire, which indicates schools as

having 1–399 students, 400–599 students, etc. Although

most of the literature uses total school population, we also

examine the effects of the total tenth grade population. We

chose this measure to more closely define which specific

sizes of cohorts, as compared to or in conjunction with total

school population, may have an effect on student engage-

ment and achievement. Specifically, we intend for this

measure to more closely reflect a tenth grader’s actual

school experience. Much of this experience is conditioned

by course sequencing and its organization by grade-level

cohorts (Stevenson et al. 1994). Consequently, as groups of

adolescents proceed through similar course experiences

bounded by grade-level, this mechanism serves as the

primary vehicle through which peer relations develop and

endure (Monk and Haller 1993). Because of the importance

of grade-level cohorts in shaping both adolescents’ aca-

demic and social lives, we created a composite measure for

cohort size using tenth grade population. Most literature on

school size and engagement is on whole schools or class-

rooms, the group excluded from these studies is grade-level

cohorts (our focus). Specifically, the label ‘‘small’’ reflects

a tenth population of under 200 students, size ‘‘moderate’’

denotes a cohort of 200–299 students, ‘‘moderately large’’

represents a group of 300–399 students and ‘‘large’’ rep-

resents a population of 400 or more students.

We do not analyze classroom sizes for two reasons. Pri-

marily, classroom size is a measure used with children, not

adolescents, to measure engagement. Also it would be a

premature jump to smaller units from the aggregate mea-

surement of school size. To move too quickly from school

size to classroom size eliminates a potential confounding

factor of cohort size. Additionally, as noted above, much

research supports the theory that grade-level groups are also

significant (as compared to class-level). For example,

Hallinan and Sorenson (1985) find that though ability groups

are significant in student friendship networks, over time

these groups overlap into larger-grade-level formations.

Control Variables

We also include a set of student-level controls in our models,

with variables for students’ sex, race, and previous grade

retention taken from questions asked in the student question-

naire, along with measures of parental education and

economic status taken from the parent interview. The variables

used as controls were selected because they have been

repeatedly shown to affect a variety of school-based outcomes.

We use a measure of whether the student had been

retained at least once during the schooling career prior to

the end of eighth grade as a proxy for age. Because the

sample was chosen based on attendance in a particular

grade, age and previous retention are highly correlated,

preventing inclusion of both measures in our models. The

retention variable has been shown to be a more powerful

predictor of academic and behavioral difficulties. The

measure, taken from the parent interview, is dichotomous,

equal to one if the student has been retained previously.

Analytic Procedure

Because these data are nested, with a group of students

clustered within a group of schools, a multilevel analysis

strategy is required (Bryk and Raudenbush 1992). Thus, we

utilize multilevel regression of the various outcomes on the

control factors that allows us to see if the size of school or

cohort is significantly related to student outcomes. Moreover,

use of this model controls for unobserved factors at the school

level that may account for differences in students’ outcomes.

Results

Our results reflect a variety of relationships between school

engagement, student cohort size, and standardized mathe-

matics scores. In Table 1, we present statistics for each of

the variables used in this analysis. The composite

engagement measure, as well as all created measures

168 J Youth Adolescence (2010) 39:163–176

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contained in it, was viable as a predictor of the measured

outcomes. We therefore present both the specific dimen-

sions of engagement as well as the composite measure.

In this sample, *46% of students are male and *18% of

students have been held back at some point in their scho-

lastic career. Of parents of the students in this sample,

*37% have some college while nearly 30% of the same

population holds at least a bachelor’s degree. Said another

way, about two-thirds of students’ parents in this sample

have at least 2 years of post-secondary education. Approx-

imately one quarter of the population is African–American.

On a standardized scale (0–1.80) half of the students score

below a .7 with regard to socio-economic status (SES) while

the top 25% appear above the 1.18 mark.

Predicting Student Engagement

In the first stage of our analysis, we examine the contours

of school engagement, estimating a series of models that

predict our measure of engagement from a set of socio-

demographic characteristics of students. Results from these

models are presented in Table 2.

The results in Table 2 show that all included socio-

demographic characteristics are significantly related to stu-

dent engagement (p \ .001). Students who have been

previously retained or who are Hispanic have lower levels of

engagement while female students, African–Americans,

those whose mothers have more than 2 years of college and

from higher-SES families have higher levels of engagement.

In general, these results confirm patterns consistent in recent

research (e.g., Shernoff and Schmidt 2007).

School Size on Engagement and Achievement

Our first examination is of the effects of school size and

cohort size on engagement. As noted earlier, there are

many exemplary studies on the effect of school size on

students. We examine both engagement and math

achievement as outcomes here, although we find greater

Table 1 Descriptive characteristics (N = 10,946)

Variable Mean (SD) Percent in low categorya Percent in higher category Range bottom

A Female – 45.99 50.95 –

B Age (held back) – 82.03 17.97 –

C Parent’s education (some college) – 62.61 37.39 –

D Parent’s education (Ccollege) – 70.67 29.33 –

E Black – 75.72 24.28 –

F Hispanic – 96.02 3.98 –

G SES 0.493 (0.233) Bottom 50% \ 0.693 Top 25% [ 1.183 0.00

H Teacher experience 0.011 (0.950) Bottom 25% \ -0.917 Top 25% [ 0.785 -1.390

I Delinquent behavior 0.068 (1.133) Bottom 25% \ -0.580 Top 25% [ 0.314 -0.796

J Academic friends -0.025 (1.049) Bottom 25% \ -0.947 Top 25% [ 0.617 -3.229

K Educational motivation -0.025 (1.045) Bottom 25% \ -0.680 Top 25% [ 0.814 -3.683

L Teacher’s beliefs about ability -0.139 (1.064) Bottom 50% \ -0.458 Top 25% [ 0.542 -3.429

M School preparedness -0.009 (1.027) Bottom 25% \ -0.179 Top 25% [ 0.664 -2.677

N Parental involvement -0.146 (0.949) Bottom 50% \ -0.887 Top 25% [ 0.548 -0.8874

H–N School engagementb -0.163 (0.628) -3.194

a For dichotomous variables onlyb The variable ‘‘school engagement’’ is a composite derived variables H–N

Table 2 Multilevel regression analysis of engagement, with demo-

graphic characteristics only (N = 10,946)

Variable Engagement

Female 0.114***

(SE) (0.010)

Age (held back) -0.171***

(0.010)

Parent education (some college) 0.103***

(0.010)

Parent’s education (Ccollege) 0.092***

(0.010)

Black 0.075***

(0.010)

Hispanic -0.086**

(0.030)

SES 0.157***

(0.020)

Constant -0.332***

(0.020)

R2 0.2285

* p \ .05, ** p \ .01, *** p \ .001

J Youth Adolescence (2010) 39:163–176 169

123

effects on engagement. In this analysis, the school sizes are

grouped into categories based on Lee et al. (2000)’s work:

1–599 students (which we label ‘‘small schools’’), 600–999

students (‘‘moderately small schools’’), 1,000–1,599 stu-

dents (‘‘moderately large schools’’) and 1,600–2,499

students (‘‘biggest schools’’; Table 3).

Results of these models are presented in Table 3. Con-

sistent with the findings of Lee and Smith (1997), these

models show that there are significant differences related to

student engagement between schools of different sizes,

while school size is not significantly related to mathematics

achievement. Compared with students attending schools of

the smallest size (the omitted category), those in schools

with 1,000–15,999 students or with more than 1,600 stu-

dents have lower levels of engagement.

Examining differences related to individual character-

istics, we find that females have significantly higher

engagement than males (b = .290, p \ .001); however, in

the models for math achievement, females do worse than

their male peers. Students previously held back score

almost nine points lower on math evaluation than average-

aged students and also are less engaged (p \ .001). We find

expected positive effects of parental education and some

negative relationships between race and the two outcome

measures. However, it is noteworthy that African–Ameri-

can students are not significantly different in engagement

than White students.

In sum, the relationships between school size, demo-

graphic characteristics’ and student engagement, or

achievement, confirm previous research on school size.

However, examining the effects of cohort size provides

further information on how size affects students.

Cohort Size on Engagement and Achievement

In Table 4 we repeat the models of the previous table to

examine the effects of cohort size on our outcome mea-

sures. We find very similar results to school size though we

here measure the relationship between a student’s grade-

level group, or cohort, and the outcome variables. These

figures indicate that the variables for cohort size also have a

significant effect on engagement. Each of the dummies for

cohort size, included in the models predicting engagement,

is negative and significant. This indicates that students in

each of the three included cohort sizes have lower levels of

engagement than students in the smallest cohort. However,

as was the case in the models of Table 3, there are no

significant differences in mathematics achievement by

school size. The effects of the individual-level variables are

roughly the same (both in magnitude and significance) as in

the previous set of models.

Overall, the findings reported in Tables 3 and 4 show

that school size effects and cohort size effects are func-

tionally equivalent, in terms of effects on student

engagement and achievement. Therefore, in the section

following, we expand the exploration of school size into

one of cohort size, hypothesizing that it is the size of the

student’s cohort, more than the size of the school, which

has significant effects on engagement.

Examining Engagement within Cohort Size

In the next section of analysis, we examine whether the

relationship between the individual-level factors and

engagement vary by cohort size. Results of this analysis are

presented in Table 5.

The first column of Table 5 is quite similar to the results

presented in Table 4, though the model in Table 5 does not

contain the cohort size predictors of Table 4. The second

column in Table 5 contains data on the relationship of the

full set of predictors to student engagement among students

in tenth grade cohorts of fewer than 200 students. In these

Table 3 Multilevel regression analysis of engagement and math

achievement, with demographic characteristics, comparing school

sizes (N = 10,946)

Variable Student

engagement

Math

achievement

Female 0.290*** -0.746***

(SE) (.010) (.150)

Age (held back) -0.505*** -8.645***

(0.013) (0.196)

Parent education (some college) 0.308*** 3.293***

(0.012) (0.169)

Parent’s education (Ccollege) 0.191*** 5.365***

(0.014) (0.208)

Black -0.017 -7.419***

(0.020) (0.296)

Hispanic 0.311*** -5.317***

(0.028) (0.406)

SES 0.427** 5.195***

(0.024) (0.352)

Moderately-small schools -0.065 -0.821

(0.045) (0.784)

Moderately-large schools -0.135** -0.031

(0.044) (0.764)

Large schools -0.146*** -0.020

(0.030) (0.520)

Constant -0.707*** 48.542***

(0.032) (0.526)

R2 0.2664 0.2773

School sizes: Small, 1–599 students in school; Moderately-small,

600–999; Moderately-large, 1,000–1,599; Large, 1,600–2,499

* p \ .05, ** p \ .01, *** p \ .001

170 J Youth Adolescence (2010) 39:163–176

123

groups we find that previous grade retention and higher

levels of family SES are negative predictors of student

engagement. Students whose parents have higher levels of

education and female students have higher levels of

engagement in the smallest schools.

The effects of individual-level predictors in the next

largest cohort size (between 200 and 299 students) are

presented in the next column labeled ‘‘Moderate.’’ Similar

to the previous model, female students are more engaged

than males and students who have been held back are less

engaged than those who have not. There are some differ-

ences in coefficient size and significance as well. In cohorts

of moderate size, African–American students are less

engaged than their White peers. Moreover, in this model,

family socioeconomic status has a significant positive

effect on engagement.

For models of the next largest cohort size, those with

300–399 tenth grade students and labeled ‘‘Moderately

large,’’ Many of the coefficients are of a similar size and

significance level as the previous model. The most note-

worthy difference is positive influence of moderately-large

cohorts on Black and Hispanic students. Black and His-

panic students with cohorts of this size are more engaged

than their white peers. The final column of the table show

results from the models for students with the largest sized

cohorts. The dummy variables for race are not significant

in these models.

Age, here, merits closer examination as it has different

effects between groups. Being held back is highly signifi-

cant in all cohort sizes with a small to medium sized

negative effect on engagement; the largest effect is in

moderately large cohorts (b = -.851, p \ .001). The

negative relationship between age and engagement

decreases significantly in larger cohorts (those over 400

students; -.151, p \ .001).

Predicting Mathematics Scores from Engagement

The final step of our analysis examines whether and how

this measure of student engagement is related to mathe-

matics achievement in the tenth grade. Table 6 reports the

findings from a set of models predicting students’ mathe-

matics scores as a function of the same set of socio-

demographic control variables while including the predic-

tive engagement measure that was the outcome in the

models shown in Table 5.

The baseline model, presented in the table’s first col-

umn, reveals some patterns consistent with previous

research examining tenth graders’ scores on standardized

tests of mathematics. Females score significantly lower

than do their male counterparts (b = -.928, p \ .001).

The coefficient for previous grade retention is large and

significant (b = -8.562, p \ .001). Parental education is

highly predictive of mathematics performance, with stu-

dents of better educated parents as well as those whose

parents have higher socioeconomic status scoring signifi-

cantly higher. African–American and Latino students have

significantly lower scores than do their White and Asian

classmates. In the next columns, we analyze the relation-

ship between these characteristics for each of the cohort

sizes controlling for engagement. Looking across all

models, we find previous retention status and race strongly

and negatively associated with mathematics scores in

tenths grade (p \ .001). In short, students who are older

than their grade-level peers, or of a minority race, have

much lower mathematics scores than the average student,

even controlling for engagement.

There are important differences in the effect of gender

across cohorts. The differences between males and female

is non-significant in schools of the two smallest cohort

sizes, while the larger two cohort sizes show negative

Table 4 Multilevel regression analysis of engagement and math

achievement, with demographic characteristics, comparing cohort

sizes (N = 10,946)

Variable Student

engagement

Math

achievement

Female .267*** -.927***

(SE) (.010) (.141)

Age (held back) -.492*** -8.56***

(.012) (.183)

Parent education (some college) .271*** 2.900***

(.011) (.162)

Parent’s education (Ccollege) .222*** 5.394***

(.013) (.196)

Black .020 -7.269***

(.018) (.271)

Hispanic .208*** -5.685***

(.025) (.369)

SES .326*** 4.876***

(.022) (.323)

Moderately-small cohort -.127*** -.262

(.029) (.516)

Moderately-large cohort -.070* .554

(.030) (.535)

Biggest cohort -.133*** .450

(.027) (.473)

Constant -.648*** 48.864***

(.027) (.442)

R2 0.2315 0.2915

Small cohort, 1–200 students (reference group); Moderately small

cohort, 2–300 students; Moderately large cohort, 3–400 students;

Large cohort, [400 students

* p \ .05, ** p \ .01, *** p \ .001

J Youth Adolescence (2010) 39:163–176 171

123

effect on mathematics scores (b = -3.126, p \ .001;

b = -1.515, p \ .01). Not surprisingly, we find that

parental education, SES and higher engagement reflect

positively on mathematics scores (p \ .001). Here there is

a large difference between some parental college education

and parents who have completed more than a bachelor’s

Table 5 Multilevel regression analysis of engagement, with demographic characteristics and cohort size (N = 10,946)

Variable Base model Small cohort Moderate cohort Mod. large cohort Large cohort

Female .267*** .227*** .122*** .321*** .167***

(SE) (.010) (.014) (.024) (.016) (.027)

Age (held back) -.491*** -.262*** -.303*** -.851*** -.151***

(.012) (.018) (.033) (.020) (.043)

Parent education (some college) .270*** .265*** .156*** .366*** .125**

(.010) (.016) (.031) (.017) (.038)

Parent’s education (Ccollege) .218*** .256*** .050 .416*** .175***

(.013) (.019) (.034) (.024) (.039)

Black .019 .002 -.169*** .292*** -.004

(.018) (.025) (.037) (.045) (.039)

Hispanic .203*** -.092 .047 .530*** -.071

(.025) (.067) (.052) (.038) (.052)

SES .321*** -.147** .196*** .737*** .214***

(.022) (.043) (.046) (.037) (.047)

Constant -.723*** -.390*** -.369*** -1.146*** -.508***

(0.023) (.039) (.055) (.042) (.058)

R2 0.2285 0.1819 0.0850 0.4005 0.0973

Small cohort, 1–200 students; Moderate cohort, 2–300 students; Moderately large cohort, 3–400 students; Large cohort, [400 students

* p \ .05, ** p \ .01, *** p \ .001

Table 6 Multilevel regression analysis of mathematics achievement, with demographic characteristics and student engagement by cohort size

(N = 10,946)

Variable Base model Small cohort Moderate cohort Mod. large cohort Large cohort

Female .928*** .341 -.587 -3.126*** -1.515**

(SE) (.141) (.251) (.452) (.195) (.469)

Age (Held back) -8.562*** -9.084*** -7.227*** -7.380*** -5.849***

(.183) (.320) (.601) (.271) (.741)

Parent education (some college) 2.903*** -1.928*** 3.292*** 5.148*** 2.018**

(.161) (.284) (.573) (.208) (.656)

Parent’s education (Ccollege) 5.404*** 2.182*** 7.185*** 7.508*** 2.968***

(.196) (.325) (.628) (.209) (.684)

Black -7.251*** -7.380*** -5.563*** -8.077*** -6.870***

(.271) (.424) (.673) (.559) (.687)

Hispanic -5.672*** -3.545* -5.128*** -6.603*** -4.597***

(.369) (1.147) (.960) (.451) (.899)

SES 4.896*** 3.812*** 6.812*** 2.921*** 4.955***

(.323) (.731) (.860) (.449) (.804)

Engagement – 2.288*** .489 1.861*** 4.024***

(.253) (.489) (.163) (.469)

Constant 49.029*** 50.188*** 45.729*** 52.187*** 51.225***

(.352) (.674) (.999) (.617) (1.048)

R2 0.2820 0.3042 0.2598 0.4453 0.3039

Small cohort, 1–200 students; Moderate cohort, 2–300 students; Moderately large cohort, 3–400 students; Large cohort, [400 students

* p \ .05, ** p \ .01, *** p \ .001

172 J Youth Adolescence (2010) 39:163–176

123

degree on student scores, though both effects are strong. It

is also evident in this table that, on average, SES is a

powerful predictor of mathematics scores. Students in the

moderate cohort size feel the advantages of SES most

strongly with an almost seven point lead in mathematics

scores.

The effects of engagement on mathematics outcomes

also vary by the size of the tenth grade class size. Looking

across the models, we see that student engagement is

positively related to mathematics achievement in all cohort

sizes, save for the column labeled ‘‘Moderate.’’ The mag-

nitude of engagement’s effect is greatest in cohorts of the

largest size.

Discussion

Our results show that very small student groups tend to

exacerbate already extant disadvantages among adoles-

cents, particularly with regard to race. Consistent with

previous research on small schools, moderately sized

cohorts appear to provide the greatest advantage for all

students. Our findings support the general literature point-

ing to beneficial school sizes of *600 students and

additionally show that student-grade cohorts begin to

exhibit negative effects when they grow beyond 400 stu-

dents. However, our most important contribution is to

highlight the diverse impact that each size has on different

students thereby calling into question policies advocating

for an optimal size as they are potentially detrimental to

certain students.

Similar to Wyse et al. (2008) findings, the results of this

study raise questions about the implications of the return on

the investment from these smaller school environments.

The impact of high school size on both student engagement

and various academic outcomes is a pressing educational

policy concern. In light of the establishment of small

schools across the United States, particularly in historically

under-achieving school districts in urban areas, the

importance and relevance of this topic has grown. How-

ever, few studies have systematically and directly

investigated the relationships between size, engagement,

and outcomes. Our analysis finds that smaller cohorts are

associated with higher levels of student engagement;

however, these differences in engagement do not appear to

translate consistently into benefits for student achievement.

In this analysis we find that, overall, sex’s effect is more

significant when examined within larger cohorts; females

appear to have an advantage over males with regard to

engagement. However, when controlling for engagement

their sex becomes a disadvantage with regard to mathe-

matics achievement. We find that being held back

negatively affects both engagement and achievement in all

cohort and school sizes, while parental education is posi-

tively related to these outcomes.

Most interesting are our results with regard to mathe-

matics achievement, engagement and race. We find that

although, predictably, Black and Hispanic students are

expected to score several points lower than their non-

Black, non-Hispanic peers when controlling for engage-

ment, these same Black students do have small positive

relationships with engagement, providing further evidence

of what is referred to as the ‘‘engagement-achievement

paradox’’ (Shernoff and Schmidt 2007); these relationships

vary by both school and cohort size. Interestingly we see

these small positive effects when looking only at cohort

sizes. Similar to the cohort size analysis, the negative

effects of being Black which appear in both the school size

and math score analyses are smallest in moderately sized

schools and cohorts.

Although we are unable to examine the possibility in

these data, future research should focus on potential effects

of adolescent peer groups. Related to the findings of

Goldsmith (2004), it may be that larger cohort sizes pro-

vide diverse peer group options that may serve to mediate

racial differences. Unfortunately, Hispanic students reflect

traditional expectations and have a negative relationship

with both achievement and engagement (e.g., Shernoff and

Schmidt 2007). As with African–American students, this

effect is most notable in the smallest cohort size and it

could be said that in this cohort the minority students stand

alone as minorities as opposed to having both the support

and peer choice available in larger tenth grade student

bodies.

In addition to the ways in which students function within

peer groups, social scientists also look to other individual-

level characteristics that may influence academic engage-

ment such as economic standing, parental education, and so

forth. Yet our knowledge base of how individual charac-

teristics interact with environment is lacking. Our analysis

suggests that no school or cohort size will optimize out-

comes for all students.

As with most studies there are limitations within this

work that require acknowledgement. Primarily, we must

point out that this data is cross-sectional in nature. Using

cross-sectional data is useful for descriptive analyses but

does not allow the illumination of causal relationships or

the exploration of change in outcome variables over time

(Singleton and Straits 2005). Additionally, we highlight a

concern about the use of observational data where results

are often distorted in non-experimental data by what is

called ‘‘selectivity bias.’’ Such bias is potentially severe

because it risks a misrepresentation of students’ outcomes

caused by unobserved differences in background, envi-

ronment, or personal traits as potentially being caused by

the size of the school or cohort. As a result, we have used

J Youth Adolescence (2010) 39:163–176 173

123

statistical controls for student characteristics in order to

better approximate an ‘‘apples to apples’’ comparison.

However, there remains a possibility that these students

differ in ways not recorded by the available data though

this cannot be controlled by even the most sophisticated

methods. Finally, this analysis lacks knowledge about the

technical core within the sample. Though we are able to

measure general statistics such as a teacher’s number of

years in the classroom, more extensive data on teacher

quality or effectiveness is not present in this data set.

Though much has been written on teacher quality and

qualifications, conclusions about the value of teacher

qualification data are mixed leaving the use of such sta-

tistics unreliable (Kennedy 2008). More qualitative

evaluations of teacher quality are even less consistent as it

is difficult to guarantee inter-rater reliability throughout

evaluations of subjective measures. These three limitations

are not sufficient to discount this study though they are

integral as a cautionary note to both academics and lay

persons in interpreting these results.

There is a great deal still to learn about the ways that

school structure may affect both student engagement and

thereby student achievement. Our results vary on each

student body size and demand a closer analysis of each

group as it effects the engagement and achievement of

students with various socio-demographic characteristics. In

addition, it is likely that there are other school character-

istics that may affect our outcomes such as neighborhood,

school resources and others. Extant research shows that

what makes smaller schools and cohorts successful is not

just their size but the resources of the parents and com-

munities in which they are located (Passmore 2002). Large

high schools, particularly those in urban areas, do not have

the same resources as small schools, whose social capital

both in, and out, of school can reinforce norms that aligned

with the goals of schooling (Noguera 2003). However,

larger schools and cohorts have other resources, such as a

larger variety of extracurricular and course options, which

may provide different avenues for greater levels of

engagement with the school environment. A reduction in

size does not guarantee that students in those schools will

experience the same benefits as students in smaller schools

and cohorts in other locations.

Our analysis highlights the importance of an under-

standing of adolescent behavioral processes when

evaluating the most appropriate school forms for this

unique population. We do not find consistent benefits of

smaller schools for all types of students. Policy makers

must tread carefully as they jump on new trends in reform,

carefully evaluating the evidence that best matches their

students’ demographics not just the general population.

Small schools for adolescents are not a one size fits all

solution and must be carefully constructed in each locale to

carefully reflect both individual capacity and needs.

Appendix

See Table 7.

Table 7 Multilevel regression

analysis of engagement with

demographic characteristics, by

school size (N = 10,946)

School Sizes: Small, 1–599

students in school; Moderately-

small, 600–999; Moderately-

large, 1,000–1,599; Large,

1,600–2,499

* p \ .05, ** p \ .01,

*** p \ .001

Variable Small

school

Mod. small

school

Mod. large

school

Large

school

Female -.011 .313*** .373*** .101*

(SE) (.031) (.017) (.015) (.043)

Age (Held back) -.166*** -.270*** -.744*** -.190**

(.041) (.020) (.019) (.065)

Parent Education (some college) 0.04 .390*** .405*** .044

(.036) (.018) (.016) (.060)

Parent’s Education (Ccollege) .206*** .278*** .162*** .094

(.043) (.020) (.022) (.062)

Black 0.064 -.087** -.092** -.028

(.066) (.026) (.034) (.063)

Hispanic -.178 -.050 .473*** -.094

(.114) (.066) (.034) (.077)

SES 0.016 -.164** .719*** .144*

(.082) (.048) (.032) (.071)

Constant -.012 -.598*** -1.127*** -.268**

(.072) (.048) (.036) (.093)

R2 .0423 .3119 .3875 .0684

174 J Youth Adolescence (2010) 39:163–176

123

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Author Biographies

Christopher C. Weiss directs the Quantitative Methods in the Social

Sciences (QMSS) M.A. Program at Columbia University, where he is

also affiliated with Columbia’s Institute for Social and Economic

Research and Policy and the Robert Wood Johnson Foundation

Health and Society Program. His primary research interests center on

the influence of organizations and institutions on children and

adolescents, and environmental influences on obesity.

Brian V. Carolan is an Associate Professor of Education at the

College of Staten Island, The City University of New York. He

received his doctorate in sociology of education from Teachers

College, Columbia University. His major research interests include

school organization and social networks.

E. Christine Baker-Smith is a student in the Leadership, Policy and

Politics EdM program at Teachers College, Columbia University and

a graduate of Stanford University’s School of Education. She is the

Program Coordinator of both the Quantitative Methods in the Social

Sciences and the Oral History Master’s degree programs at Columbia.

Her research interests are focused on sociological issues of inequality

and social stratification.

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