J Youth Adolescence (2010) 39:163–176 DOI 10.1007/s10964-009-9402-3

EMPIRICAL RESEARCH

Big School, Small School: (Re)Testing Assumptions about High School Size, School Engagement and Mathematics Achievement Christopher C. Weiss Æ Brian V. Carolan Æ E. Christine Baker-Smith

Received: 28 August 2008 / Accepted: 13 February 2009 / Published online: 3 March 2009 Ó 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-representative 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 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: [email protected] 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: [email protected] 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: [email protected]

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 advocating for the restructuring of secondary schools into smaller educational units. Examples of these efforts include the Coalition of Essential Schools and the Carnegie Foundation’s joint initiative, which focuses on more personalized teaching and learning (National Association of Secondary School Principals 1996; Sizer 1992); the Annenberg Foundation’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).

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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 student 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 anecdotal 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 mechanism 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 conclusions are currently applicable. This study addresses these limitations and contributes to the growing body of research on high school size, engagement 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 compared 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

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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 engagement 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 operationalization, 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 communal 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 functioning as intended (Battistich et al. 1995; Ravitch 2006). This may simply be a function of the problems of scalability 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–

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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 community, 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 curriculum 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 outcomes (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 disaffection 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 psychological 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

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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, Roderick’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 AinsworthDarnell (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

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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 (Burkam 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 mathematics 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’ gradelevel cohort. Moreover, because student engagement is a concept that encompasses both sociological and psychological 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 students’ 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

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(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 probability 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 sophomores. 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, academics, 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 comprehensive 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 contemporary 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.

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Measures

Academic Friend

School Engagement

Third, academic friend measures the importance of scholastic 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 important’’ (a = .617).

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 student 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.

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 education is necessary for future work attainment, if the student will learn skills directly related to future employment 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 variables used were reversed to provide a consistent measurement (a = .572). School Preparedness

Teacher Experience First, teacher experience consists of two variables that measure the teacher’s years of experience teaching mathematics 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).

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

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).

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 associations. These items are measured on a binary scale of with answers being yes or no scored with no = 0 and yes = 1.

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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 difficulty second-stage form, depending on the student’s number of correct answers in the routing test. Test specifications 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 criterionreferenced (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 engagement 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’ academic 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 classrooms, 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’’ represents a population of 400 or more students.

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We do not analyze classroom sizes for two reasons. Primarily, 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 measurement 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 questionnaire, 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 mathematics 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

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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 L

Educational motivation Teacher’s beliefs about ability

-0.025 (1.045) -0.139 (1.064)

Bottom 25% \ -0.680 Bottom 50% \ -0.458

Top 25% [ 0.814 Top 25% [ 0.542

-3.683 -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 engagement

b

-0.163 (0.628)

-3.194

a

For dichotomous variables only

b

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

contained in it, was viable as a predictor of the measured outcomes. We therefore present both the specific dimensions 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 scholastic 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. Approximately 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.

Table 2 Multilevel regression analysis of engagement, with demographic characteristics only (N = 10,946) Variable

Engagement

Female

0.114***

(SE) Age (held back)

(0.010) -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**

Predicting Student Engagement

(0.030) SES

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 sociodemographic characteristics of students. Results from these models are presented in Table 2. The results in Table 2 show that all included sociodemographic characteristics are significantly related to student 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).

0.157*** (0.020)

Constant

-0.332*** (0.020)

R2

0.2285

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

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

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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) Age (held back)

(.010)

(.150)

-0.505***

-8.645***

(0.013) Parent education (some college)

0.308*** (0.012)

Parent’s education (Ccollege)

0.191*** (0.014)

Black

-0.017 (0.020)

Hispanic

0.311*** (0.028)

SES

0.427**

Moderately-small schools

(0.024) -0.065

Moderately-large schools

-0.135**

Large schools

-0.146***

Constant

-0.707***

(0.045) (0.044) (0.030)

R

2

(0.196) 3.293*** (0.169) 5.365*** (0.208) -7.419*** (0.296) -5.317*** (0.406) 5.195*** (0.352) -0.821 (0.784) -0.031 (0.764) -0.020 (0.520) 48.542***

(0.032)

(0.526)

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

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 students (‘‘moderately large schools’’) and 1,600–2,499 students (‘‘biggest schools’’; Table 3). Results of these models are presented in Table 3. Consistent 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 students have lower levels of engagement. Examining differences related to individual characteristics, 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

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almost nine points lower on math evaluation than averageaged 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–American students are not significantly different in engagement than White students. In sum, the relationships between school size, demographic 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 measures. We find very similar results to school size though we here measure the relationship between a student’s gradelevel 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 functionally 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

J Youth Adolescence (2010) 39:163–176

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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) Age (held back)

(.010) -.492*** (.012)

Parent education (some college)

.271*** (.011)

Parent’s education (Ccollege) Black

.222***

Moderately-small cohort

(.162) 5.394*** (.196)

.020

-7.269***

.208*** (.025)

SES

(.183) 2.900***

(.013) (.018) Hispanic

(.141) -8.56***

.326*** (.022) -.127***

(.271) -5.685*** (.369) 4.876*** (.323) -.262

(.029)

(.516)

Moderately-large cohort

-.070*

.554

(.030)

(.535)

Biggest cohort

-.133*** (.027)

Constant R

2

-.648***

significance level as the previous model. The most noteworthy difference is positive influence of moderately-large cohorts on Black and Hispanic students. Black and Hispanic 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 significant 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).

.450 (.473) 48.864***

(.027)

(.442)

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

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 differences 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

Predicting Mathematics Scores from Engagement The final step of our analysis examines whether and how this measure of student engagement is related to mathematics achievement in the tenth grade. Table 6 reports the findings from a set of models predicting students’ mathematics scores as a function of the same set of sociodemographic control variables while including the predictive engagement measure that was the outcome in the models shown in Table 5. The baseline model, presented in the table’s first column, 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 students of better educated parents as well as those whose parents have higher socioeconomic status scoring significantly higher. African–American and Latino students have significantly lower scores than do their White and Asian classmates. In the next columns, we analyze the relationship 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

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Table 5 Multilevel regression analysis of engagement, with demographic characteristics and cohort size (N = 10,946) Variable Female (SE) Age (held back)

Base model .267*** (.010) -.491*** (.012)

Parent education (some college)

.270*** (.010)

Parent’s education (Ccollege) Black

.218***

SES

R2

-.262*** (.018) .265*** (.016) .256*** (.019)

.019

.002

.203*** (.025) .321*** (.022)

Constant

.227*** (.014)

(.013) (.018) Hispanic

Small cohort

-.723***

Moderate cohort .122*** (.024) -.303*** (.033) .156*** (.031) .050 (.034) -.169***

Mod. large cohort .321*** (.016) -.851*** (.020) .366*** (.017) .416*** (.024) .292***

(.025)

(.037)

(.045)

-.092 (.067)

.047 (.052)

.530*** (.038)

-.147** (.043) -.390***

.196*** (.046) -.369***

.737*** (.037) -1.146***

Large cohort .167*** (.027) -.151*** (.043) .125** (.038) .175*** (.039) -.004 (.039) -.071 (.052) .214*** (.047) -.508***

(0.023)

(.039)

(.055)

(.042)

(.058)

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

Female (SE) Age (Held back)

.928***

Small cohort .341

Moderate cohort

Large cohort

-3.126***

-1.515**

(.141)

(.251)

(.452)

(.195)

(.469)

-8.562***

-9.084***

-7.227***

-7.380***

-5.849***

(.183)

(.320)

(.601)

(.271)

(.741)

3.292***

5.148***

2.018**

-1.928***

-.587

Mod. large cohort

Parent education (some college)

2.903*** (.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***

Hispanic

(.271) -5.672***

(.424) -3.545*

(.673) -5.128***

(.559) -6.603***

(.687) -4.597***

(.369)

(1.147)

(.960)

(.451)

(.899)

3.812***

6.812***

2.921***

4.955***

(.323)

(.731)

(.860)

–

2.288***

SES

4.896***

Engagement Constant R

2

.489

(.449)

(.804)

1.861***

4.024***

(.253)

(.489)

(.163)

(.469)

49.029***

50.188***

45.729***

52.187***

51.225***

(.352)

(.674)

(.999)

(.617)

(1.048)

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

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

123

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

J Youth Adolescence (2010) 39:163–176

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 magnitude 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 adolescents, 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 pointing to beneficial school sizes of *600 students and additionally show that student-grade cohorts begin to exhibit negative effects when they grow beyond 400 students. 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. However, 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 mathematics achievement. We find that being held back negatively affects both engagement and achievement in all

173

cohort and school sizes, while parental education is positively related to these outcomes. Most interesting are our results with regard to mathematics achievement, engagement and race. We find that although, predictably, Black and Hispanic students are expected to score several points lower than their nonBlack, non-Hispanic peers when controlling for engagement, 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 provide 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 individuallevel characteristics that may influence academic engagement such as economic standing, parental education, and so forth. Yet our knowledge base of how individual characteristics interact with environment is lacking. Our analysis suggests that no school or cohort size will optimize outcomes 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, environment, or personal traits as potentially being caused by the size of the school or cohort. As a result, we have used

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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 statistics 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 characteristics 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

Table 7 Multilevel regression analysis of engagement with demographic characteristics, by school size (N = 10,946)

just their size but the resources of the parents and communities 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 understanding 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.

Variable

Small school

Female

-.011

(SE)

(.031)

Age (Held back)

-.166*** (.041)

Parent Education (some college)

0.04 (.036)

Parent’s Education (Ccollege) Black

School Sizes: Small, 1–599 students in school; Moderatelysmall, 600–999; Moderatelylarge, 1,000–1,599; Large, 1,600–2,499 * p \ .05, ** p \ .01, *** p \ .001

123

.206***

Mod. small school .313*** (.017)

-.744*** (.019)

.390***

.405***

(.018) .278***

.162***

(.020)

(.022)

0.064

-.087**

-.092**

(.066)

(.026)

(.034)

(.114)

(.066)

SES

0.016

-.164**

(.082)

(.048)

R

(.016)

(.043)

-.178

2

.373*** (.015)

-.270*** (.020)

Hispanic

Constant

Mod. large school

-.012

-.050

-.598***

.473*** (.034) .719***

Large school .101* (.043) -.190** (.065) .044 (.060) .094 (.062) -.028 (.063) -.094 (.077) .144*

(.032)

(.071)

-1.127***

-.268**

(.072)

(.048)

(.036)

(.093)

.0423

.3119

.3875

.0684

J Youth Adolescence (2010) 39:163–176

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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.