An Examination of the Relationship Among High School Size,

Social Capital, and Adolescents’ Mathematics Achievement

Brian V. CarolanMontclair State University

In an effort to enhance both adolescents’ social capital and increase achievement, public school districts across theUnited States have created small high schools. Using data derived from a longitudinal and nationally representativestudy of U.S. high school students, the Educational Longitudinal Study of 2002, results show that when adolescents’parents know their friends’ parents math achievement is significantly predicted. This association, however, is nonsig-nificant when conditioned on standard measures of prior achievement and family background, among others. In addi-tion, while this relationship is also strong and significant within small high schools, it, too, is eliminated whenconditioned on select confounding variables. These findings are discussed in terms of current efforts to improveachievement through reductions in school size.

For the last 20 years, reformers have promoted anumber of initiatives that restructure high schoolsinto smaller educational units. Examples includethe Annenberg Foundation’s emphasis on reducingadolescents’ alienation in schools (Chicago Annen-berg Challenge, 1994), and the U.S. Department ofEducation’s Small Learning Communities (SLC)Program, which continues to award discretionarygrants to districts to improve achievement throughreductions in high school size (No Child LeftBehind Act, 2001). Most prominent among theseinitiatives is the one promoted by the Bill andMelinda Gates Foundation, which over the past10 years has invested over $2 billion dollars to sup-port the creation of 2,000 small high schools nation-wide, particularly in urban districts that serveyouth and adolescents of color (Hursh, 2011). NewYork City reflects this broad trend. The nation’slargest school district opened 12 new small highschools in September 2011, bringing the number ofsuch schools created in the past 5 years to morethan 200 (New York City Department of Education,2011).

Embedded in the larger school choice and mar-ket-based reform movement (Kafka, 2008), smallschools are thought to facilitate greater “connected-ness” among adolescents, which is associated withincreases in achievement, rates of graduation, andthe likelihood of postsecondary attendance (Cotton,1996a,b). That is, one of the primary mechanismsthrough which smaller high schools are believed to

benefit adolescents is through enhanced socialcapital, a concept first invoked by Coleman (1988)to explain differences in student learning acrossschool sectors (Catholic vs. public). However, thesereforms are based more on theory and anecdotalevidence, while empirical evidence linking smallschool size to better outcomes through socialcapital is thin.

A small number of rigorous reviews and studieshave linked high school size with adolescents’academic performance (e.g., Lee & Smith, 1997;Leithwood & Jantzi, 2009). Many of these haveimplicitly suggested that social capital is the proxi-mate mechanism of this benefit; however, empiricalwork on this topic has three key limitations. First,measures of social capital have been restricted tostructural dimensions, neglecting to fully captureand account for the effects of the content of adoles-cents’ social networks. Second, a number of studieshave tested whether social capital explainsdifferences in achievement across school sectors(e.g., Morgan & Sørensen, 1999; Morgan & Todd,2009). However, even though reformers promotereductions in school size based on the idea thatthey promote more favorable social relationships,few empirical studies have explicitly modeled thisrelationship with a theoretical and analytical focuson adolescents in the public sector. Moreover,methodologically, studies that have measured theeffects of social capital on achievement havetypically used less current observational data (e.g.,the National Longitudinal Study of 1988), limitingthe extent to which conclusions are applicable.

Requests for reprints should be sent to Brian V. Carolan,Department of Educational Foundations, Montclair State Univer-sity, Room UN 2161, 1 Normal Ave., Montclair, NJ 07043. E-mail: carolanb@mail.montclair.edu

© 2012 The Author

Journal of Research on Adolescence © 2012 Society for Research on Adolescence

DOI: 10.1111/j.1532-7795.2012.00779.x

JOURNAL OF RESEARCH ON ADOLESCENCE, ***(*), 1–13

PURPOSE

This study addresses these limitations and contrib-utes to the growing body of research on socialcapital, high school size, and achievement in threeways. First, measures of social capital are createdin a manner that captures the structure and contentof parents’ and their adolescents’ social relation-ships around school. Next, these measures areemployed in hierarchical linear models that alloweffects on achievement to vary across U.S. publichigh schools of different sizes. Third, this studyemploys longitudinal data from the most recentnationally representative study of a cohort of highschool students, the Education Longitudinal Studyof 2002 (ELS) (Ingels, Pratt, Rogers, Siegel, & Stutts,2005).

BACKGROUND

One aim of the current wave of high school reformis to create schools where adolescents are better“connected.” The assumption is that this “connect-edness” improves performance by facilitating com-munication, the exchange of socioemotionalsupport, and reinforcing norms—a notion that isreferred to as social capital (Lin & Erikson, 2011).Unfortunately, in part because of the popularity ofthis notion, social capital is often discussed invague terms that frequently do not make explicitthe mechanisms through which relationships affectacademic achievement (Maroulis & Gomez, 2008).Kadushin (2004) makes a larger point, stating thatsocial capital has become a “potpourri” of conceptsand theoretical orientations and a term that is“widely thrown about.” Nowhere is this ambiguitymore apparent than in the debate surrounding highschool size.

High School Size and Social Capital

Size as a structural characteristic of schools hasreceived much attention from developmentalresearchers (see e.g., Eccles, Lord, & Midgely, 1991),with the particular dimension of size varying basedon the level of schooling studied. While research onelementary school size has generally focused on sizeof classroom (Finn & Achilles, 1999), research onhigh school size has focused on the size of the aggre-gate unit (either total school or school-within-school)(Cotton, 2002). The current focus on school size, to agreat extent, stems from research on large, compre-hensive, “shopping mall” high schools (Powell,Farrar, & Cohen, 1985), whose social organization

may exacerbate the intrapsychic turbulence associ-ated with adolescent development (Eccles & Roeser,2011). Policy responses to these issues focus on cre-ating smaller schooling units to foster engagementamong adolescents, as well as between teachers andtheir students and among parents (e.g., Fine, 2005).However, among the existing knowledge base areconcerns about both the quality of studies that havebeen performed and the way in which policies haveaddressed the exposed problems.

Research concerned with the above problemshighlights the variability of the effects of schoolsize. One important qualification stems from whatBidwell and Kasarda (1980) refer to as the differ-ence between the measurable characteristics of stu-dents and schools, and the activities that happenwithin and around schools. For example, althoughrelationships generally were more positive and inti-mate in the smaller high schools studied by Lee,Smerdon, Alfeld-Liro and Brown (2000), this situa-tion did not always benefit all students, particu-larly those adolescents who preferred theanonymity of large schools due to the fact thattheir reputations or those of their families followedthem at school.

Another concern surrounding small schools istheir ability to increase achievement by creating amore communal climate. If schools are successfulin strengthening the sense of community anddeveloping a positive school climate, but not ableto raise achievement at the same time, it wouldappear that the reform might not be functioning asintended (Ravitch, 2006). This may simply be afunction of the problems of scalability and replica-bility. When schools are the unit of innovation,effective change should be located in the schoolitself and be specific to the school, each of which islikely to have a unique organizational characterand student population (Stevenson, 2000).Therefore, simply creating smaller schools andtransferring students into them from larger schoolsmay not produce the desired effect (Wyse, Keesler,& Schneider, 2008).

In general, the research on the appropriate sizeof school unit for student benefit has yielded incon-sistent results; there is little agreement about whatspecific size works best for adolescents. Garbarino(1980), echoing Barker and Gump (1964), describedthe advantages for high schools with more than500 students, while Goodlad (1984) advocated forschools between 500 and 600 students (see Leeet al., 2000 for a review of this literature). Lee andSmith (1997) concluded that learning was greatestin middle-sized schools (i.e., 600–900 students)

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compared with larger or smaller schools. They alsofound that learning was more equitably distributedin smaller schools; school size has important effectson learning; many high schools should be smallerthan they currently are; and high schools can betoo small. It could also be that school size itselfyields no benefits, but appears to, given that schoolsize is a feature of schools that is often correlatedwith a number of other factors that predictachievement.

While some have offered specific recommenda-tions for size, others (e.g., Meier, 1998) have usedqualitative criteria, such as sense of community, todefine what a “small school” is. Such authors pre-fer instead to describe size in relation to a school’sability to provide collaborative opportunities forfaculty and possibility for personalization andsafety for others within the school. The work donein this vein reflects the ambiguity that limits theempirical utility of social capital in educationalresearch. Consequently, although such studies arevaluable in generating rich accounts of trust, col-laboration, adherence to norms, and educationaloutcomes of interest, they usually stop short ofunraveling the relational mechanisms responsiblefor the associations.

Network Structure

The debate surrounding high school size is implic-itly grounded in Coleman’s (1988), which first out-lined those relational mechanisms in an attempt toexplain differences in adolescents’ achievementacross school sectors. As noted by Morgan andSørensen (1999), Coleman delineated two types ofsocial capital that he believed combined to giveCatholic school students a learning advantage: theideology of the Catholic church, and social closure.

Social closure has received considerable atten-tion from adolescent development researchers andeducational policy makers and consequently ismore widely discussed in the literature on schoolreform (e.g., Carbonaro, 1998, 1999; Horvat, Wein-inger, & Lareau, 2003; John, 2005). Social capital, inthe form of social closure, refers to a pattern ofsocial relations—a social network structure—inwhich those to which one is connected are alsoconnected to each other. In such a network, forexample, one might infer that a student would bemore likely to help another student if their parentsknow each other. The logic of this hypothetical sit-uation is an example of the presumed effects ofsocial closure and closely resembles the intuitionundermining small school reform. For example,

small school advocates tout the “strong personalbonds” associated with smallness, where adoles-cents “feel a greater sense of engagement, belong-ing, and personal value when their classmates andteachers get to know them” (WestEd, 2001, p. 2).These strong personal bonds are embedded insocial networks that exhibit high social closure;those in which everyone is connected in a way thatone’s behavior or attitudes cannot escape the obser-vation or influence of others. The image here is ofdense, redundant network where the parents ofadolescents’ friends are also likely to be friends. Infact, this image is consistent with Coleman’s argu-ment that when the parents of a group of adoles-cents all know each other, valuable social capitalresources accumulate in the relationships amongthem that can contribute to achievement.

The “dark side” of social closure, however, isthat a dense network potentially limits the diver-sity of information, as well as one’s freedom topursue ideas or demonstrate behaviors outsidegroup norms (Gargiulo, Ertug, & Galunic, 2009).This gives rise to an alternative network mecha-nism associated with social capital, Burt’s (1992)related notions of structural holes and brokerage.Burt’s formulation is that those who bridge net-work holes between different groups of peopleenjoy informational and control benefits of that net-work position. In other words, people in networkswith less closure are thought to have access to dif-ferentiated streams of information and a greaterfreedom to act. Citing evidence from several popu-lations of business managers, Burt (2004) hasamassed an impressive body of evidence for theassociation between individuals with networks richin brokerage opportunities and individual perfor-mance. Compared with Coleman’s hypothesizedmechanism of social capital in the form of socialclosure, Burt’s emphasis on structural holes andthe brokerage opportunities they provide hasreceived little attention in educational researchprobably due to the context (business) from whichhis evidence has been drawn. Consequently, hisalternative mechanism has had minimal directimpact on policies that address the social organiza-tion of schools.

Network Content

Focusing solely on the pattern of relationships,however, neglects the critical role of the resourcesmade available by others in one’s network. This isreferred to as network content, which leads to dif-ferences in performance due to indirect influence,

SCHOOL SIZE AND SOCIAL CAPITAL 3

information, or assistance from others in one’s net-work. Using the example introduced earlier, anadolescent may be more likely to help another ado-lescent if their parents know each other. But oneadolescent has to have something of value to offerthe other if that relationship is to ultimately yieldbenefits. Burt (1992, 1997) refers to this as networkcontent: the valuable behavioral, informational, andattitudinal resources to which one has access.

This general idea has been formalized in net-work models of social influence or contagion andhas been used to estimate the diffusion of attitudesand behaviors, including teachers’ adoption oftechnology (Frank, Zhao, & Borman, 2004) andteachers’ susceptibility to reform efforts (Penuel,Riel, Krause, & Frank, 2009). Specifically, thesemodels posit that one’s behaviors or attitudes are afunction of one’s individual characteristics, thebeliefs and characteristics of others to whom one isconnected, and a set of non–network-related attri-butes specific to the individual.

The most direct application to student achieve-ment is to model a student’s performance as afunction of the mean beliefs and characteristics ofhis or her friends and the individual characteristicsof the student, such as previous academic achieve-ment and demographic traits. This is the approachimplicit in many standard peer effects modelsemployed by sociologists and economists. Suchmodels can help distinguish between differences inperformance attributable to other determinates ofacademic achievement, such as family backgroundor teacher quality. They are limited, however, inhelping one differentiate between the mechanismsthat are potentially responsible for average peereffects. To draw further inference about the mecha-nism, one must go beyond average peer effects andexamine what one might think of as the adoles-cent’s location in a broader social structure (Marou-lis & Gomez, 2008). Therefore, both networkstructure and content must be considered wheninvestigating the role of social capital in influencingacademic achievement, or any other desired return(Lin, 1999).

Summary Statement and Hypotheses

The social capital made available to an adolescentcan be viewed as arising from mechanisms thatcan be measured by their network’s structure andcontent (Maroulis & Gomez, 2008). With regard tonetwork structure, adolescents located in dense,norm-enforcing networks may leverage the benefitsof increased trust and conformity that come from

one’s parents knowing other parents. Therefore, ifsocial closure were the primary source of socialcapital, then one would expect a significant posi-tive association between adolescents’ parentsknowing their friends’ parents and academicachievement. With respect to network content, net-work data can be used to devise aggregate mea-sures of peer traits to associate with individualstudent outcomes. All else being constant, onewould expect that desirable peer traits, such as theimportance that one’s friends attach to grades,would have a significant positive association withan outcome, for example, achievement.

Theory and prior empirical work, however,strongly suggest that the effects of network struc-ture and content vary by organizational context.Therefore, the effects of network structure and con-tent on achievement are likely to differ whenexamined across schools of different sizes. Theimportant issue is unlikely to be school size per se.Rather, school size facilitates or constrains how stu-dents and adults relate to one another and the pat-tern of relationships that surrounds adults’ effortsto facilitate the academic development of the ado-lescents they serve.

To investigate the associations suggested by theliterature on social capital, high school size andstudents’ achievement, two research questions areasked. First, what is the relationship among stu-dents’ parents knowing other parents, friends’grade importance, and achievement? Second, howdo these relationships vary across high schools ofdifferent sizes? Based on previous research, it ishypothesized that (1) students’ parents knowingother parents and their friends’ grade importancewill be significantly related to students’ achieve-ment, and (2) this relationship will significantlyvary according to the size of students’ highschools.

METHOD

Data and Sample

To address these questions, data were drawn fromthe 2002 (base-year) and 2004 (first follow-up) sur-veys of the Educational Longitudinal Study of 2002(ELS) (Ingels et al., 2005). Conducted on behalf ofthe National Center for Education Statistics (NCES)of the U.S. Department of Education, ELS is a longi-tudinal, multilevel study that followed a nationallyrepresentative cohort of students from the time theywere high school sophomores through the rest oftheir high school careers. The analytic sample cho-

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sen can be generalized to 10th grade studentsenrolled in a public high school in the United Statesin 2002. The sample includes only those base-year(10th grade) respondents who: (1) two years later,were in the same school, either in-grade or out-of-grade; (2) had a valid score on the dependent vari-able; and (3) had a nonzero longitudinal weight. Inaddition, given the focus of recent public schoolreform efforts and the emphasis on reductions inoverall school size, the sample excludes those stu-dents who attended a Catholic or other privateschool. The models of achievement are thereforeestimated on a final sample of 9,647 students (49%male, 66% White or Asian) meeting the abovecriteria, nested in 579 U.S. public high schools.

Variables

Student- and school-level variables used for theanalysis most of which are self-explanatory, arepresented in Table 1. Confounding variablesinclude indicators for held back, male, highestmath course taken as of 12th grade, and continuousmeasures for student expectations and parent aspi-rations, among others. The variables serving asmeasures of social capital, however, require furtherexplanation. To examine the social capital hypothe-ses developed by Coleman (1988), ELS includes ashort set of sociometric questions. Students werefirst asked to list the names of their three closestfriends in school. Next, for each friend, studentswere asked to rate the importance of getting goodgrades. These responses are coded on a scale of 0to 3 for each listed friend. The variable friends’grade importance was computed as the sum for eachstudent’s responses to this last prompt, which hada possible range from 0 to 9. This composite vari-able thus serves as a measure of the network con-tent of students’ first-order contacts.

The structural dimension of Coleman’s socialcapital explanation is also captured through a com-posite variable created from two different items onthe 10th grade student questionnaire. The construc-tion of this variable attends to the criticism offeredby Hallinan and Kubitschek (1999) of Morgan andSørensen’s (1999) measure and parallels the morerecent measure employed by Morgan and Todd(2009). The characteristics of students’ and parents’social networks used for the analysis were alsotaken from the base-year 10th grade survey. On thestudent questionnaire, each respondent was askedto answer questions about each of their three clos-est friends in their present school. The primaryexplanatory variable, labeled parents know parents,

is the sum of whether students indicated that theirparents knew their nominated friends’ parents,which had a possible range of 0 to 3.

Two other variables derived from the base-yearparent questionnaire have also been included toreflect the richness of Coleman’s conceptualizationand his emphasis on the role of parents’ networksin adolescents’ lives. First, parents have adequate sayreflects the level of parents’ agreement on a four-point scale with the statement, “Parents haveadequate say in setting school policy.” Likewise,parents work together reflects the level of parents’agreement with the statement, “Parents worktogether in supporting school policy.”

TABLE 1Weighted Descriptive Statistics

Variable % or Mean (SD)

SES �0.03 (0.72)White or Asian 0.66Male 0.49Parent aspirations 5.38 (1.26)Student expectations 5.32 (1.37)Held back 0.1010th grade math score 38.10 (11.84)Thinks math is fun 2.23 (0.83)Hours/week on math homeworkoutside school

2.71 (3.12)

Highest Math Course Taken as of 12th GradeNo math/math is otherc 0.01Prealgebra or general math 0.04Algebra I 0.06Geometry 0.14Algebra II 0.30Trigonometry/precalculus/calculus 0.45School mean SES �0.05 (0.37)Northeastc 0.17Midwest 0.24South 0.37West 0.21Urbanc 0.25Suburban 0.51Rural 0.23

Social capital variablesParents work togethera 2.25 (0.63)Parents have adequate saya 2.44 (0.71)Parents know parents 1.87 (1.03)Friends’ grade importance 7.26 (1.31)

School size variablesb

Small 0.20Moderate 0.19Moderately large 0.30Large c 0.31

Notes. N = 9,647. SD reported for continuous variables only.aOriginal variable has been reverse-coded so that higher

scores are more favorable.bSee text for how the school size indicator variables have been

defined.cReferent category in subsequent regression models

SCHOOL SIZE AND SOCIAL CAPITAL 5

In addition to the school-level variables thatmeasure social capital, the other covariate ofinterest, school size, is measured by the indicatorvariables small (<600 students), moderate (600–999students), and moderately large (1000–1599 stu-dents), with large-sized schools (>1599 students)serving as the referent category.

The outcome variable, 12th grade math score, isa student’s score on the ELS mathematics assess-ment, administered during the first follow-up andbased on item response theory (IRT) (M = 48.05,SD = 15.13). This assessment maximizes the accu-racy of measurement that could be achieved in alimited amount of testing time while minimizingfloor and ceiling effects by matching sets of testquestions to initial estimates of students’ achieve-ment. Most items were multiple-choice, with aboutten percent being open-ended. Importantly, stu-dents’ scores on the base-year mathematics assess-ment are employed as a statistical control. Checkson the reliability and validity of the mathematicsassessments are reported in Ingels et al. (2005),Appendix J.

Analytic Procedure

Hierarchical linear models are estimated to assessthe relationship among the measures of social capi-tal, school size, and students’ mathematics achieve-ment in 12th grade. These models employ data atthe student- and school-levels, as well as cross-level interactions, in a single analysis and include afull set of relevant controls for potential confound-ing covariates. As ELS is a longitudinal, multi-stagestratified random sample of students nested inschools, these models also control for its complexdesign and include the appropriate student-levelpanel weight. The restricted maximum likelihood(REML) algorithm is used to compute estimates forall covariates, with standard errors calculated bythe Taylor series linearization method. To easeinterpretation and reduce collinearity, all continu-ous student- and school-level covariates are enteredas grand-mean centered (M = 0, SD = 1) fixedeffects (Bryk & Raudenbush, 1992).

RESULTS

First, a null model was run to provide a baselineagainst which subsequent models are compared.The results of the null model show that the vari-ance estimate for 12th grade mathematics IRTscores for schools is 40.38 and the student-levelvariance is 189.11. Using these two values, the

intraclass correlation (ICC) is approximately 18%.Therefore, most of the variability in math achieve-ment is due to differences between students; how-ever, the amount of variability attributable toschools substantiates the use of HLM.

Main Findings

Table 2 reports the coefficients of primary interestfrom four separate hierarchical linear models, andprovides a test of the first hypothesis. The mainhypothesis resulting from Coleman’s work is thatclosed functional communities lead to increases instudents’ achievement. Model 1 is an initialattempt to evaluate this claim, as it predicts stu-dents’ 12th grade math scores from the two vari-ables that measure network structure and content.The relevant predictor for network structure is par-ents know parents, which captures the structural

TABLE 2Coefficients From the Regression of 12th Grade Math Score onParents Know Parents, Friends’ Grade Importance, and School

Size

Variables Model 1 Model 2 Model 3 Model 4

FIXED EFFECTSIntercept 48.02 49.27 49.83 41.65

Small school �2.18(0.94)**

�2.76(0.95)**

�0.24(0.40)

Moderateschool

�3.25(0.95)***

�3.66(0.96)***

�0.68(0.36)

Moderatelylarge school

�0.46(0.83)

�0.86(0.85)

�0.04(0.31)

Parents knowparents

0.52(0.16)***

0.62(0.18)***

�0.17(0.12)

Friends’ gradeimportance

�0.21(0.16)

�0.20(0.17)

�0.19(0.12)*

Parents worktogether

�0.07(0.14)

Parents haveadequate say

0.37(0.14)**

Confoundingvariablesa

10th grademath score

11.55(0.15)***

RANDOM EFFECTSSD (constant) 6.30 6.31 6.22 0.96SD (residual) 13.60 13.71 13.58 5.81

Notes. N = 9,647 students in 579 schools. Values in parenthe-ses are standard errors estimated using the Taylor series lineari-zation method. All continuous variables entered as grand meancentered. aAdditional controls for confounding variablesinclude SES, school mean SES, parent aspirations, student expec-tations, thinks math is fun, number of hours on math homeworkper week, and indicators for highest math course taken, Whiteor Asian, male, region, urbanicity, and ever held back.

*p < .10; **p < .05; ***p < .001 (two-tailed tests).

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aspect of Coleman’s social capital explanation.According to this explanation, social capital existswhen adolescents’ close friends attend the sameschool and the parents of these friends all knoweach other. The second predictor, friends’ gradeimportance, captures a student’s network content,the attitudinal resources that are aligned with thenormative expectations of school. Both are pre-dicted to have a positive, significant relationship tothe outcome.

This model provides contradictory evidence insupport of the first hypothesis. As parents know par-ents increases one standard deviation unit, students’12th grade math scores increase about one-halfpoint (β = 0.52, z = 3.22, p = .001). Therefore, eachstandard deviation of parental closure is associatedwith 0.04 standard deviations increase on the 12thgrade math score (i.e., 0.52* .1.03/15.13 = 0.04,where 1.03 is the standard deviation of parents knowparents and 15.13 is the standard deviation of the12th grade math test scores). While statistically sig-nificant in the conventional sense, the substantiveimportance of this coefficient is minimal. Thecoefficient in Model 1 for friends’ grade importance isneither large nor statistically significant.

Model 2 replaces school size indicator variablesfor the network structure and content variables.Thus, this model serves as the baseline estimate ofthe school size effect. Similar to recent analyses ofELS data by Weiss, Carolan, and Baker-Smith(2010) and Wyse et al. (2008), and contrary to pop-ular perception, the coefficient for small highschools is negative and significant (b = �2.18,z = �2.31, p = .021), as it is for the other two schoolsize indicators as well. This finding hints that thehype surrounding small high schools (i.e., thosewith <600 students)—or any other school-levelreform that focuses on size, for that matter—needsa much closer and critical public evaluation.

Model 3 includes the network structure and con-tent variables with the three school size indicatorsin a single model. The measure of network content,parents know parents, retains a small, positive, sig-nificant relationship with the outcome variable,12th grade math score (b = 0.61, z = 3.48, p = .001).Once again, the measure of network content,friends’ grade importance, does not significantly pre-dict 12th grade math score, and small and moder-ately sized schools are significantly associated withsmall decreases in achievement.

The final model in Table 2, Model 4, providesthe most direct test of the first hypothesis. Thismodel includes a full set of student- and school-level controls, as well as two others that indirectly

reflect a student’s parents’ interaction with schools.There are four interesting conclusions to be drawnfrom this full model. First, the variable parents knowparents is no longer significantly related to 12thgrade math scores (b = �0.16, z = �1.38, p = .166).This result parallels Morgan and Todd’s (2009) con-clusion that network structure in this observedform is not associated with increases in students’achievement in public schools, suggesting thatparental monitoring of adolescents’ behavior doesnot balance some of the costs associated withparental closure. Second, only the indicator formoderately sized schools has a marginally signifi-cant, albeit negative, relationship to 12th grademath scores (β = �0.68, z = �1.90, p = .06). How-ever, this finding has relatively minimal practicalsignificance; students in moderately sized schoolsare predicted to score 0.04 standard deviations lessthan students in large schools, the referent cate-gory. A third noteworthy finding from this modelis that, net any other predictor, parents have adequatesay is significantly related to 12th grade mathscores (b = 0.36, z = 2.57, p = .010), while the coeffi-cient for parents work together is both small andnonsignificant. These results, together with thenonsignificant coefficient for parents know parents,suggest that the influence of parents on adoles-cents’ educational outcomes does matter, but notnecessarily in the manner or degree in which Cole-man’s theory has posited.

The fourth finding from the full model is thatprior math scores and curriculum experiences havea significant and sizable relationship with 12thgrade math scores. For example, the coefficient for10th grade math score (b = 11.54, z = 75.12,p < .001) indicates that a one standard deviationincrease is associated with .76 standard deviationson the 12th grade test. This is by far the strongestpredictor of 12th grade math scores. A secondstrong predictor is students’ highest math coursetaken for half a year or more, asked during the firstfollow-up when most students were the 12th grade(not reported in Table 2). Compared with studentswho had no math course (the referent category),students who took Algebra II were predicted toscore about 3.5 points higher (b = 3.48, z = 1.66,p = .036) and students who took trigonometry, pre-calculus, or calculus were predicted to score 7points higher (b = 7.08, z = 1.66, p < .001).Together, these findings suggest that rather thanfocusing on efforts to either generate closureamong parents or reductions in school size, schoolsmight be better off redirecting their efforts towardhigher achievement earlier in adolescents’ school

SCHOOL SIZE AND SOCIAL CAPITAL 7

lives and more rigorous, higher-level curricularexperiences for a larger number of students.

As Model 4 includes these, as well as other stu-dent- and school-level control variables, it providesthe most direct test of the first hypothesis. As aresult of this model, the null hypothesis cannot berejected. This suggests that network structure andcontent in its observed forms (parents knowing par-ents and friends’ grade importance) do not signifi-cantly predict students’ 12th grade math scores.

Variation in Effects Across Differently SizedHigh Schools

However, the size of adolescents’ high school con-ditions the social capital that is available to them.For example, if the typical 12th grade adolescenthas two named friends, those students will bemore closely linked to other students in the schoolif the school has 500 students than if the school has2,000 students. Therefore, one would expect theeffects of network structure and content to vary byschool size. Theory and popular convention sug-gest that the benefits of network structure and con-tent would be more easily reaped in smallersettings, where the monitoring of adolescents’behavior by parents and the positive influences ofpeers’ positive attitudes toward achievement areless likely to be canceled out by larger countervail-ing forces as the level of aggregation (school size)increases (Wilkinson et al., 2000).

Table 3, Model 5 provides a first test of this sec-ond hypothesis. This model includes the maineffects of network structure (parents know parents),network content (friends grade importance), andschool size indicators. In addition, this modelincludes cross-level interactions between the net-work variables and school size indicators, whichestimate whether and to what degree the networkvariables vary across differently sized high schools.Similar to Model 2, the main effects of the schoolsize indicator variables, relative to the referent cate-gory of large schools, are negative and small. Forexample, students in small schools are predicted toscore about 3 points less than students in largeschools (b = �2.90, z = �3.01, p = .003). Neithernetwork variable has a significant main effect.However, the cross-level interactions predict 12thgrade math scores in ways that do align with Cole-man’s theory. The interaction between the variablesparents know parents and small schools is positiveand significant (b = 1.14, z = 2.21, p = .027), as isthe interaction between parents know parents andmoderate schools (b = 1.05, z = 2.02, p = .044).

These two coefficients indicate that in settings withless than 1000 students (small and moderatelysized schools), the benefits of parents know parentsare more likely to be actualized. Although notdirectly measured, this result is likely the byprod-uct of network density. The parents of students’friends will be more closely linked to other parentsin the school if the school has, for example, 300students than if the school has 3,000 students.

The main effect of friends’ grade importance, theobserved measure of network content, and its inter-action with the school size indicators, does not,however, support the network content componentof Coleman’s social capital explanation. The maineffect of friends’ grade importance in Model 5 is smalland nonsignificant (b = 0.08, z = 0.26, p = .792).This main effect, similar to the main effects

TABLE 3Coefficients From the Regression of 12th Grade Math Score onthe Interactions Between Parents Know Parents, Friends’ Grade

Importance, and School Size

Variables Model 5 Model 6

FIXED EFFECTSIntercept 49.74 41.58Parents know parents 0.15 (0.31) �0.35 (0.21)*

Friends’ grade importance 0.08 (0.31) �0.16 (0.21)Small school �2.90 (0.96)** �0.35 (0.41)Moderate school �3.67 (0.96)*** �0.63 (0.37)*

Moderately large school �0.79 (0.85) �0.06 (0.31)Parents knowparents9small school

1.14 (0.52)** 0.45 (0.33)

Parents knowparents9moderate school

1.05 (0.52)** �0.01 (0.35)

Parents knowparents9moderately large

0.22 (0.44) 0.29 (0.29)

Friends’ gradeimportance9small school

�0.83 (0.48)** �.15 (0.32)

Friends’ gradeimportance9moderateschool

�0.54 (0.49) �0.18 (0.32)

Friends’ gradeimportance9moderatelylarge

�0.02 (0.44) 0.16 (0.29)

Parents work together �0.08 (0.14)Parents have adequate say 0.37 (0.14)**

Confounding variablesa

10th grade math score 11.54 (0.15)***

RANDOM EFFECTSsd (constant) 6.25 0.97sd (residual) 13.57 5.81

Notes. N = 9,647 students in 579 schools. Values in parenthe-ses are standard errors estimated using the Taylor series lineari-zation method. All continuous variables entered as grand meancentered. aAdditional controls for confounding variablesinclude all those used in Model 4.

*p < .10; **p < .05; ***p < .001 (two-tailed tests).

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reported in Models 1 and 3, indicates that there isno significant relationship between a student’s net-work content consisting of friends who value goodgrades and a student’s 12th grade math score. Thesemodels may point to a broader example of the atti-tude-achievement paradox (Downey, Ainsworth, &Qian, 2009; Mickelson, 1990), which posits thatabstract ideas, such as the importance that oneattaches to grades, are disassociated from the con-crete ideas and understandings that lead to gains inschool-specific content areas such as mathematics.While few would argue against being friends withthose who value getting good grades, this character-istic of network content does not necessarily trans-late into an adolescents’ ability to demonstrate mathproficiency. Achievement growth in mathematics isvery sensitive to a number of school-based factors—the quality of instruction and curriculum, for exam-ple (Zvoch & Stevens, 2006). The attitudes that anadolescent’s friends have toward getting goodgrades in school cannot serve as a substitute forthese critical, school-specific factors.

The interaction between friends’ grade importanceand small schools reported in Model 5 results in amarginally significant, negative coefficient (b =�0.83, z = �1.71, p = .087), which can be explainedalong the same lines. While small schools may havean easier time constructing and maintaining a favor-able normative climate in which an adolescent’sfriends value getting good grades, they may have amore difficult time providing high-quality instruc-tional and curricular experiences for a varied stu-dent body. For example, small schools have fewermath teachers, limiting the range and depth ofinstructional expertise and content knowledge.Given these constraints on capacity, small schoolsmay be able to create communal climates andincrease students’ engagement with school, butmight not be able to influence achievement in waysthat match its normative expectations (Kahne,Sporte, de la Torre, & Easton, 2006).

However, the estimates reported in Model 5 donot control for student- and school-level covariates,thereby seriously limiting the model’s internalvalidity and the extent to which any meaningfulconclusion can be drawn. These estimates can beconsidered unconditional associations. Model 6offers a separate attempt to identify and estimatethe causal effect of the interactions between thenetwork variables and school size indicators afteradjusting for the same confounding variablesemployed in Model 4. Therefore, Model 6 offersthe most direct and appropriate test for the secondhypothesis.

Based on the results of Model 6, there is no sup-port for the hypothesis that the effects of networkstructure (parents know parents) and network content(friends’ grade importance) systematically vary by highschool size. The main effects of the network variablesand school-size indicators, as well as the interactionsbetween them, are all statistically nonsignificant.Despite these null findings, there are three notewor-thy results from this final model that have implica-tions for Coleman’s social capital hypothesis. First,the coefficient for parents have adequate say is positiveand significant (b = 0.37, z = 2.58, p = .010), indicat-ing that the positive influence of parents on students’achievement need not be actualized directly througheffective and dense parent networks. This findingalone has implications for school policies that mightbe advantageous for adolescents in the later stage oftheir secondary school careers. Second, prior mathachievement and curricular experiences really mat-ter. Here, too, there are implications for school policythat may be unrelated to reductions in school size orparental networks.

The third noteworthy result is derived from theestimates of the models’ random effects, reportedat the bottom of Table 3. The total residual vari-ance for this Model 6 is estimated at 34.60(0.962 + 5.812). It follows that R2 for this model is.85, so 85% of the variance is explained by the co-variates in this full model (229.38 � 34.60/229.38,where 229.38 is the total variance from the nullmodel). The R2 for Model 5, the interaction modelwithout any student- or school-level controls, isestimated at .03. Therefore, the inclusion of theseconfounding variables accounts for an approxi-mately 82% change in R2. These confounding vari-ables have little to do with characteristics ofadolescents’ networks or school size. They aremostly related to background characteristics thatcannot be easily manipulated (e.g., SES) by school-level reform. Other confounding covariates, such asprevious math achievement, highest math coursetaken, and hours per week on math homeworkoutside of school, are more malleable from a policyperspective, but pose their own set of implementa-tion challenges. This calls into question the degreeof change in students’ achievement that one canexpect from school reform initiatives.

DISCUSSION

Summary of Main Findings

Social capital is a useful concept as it focuses ana-lytic attention on the resources that are exchanged

SCHOOL SIZE AND SOCIAL CAPITAL 9

within and across relationships bounded by socialstructures. However, this utility is often limited byan under-specification of the relational mechanismsand resources that are presumed to be associatedwith favorable effects.

The primary contribution of this work is to defineand disentangle these two network dimensions—structure and content—in an effort to judge the con-cept’s potential for offering concise empirically-grounded explanations of the effects of parents andpeers on students during middle adolescence. Thisresearch concludes that network structure (as mea-sured by parents know parents) and network content(as measured by friends’ grade importance) alone haveno average association with adolescents’ mathemat-ics scores after accounting for student- and school-level characteristics. This finding echoes McNeal’s(1999) conclusion that parent involvement and mon-itoring, at least when it comes to adolescents andtheir development, may have a greater influence onbehavioral (e.g., attendance, discipline) than on cog-nitive outcomes such as achievement, particularlyin content areas that are as school-specific as mathe-matics. It also corroborates Morgan and Todd’s(2009) analyses of ELS data, where math gainsbetween the 10th and 12th grades were not signifi-cantly related to their school-level network measureof parents know parents, after conditioning on a num-ber of background characteristics.

But how can these null results be reconciledwith Coleman’s theory? One possible line of expla-nation focuses on the fact that Coleman’s theoriz-ing emphasized how voluntary communities ofchoice (e.g., Catholic schools) may be more likelyto become functional communities than the invol-untary residential communities that typically sur-round public schools. The claim emanating fromthis proposition is that adults who share normsand attitudes about their children’s schools aremore likely to agree on how best to oversee adoles-cents and their schools in order to induce students’learning. This claim could not be directly testedusing the ELS data, as there were relatively fewpublic schools of choice at the time of the initialbase-year sampling. However, the results provideno support for the claim that voluntary communi-ties will emerge with effective and dense parentalnetworks if school choice programs, which moreoften than not involve the creation of small schools,are scaled up. As noted by Morgan and Todd(2009), the Catholic schools on which Coleman’stheory was founded have strong ties to a norm-reinforcing institution—the Catholic Church—thathas no analog in the public sector. Without this

other source of norms and attitudes, it is unsafe toassume that a rather significant shift in the waythat parents relate to each other in the publicschool sector will materialize as a consequence ofsmall school reform, even as part of a broaderschool choice movement. For example, despite thefavorable outcomes reported in an intensive studyof six Gates-funded small schools in New YorkCity, teachers reported that getting parentsinvolved was a “challenge” (Academy for Educa-tional Development, 2010).

Limitations

Although this work directly challenges social capi-tal as a possible explanation of students’ achieve-ment, particularly for adolescents in small publicschools, four notes of caution about the analysisand its conclusions are warranted. First, becausethe outcome of interest, 12th grade math score, isone that is very much dependent on the coursesthat students take, who has taught it, and how thatmaterial has been taught, this work is inhibited bythe exclusion of variables that capture these criticalfactors. Although ELS does contain information onstudents’ course-taking patterns, which wereaccounted for in the models, and teachers’ back-ground characteristics, it does not contain measuresof how the material has been taught. This latterfactor is arguably the most critical when it comesto achievement in mathematics.

Second, students are not randomly assigned toschools in the ELS, and so these data have thesame potential selection bias as all other observa-tional studies. To limit the magnitude of this biasand increase internal validity, this study employsthe standard strategy of controlling for confound-ing variables that have been associated with stu-dents’ academic achievement in previous research(Schneider, Carnoy, Kilpatrick, Schmidt, & Shavel-son, 2007). As with all analyses based on observa-tional data, caution must be exercised ininterpreting any significant estimates as causal; it isthrough the accumulation of similar estimates fromstudies with varying data and alternative method-ologies that a conclusion that estimated effects arecausal becomes substantiated.

Third, the social capital variables lack informationabout students’ and parents’ second-order contacts.In other words, to more accurately reflect networkproperties, information about not only students’friends needs to be collected, but also informationabout the friends’ friends. Obviously, this increasesboth the costs and difficulties associated with

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collecting network data and few large complexsurveys manage to do so. A notable exception is thein-school network component of The NationalLongitudinal Study of Adolescent Health, alsoknown as ADD Health (Harris et al., 2009).

Finally, related to the previous point is that thesocial capital variables were seriously constrainedby the number of first-order contacts (three) thatstudents could list. Recall that parents know parentswas derived from a battery of survey questionsmeasuring social networks. The upper limit of thisname generator is different than the one used inNELS, where students’ parents listed up to fivefriends (though they did not need to be in thesame school). Given that previous researchsuggests that close adolescent friendship groupshave about five members (Cotterell, 1996), it iscurious as to why ELS limited the number ofnominations to three.

Implications

These limitations notwithstanding, the null resultsreported in these analyses have implications forboth researchers and policy makers. First, theseresults may be less of a refutation of Coleman’stheory and more of an issue regarding the concep-tualization and measurement of the network vari-ables themselves. Hallinan and Kubitschek (1999,p. 687) note that “even when parents in sociallyclosed networks share academic norms, the effectsof these shared norms on their adolescents’ schoolperformance may be negligible.” This is becausethe impact of parents’ academic norms depends onfactors such as the intensity, content, length, andinteraction pattern of parents’ friendships. Unlessparents frequently interact about school matters,these shared norms are unlikely to have a mean-ingful impact on students’ outcomes. The two pri-mary explanatory variables, parents know parentsand friends’ grade importance, neglect to capture allthese critical characteristics. While the variable par-ents have adequate say significantly predictsincreases in 12th grade math scores in the fullmodels (Models 4 and 6), its measurement wasdisassociated from parents’ friendship patterns.Nothing in this measure indicates that the otherparents with whom a parent interacts share thisperception about having an adequate say in settingschool policy. The larger point is that researchmust continually move forward in a way thatdevelops more appropriate and robust measuresthat better capture the complexity of Coleman’soriginal conjecture.

Similarly, these results say nothing about thepossible benefits of structural holes (the brokeragehypothesis) for adolescents that attend publicschools. The variable parents know parents is aninappropriate measure for this construct because itis directly derived from the number of friendsnamed by each student respondent. Therefore,while high scores on this variable indicate socialclosure, low scores on this variable do not suggestthe presence of social capital in the form of broker-age. In addition, friends in school may be an ade-quate measure of a construct such as a student’ssociability, but it says little about the connectivityamong that student’s named peers. To more accu-rately capture this structural property, one wouldneed sociometric data on which a student’s con-straint score (Burt, 1992) could be calculated (e.g.,see Maroulis & Gomez, 2008). This would enable amore appropriate and direct test of the effects ofthis alternative hypothesis.

The second implication is that school size is astructural attribute whose importance for adoles-cents is over-hyped. The most consistent finding inthis study is that there is no one-size-fits-all pre-scription for school size reform. Policy makersshould proceed more carefully as they adopt newtrends in reform, carefully weighing the evidencethat best matches individual adolescents’ develop-mental trajectories and schools’ demographics, notjust the general population. Small schools for ado-lescents are not a one-size-fits-all solution and mustbe cautiously constructed in each locale to carefullyreflect both a school’s and individuals’ capacitiesand needs (Weiss et al., 2010).

Finally, this research reaffirms the conclusion ofMorgan and Sørensen (1999, p. 675) that “theextrapolation to the public sector of any positiveeffects of parental social closure is unwarranted.”The data reported here would also extend this cau-tion to efforts aimed at promoting desirableachievement norms among adolescents. Thesenorms must be matched by a school-wide commit-ment to provide instruction and curricula that ade-quately support these attitudes. In addition, thereis still much to be learned about which networkstructures among teachers, parents, and adolescentsbest promote the development and reproduction ofthese norms.

This final point should give pause to reformerswho promote efforts to increase students’ achieve-ment through larger endowments of social capital.Despite a large body of evidence from numerousempirical settings regarding the benefits of socialcapital, there is still much unknown about how the

SCHOOL SIZE AND SOCIAL CAPITAL 11

structure and content of adolescents’ social net-works are associated with academic outcomes.Moreover, the results reported here indicate thatthe positive relationship between parents know par-ents and math achievement diminishes after con-trolling for a relatively small number ofconfounding variables. The diverse impacts ofschool size and social capital require a morenuanced understanding of how properties of socialnetworks may or may not benefit adolescents whenit comes to school.

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