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record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd leader nam a22 u 4500 controlfield tag 008 c20009999azu 000 0 eng d datafield ind1 8 ind2 024 subfield code a E1100166 0 245 Educational policy analysis archives. n Vol. 8, no. 22 (May 10, 2000). 260 Tempe, Ariz. : b Arizona State University ; Tampa, Fla. : University of South Florida. c May 10, 2000 505 Influence of scale on school performance : a multilevel extension of the Matthew principle / Robert Bickel [and] Craig Howley 650 Education x Research v Periodicals. 2 710 Arizona State University. University of South Florida. 1 773 t Education Policy Analysis Archives (EPAA) 4 856 u http://digital.lib.usf.edu/?e11.166 xml version 1.0 encoding UTF8 standalone no mods:mods xmlns:mods http:www.loc.govmodsv3 xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.loc.govmodsv3mods31.xsd mods:relatedItem type host mods:identifier issn 10682341mods:part mods:detail volume mods:number 8issue 22series Year mods:caption 20002000Month May5Day 1010mods:originInfo mods:dateIssued iso8601 20000510 PAGE 1 1 of 32 Education Policy Analysis Archives Volume 8 Number 22May 10, 2000ISSN 10682341 A peerreviewed scholarly electronic journal Editor: Gene V Glass, College of Education Arizona State University Copyright 2000, the EDUCATION POLICY ANALYSIS ARCHIVES. Permission is hereby granted to copy any article if EPAA is credited and copies are not sold. Articles appearing in EPAA are abstracted in the Current Index to Journals in Education by the ERIC Clearinghouse on Assessment and Evaluation and are permanently archived in Resources in Education The Influence of Scale on School Performance: A MultiLevel Extension of the Matthew Principle Robert Bickel Marshall University Craig Howley Ohio University and AEL, Inc.AbstractIn this study, we investigate the joint influence o f school and district size on school performance among schools with eighth gra des (n=367) and schools with eleventh grades in Georgia (n=298). Sc hools are the unit of analysis in this study because schools are increasi ngly the unit on which states fix the responsibility to be accountable. Th e methodology further develops investigations along the line of evidence suggesting that the influence of size is contingent on socioeconomic st atus (SES). All previous studies have used a singlelevel regressio n model (i.e., schools or districts). This study confronts the issue of cr osslevel interaction of SES and size (i.e., schools and districts) with asingleequationrelativeeffects model to interpret the joint influence of school and district size on school performance (i.e ., the dependent variable is a schoollevel variable). It also tests the equity of schoollevel PAGE 2 2 of 32outcomes jointly by school and district size. Georg ia was chosen for study because previous singlelevel analysis there had revealed no influence of district size on performance (measured at the district level). Findings from this study show substantial crosslev el influences of school and district size at the 8th grade, and weak er influences at the 11th grade. The equity effects, however, are strong at both grade levels and show a distinctive pattern of size interactions Results are interpreted to draw implications for a "structuralist" view of school and district restructuring, with particular concern for schoolin g to serve impoverished communities. The authors argue the imp ortance of a notion of "scaling" in the system of schooling, adv ocating the particular need to create smaller districts as well as smaller schools as a route to both school excellence and equity of school outcome s. Research on the role of school and district size as an influence on school performance has a long history and a large literatu re (see, for example, Barker & Gump, 1964; Guthrie, 1979; McDill, Natriello, & Pallas, 1 986; Smith & DeYoung, 1988; Fowler, 1991; Walberg & Walberg, 1994; Khattari, Ri ley & Kane, 1997; Stiefel, Berne, Iatarola, & Fruchter, 2000). The varying methods us ed to study the issue have, of course, generated conflicting results (Rossmiller, 1987; Ca ldas, 1993; Lamdin, 1995; Rivkin, Hanushek & Kain, 1998). In consequence, size has of ten been relegated to the status of an obligatory but uninteresting control variable. N ot infrequently, it has simply been ignored altogether (Barr & Dreeben, 1983; Burtless, 1996; Gamoran & Dreeben, 1986; Farkas, 1996; Wyatt, 1996; Hanushek, 1997, 1998). A recent school effectiveness review by eleven productionfunction virtuosos, for example, devoted just three of its 396 pages to school size (Betts, 1996, pp. 166168) Consequences of variability in school size, moreover, were, in passing, judged to be uncertain. District size is considered even less interesting than school size b y most researchers interested in school performance. The study reported here, by contrast, build s on a line of evidence that has related the size of both districts and schools to aggregate student achievement. Previous research developing this line of evidence, however, has constructed only singlelevel analyses (schools or districts). The present study deploys a multilevel method (Boyd & Iversen, 1979; Iversen, 1991) to link effects at th e two levels. In other words, this new work constitutes a first step from an empirical con sideration of "size effects" toward an empirical consideration of "scale effects" (cf. Gut hrie, 1979).School System Scale: A Timely Issue A great deal of skepticism exists about the role of size as a structural condition of US schooling. Educators have generally disparaged t he role of structure and focused attention on the role of process. This focus of int erest is easy to fathom. Both school teachers and administrators devote themselves to th e processes of teaching and administration; the structural features of their pr actices are, for the most part, tacit. Teachers and principals encounter schools and distr icts as the particular stages on which they personally enact their work and deploy profess ional processes. Whatever structural variety might distinguish one such "stage" from the next, teachers and principals do not often personally experience it. Superintendents, by contrast, are better positioned to develop a sense of structural differences among sch ools and districts, but such an PAGE 3 3 of 32appreciation might be almost as exceptional among s uperintendents as it is among other educators, since process also consumes most of a su perintendent's time. This propensity to focus on process has a p hilosophical dimension, as well. A structuralist view confines free will to an apparen tly smaller range of influence as compared to a view that privileges process. Educati on, and the culture of education, pays considerable homage to free will (cf. Bruner, 1996) In the grandest tradition, education is seen as the route to a "larger life" open to eve ryone equally (e.g., Prichard Committee, 1990). James Coleman was among the first to point o ut that equal educational opportunity was more problematic than previously im agined, of course, and due to structural reasons. The school effectiveness litera ture ensued and dramatically valorized process as the profession's response to a sociologi cal perspective on structure; school reform has had a procedural focus ever since (cf. D orn, 1998). Recent research and current events, however have combined to challenge the conventional disposition to privilege process over structure. First, nearly a decade of research on school size (in particular) has develop ed a preponderance of evidence to suggest that smaller school size would improve scho oling in impoverished communities (Howley, 1989; Irmsher, 1997; Raywid, 1999). Second schoolshooting tragedies have curiously and sadly brought the issue of school siz e to popular attention. Possibly as a result of these awful events, the US Secretary of E ducation and the Governors of Georgia and North Carolina have recently spoken in favor of small schools. Surprisingly, the Secretary praised the resistance of rural communities that have fought fiercely for decades to preserve their small school s in the face of consolidation (Riley, 1999). It has, of course, been a losing battle, wit h some fortunate exceptions. The recent attention has not even begun to challenge the privileged position that process enjoys, of course, and many observers conti nue to believe that administrative arrangements like "schoolswithinschools" and "hou ses" can replicate the processes presumed to characterize small scale. Both Mary Ann e Raywid (1996) and Deborah Meier (1995) argue persuasively that the conditions of smallness entail characteristics tantamount to structural difference: separate admin istration, separate budgets, distinctive authority, unique cultures, and so forth. Simulatio ns, it turns out, have difficulty reproducing these structural features of small scal e. Nonetheless the rhetorical change is itself dramatic. No longer does size appear merely as a footnote to effectiveness studies or as a container of essentially interesting processes, but as a distinct phenomenon. School siz e now matters in discourse, anyhow. School district size, however, continues to be regarded as a much less interesting issue than school size. The size of a district woul d seem to have no direct and little if any net influence on student achievement. As a vari able, district size seems quite remote from student learning. Thus, most studies have cons idered district size almost purely as an administrative issue bearing on resource allocat ion (e.g., Bidwell & Kasarda, 1975; Meyer, Scott, & Strang, 1987). There have been a fe w exceptions within these studies, of course. Bidwell and Kasarda (1975) studied distr ict size and concluded its influence on school performance was complex and contradictory : The total effects of [district] size were slight be cause its consequences for output, transmitted mainly by the structural and st aff qualifications variables, were of roughly equal strength in a posi tive and in a negative direction.... It was associated with wellqualified staff and low administrative intensity (and, therefore, we have a rgued, with minimal diversion of human resources away from frontline t asks). But large size also meant more students to teach and thus higher r atios of students to PAGE 4 4 of 32teachers. (p. 69) However, beginning with a 1988 study (Fried kin & Necochea, 1988), a new line of evidence has developed the hypothesis that the infl uence of both school and district size on aggregate performance is contingent on socioecon omic status. The direction of the effect has implicated small size (of schools and di stricts separately analyzed) as productive for the performance of schools or distri cts serving more impoverished communities, but larger size as productive for more affluent communities. Howley (1996) replicated the California study in West Virg inia and reported similar results. Recent work (to be considered shortly) has extended the singlelevel findings to Georgia, Montana, Ohio, and Texaswith nearly ident ical results.Relevant Literature Researchers' tendency to overlook the inter action of school and district size with other variables (such as poverty) may be a disablin g limitation of most studies that investigate the influence of school and district si ze on achievement, including quite recent efforts (e.g., Stiefel et al., 2000; Mik & F lynn, 1996; Riordan, 1997). This oversight tends to perpetuate the view that one siz e must fit all circumstances, or that some universally "best size" must exist (e.g., Lee & Smith, 1997; Stevenson, 1996). On this dubious view, sizerelated benefits and sizer elated costs are inadvertently construed as being enjoyed equally by all students (Conant, 1 959; Haller, 1992; Haller, Monk, & Tien, 1993; Hemmings, 1996). Stiefel and colleagues (2000), using a somewhat more refreshing approach, recently found that small regu lar 912 high schools have a budgetpergraduate that is no greater than the budgetper graduate of other 912 high schools, and, in some cases a much cheaper budgetpergradua te. (The Berne study, however, uses a small sample of schools from a single large city (n=121) and leaves aside the question of the difference between budgeted and act ual costs. The conclusions about small school size, unfortunately, rest on data from just 19 small high schools, of which only 8 are "regular" schools!) Within the past decade, however, a growing body of empirical research has held that size is negatively associated with most measur es of educational productivity. These conclusions encompass measured achievement levels, dropout rates, grade retention rates, and college enrollment rates (e.g., Walberg & Walberg, 1994; Stevens & Peltier, 1995; Fowler, 1995; Mik & Flynn, 1996). The drift o f the past decade of this research, then, is to portray the optimal or best size as som ewhat smaller than it was after James Conant proposed 400 students as the absolute minimu m size for a suitably "comprehensive" high school (Conant, 1959; Lee & Sm ith, 1997). Seldom have policy makers or researchers as ked "Better for whom?" or "Better for what?" or "Better under what conditions?" Asking su ch questions, of course, may be seen as leading to unbearable complications. Again, in this welter of interest, indifference, and outright evasion, the role of dis trict size is seldom considered, though both Herbert Walberg's (urban) and John Alspaugh's work (rural) remain notable exceptions (e.g., Alspaugh, 1995; Walberg & Walberg 1994). SizebySocioeconomic Status Interaction Effects The joint or interactive, rather than indep endent, effects of size and socioeconomic status (SES), may also have contributed to renewed interest in smaller schools and PAGE 5 5 of 32districts. If smaller schools and districts are sho wn to benefit some settings, the new conventional wisdom (i.e., "smaller is better") gai ns support. Specifically, interaction effects reported in some studies suggest that the wellknown adverse consequences of poverty are tied to school size and, to some extent to district size, in substantively important ways. In brief, as size increases, the mean achievement of a school or district with lessadvan taged students declines. The greater the concentration of lessadvantaged students atten ding a school, the steeper the decline. Investigations of the interaction hypothesi s are relatively new, and multiple replications have only recently been undertaken and completed (see Howley & Bickel, 1999, for a recent synthesis of results in four sta tes). Replications are important because without them, confidence in findings would be compa ratively weak; research done in other locations could well yield different, and per haps sharply conflicting, results. The additional replications, however, now e xtend the scope of findings to Georgia (Bickel, 1999a), Montana (Howley, 1999a), Ohio (How ley, 1999b), and Texas (Bickel, 1999b). Previous work concerned California (Friedki n & Necochea, 1988); Alaska (Huang & Howley, 1993, in a study in which students were the unit of analysis), and West Virginia (Howley, 1996). These states represen t considerable variety salient to the structure and operation of schooling in the United Statesrural and urban mix, ethnic mix, magnitude of influence of State Education Agen cy, district organization types, school and district size, and funding inequity (How ley & Bickel, 1999). The schoollevel findings in these singlel evel analyses are robust. In every study, an interaction effect has been confirmed. The effec t varies from very strong (California, Georgia, Ohio, Texas, and West Virginia) to weak, ( Montana ) (Note 1). The overall conclusion is that smaller schools help maximize ac hievement for schools serving impoverished communities, but that larger schools s erve the same function for more affluent communities. Robust districtlevel interaction effects, however, were discovered in the four recent studies only in Ohio. Somewhat weaker direct negative effects of district size were reported for Texas; still weaker direct and in teractive effects were evident in Montana. No districtlevel interactions were found in the Georgia study (Bickel, 1999a). The recent findings about districtlevel effects di ffered from the earlier findings for California and West Virginia, where substantial dis trictlevel interactions were evident (Friedkin & Necochea, 1988; Howley, 1996).Equity Effects In addition to reviving interest in school size as a variable of importance in educational research, this work has begun to sensit ize researchers, policymakers, journalists, and (perhaps most notably) citizens to equity concerns associated with school size. Onesizefitsall is no longer a unani mous judgment. Some researchers and policymakers have indeed begun to ask, "Bestsizef orwhom?" (Henderson & Raywid, 1994; Devine, 1996). In the five replications of the Friedkin an d Necochea work (i.e., West Virginia, Georgia, Montana, Ohio, and Texas) Howley and Bicke l also hypothesized equity effects of size. This hypothesis proceeds logically from co nfirmation of the interaction hypothesis. Namely, if small size improves the odds of academic success in small schools and districts (a sort of "excellence effect of size), then the usual relationship between SES and performance must be to some extent disrupted in them as compared to larger schools and districts. Simple zeroorder co rrelational analysis was used to measure the magnitude of relationship between SES a nd achievement in smaller versus PAGE 6 6 of 32 larger units (schools or districts divided at the m edian in these separate data sets). The equity effects of size are more consist ent and more impressive, in fact, than the excellence effects. At all grade levels, in all fiv e states, for both schools and districts, for a variety of alternative measures of SES, and for q uite different sorts of achievement tests (i.e., both criterionreferenced and normref erenced), the amount of variance in achievement associated with SES is substantially re duced in smaller units. In most cases, the magnitude of the relationship (Note 2) among th e smaller units is about half what it is among the larger units (Howley, 1996; Howley & B ickel, 1999). The Challenge of CrossLevel Interactions Although the "excellence effects" of school size and the "equity effects" of both school and district size seem clear from the analys es reported by Howley and Bickel (1999), failure to confirm interaction "excellence effects" for districts in some states is intriguing. The line of evidence about school and d istrict size has not, however, thus far included examinations of possible links between sch ool size and district size. As a result, if unacknowledged multilevel contextual ef fects were present, previous studies would have ignored some portion of the structural i nfluence of size on achievement. If the cultivation of high levels of achievement is a complex matter dependent on multiple influences, then we ought to suspect the existence of crosslevel influences. Further, discovery of such crosslevel infl uences could be considered evidence that a structural notion of organizational scale was rel evant to the enterprise of schoolingmost particularly to the cultivation of a cademic achievement. If such crosslevel relationships existed, administrators a nd policy makers would be well advised to coordinate their view of school size wit h a view of district sizeand eventually with classroom size, and individual stud ent performance, at one end of the spectrum, and size of the state and even national s ystems at the other end. The phenomenon of scaling could be seen as a structural characteristic of state school systems (see Thitart & Forgues, 1995, for an inter esting discussion of scaling as a feature of nonlinear dynamic systems in a chaotic s tate).Methods The present study addresses these issues by extending the consideration of "excellence effects" and "equity effects" of school and district size to a multilevel analysis with crosslevel interaction terms. We cho se to examine these relationships with the data for Georgia precisely because no effects o f district sizeeither direct or interactivehad been discovered in the singlelevel analyses conducted by Bickel (1999a). On the basis of districtlevel effects tha t are inconsistently evident across states, we hypothesize the presence of crosslevel interact ions that could not be detected in the previous singlelevel analysis. The Georgia dataset on which all analyses in this r eport are based is available for download here in any one of three formats: SPSS (409K filesize) Excell (1.65M) or ASCII text (460K). PAGE 7 7 of 32 We might as easily have chosen any of the o ther states, but the use of individual states is advisable for two reasons, the first theo retical and the second practical. First, from the perspective of scale, each state constitut es a uniquely structured system. In this sense, combining dissimilar states is more likely t o misrepresent reality than to provide a fuller picture of it. Second, since comparable achi evement measures are not available for schools and districts across the four states for wh ich we have assembled recent data, the merging of data sets would necessarily inflate meas urement error. A SingleEquation RelativeEffects Model To study further previously identified equi ty effects, we specifically ask, in this twolevel analysis, if there are crosslevel intera ction effects that remain significant in regression equations constructed to include school and district size, as well as school and district SES, and which also control for the propor tion of students who are African American, the proportion of students from ethnic mi norities, and pupilteacher ratio (a proxy for class size). Our focal interaction terms are the products of (1) district size and school SES and (2) school size and district SES. Ou r model also includes the two original interaction terms: (1) the product of dist rict size and district SES and (2) the product of school size and school SES. We use a procedure developed by Boyd and Iv ersen (1979) and Iversen (1991). It employs ordinary least squares estimates (Note 3) o f partial regression coefficients for schoollevel variables, districtlevel variables, a nd schoolbydistrict interactions in the same equation. In effect, we are combining schooll evel and districtlevel regression models, and including schoolbydistrict interacti ons, which reflect variability in districtlevel effects from school to school (Bryk & Raudenbush, 1992, pp. 7074). The dependent variables in these equations are always s choollevel performance measures. We adopt the singleequation relativeeffec ts version of the model, since schoollevel and districtlevel variables are likel y to be closely correlated. In this model, schoollevel variables are centered with respect to their group means (i.e., district means) and districtlevel variables are centered wi th respect to the grand mean. Centering all independent variables in this way hel ps to avoid inflated estimates of standard errors due to multicollinearity (Cronbach, 1987). Centering also enables us to unambiguously partition the percentage of variance in a dependent variable accounted for by each set of independent variables in our mul tilevel models (Iversen, 1991). Four such distinct sets of independent variables exist i n our model: (1) the set of individuallevel (school) variables, (2) the set of grouplevel (district) variables, (3) the set of singlevariable interactions by level (e.g., the product of school size and district size), and (4) a set of within and crosslevel inte ractions of different variables. Within the fourth set of variables are found the focal int eractions of this studythe two crosslevel interactions of SES and size: (1) the p roduct of district size and school SES and (2) the product school size and district SES. Examination of residuals plotted against th e independent variables shows that the residuals are not uniformly distributed with respec t to SPANSIZE for the 8th grade outcome measures. The same is true for FREEPCT when using the eleventh grade outcome measures. As a result, we used weighted lea st squares to remedy these departures from homoscedasticity, thereby restoring the efficiency of the estimators (Gujurati, 1995, pp. 381390).Data Sources and Variables PAGE 8 8 of 32 Official representations describe Georgia a s a state with an educational system encompassing approximately 1800 public schools (e.g ., Georgia Department of Education, 1999). The data set we are using, for sc hool year 199697, contains complete information on 1626 regular public schools. For thi s study we selected for analysis data about the universe of schools with grade 8 or grade 11 test scores. Grade 8 is the grade level in Georgia with scores prior to the wave of e arlyschool leaving that transpires at the high school level (generally grade 10), whereas grade 11 data portray the relationships that prevail subsequent to this toof amiliar exodus. The choice of these grade levels for analys is is therefore strategic. First, students from impoverished backgrounds become dropouts more frequently than students from more affluent backgrounds. Second, this being the c ase, the demography of schooling at grade 11 will differ somewhat from the demography a t grade 8, namely in the fact that the proportion of impoverished students will have d eclined. Third, the probable effect of these changed conditions, we hypothesize, will be t o weaken grade 11 results. The reason for this inference is that if smaller sizes positively influence achievement in impoverished schools, demographic changes in larger schools serving impoverished students will, in effect, cast off the cause of the ir negative influenceby removing disproportionate numbers of impoverished students. (Note 4) Dependent variables Dependent variables are schoollevel percentile r ank scores for eight subtests of the widely used Iowa Test of Basic Skills (grade 8) and schoollevel percentage of students passing the first administra tion of the Georgia High School Graduation Test (grade 11). Schoollevel means vary dramatically with both tests, from as low as the first percentile to as high as 93rd f or the ITBS and from 11 to 100 percent passing (on the grade 11 Graduation Test). Seven of the ITBS subtests are designed to measure achievement in reading comprehension, mathematics, reading vocabulary, soc ial studies, language arts, science, and research skills. The eighth subtest is a compos ite measure, intended to provide a global gauge of achievement. The High School Graduation Test is used in this study because the ITBS is not administered above grade 8 in Georgia. The Graduati on test gauges achievement in English, mathematics, social studies, and science. In addition, students receive a composite score. First administration passing perce ntages for the five scores are used as our outcome measures for the eleventh grade. Independent variables Our main predictor variables, (each measured at t he school level, at the district level, and as the interactio n between the school and district level) include the following: (1) number of students per g rade level in thousandstudent units as our measure of size (SPANSIZE); (2) proportion o f all students eligible for free or reducedprice meals (FREEPCT); (3) proportion of Af ricanAmerican students (BLACKPCT); (4) proportion minority (i.e., nonwhite ) students (MINORPCT); and (5) studentteacher ratio (S/SRATIO), a proxy for class size. We include studentratio, in particular, to address the possibility that any fin dings might principally be the result of differences in class size, rather than differences in school or district size. In order to test for the existence of cross level interactions between size and SES, we include four interaction terms: (1) school SPANS IZE by school FREEPCT, which is the same as the schoollevel interaction term that had proven significant in previous singlelevel analyses; (2) district SPANSIZE by dis trict FREEPCT, which is the same as the districtlevel interaction term that had proven nonsignificant in previous singlelevel analyses of Georgia data; (3) district SPANSIZE by school FREEPCT, which is one crosslevel interaction term of interest in this mu ltilevel analysis; and (4) school SPANSIZE by district FREEPCT, the other crosslevel interaction term of interest in the PAGE 9 9 of 32present study.Results Tables 1 and 2 provide descriptive statisti cs (means and standard deviations) for our dependent and independent variables for grade 8 and 11, respectively. SPANSIZE, at both the school and district level is measured in u nits of 1,000 students. A standard deviation of ".NNN," in the case of district size, for instance, is therefore equivalent to the product of ".NNN" and 1,000. Tables 3 through 1 0 report regression results (Note 5) for the eight achievement measures that predict sch ool performance at the 8th grade level. The first panel in each table apportions exp lained variance in three columns to (1) individuallevel (schoollevel), (2) grouplevel (d istrictlevel), and (3) individualbygroup (school by district) interactio ns. The second panel reports, in a single column, the variance attributable to interac tions among SES and size variables, at both levels (i.e., individual and group), yielding the four interaction terms specified in the concluding paragraph of the methods section. In the reporting of results below, only sel ected tables are presented, which nonetheless convey the findings from the complete s et of analyses. The complete set of tables in Rich Text Format can be downloaded from t his point. Table 1 Descriptive Statistics: Grade 8Dependent Variables Schools MeanSt. Dev.READING COMPREHENSION47.0212.88MATHEMATICS 52.2612.42READING VOCABULARY43.8215.05LANGUAGE ARTS 54.2012.72SOCIAL STUDIES 51.3112.04SCIENCE 51.0713.88RESEARCH SKILLS53.0112.60COMPOSITE51.2513.71 Independent Variables Mean/(St. Dev.) DistrictsSchoolsSPANSIZE0.2190.259 (0.101) (0.124)FREEPCT48.1845.28 (17.48) (22.93)BLACKPCT34.4737.29 (25.25)(29.66) PAGE 10 10 of 32MINORPCT 2.914.14 (4.22)(5.41)S/RRATIO 16.1316.25 (1.51)(1.86) N=158N=367Table 2 Descriptive Statistics: Grade 11Dependent Variables Schools MeanSt. Dev.ENGLISH92.875.18MATHEMATICS85.339.77SCIENCE 70.6615.22SOCIAL STUDIES75.1412.97 COMPOSITE 63.8916.41 Independent Variables Mean/(St. Dev.) DistrictsSchoolsSPANSIZE0.2330.280 (0.139)(0.114)FREEPCT 48.1833.49 (19.76)(21.26)BLACKPCT 35.42 38.03 (25.30)(29.80)MINORPCT 2.533.84 (3.42)(5.03)S/RRATIO17.03 17.74 (2.34)(3.15) N=155N=298 Recall that previous singlelevel analyses reported statistically significant and negative SPANSIZE by FREEPCT interaction effects. T hese conspicuous effects meant that as school (and in some states, district) size increased, the mean achievement costs associated with lessadvantaged students increased. Tables 1 through 8 again confirm interaction effects, but the interactions portrayed there are quite clearly shown to represent a complex phenomenon that escaped notice in singlelevel analyses. These more complex effects were predictably masked in the earlier singlelevel analyses, since those analyses examined schools and districts separ ately. The following written report of the findings may be difficult to follow, but the Ta bles themselves actually picture a PAGE 11 11 of 32consistently complex set of relationships prevailin g between schools and districts as those complex relationships influence schoollevel performance. We encourage readers to refer to the Tables as they read the following d iscussion. Eighth Grade "Excellence Effects" Combining schools and districts in a multil evel analysis, the singlelevel SPANSIZE by FREEPCT interaction effects that were s o conspicuous in the previous singlelevel research are not evident at all at th e 8th grade. However, several interesting (and uniquely specified) singlelevel and crosslev el interactions are present in the equations. Overall this means that the effects of s ize on achievement depend on multiple influences, and not merely schoolor districtleve l SES. One size is shown more clearly than ever before not to fit all cases, and, at the same time, these results suggest that the influential features of circumstance vary to such a n extent that each setting can be understood as unique. We present this conclusion pr ematurely in order to help readers take a wider perspective on the presentation of det ailed findings that follows. Single Variables Within and Across Levels. First let us consider the results given in panel 1 of Tables 3 through 10 (the unique influenc e of single variables at each of two levels separately and then jointly across levels). We will interpret the results of Table 10 (composite achievement) only, as the results given there can be viewed as not only encompassing the generality of the findings reporte d in Tables 3 through 9, but as representing a summative indicator of school perfor mance. Readers are, however, directed to those other Tables to observe the somew hat variant results among the various ITBS subtests. We will first consider the single va riables as unique schoollevel and districtlevel influences (Note 6): (1) Both FREEPCT () and BLACKPCT () exhibit uniqu ely significant (p <.001 and < .01, respectively) schoollevel influen ces in the equation, accounting for 26.4% of the variance in schoolleve l performance. Neither SPANSIZE nor S/RRATIO (our proxy for class size) sh ow any net direct influence at the school level.(2) FREEPCT () and MINORPCT (+) exhibit uniquely s ignificant (p<.001 and p<.01, respectively) districtlevel influences in the equation, accounting for 31.3% of the variance in school performance.Table 10 Weighted Regression Results with Corrected Standard Error Grade 8: Composite ScoreUnstandardized and (Standardized) Regression Coeffi cients IndividualLevelGroupLevel Individual by Group InteractionsSPANSIZE 6.40127.174308.619** (.050)(.094)(.167) PAGE 12 12 of 32FREEPCT0.401***0.340***0.005* (.418)(.460) (.098)BLACKPCT0.119**0.0010.003** (.207) (.002 ) (.212)MINORPCT 0.1120.347**0.014 (.040)(.121)(.033 )S/RRATIO0.4570.6600.460 (.043)(.060) (.068)Variance Explained26.4% 31.3% 10.8% WithinLevel and CrossLevel InteractionsSCHOOL SPANSIZE by SCHOOL FREEPCT0.141 (.024)DISTRICT SPANSIZE by DISTRICT FREEPCT0.332 (.023)DISTRICT SPANSIZE by SCHOOL FREEPCT4.304*** (.211)SCHOOL SPANSIZE by DISTRICT FREEPCT1.046*** (.237)Variance Explained10.7% Residual Intraclass Correlation .056 School/District Ratio 2.32 Standard Error Inflation 6.88% (Corrected)Partial Derivatives for Y with Respect to (1) SCHOO L SPANSIZE and (2) DISTRICT SPANSIZE Y wrt 1 = 308.619 x (DISTRICT SPANSIZE ) 1.046 x (DISTRICT FREEPCT) Y wrt 2 = 308.619 x (SCHOOL SPANSIZE) 4.304 x ( SCHOOL FREEPCT) *p <.05** p <.01***p <.001 These two singlelevel results show that a substantial portion of the variance in school performance (i.e., mean ITBS percentile rank in a school) actually is accounted for by districtlevel influences. Poverty contribut es a negative influence that is about 4 times the magnitude of the positive influence of MI NORPCT. The direct influence of district size and district student teacher ratio, w e note, are once again nonsignificant. We next consider the individual by group in teractions reported in column 3 of panel 1 (Table 10). This column reports crosslevel inter actions for each of the major variables separately. That is, these reported interactions co mpute the interactive (joint) influence of SPANSIZE, FREEPCT, BLACKPCT, MINORPCT, and S/SRA TIO at the two levels. Results, which account for a unique 10.8% of the va riance in schoollevel performance, are summarized as follows: (1) The unique interactive influence, across levels of SPANSIZE () is highly significant (p<.001). PAGE 13 13 of 32(2) The unique interactive influence, across levels of FREEPCT () is somewhat significant (p<.05).(3) The unique interactive influence, across levels of BLACKPCT () is also significant (p<.01).(4) There is no unique interactive influence, acros s levels, of MINORPCT or S/RRATIO. To interpret these interactive results, rec all that all independent variables are centered for the regression analyses. Values of the variables that fall below the mean are negative and values that fall above the mean are po sitive. The product of two negative values at the district level (e.g., low district po verty) and school level (small school size) will yield positive values of the interactive varia ble, just as the product of positive values at both levels will yield positive results. In this Georgia data set, the existence of small schools in small districts, and the existence of la rge schools in large districts are conditions uniquely associated with lower school pe rformance. (Note 7) Similar inferences can be drawn in the case of FREEPCT (tho ugh the influence here accounts uniquely for less than 1% of school performance) an d BLACKPCT. It is crucial for readers to keep in mind that the influences on scho ol performance discussed thus far are not interpretable in isolation from the totality of size influences. This research is developing a model of crosslevel influence of size on school performance. In this model, however, we can see that singlevariable inf luences within and across levels account for almost 70% of the variance in school pe rformance. Variables Interacting Within and Across Levels The single variableswhether uniquely at different levels, or jointly across lev elspresent a substantial but still incomplete view of influences on school performance These influences, in this analysis, are completed by an analysis of interactions betwee n variables, both within and across levels. We turn next, therefore, to a consideration of these influences, given in the second panel of Tables 3 through 10. Again, discuss ion centers on Table 10 (composite achievement) which, in the case of interactions bet ween pairs of focal variables (SES and size), very closely parallels results presented in Tables 3 through 9. We observe the following results (again, directionality is given p arenthetically): (1) The singlelevel interactions of FREEPCT and SP ANSIZE, whether schoolor districtlevel influences, are not stati stically significant. (2) The interaction () of SPANSIZE as a districtl evel influence and FREEPCT as a schoollevel influence is highly sign ificant (p<.001). (3) The interaction () of SPANSIZE as a schoolle vel influence and FREEPCT as a districtlevel influence is highly sig nificant (p<.001). The two significant interactions together a ccount for an additional 10.7% in the variation of school performance. Thus, the twoleve l model accounts for 79.2% of the variance in the performance of Georgia schools with an 8th grade. In other words, just 20% of the variance in school performance is the re sult of other influencesincluding school processes (such matters as curriculum and in struction). The first interaction, the statistically si gnificant and negative interaction of districtlevel SPANSIZE by schoollevel FREEPCT, s hows two things. First, as district PAGE 14 14 of 32sizes increase, the mean achievement cost associate d with increases in the proportion of lessadvantaged students at the school level increa ses as well. (Note 8) Secondas in the previously reported singlelevel analysesthe c onverse also pertains: As district sizes decrease (negative values of district size as a centered variable), the mean achievement cost associated with decreases in the p roportion of lessadvantaged students (i.e., negative values on schoollevel poverty) at the school level increases as well. In other words, more affluent schoolcommunities appea r to be better served by being in larger districts, but less affluent schoolcommunit ies appear to be better served by being in smaller districts. Put most simply, district pov erty and large school size are shown to jointly hurt predicted schoollevel performance, ju st as district affluence and small school size are shown to do. The relationship is in teractiveit cuts two ways. The second interaction, the statistically s ignificant and negative interaction of school SPANSIZE by district FREEPCT follows the pre ceding interpretation. First, as school sizes increase, the mean achievement cost as sociated with increases in the proportion of lessadvantaged students at the distr ict level also increases. Second, as above, the converse is true as well: As school size s decrease, the mean achievement cost associated with being in a district with decreases in the proportion of lessadvantaged students also increases. The simple form of this st atement, again, would be: school poverty and large district size are shown to hurt p redicted schoollevel performance, just as school affluence and small district size are sho wn to do. Again, this interactive relationship cuts two waysEleventh Grade "Excellence Effects" Tables 1115 present the regression results using the five eleventh grade outcome measures. As predicted, the 11th grade results are less consistent than the 8th grade regressions (Tables 3 through 10). Interestingly, t he crosslevel interaction of school SPANSIZE by district FREEPCT is highly statisticall y significant, alone accounts for as much as 15% of the variance in schoollevel perform ance, and exhibits the expected negative sign in each equation. As with the 8th gra de results, this means that as school sizes increase, the mean achievement cost associate d with being in districts with increasingly lessadvantaged students also increas es. As before, large schools in lowincome districts encounter a decided achievemen t disadvantage. Overall, the 11th grade "excellence" effects of size are considerably muted, and they leave their mark most particularly with the crosslevel interaction of SPANSIZE and FREEPCT. (Note 9) In general, the 11th grade results account for less variance than the 8th grade results. In the case of the composite score (Table 15), for instance, the model explains about 50% of the variance in schoollevel performan ce. The greatest proportion of variance accounted for by our model appears for mat hematics (about 66%); the low is English (less than 30%). Mathematics, we observe, i s a highly differentiated school subject at the highschool level, with the first co urse in algebra serving in the famous "gatekeeper" role (Silva & Moses, 1990) (Note 10). In other words, structural influences (poverty, race, size and the interactions among the m) might exert a stronger influence on school performance than they would in less differen tiated subjects such as English.Table 15 Weighted Regression Results with Corrected Standard Error Grade 11: Composite Score PAGE 15 15 of 32Unstandardized and (Standardized) Regression Coeffi cients IndividualLevelGroupLevel Individual by Group InteractionsSPANSIZE3.68819.952133.985 (.027)(.052)(.100)FREEPCT0.413***0.187*0.003 (.304)(.222) (.066)BLACKPCT0.257***0.116**0.001 (.378)(.206)(.067)MINORPCT 0.3210.2620.015 (.088)(.070)(.028)S/RRATIO 0.915*0.1450.096 (.126)(.013)(.060)Variance Explained28.7%10.0%1.7% WithinLevel and CrossLevel InteractionsSCHOOL SPANSIZE by SCHOOL FREEPCT0.281 (.075)DISTRICT SPANSIZE by DISTRICT FREEPCT0.423 (.025)DISTRICT SPANSIZE by SCHOOL FREEPCT0.027 (.001)SCHOOL SPANSIZE by DISTRICT FREEPCT1.357*** (.456)Variance Explained 8.0% Residual Intraclass Correlation .066 School/District Ratio 1.96 Standard Error Inflation 5.96% (Corrected)Partial Derivatives for Y with Respect to (1) SCHOO L SPANSIZE and (2) DISTRICT SPANSIZE Y wrt 1= 1.357 x (DISTRICT SPANSIZE) Y wrt 2 = COEFFICIENTS NOT STATISTICALLY SIGNIFICAN T*p <.05** p <.01***p <.001Interpreting the Effect Sizes of Size The regression equations provide a prospect ive tool with which to estimate the effects of projected changes in size (of schools an d districts) on school performance in PAGE 16 16 of 32Georgia relevant to the independent variables that describe a school's context. In order to interpret these predicted effects of size on school performance, we adapt the technique pioneered by Friedkin and Necochea (1988). Those researchers differentiated their regr ession equations in order to infer a rate of change in achievement attributable to size, relativ e to a school's or district's poverty level. Their procedure found the partial derivative (Note 11) of school or district performance with respect to socioeconomic status. T he partial derivative was then evaluated to find the rate of achievement change as sociated with changes in school or district size for schools or districts of a certain SES. This is the technique also used in the work recently reported by Bickel and Howley (e. g., Howley & Bickel, 1999). Since our goal here is to provide a fuller quantitative account of the relationship between size and SES we have computed partial deriv atives of the regression equations that give the rate of change in the dependent varia ble (school performance) with respect to size (school or district), holding poverty (FREE PCT) constant (at two levels of influence). It is important to remember that the de pendent variable in the partial derivatives represents a rate: change in school per formance per change in size. Because this is a twolevel analysis, howev er, two equations are necessary. One equation describes the predicted influence of chang es in school size on school performance, and in this analysis that rate turns o ut to be a function of districtlevel variables. The other equation describes the predict ed influence of changes in district size on school performance (in this case as a function o f schoollevel variables). Think of this relationship as follows: Y wrt 1 is a rate of chang e in school performance per change in the size of a school. But this rate, in crosslevel analysis, depends on districtlevel characteristics. Y wrt 2 is a rate of change in sch ool performance per change in district size; this rate depends in crosslevel analysis on schoollevel characteristics. Both equations can be standardized to give rate of chang e in standard deviation units if desired. Of most importance to this analysis, howeve r, is the prediction of total change resulting from the joint influence of variables at both levels. Computing this rate of change requires that the two partial derivatives be combined. To effect this combination, we calculate the total differential. The total diff erential predicts the magnitude of influence of changes in size (of both schools and d istricts) on school performance (which is always the dependent variable in these an alyses), all else equal. Let us begin by explaining the partial derivatives. In the immediat ely subsequent section, however, we provide an explanation of and illustrate the use of the total differential, as it constitutes the most important interpretation of size effects j ointly interaction with poverty. Partial Derivatives. In Tables 315 we report two partial derivatives, o ne for each level of influence (school and district) separately Partial derivatives give the rate of change in a dependent variable produced by focal va riables (SPANSIZE and FREEPCT, in the present case), holding constant all other va riables (i.e., BLACKPCT, MINORPCT, and S/RRATIO). Readers need to understand how they may use these additional equations. (Note 12) We will use the 8th grade composite statistics (Table 10) to illustrate our procedure, and we explain both th e creation of partial derivatives and the calculation of the total differential. First, takin g the partial derivative of Y with respect to SPANSIZE at the school level ("Y wrt 1" in Table 10) tells us that the rate of change in Y with respect to SCHOOL SPANSIZE, holding const ant the other independent variables, is equal to:f x1'(y) = [(308.619)(DISTRICT SPANSIZE)] [(1.0 46)(DISTRICT FREEPCT)] PAGE 17 17 of 32 Similarly, using the same outcome measure, taking the partial derivative of Y with respect to SPANSIZE at the district level tells us that the rate of change in Y with respect to DISTRICT SPANSIZE, holding constant the other independent variables, is equal to:f x2'(y) = [(308.619)(SCHOOL SPANSIZE)] [(4.304 )(SCHOOL FREEPCT)] The first partial derivative enables us to see that, all else equal, if we increased the value of DISTRICT SPANSIZE by, say, one quarter sta ndard deviation unit (.025 = .25 x .101), the predicted outcome measure would decrea se by 7.7 points. Similarly, if DISTRICT FREEPCT were increased by one quarter stan dard deviation unit ( 4.4 = .25 x 17.5), the outcome measure would decrease by 4.6 points. These effects, of course, are additive, and changes of equal magnitude, but in th e contrary directions, would yield no net effect. The second partial derivative enables us to determine the effect on 8th grade composite scores of an increase or decrease in SCHO OL SPANSIZE and SCHOOL FREEPCT. A one quarter standard deviation unit incr ease in SCHOOL SPANSIZE (.031 = .25 x .124) yields a 9.6 point decrease in the outcome measure. A one quarter point standard deviation unit increase in SCHOOL FR EEPCT (5.73 = .25 x 22.9) yields a 24.7 point decrease in the outcome measure.The Total Differential Information about the composite relationshi p between size and achievement is provided by the total differential. The total diffe rential (dy) is the sum of the products of the partial derivatives and their differentials, dx1 and dx2, where dx1 represents a change in SCHOOL SPANSIZE and dx2 represents a change in DISTRICT SPANSIZE. The total differential, then, is the sum of the changes in measured achievement due to changes in SCHOOL SPANSIZE and DISTRICT SPANSIZE, c ontingent on SCHOOL FREEPCT and DISTRICT FREEPCT (all else equal):dy = {[fx1'(y)]( dx1)} + {[fx2'(y)](dx2)]} The values of dx1 and dx2 represent proportional changes (e.g., .10 or +.10) in school or district size (SPANSIZE). To illustrate t he calculation of the total differential, we computed hypothetical values of dx1 and dx2 tied to reallife values in the Georgia data set. We divided the SPANSIZE into the differen ce between SPANSIZE and the difference between the value of SPANSIZE for cases n + 1 and case n. That is, using the subsequent case in the data set as a reference poin t, we inferred rates change for school and district size in the subject case. This procedu re produces arbitrary changes, but these arbitrary changes vary only within the range of var iation that the Georgia school system exhibits. In keeping with Dowling's (1980) admonition that differentials should be realistically small, we then eliminated cases with values for dx1 or dx2 greater than onehalf standard deviation above or below their me an. (Note 13) The absolute value of dx1 for all remaining cases was less than .068, and th e absolute value of dx2 was less than .026. We then randomly selected ten of the rem aining schools for inclusion in Table 16. PAGE 18 18 of 32Table 16 Total Differential: Illustrative Values for Randoml y Selected CasesGrade Eight Composite ScoresDISTRICT SPANSIZE SCHOOL SPANSIZE DISTRICT FREEPCT SCHOOL FREEPCTdx1dx2dy .0829.083579.4771.79.047.0108.47.1562.218773.3874.90.019.0096.08.2285.342720.210.90.066.0056.63.1541.152724.3427.10.013.0183.88.1437.277070.8466.20.029.0138.21.1469.149761.0759.70.108.00613.62.1311.127066.6061.20.005.0103.56.1825.212029.7619.50.000.0050.70.0980.094455.8948.10.062.0132.46.1566.306047.3955.30.016.0166.75 Notes. Values of variables are given uncentered Equations are derived from and computed with centered values.Total differential computed as: dy = {[f x1 ¢ (y)]( dx 1 )} + {[f x2 ¢ (y) ] (dx 2 )]} Values of partial differentials, dx 1 and dx 2 computed as follows(cases selected for dx £ 0.5 s ): [(SPANSIZE case(n + 1) ) (SPANSIZE case(n) )] / (SPANSIZE case(n) ) The first four columns in Table 16 describe the focal variables (district and school size and subsidized meal rates). The fifth and sixt h columns provide the (hypothetical) proportional changes in school size (dx1) and district size (dx2). The values of the total differentialthe predicted change in each school's mean Composite Test score attributable to these composite changes in sizecon tingent on these proportional changes in school and district size appear in the c olumn headed "dy" ("total differential"). Observe that Table 16 illustrates the inver se relationships between school performance (8th grade composite, in this case) and changes in SPANSIZE at both the school level and the district level. The first two cases, for instance, show a positive influence of joint school and district size in a un iformly impoverished school and district. Case seven shows the decline in similar c ircumstances of a joint increase in size. And case nine shows the somewhat more modest increa se in test scores resulting from a joint reduction in school size and increase in dist rict size. PAGE 19 19 of 32Eighth and Eleventh Grade "Equity Effects" Most people understand inequity in school f inance. Affluent communities almost always enjoy betterfunded schools, and improvement s in financial equity would require that schools in impoverished communities be much be tter funded than they are. In other words, mitigating financial inequity requires that we break the link between poverty and school finance. Some educators (we among them) beli eve that no ethical principle justifies the privilege enjoyed by more affluent ci tizens in this regard. Why should the rich enjoy the bestfunded schools? The rich common ly argue that it is their right, and the argument prevails. Inequity in achievement presents much the s ame case. Which children, in general, enjoy the highest achievement? More affluent childr en do. Some observers, of course, believe that since the constructs "affluence" and ability" correlate well, this state of affairs is actually very fair. The rich might well argue that inequity of outcomes in their favor is also their right. Others (we among them) n ote thatamong affluent and impoverished people alikea great range of abilitie s exists, and that in all adult occupations a similarly great range of abilities pe rsists. On this view, the low achievement of impoverished children is not nearly so fair as it at first might seem (e.g., Gardner, 1983). In this view, public schooling can and should do much more to nurture the learning of impoverished students, in particula r among all students. As with financial equity, equity in achievement means breakingor at least substantially mitigatingthe prevailing bond between SES and achievement. (Note 14)Table 17 MultiLevel Georgia Equity EffectsaLarger v. Smaller Schools and Districts with Grades 8 and 11 Composite Grade 8 Grade 11 Districts LbS L S Schools L .76.72 L .77.74 S .63.35 S .54.16 Reading Comprehension (8)/ English (11) Grade 8 Grade 11 Districts L S L S Schools L .84.74 L .69.59 S .71.36 S .28.16 Mathematics PAGE 20 20 of 32 Grade 8 Grade 11 Districts L S L S Schools L .71.59 L .72.65 S .46.29 S .48.25 Science Grade 8 Grade 11 Districts L S L S Schools L .82.73 L .73.71 S .70.37 S .46.27 Notes:a) Variance (R2) in school performance attributable to schoollevel subsidized meal rates.b) L = Larger half; S = Smaller half. Table 17 gives the variance in achievement associated with SES in four groups by the medians of district size and school size (2 gra des and 4 tests). Within each panel, by grade level, we report the observed variances proce eding left to right and top to bottom in each of the 8 contrasts for: (1) large schools i n large districts, (2) large schools in small districts, (3) small schools in large distric ts, and (4) small schools in small districts. In each of these 8 (2 grade levels by 4 tes ts) fourway contrasts, large schools in large districts show the highest proportion of vari ance in achievement associated with SES: between 71% and 84%, whereas the lowest propor tion of variance is exhibited among small schools in small districts: between 16% and 27%. Moreover, the order of declining variance follows an identical pattern in each of the 8 contrasts: largelarge, largesmall, smalllarge, and smallsmall. In 6 of 8 cases, the largest magnitude of decline within the evident sequence (largelarge, l argesmall, etc.) of decreasing variance comes in the change from small schools in large districts to small schools in small districts. In other words, Table 17 suggests that the predicted equity effect of reducing district size but not school size would be practica lly significant; the predicted equity effect of reducing school size but not district siz e would also be practically significant and perhaps somewhat larger; and the combined strat egy of reducing both school and district size would be predicted to yield substanti al equity and excellence effects (given the previous multilevel regression analyses). Some rural states (e.g., Montana; see Howle y 1999b) structure their school systems in just this way. That is, such systems have chosen to sustain small schools within small districts. The Montana system doubtless has plenty of room for "improvement," but on the terms of accountability (and the value of more equal outcomes), Montana is an exemplar. Please note that Montana has a substantia l American Indian population (13%), whose children also attend small, predominan tly public, schools and districts. In rural areas, the phenomena of school and district size seem mutually dependent; larger rural schools often prevail in larger rural districts (e.g., as in West Virginia; see PAGE 21 21 of 32Howley, 1996). District "reorganization" has often been a first step toward eliminating small schools (DeYoung, 1995; Peshkin, 1982). This strategy would be predictably harmful to the achievement of students in impoveris hed rural communities. In the southeast US a single high school now often serves entire rural counties, covering large geographical areas. The situation in urban areas is equally bad though in somewhat different ways. The huge bigcity districts were created, not just to i mprove schools, but to destroy a resource (school jobs) that could be controlled by ward politics. Usually portrayed as a "progressive" change, an important motive of city f athers was to wrest power back from the hands of workingclass urban communities (e.g., Tyack, 1974; Erie, 1988). Today, most urban districts are nightmares and wildernesse s of bureaucracy and outright fear (e.g., Devine, 1996). Jobs are as much a political issue as ever in many large cities, but the power to dispense them has shifted to the nexus between political regimes and school bureaucracies, with the bureaucracy often in the better position. No wonder so many thoughtful educators champion the reestablish ment of smaller schools in cities (e.g., Meier, 1995; Klonsky, 1995). Difficult as it is, in both rural and urban locales, to defend or reestablish small schools, that task leaves the structural challenge incomplete. Seldom are reductions in district sizeespecially in the case of large city districtsseriously considered. Our principal "clear and simple" recommenda tion therefore is to suggest the wisdom, of reorganizing districts that are now far too large. Policy makers should start imagining ways to recreate districts that are ever ywhere sufficiently small to respond well to students, families, and (especially) commun ities. One way to enable this decision making might be for communities to enjoy the right to charter public school districts as well as public schools (and, naturally, to receive the requisite statelevel support to succeed). The policy issues are surely difficult, b ut no more difficult than those that have already led to the counterproductive structuring th at presently prevails. To do nothing or little leaves the burden of coping with the enormit y to impoverished students, families, and communitiesexactly where it currently rests.Misuse of the findings Our findings cannot be interpreted to warra nt the construction of huge schools, however, even for relatively comfortable communitie s; in general, we advise an upper limit of about 250 students per grade for 912 high schools and about 100 students per grade for elementary schools and these ruleofthu mb upper limits apply to communities where the poverty rate is zero (Howley, 1997; but see Irmsher, 1997, and Raywid, 1999, for quite similar recommendations bas ed on recent reviews of the literature). Recently we learned that our research was b eing used to help justify construction of a school in a semirural area of an eastern state p roposed to house 2,000 elementary students in grades 36. In view of extant and easil y accessible research syntheses such as those by Irmsher and Raywid, proposals to create sc hools of this sizeparticularly elementary schoolsare, we believe, capricious and professionally irresponsible. We are unhappy (but not surprised) to learn that our work has been deployed to support such proposals; but we also understand the role that bad statelevel policy plays in shaping such decisions as this (see Purdy, 1997, for a clear example in a rural state where the state influence is heavyhanded). The adm inistration in this district experienced considerable angst when community membe rs there contacted us and we voiced our objections to the misuse of our research publicly. In fact, however, we are PAGE 22 22 of 32used to being contacted by community members resist ing such efforts and equally used to not hearing from members of our own profession a s they make construction plans. Despite uproar in the community and defeat of the b ond issue, plans for the megaschool (to be organized in "houses") apparently continue. The superintendent in this case has reportedly vowed revenge on the interfering outside researchers! We regret the angst that emerges in these situations, but we believe the pre sent study provides evidence to support our evolving position on the issues.Conclusion Small size is good for the performance of i mpoverished schools, but it now seems as well that small district size is also good for t he performance of such schools in Georgia, where district size, in singlelevel analy ses, had revealed no influence. Because of the consistency of schoollevel findings in prev ious analyses, we strongly suspect that the Georgia findings characterize relationships in most other states. This claim can, of course, only be evaluated by additional replication s, and we hope other researchers will see merit in such work. The equity effects reported here, however, extend the evidence of the previous singlelevel studies to the interaction of school a nd district size. Larger schools in larger districts seem to propagate inequality of outcomes by comparison to smaller schools and smaller districts. In fact, smaller schools in larg er districts demonstrate a useful equity effect, as well. For large schools in smaller distr icts, however, the improvements in equity might be so slight as to be called "negligib le." The equity effects are so striking, and app ear so instrumental in association with the "excellence" effects of small size in impoverished communities, that further investigation into this mitigating influence would seem crucial. How does the principle evident in the findings apply to individual student s? In what settings? To what extent? What structural features of small size enable such an effect? How do impoverished students fare in schools that are, overall, rather affluent? Is an overall upper limit to school size and district size worth establishing by policy? How should such upper limits be set? What policies can succeed in recreating sma ller districts in big cities and the rural southeast? These are interesting and important questio ns, we think, but the conclusions of this study would seem to require rather wide debate and reconsideration of the size issue, across the spectrum of poverty and wealth, and not just in the case of impoverished communities. We note that America's elite sends its children to Andover and Exeter and other such fine high schools, where enrollments sel dom exceed 1,500. What do they know that the rest of us have yet to learn, we wond er?NotesThis work is based on research funded by the Rural School and Community Trust. It does not, however, represent the opinions or positi ons of the Rural Trust. We are grateful for the support; the errors and opinions a re our own. The two authors are equal contributors to the work reported here. Unlike the other states, Montana has retained many small schools, and this historic decision is a likely cause for the weak interaction effects. Bickel (1999b) also reported no interaction effect among the 132 Texas schools that house all students 1. PAGE 23 23 of 32in grades K12 (cf. Franklin & Glascock, 1998).Magnitude of the relationship was measured as the p roportion of variance in achievement associated with SES. 2. See the Appendix for a discussion of the problem th at intraclass correlation poses to the use of ordinary least squares regression. Th e Appendix describes the conditions needed to use OLS in multilevel analysi s and shows that our data set meets these conditions. 3. This logic, of course, is also supported by the fin dings previously reported for the singlelevel fourstate analyses, in which reported effects are strongest at grade 8 or 9, and always weaker at grades 11 or 12. On the basis of past experience, then, we would have reason to suspect similar results. 4. We report statistical significiance levels as a gau ge to practical significance. Because the data set includes practically all schoo ls in Georgia, the relationships that emerge are those that prevail, and, we maintai n, should not be considered as subject to sampling error. 5. Directionality of the influence is given in parenth eses following the variable name. The effect of centering is not reflected in Tables 1 and 2. 6. These findings are conceptually consistent with pre viously reported schoollevel analyses, which found that, among impoverished comm unities, smaller schools reduced the achievement costs of poverty and that l arge ones magnified such costs; but the converse was true as well, in those cases: Among affluent communities, smaller schools increased the achievem ent costs of affluence and larger ones reduced such costs. 7. "Mean achievement costs" represent declines in pred icted achievement. Therefore, another way to put this interactive relationship is this: (1) as poverty and district size continuously increase, predicted school perfor mance continuously declines; and (2) as poverty decreases and district size decr eases, predicted performance also continuously declines. 8. We might also observe that other crosslevel intera ctions appear significantly in the equations reported in several of these Tables: Table 11 (math: FREEPCT), Table 13 (social studies: SPANSIZE), and Table 14 ( science: BLACKPCT). Crosslevel structural influences are weak at the 1 1th grade but still evident. 9. Robert Mosess "Algebra Project" construes algebra as the course that governs access to the academic track in life; failing algeb ra, or never taking it in the first place, marks one as academically inept. 10. A "derivative" can be understood as the calculus to ol for determining the "slope" of a curved line (which, in geometrical terms, is t he tangent of the curve at a given point). The slope of such a line is constantly chan ging (just as the effects of school or district size, or their joint effects, constantl y change with respect to poverty levels), and the derivative provides the formula fo r calculating this changing slope. To find this changing rate, one "takes the d erivative" of the formula that describes the line. A partial derivative holds one variable constant during differentiation (the process of "taking the derivat ive") so that the influence of that variable can be subsequently evaluated. This proces s of "holding an influence constant" is similar to calculating a partial corre lation coefficient. 11. Consult Howley (1996) for a complete description of the derivation of partial derivatives in the singlelevel analyses. 12. Dowlings counsel is important because we are deali ng, in using calculus techniques that estimate changing rates, with how t hese rates of change at the margin (i.e., the usual addition or loss of a few s tudents) under normal conditions, 13. PAGE 24 24 of 32and not, in fact, in such catastrophic alterations as are produced by consolidations of two or more schools (where size may well increas e by hundreds of students). Calculus is the mathematics of smooth curves and no t of disruption and disjunction.In practical terms, one is unlikely to break the bo nd completely, because the negative effects of poverty can be eliminated only when a society finds them intolerable and actively cultivates the wellbeing of the poor. Even in the current economic boom, however, such a realization has not overtaken the US, and in general, the gap between the affluent and the impov erished is growing ever wider here. 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Organization Science, 6 (1), 1931. Tyack, D. (1974). The one best system: A history of American urban ed ucation Cambridge, MA: Harvard University Press.Walberg, H., & Walberg, H. (1994). Losing local con trol of schools. Educational Researcher, 23 1926. Wyatt, T. (1996). School effectiveness research: De ad end, damp squib, or smoldering fuse? Issues in Educational Research, 6 79112.About the AuthorsRobert BickelMarshall University Email: bickel@marshall.edu PAGE 29 29 of 32Robert Bickel is a Professor of Advanced Educationa l Studies at Marshall University. His recent research is concerned with school size a s a variable which moderates the relationship between social class and measured achi evement, evaluation of early childhood interventions, and with contextual factor s which occasion the atrisk designation.Craig HowleyAppalachia Educational Laboratory, Inc.Ohio University75619 Lively Ridge RoadAlbany, OH 45710Phone: 740.698.0309 Email: howleyc@ael.org Craig Howley is an education writer and researcher based in Albany, Ohio, and affiliated parttime with Ohio University (Athens, OH) and AEL Inc. (Charleston, WV). Ongoing scholarly projects in which he is a partner include the following: research on school and district size, rural school busing, and three book projects: an examination of small rural high schools (for Appalachian Educational Laborator y), an extended interpretation of developmentalism as a school ideology (with Aimee H owley) and a textbook (with A. Howley and Ohio University colleagues) on school ad ministration.Appendix Ordinary Least Squares Regression and theProblem of Intraclass Correlations One of the assumptions of ordinary least sq uares estimators is that residuals are not correlated. However, in a multilevel analysis this assumption may be erroneous. The reason is that firstlevel observations are located within the groups that constitute the second level of analysis. Grouping of firstlevel o bservations (schools) into districts may well mean that schools within a district are more l ike each other than they are like schools in other districts. The consequence is intr aclass correlation, or covariance among residuals for schools in the same district (see Kre ft & de Leeuw, 1998, pp. 910). This observation yields the primary objecti on to traditional contextual models such as ours. Through uncritical use of ordinary least s quares, the magnitude of standard errors of regression coefficients may be underestim ated and alpha levels artificially inflated (Goldstein, 1995). The observation holds e ven though ordinary least squares estimators remain unbiased (Barcikowski, 1981). In the present study, intraclass correlatio ns, which vary by outcome measure and grade level, range in magnitude from .048 to .101. The number of groups or districts is 158 for the 8th grade and 155 for the eleventh grad e. With 367 schools reporting 8th grade test scores, and 298 reporting eleventh grade scores, the relative number of secondlevel observations is large, indeed (Goldste in, 1995). We conclude that intraclass correlation is a negligible problem. Given this confluence of circumstancessmall intraclass correl ations and large numbers of districts relative to the number of schoolsordinar y least squares will yield estimates which are unbiased and will provide such estimates with very little inflation of PAGE 30 30 of 32 regression coefficient variances (Singer, 1987). Fu rthermore, using a procedure presented by Singer (1987), we have calculated the remaining modest inflation of regression coefficient variances, standard errors, and resulting tvalues. We compensated for this statistical artifact when running tests of significance, reducing the magnitude of the affected statistics by the amount they are infl ated due to intraclass correlation.Copyright 2000 by the Education Policy Analysis ArchivesThe World Wide Web address for the Education Policy Analysis Archives is epaa.asu.edu General questions about appropriateness of topics o r particular articles may be addressed to the Editor, Gene V Glass, glass@asu.edu or reach him at College of Education, Arizona State University, Tempe, AZ 8 52870211. (6029659644). The Commentary Editor is Casey D. C obb: casey.cobb@unh.edu .EPAA Editorial Board Michael W. Apple University of Wisconsin Greg Camilli Rutgers University John Covaleskie Northern Michigan University Alan Davis University of Colorado, Denver Sherman Dorn University of South Florida Mark E. Fetler California Commission on Teacher Credentialing Richard Garlikov hmwkhelp@scott.net Thomas F. Green Syracuse University Alison I. Griffith York University Arlen Gullickson Western Michigan University Ernest R. House University of Colorado Aimee Howley Ohio University Craig B. Howley Appalachia Educational Laboratory William Hunter University of Calgary Daniel Kalls Ume University Benjamin Levin University of Manitoba Thomas MauhsPugh Green Mountain College Dewayne Matthews Western Interstate Commission for HigherEducation William McInerney Purdue University Mary McKeownMoak MGT of America (Austin, TX) Les McLean University of Toronto Susan Bobbitt Nolen University of Washington Anne L. Pemberton apembert@pen.k12.va.us Hugh G. Petrie SUNY Buffalo Richard C. Richardson New York University Anthony G. Rud Jr. Purdue University PAGE 31 31 of 32 Dennis Sayers Ann Leavenworth Centerfor Accelerated Learning Jay D. Scribner University of Texas at Austin Michael Scriven scriven@aol.com Robert E. Stake University of IllinoisUC Robert Stonehill U.S. Department of Education David D. Williams Brigham Young UniversityEPAA Spanish Language Editorial BoardAssociate Editor for Spanish Language Roberto Rodrguez Gmez Universidad Nacional Autnoma de Mxico roberto@servidor.unam.mx Adrin Acosta (Mxico) Universidad de Guadalajaraadrianacosta@compuserve.com J. Flix Angulo Rasco (Spain) Universidad de Cdizfelix.angulo@uca.es Teresa Bracho (Mxico) Centro de Investigacin y DocenciaEconmicaCIDEbracho dis1.cide.mx Alejandro Canales (Mxico) Universidad Nacional Autnoma deMxicocanalesa@servidor.unam.mx Ursula Casanova (U.S.A.) Arizona State Universitycasanova@asu.edu Jos Contreras Domingo Universitat de Barcelona Jose.Contreras@doe.d5.ub.es Erwin Epstein (U.S.A.) Loyola University of ChicagoEepstein@luc.edu Josu Gonzlez (U.S.A.) Arizona State Universityjosue@asu.edu Rollin Kent (Mxico)Departamento de InvestigacinEducativaDIE/CINVESTAVrkent@gemtel.com.mx kentr@data.net.mx Mara Beatriz Luce (Brazil)Universidad Federal de Rio Grande do SulUFRGSlucemb@orion.ufrgs.brJavier Mendoza Rojas (Mxico)Universidad Nacional Autnoma deMxicojaviermr@servidor.unam.mxMarcela Mollis (Argentina)Universidad de Buenos Airesmmollis@filo.uba.ar Humberto Muoz Garca (Mxico) Universidad Nacional Autnoma deMxicohumberto@servidor.unam.mxAngel Ignacio Prez Gmez (Spain)Universidad de Mlagaaiperez@uma.es Daniel Schugurensky (ArgentinaCanad)OISE/UT, Canadadschugurensky@oise.utoronto.ca Simon Schwartzman (Brazil)Fundao Instituto Brasileiro e Geografiae Estatstica simon@openlink.com.br PAGE 32 32 of 32 Jurjo Torres Santom (Spain)Universidad de A Coruajurjo@udc.es Carlos Alberto Torres (U.S.A.)University of California, Los Angelestorres@gseisucla.edu 