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Educational policy analysis archives.
n Vol. 3, no. 18 (November 15, 1995).
Tempe, Ariz. :
b Arizona State University ;
Tampa, Fla. :
University of South Florida.
c November 15, 1995
Matthew principle : a West Virginia replication? / Craig Howley.
Arizona State University.
University of South Florida.
t Education Policy Analysis Archives (EPAA)
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1 of 25 Education Policy Analysis Archives Volume 3 Number 18November 15, 1995ISSN 1068-2341A peer-reviewed scholarly electronic journal. Editor: Gene V Glass,Glass@ASU.EDU. College of Educ ation, Arizona State University,Tempe AZ 85287-2411 Copyright 1995, the EDUCATION POLICY ANALYSIS ARCHIVES.Permission is hereby granted to copy any a rticle provided that EDUCATION POLICY ANALYSIS ARCHIVES is credited and copies are not sold.The Matthew Principle: A West Virginia Replication? Craig Howley ERIC Clearinghouse on Rural Education and Small Sch ools Appalachia Educational Laboratory Charleston, West Virginia firstname.lastname@example.org This article does not represent the official views of AEL, Inc., or of the entities from whom it holds contracts or whom it may serve, nor did AE L support the research on which it is based. Rather, it is based on research conducted as part o f the author's degree work in the cooperative doctoral program in educational administration join tly operated by West Virginia University, Marshall University, and the West Virginia Graduate College. Acknowledgements: I wish to thank three anonymous reviewers for helpf ul hints about organization and expression, Jack Sanders for bringing Friedkin and Necochea's work to my attention seven long years ago, and Susan Voelkel for at the eleventh ho ur pointing out a range error in one of the tables. Finally, my committee has been consistently supportive and helpful: Richard Hartnett (chair), Paul Leary, Ermel Stepp, Linda Spatig, and Edwina Pendarvis. The epigraph given at the beginning of this article came to my attention in a paper by William Turner, of Massey University, New Zealand. In a lecture titled "Does Reading Recovery Work?", Turner (1989, p. 2) noted: "I'm particularl y concerned about children who encounter difficulty in acquiring literacy skills, especially in light of recent research on the consequence of literacy acquisition for further cognitive growth a nd academic achievement. The effects appear to be profound: relatively small differences in readin g ability and literacy-related knowledge and skills at the beginning of school often develop int o very large generalised differences in school-related skills and academic achievement. The se effects have been referred to as Matthew
2 of 25effects, or rich-get-richer and poor-get-poorer eff ects, after a passage from the Gospel according to Matthew." At this point in the paper, Professor Turner supplies the apt passage. Abstract: This study extends and interprets a regression tech nique used to examine the possible role that socioeconomic status may have in regulating th e effects of school and district size on student achievement. The original study (Friedkin & Necoche a, 1988), with data from California, confirmed an interaction between size and SES such that large schools benefitted affluent students, whereas small schools benefitted impoveri shed students. This replication applies the model to a very different state, West Virginia. Res ults are similar, except that the pattern of effects is shown to derive in part from the fact th at in West Virginia impoverished students were shown actually to attend small schools in 1990. Sma ll schools are shown to disrupt the usual negative relationship between socioeconomic status and student achievement. These results would be cause for celebration except that since 19 88 West Virginia has, under a successful consolidation scheme facilitated by the state, clos ed nearly 20 percent of its schools, most of them small schools that had served rural communitie s in this mostly rural state. The discussion interprets findings with respect to this context an d interprets the practical significance of studying structural variables such as those used in the stud y. The Matthew Principle: A West Virginia Replication? For whosoever hath, to him shall be given, and he s hall have more abundance: but whosoever hath not, from him shall be taken away ev en that he hath. (Matthew, 13:12) This epigraph, revealed two thousand years ago, cap tures something of how the world works, the social world certainly, but perhaps also the natural world. For instance, Jesus's remarks may allude to the principles related to ite rative processes that are now understood to account for much that natural science previously fo und obscure. In these "chaotic" processes small differences in initial states lead to great d ifferences in final states. But when the differences in initial social states a re great, it should come as no surprise that differences in final social states can be dramatic. After two millennia, we can say with more certainty than ever that it takes money to make mon ey, and also, that in this chaotic process of making money, the rich get richer and the poor get poorer. Schooling plays a part in postindustrial societies. As Christopher Jencks and others have noted, education has become the route through which the affluent bequeath their adv antages to their children; education mostly bestows its advantages on the advantaged (cf. Jenck s et al., 1979). This article derives from doctoral research in whic h the findings tend to confirm both the wisdom of the Gospel and the wisdom of common sense The inferential statistics reported here (regression analysis), however, are conventional, s traightforward, and solidly based on the literature about two key structural variables relat ed to schooling: socioeconomic status (a most salient contextual variable) and school size (a mos t salient institutional variable). The analysis replicates a widely cited California study (Friedki n & Necochea, 1988), which, curiously, has not been replicated until now. Its findings may have be en too disturbing, or its methodology may have been a little obscure. The original study found that socioeconomic status systematically influences the effects of school and district size on aggregate student achie vement. Large schools and districts (in California) benefit affluent students moderately, w hereas small schools and districts benefit impoverished students to an even greater extent tha n the large schools benefit the affluent. The
3 of 25opposite relationship is true as well: Large school s compound the negative effects of being impoverished, whereas small schools reduce the adva ntages that affluence normally brings. Small schools, on this basis, might not serve stude nts from affluent communities particularly well, at least on average. Despite the fact that the original study is widely cited, its possible and likely implications have not been considered. Should school buildings b e designed to match the socioeconomic characteristics of communities? What should happen to buildings as residential areas change character? Should all schools be smaller, as school reformers like Sizer, Boyer, Husen, and Sergiovanni recommend (despite the apparent damage done to the achievement of affluent students)? The implications would indeed challenge the capacity of educational policymaking, which is so often a matter of compromise, if not ex pedience. Complicated findings are not usually appreciated. The most suggestive inference from the original stu dy--one that emerges in this replication as well--is that we must consider the probable fact that large schools are not just dysfunctional for impoverished students. Instead, large schools d ramatically compound the educational disadvantages confronted by impoverished students. A long line of historical research, beginning at least with Michael Katz's The Irony of Early Sch ool Reform (1968), suggests that educational policy has aimed, perhaps covertly, at improving th e advantages of affluence. Part of the irony is that we can now surmise that such improvement (for the affluent) may represent a simultaneous disservice to the disadvantaged. Such a surmise motivated this study. The original 1 988 study has been widely shared with educators, policymakers, and citizens in West Virgi nia, and, yet, beginning in 1988, the state has laid plans to close many small schools, schools tha t have served a population made historically destitute because of the way in which the state's n atural resources (salt, timber, coal, oil, and natural gas) have been developed and exploited (Whi snant, 1980). Mostly, the land is controlled by large corporations with headquarters outside the state (Appalachian Land Ownership Task Force, 1981, 1983; DeYoung, 1995; Gaventa, 1980). The fact that the state maintained many small schoo ls, however, was not considered a cause for celebration in West Virginia. Teacher sal aries were low, buildings were in bad--even illegal--shape in some places, and the race for the cachet of high-tech pioneer was one the state would lose if something did not happen to reorder p riorities and reallocate resources. Between 1988 and 1995, the state forced the closure of abou t 20 percent of its schools, mostly small schools in rural areas, the bulk of closures occurr ing after 1990. Teacher salaries rose, teachers and administrators were RIFed, new schools replaced old (with much contention and antagonism at the local level), and a variety of large-scale c omputer initiatives were undertaken by the state. For the most part, closures were resisted by local citizens and embraced by professionals. Some professionals have balked, but for the most part pr ofessionals had little choice, since the state (perhaps arbitrarily) imposed "economies of scale" that ruled out the possibility of building new schools that were small. The so-called economies of scale were really just requirements for building size; whether they were more economical or not, or on what basis, no one knew. State officials did not care; they were merely exercising power via duly constituted legal authority. As districts applied for state-supplied constructio n funds, their plans had to meet the perhaps arbitrary economy-of-scale standards. At an y rate, the state has not defended its size standards, nor has the basis of standards per se be en challenged in any court action. Indeed, I am not aware of any reputable or currently valid resea rch on which the standards could be defended. Leading researchers have roundly discredited the no tion that there is any "standard," "best," or "optimal" size for a school. The West Virginia effort is one of the more success ful consolidation efforts authorized by any state in recent years. Similarly ambitious effo rts appear to be underway at present not only in Pennsylvania, Iowa, Illinois, Missouri, and other U .S. states that operate large numbers (e.g.,
4 of 25n>300) of school districts, but also in Canadian pr ovinces, where a major recent goal in many provinces is to reduce the number of local educatio n agencies. It is past time that those who advise policymakers begin to articulate the complexity associated with issues of school size. Though state s can reallocate funds by closing schools and consolidating operations, the question is who benef its as a result, and the evidence suggests that the Matthew principle governs the outcome. This art icle is particularly aimed at researchers in the hope that related investigations will be undert aken elsewhere. For this reason, results and analysis are reported in detail. Investigators inte rested to collaborate on similar work are invited to contact me directly. Similar work might well inc lude multilevel analyses (e.g., classes within schools within districts). Other state-based replic ations are needed as well, however. Units of Analysis and Subjects The units of analysis in this investigation are sch ools and school districts in West Virginia. Four relevant hypotheses are tested with variables from West Virginia and federal data sets. The West Virginia data set provides the dependent achie vement variable at four grade levels (i.e., aggregate scores at the third, sixth, ninth, and el eventh grades). Subjects of this study include (1) the universe of West Virginia schools with a third grade, a sixth grade, a ninth grade, or an eleventh grade and, at the district level, (2) the universe of West Virginia districts. At the school level, each group of schools serving a given cohort (e.g., grades three or six) comprises a distinct, but not necessarily mutually exclusive, group. Some schools, for instance, house both third and sixth g rades; a few house third, sixth, and ninth grades; and there are a small number of K-12 school s in the state. At the district level, subjects include the universe of county districts (all distr icts are county districts in West Virginia). Definitions Variables of interest in the subsequent statistical analyses are (1) school size, (2) achievement, and (3) socioeconomic status. For the school-level analyses the following definitions apply: School size is defined as fall 1990 enrollment in t he grade-level cohort that is the subject of analysis (e.g., total number of third-gr ade students in the school or county when third grade achievement is the dependent varia ble.) This procedure is common in school-size studies (e. g., Fowler & Walberg, 1991; Friedkin & Necochea, 1988) because it controls for the effects of variation in the grade-level configurations of schools (e.g., K-4, K-6, K-8). The definition al lows one to distinguish size effects independent of the grade span encompassed by a school. Achievement is defined as composite basic skills sc ores achieved by regular education students on the spring 1991 administratio n of the Comprehensive Test of Basic Skills (CTBS) in grades three and six and the fall 1990 administration of the CTBS in grades nine and eleven, aggregated to the s chool level. Standardized norm-referenced achievement scores on group tests such as the CTBS are common measures of student learning (e.g., Anderson 1991) and are comparable to those used by Friedkin and Necochea (1988). CTBS test scores a re a perpetual concern in West Virginia (e.g., Probation, 1990; Student Test Scores, 1994; W.Va Student, 1993), in part because they are among three measures (also including attendance and dropout rates) used by the West Virginia
5 of 25Department of Education to determine a district's e ffectiveness. Districts judged to be ineffective are subject to the direct intervention of the state (Scores, 1995). Socioeconomic status is defined as the proportion o f students at each school receiving free or reduced lunches in the school-lun ch program during the fall of 1990, as reported by the West Virginia Department o f Education to the National Center for Education Statistics. Though commonly used, free and reduced-price lunch rates exhibit an increasingly weak relationship to student achievement as grade level rises, possibly because older students are less likely than younger students to file applications f or assistance. Unfortunately, at the school level, there are in West Virginia no good alternatives (se e, however, the discussion of an alternative measure used in the district-level analyses). The availability of more suitable measures of socio economic status may explain one of the anomalies of recent studies of school size effects (e.g., Plecki, 1991; Walberg & Fowler, 1987; Fowler & Walberg, 1991; Friedkin & Necochea, 1988). The best studies are frequently conducted in two very urban states, California and New Jersey. Part of the reason for this choice may well be the availability in those states of com prehensive measures of socioeconomic status at the school level (cf. Friedkin & Necochea, 1988; Walberg & Fowler, 1987). Given the weak nature of the school-level socioeconomic status var iable, then, the availability of an alternative measure for the countylevel analyses provides an additional test of the hypotheses. When the county district is the unit of analysis, t he following definitions apply: School size is defined as total enrollment in fall 1990 for the subject cohort in the county district. Achievement is defined as individual composite basi c skills scores achieved by regular education students on the spring 1991 (g rades three and six) or fall 1990 (grades six and eleven) administration of the CTBS aggregated to the county district level. Socioeconomic status is defined as the proportion o f students in each county receiving free or reduced lunches in the school-lun ch program during the fall of 1990. In addition, the county-level analyses also use an alternative measure of socioeconomic status, as follows: Alternatively, at the county level, socioeconomic s tatus is defined as the proportion of county residents at least 20 years old who, acco rding to the 1990 decennial census, had not completed high school (grade twelve ). This census measure is available because all West V irginia's school districts are county districts and the decennial census reports population statist ics at the county level in all states. Data for thi s measure were taken from the School District Data Bo ok (National Center for Education Statistics, 1995).Design This study employs bivariate correlational and mult ivariate regression analysis to test hypotheses about the relationship between size of e ducational units (schools and districts) and aggregate student achievement in West Virginia. Thi s design is appropriate for studies seeking to establish relationships between constructs operatio nalized, as in this case, with interval-level
6 of 25data. Hypotheses. Null hypotheses are used to examine the following two questions, each at the school and district level (p < .05): (1) What is th e (zero-order) relationship between school size and student achievement in West Virginia schools? a nd (2) Does socioeconomic status regulate the relationship of school size and student achieve ment in West Virginia? For the second question, the hypotheses test the significance of t he interaction term in the regression anlaysis. Ancillary analyses explore the results of hypothesi s testing in greater depth. Bivariate analysis. At the beginning of the 20th ce ntury, it was widely believed that school size would be positively related to achievement in a straightforward fashion (e.g., Cubberley, 1922; Stemnock, 1974). The assumption behind this h ypothesis was in fact quite reasonable, given the absence of data, methods, and experience needed actually to judge the situation appropriately. Why should educators of that time ha ve believed anything else but that larger schools would make possible the provision of better equipped classrooms, better prepared teachers, and more effective administration and tha t such provision would improve student learning? This commonsensical view not only preva iled through the early 1960s (e.g., Conant, 1959), it has been widely reiterated by government officials and their supporters in West Virginia (e.g., Governor's Committee on Education, 1990), in cluding the editorial staffs of both state newspapers, the Charleston Gazette and the Charlest on Daily Mail (e.g., Consolidation, The State Isn't Wrong, 1993; Consolidation, Yes, 1992; Marsh, 1992). The various traditional arguments in favor of creating large schools have only recently begun to be debunked by such organizations as the Carnegie Foundation and leading reformers (e.g. Boyer, 1995; Goodlad, 1984; Sergiovanni, 1993) This circumstance, in particular, provides a reason to report the various zero-order correlations between achievement and school size. Z ero-order correlations will be developed to test hypotheses at all four grade levels. Multivariate analysis. This study employs a very si mple regression model, with an interesting methodological innovation. The model wa s developed and applied by Noah Friedkin and Juan Necochea (1988) to a California data set. The study is widely cited in the school size literature, but it has never been replicated (Noah Friedkin, personal communication, March 2, 1994). The Friedkin and Necochea model involves regressing achievement aggregated to the school and district levels on size, SES, and the pr oduct of SES and size (interaction term). The innovation involves differentiation of the regressi on equation to provide a basis for determining possible effect sizes of size in communities of dif fering socioeconomic status. The partial derivative is calculated and standardized to obtain the net effect of size on achievement at chosen SES intervals. Any use of calculus in educational s tudies is unusual, and hence merits careful explanation, which is provided in Appendix A. Friedkin and Necochea justified their model as an a pplication of systems theory. In brief, schools (either individual schools or districts) co nstitute open systems that interact with a wider environment that may facilitate or constrain the op eration of the system. In particular, community socioeconomic circumstances have this facilitating or constraining effect on school systems. On this basis, the researchers hypothesize a previousl y unconsidered circumstance, namely, that school size (a system variable) interacts with soci oeconomic status (an environmental variable) to determine achievement (a measure of system perfo rmance). The hypothesized effects are analogous to the familiar concept of aptitudetrea tment interactions. This comparison is perhaps invidious, however, given the meritocratic ideology of American culture (in which poverty is widely believed to be the just reward of inaptitude ). Systems theory nonetheless underestimates many of t he advantages of this sort of analysis. In particular, one may ask what is the advantage of a parsimonious model over more complex production function models. A common view is that s ize represents "nothing magical" in itself,
7 of 25and that only processes count when educational impr ovement is the goal. The concluding section of this article considers related issues and offers a stronger defense of the model. Analysis Because two units of analysis are employed in this study (school and district), two separate datasets comprising the focal variables were constr ucted. Among elementary and middle schools, 21 cases (or about 3 percent) exhibited missing val ues for free and reduced-price lunch rates. Among junior and senior high schools, 13 (or about 8 percent) exhibited missing values for this variable. The reason these values are missing is th at they were not reported; in all likelihood, it was the school that originally did not report the d ata to the state. These data are regarded as sensitive information in some localities, but their absence is another indication that free and reduced-price lunch rates are weaker proxies for so cioeconomic status than one could desire. Rather than eliminate these cases, values were impu ted for schools with missing data. The per-pupil weighted mean of other schools at the sam e level (i.e., other elementary, middle, or secondary schools) in the same district was substit uted for the missing value. This procedure constitutes a form of mean substitution based on th e district mean. Substitutions accomplished in this fashion are probably less biased than would be the case had the usual method of substituting the overall mean for the variable (the most common form of mean substitution) been used. Bivariate analyses. With data set construction and data entry completed, bivariate correlations at the school and district level were calculated. Two-tailed tests of significance were applied to test the null hypothesis (p < .05). Multivariate analyses. Multivariate analysis constr ucted the specified regression equation with backward stepwise entry of variables (i.e., si ze, SES, and interaction term). Backward stepwise methodology assumes that the original mode l will apply and adjusts the model on the basis of characteristics of the data (e.g., a weakn ess of the SES variable) that might reduce the significance of one or more of the variables of int erest. Statistically nonsignificant variables are available for removal from the equation at each ste p. The full model as given in the Friedkin and Necoche a study (see equation 1 in Appendix A) was entered and then variables were removed in d escending order of nonsignificance (probability level for removal was p =.05 or greate r). Variables were removed in this fashion until all remaining variables left in the equation were significant at p <. 05. A significant interaction term (i.e., a statistical ly significant regression coefficient for the product of size and socioeconomic status) in any eq uation indicated the presence of an interaction effect of size and socioeconomic status In practical terms, any resulting regression equation would be available and eligible for differ entiation (see equation 2 in Appendix A) to assess an interaction effect at varying levels of s ocioeconomic status if one of the following three conditions were to apply: (1) all three independent variables were significan t; or (2) (a) the size or the socioeconomic status term and (b) the interaction term were significant; or (3) only the interaction term were significant. In the first of these three cases, both size and so cioeconomic status variables would exhibit a direct effect as well as an interactive effect of size on achievement. In the second of these cases, only one variable would show a direct effect but th e two variables together would show an interactive effect of size on achievement. In the t hird case, neither variable would exhibit a direct effect, but the data would exhibit an interactive e ffect. If the interactive term were not significant
8 of 25 in regression analyses, then the null hypothesis wo uld be confirmed. For the regression analyses, a final criterion appl ied. The partial derivative was calculated if the regression equation had an F-value significa nt at p = .05 or less. To transform the equations into a form that gives effect sizes, both sides of the equation are multiplied by the ratio of the standard deviation of size to the standard deviatio n of achievement (see equation 3 in Appendix A). For all regression equations meeting these criteria illustrative values of effect size are calculated for different socioeconomic levels. The effect sizes for different levels of socioeconomic status serve to translate the finding s given by the final form of the regression equations into a more easily interpretable form.Bivariate Results Two hypotheses present the relationships to be test ed in bivariate analysis, as follows: (1) The zero-order relationship between school size and student achievement in West Virginia schools is not statistically differen t from zero at p < .05. (2) The zero-order relationship between school size and stu dent achievement in West Virginia schools, aggregated to the district level, is not statistically different from zero at p < .05. Each hypothesis entails the correlation of the spec ified achievement and size variables at three grade levels--3, 6, 9 and 11. Box-plot analys es were completed prior to testing the hypotheses, and outliers on size identified by the box-plot analysis were eliminated from the subsequent bivariate analyses. Identified outliers in all cases were those which exhibited largest enrollments. No outliers were identified in box plo ts among units of analysis exhibiting the smallest enrollments. For any analysis at the schoo l level, no more than five cases were dropped. For the county-level analyses only the state's larg est district--among the top 100 districts in the nation on enrollment--was eliminated. School-level results. Table 1 presents the results of the test of the school-level hypothesis (hypothesis 1). Table 1 Zero-order Correlation of Size and Achievement (Sch ool-Level Analysis) StatisticGrade 3 a6 b9 c11 c r .11(.01).03(.44).00(.96).18(.07) N 628508196106 Note. Two-tailed tests of significance; p < .05;significance levels are given in parentheses.a Outliers (grade 3 enrollment > 140) removed.b Outliers (grade 6 enrollment > 382) removed.c No outliers. The data reported in Table 1 confirm the null hypot hesis at the school level (hypothesis 1)
9 of 25 in three cases out of four. The third grade results are the only statistically significant correlation at p < .05. The preponderance of the literature ind icates that the zero-order correlation between size and achievement is near zero, not that it is a lways zero, and the third grade correlation is within the range of values typically reported in th e literature. In the regression results reported in a subsequent section of this study, the correlation between size and achievement at the third grade level does not remain significant once the so cioeconomic status variable enters the equation. At the school level, therefore, the null hypothesis (hypothesis 1) is accepted. In West Virginia schools, among regular education students, the zero-order correlation between achievement and school size is assessed as neither practically nor statistically significant. County-level results. Table 2 presents the results of the test of the county-level hypothesis (hypothesis 2): Table 2 Zero-order Correlation of Size and Achievement (Cou nty-Level Analysis) StatisticGrade 3 a6 b9 c11 c r .08(.58).03(.85)-.15(.27)-.03(.82) N 54545454 Note. Significance level = .05; two tailed tests of significance;significance levels are given in parentheses.aOutliers (Kanawha County) removed. The data reported in Table 2 indicate confirmation of the null hypothesis at the county level (hypothesis 1) in all cases. The observed mag nitudes of the nonsignificant relationships are, one may note, similar to those reported for the sch ool-level analysis. In West Virginia districts, among regular education students, the zero-order co rrelation between achievement and district size is assessed as neither practically nor statist ically significant. Multivariate Results Two hypotheses (i.e., hypotheses 3 and 4) present t he relationships to be explored in multivariate analysis, as follows: (3) In regression analysis, the multiplicative term signifying the interaction of socioeconomic status and school size is not statist ically significant at p < .05. (4) In regression analysis, the multiplicative term signifying the interaction of socioeconomic status and school size, aggregated to the district level, is not statistically significant at p < .05. As with the bivariate analyses, each hypothesis ent ails the correlation of the specified achievement and size variable at three grade levels --3, 6, 9 and 11. Elimination of size outliers was the same as for the bivariate analyses. At the county level, as previously discussed, the analysis employed an alternative measure of socioec onomic status. The alternative measure was employed because of concerns about the adequacy of using free and reduced-price lunch rates as
10 of 25 a test of the Friedkin and Necochea model. Such an alternative, as previously noted, was available only for the district level analyses. Usi ng census data aggregated to the zip code level was attempted but proved to be an inadequate proxy for SES. School-level results. Table 3 presents the final eq uations derived from the backward stepwise regression analysis at the school level (w ith free and reduced-price lunch rates for the school, grade-level enrollment at the school, and t he product of the two values as independent variables and the school's CTBS expanded scale scor es for the relevant grade level as the dependent variable; see "equation 1" in Appendix A. .) For the backward stepwise analysis, all three variables in the Friedkin and Necochea model were entered first, followed by backward stepwise removal of the least significant variable. Removal was stopped when all remaining variables were significant at p < .05. Table 3 Summary of Hierarchical Regression Analysis for the Friedkin and Necochea Model Among Regular, Operational West Virginia Schools Grade Level Variables in the Equation BSE 3a Free/Reduced Lunch-66.473.61-.37 6b Free/Reduced LunchInteraction Term 25.28 -.094 2.84 .04 .37 -.10 9c Free/Reduced LunchInteraction Term -15.97 -.072 4.01 .03 -.28-.15 11d Grade 11 EnrollmentInteraction Term .026 -.084 .01.04 .33 -.21 Note. All regression coefficients significant at p < .05; all equations have F significant at p < .05; residuals are normally dist ributed; Durbin-Watson statistics vary between 1.6 and 1.8.a Outliers removed (grade 3 enrollment > 140); N = 628; R2 = .14. b Outliers removed (grade 6 enrollment > 382); N = 508; R2 = .17. c No outliers identified by boxplot; N = 196; R2 = .13. d No outliers identified by boxplot; N = 106; R2 = .09. Table 3 provides evidence that a practically and st atistically significant interaction effect operates in West Virginia at the school level to re gulate the effects of school size. The interaction term is significant in three of the four grade-leve l equations (i.e., at grades 6, 9, and 11). At the third grade level, of the three independent variabl es only free and reduced-price lunch rates remain statistically significant after application of the backward stepwise analysis to the basic model. With these data, then, no interaction effect is detectable at the third grade level. At the sixth and ninth grade levels, however, free and red uced-price lunch rates and the interaction term are both statistically significant. At the eleventh grade level free and reduced-price lunch rates are eliminated by the backward stepwise analysis, and g rade 11 enrollment and the interaction term remain in the equation as statistically significant Based on this analysis, hypothesis 3 is accepted fo r three of the four grade levels examined. Three of the four equations, therefore, a re eligible for calculation of the partial
11 of 25 derivative to assess the effect of school size on s tudent achievement at differing levels of socioeconomic status. The relevant effect sizes app ear in Table 4. Please note that the SES values given are the values of y used in the differentiate d equations given in footnotes a-c; they do not represent observed data values. Table 4 Zero-order Correlation of Size and Achievement (Sch ool-Level Analysis) GradeFree & Reduced Lunch Rate 5% 15% 25% 35% 45% 55% 65% 75% 85% 95% 6 a -.01-.02-.03-.03-.05-.06-.07-.08-.09-.11 9 b -.03-.09-.15-.20-.26-.32-.37-.43-.50-.56 11 c +.28+.17+.06-.04-.15-.26-.37-.48-.58-.69 Note. At grades 6 and 9 there is no direct effect o f size. Only at grade 11 there is a direct, positive effect of size (in addition to an interactive effect), and this difference between the equations for grades 6 and 9 versus gra de 11 accounts for the positive effects among schools with low rates of free and re duced-price lunches. Effect size equations, which follow in footnotes a-c, are based on equation 3 as given in Appendix A; that is, effect size (a + cy)(s(x)/s(z) ), where x denotes the size variable and z denotes the achievement variable. Values are calculated from the partial derivative of the regression equation and are not o bserved values. a es = (-.094y)(21.73/18.33)b es = (-.072y)(78.23/9.64)c es = (.026 .084y)(112.24/8.75) In Table 4 the disparate effects of size on student achievement can be assessed at three grade levels. Because size does not exert a direct effect on achievement at grade 6 and grade 9, the interaction effect (which exhibits a negative d irectionality, cf. Table 3) varies from near zero when socioeconomic status is high (i.e., when value s of the free and reduced-price lunch variable are low) to negative when socioeconomic status is l ow (i.e., when values of the free and reduced-price lunch variable are high). At grade 6 this negative effect on achievement is t antamount to depressing achievement scores by 1/10 of a standard deviation unit of achi evement for every change in a standard deviation unit of size. This is a comparatively mod est negative effect. At grade 9, however, the negative effect on achieve ment is greater than 1/2 standard deviation unit of achievement for every change in a standard deviation unit of size. That is, at grade 9, the effect of large school size on student s in the poorest communities is negative and substantial. Gene Glass, the originator of meta-ana lysis, provides a rough rule-of-thumb for assessing the practical implications of achievement effect sizes. According to him, an effect size of 1.0 is equivalent to about one year of learning. Thus, the negative effects of size among the very poorest communities are equivalent to about a half year at grade nine. At grade 11 the data exhibit a direct as well as an indirect effect of size. The net result is a positive correlation of size and achievement at hig h levels of socioeconomic status and a negative correlation of size and achievement at low levels of socioeconomic status. These results more nearly parallel those discovered by Friedkin a nd Necochea (1988) in California. The
12 of 25positive effect in communities where socioeconomic status is high is equivalent to approximately 1/4 of a year of learning and the negative effect i n communities where socioeconomic status is low is about 2/3 of a year of learning. The finding most salient to the hypothesis, however is not so much the discovery of a significant interaction effect, but the pattern of relationships that characterizes students' experience at increasing grade levels. The data rev eal neither a direct nor indirect effect of size at grade 3, once socioeconomic status is accounted for A modest indirect effect appears at grade 6 and by grade 9 appears to strengthen. At grade 11 t he size effects are stronger still, with the final regression equation exhibiting both direct as well as indirect (interaction) effects, a combination of effects that not only debase student achievement in communities where socioeconomic status is low, but now appear to enhance student achieveme nt in communities where socioeconomic status is high. The data suggest a pattern of incre asingly strong size effects that systematically benefit advantaged students and systematically hand icap disadvantaged students. An important caveat applies to the school-level ana lyses. The comparative weakness of the socioeconomic status variable (free and reduced-pri ce lunch rates) may obscure somewhat the relationships that actually exist among the variabl es. Free and reduced-price lunch rates account for a comparatively small proportion of variance in the dependent variable, and hence, the amount of variance explained by the school level eq uations is a good deal less than might reasonably be expected from a strong SES variable. In the present analysis, the proportion of variance in the dependent variable explained by the independent variables in the regression equations varies from 9 percent (at grade 11) to 17 percent (at grade 6). In the Friedkin and Necochea study, by contrast, the proportion of vari ance explained by the school-level equations varied from 32 to 45 percent (Friedkin & Necochea, 1988, p. 245), with the greater amounts of variance explained at the higher, rather than lower grade levels. Their zero-order correlations (between size and achievement at the school level) varied from r = .55 to r =3D .67 (Friedkin & Necochea, 1988, p. 243). In this study, the zeroo rder correlations were about half that magnitude and the correlations, moreover, were stro nger at lower, rather than higher, grade levels (i.e., the reverse of the pattern observed in the F riedkin and Necochea data). Despite these difficulties with the school-level an alysis, the data are sufficient to permit rejection of the null hypothesis as given in hypoth esis 3. It appears at the school level that socioeconomic status regulates the influence of sch ool size on the achievement of regular education students enrolled in 1990 in West Virgini a schools at grades 6, 9, and 11. The data also suggest that size effects related to socioeconomic status may be cumulative, a finding that is consistent with other research th at documents a widening achievement gap between disadvantaged and advantaged students over the course of 13 years of schooling. If this is the case, then one would expect that the associa tion of achievement and socioeconomic status at grade 11 would be comparatively weak in small sc hools and comparatively strong in large schools (cf. Huang and Howley, 1993). One way to assess such a hypothesis is to measure t he strength of that association in senior high schools as compared with all other schools ser ving students in grade 11. In the 1990 data set 42 schools had low grades greater than 9 (i.e., the y housed only grades 10-12), and 106 schools had low grades less than or equal to 9 (i.e., they housed grades K-12, 5-12, 7-12, and so forth). Table 5 presents the results of a subsidiary bivari ate analysis to assess the comparative strength of the association between socioeconomic status and achievement among eleventh grade students attending these two different groups of schools. Table 5 Eleventh Grade Achievement, Socioeconomic Status, a nd School Size in Two Groups of West Virginia Schools Serving 11th Grade Students
13 of 25 Grade SpanMeans and Correlations Achievement Mean a Free/Reduced Lunch Mean a Grade 11 Enrollment a ry,z 10-12 b 754.1 (728.8-772.4) .19 (.01-.75) 253.1 (75599) -.51** other c 753.2 (720.8-774.3) .40 (.00-.96) 114.6 (1327) -.11(ns) Note. ry,z is the correlation of grade 11 achieveme nt with free and reduced-price lunch rate; ** = p < .01; ns = not significant (one -tailed tests of significance, as a negative relationship is expected).a Ranges for values of the variables given in par entheses. b n = 42 (listwise)c n = 106 (listwise) A principal threat to analyses such as that present ed in Table 5 is the possible effect of restricted range in one of the correlated variables Both variables exhibit similar ranges, and, in fact, the "other" cohort--where restricted range wo uld threaten the results most seriously (given the hypothesis of an attenuated relationship betwee n achievement and socioeconomic status)--exhibits a range that is somewhat greater than the range for senior high schools (grade 10-12 schools). The data in Table 5 suggest that in these data the association between socioeconomic status and the achievement of eleventh grade studen ts is greater in senior high schools than it is in those schools serving eleventh grade students th at have other sorts of grade-span configurations. Further, the data suggest that the poverty rate prevalent among the "other" schools is about twice that prevalent among the sen ior high schools. The same caveat applies here, however, that applied in the case of the obse rved difference in range for the free and reduced-price lunch variable: The free and reducedprice lunch rates for the senior high schools may represent, to an unknown degree, underestimates of the percentage of students actually eligible to receive free or reduced-price lunches a t the senior high level. In any case, Table 5 shows that the average enrollm ent of grade 11 students in the "other" schools is about half that in the senior high schoo ls. Moreover, average eleventh grade achievement in the two schools is about the same (e ven in view of likely differences in socioeconomic status). These results are consistent with previous research (e.g., Fowler & Walberg, 1991; Friedkin & Necochea, 1988; Huang & H owley, 1993). In order to eliminate the difficulties posed by the possibility that eligibility for free and reduced-price lunches is underestimated among stude nts attending senior high schools, the analysis reported in Table 5 was repeated for stude nts in grade 6. Two groups were selected for comparison on the basis of their relationship to th e grade 6 enrollment mean. The group of larger schools comprised all those schools with enrollment greater than 1/2 standard deviation above the mean, whereas the smaller schools group compris ed all those schools with enrollment less than 1/2 standard deviation below the mean. The mea n was 46.5 student and the standard deviation 40.7. The larger schools group, then, had grade 6 enrollments greater than 67, whereas the lower schools group had grade 6 enrollments les s than 26. Table 6 reports the results of this analysis. Table 6
14 of 25 Sixth Grade Achievement, Socioeconomic Status, and School Size in Two Groups of West Virginia Schools Serving 6th Grade Students Size GroupMeans and Correlations Achievement Mean a Free/Reduced Lunch Mean a Grade 6 Enrollment a ry,z Larger b Schools 718.9 (696.5738.9) .33 (.09-.73) 111.3 (68-383) -.49** Smaller c Schools 716.1 (681.7755.6) .48 (.10-.95) 17.3 (2-25) -.22* Note. ry,z is the correlation of grade 6 achievemen t with free and reduced-price lunch rate; ** = > p < .01; => p < .05 (one-taile d tests of significance, as a negative correlation is expected).a Ranges for values of the variables given in par entheses. b n = 91 (listwise)c n = 165 (listwise) The results given in Table 6 are very similar to th ose given in Table 5, though with the grade 6 analysis, the free and reduced-price lunch rate in the smaller schools group is 50 percent greater than the rate in the larger schools group ( rather than 100 percent greater, as in Table 5). In Table 6, as in Table 5, achievement is approximatel y the same (the standard deviation of achievement among the entire group is 12.0), and th e relationship between the socioeconomic variable and the achievement variable is much weake r in the smaller schools group as compared to the larger schools group (accounting for 4 perce nt of the variance in the smaller schools group, as opposed to 25 percent of the variance among the larger schools group). A final question to ask relevant to the school-leve l regression analyses concerns the size of schools serving students of varying socioeconomic s tatus. The practically and statistically significant interaction terms do not specify for us the source of the relationship. Rather, the interaction is the result of an overall pattern inh erent in the data. A similar result could be obtained as the result of impoverished students bei ng served preponderantly in large schools, or in small schools, or, more likely, in some combinat ion of small and large schools. Given the evident similarity of the school-level an alyses with those reported by Friedkin and Necochea in 1988 in California (where large urb an schools serve a large proportion of impoverished communities), it is important to devel op evidence that might suggest if the same, or a different, pattern prevails in West Virginia. West Virginia operates many small schools (precisely why consolidation has been promoted with such vigor), so that one would hypothesize that the source of the pattern observed has somethi ng to do with these many small schools. The practical question is simple: Do small schools in W est Virginia enhance or detract from the achievement of impoverished students? A meaningful way to make this assessment in the con text of the school-level regression analyses is to relate the data to the free and redu ced-price lunch rate intervals used to interpret school-size effects in Table 4 (i.e., 5%, 15%, and so on). This analysis is easily accomplished by recoding the free and reducedprice lunch data int o 10 groups, 0-5%, 5-15%, 15-25%, and so on, with the final group comprising rates in excess of 85% (the highest observed value is 96 percent). Table 7 gives the average enrollment (grade cohort size) for each such interval at each grade level; the note also gives the zero-order correlati on of free and reduced-price lunch rates and grade cohort size (twotailed tests of significanc e).
15 of 25 Table 7 Average Grade-Level Enrollment by Socioeconomic Sta tus Groups (Free and Reduced-Price Lunch Rates) Among Four Gra de Levels Grade Level a Free and Reduced-price Lunch Groups (Upper Bound) 5% 15% 25% 35% 45% 55% 65% 75% 85% 95% 3 b 43 (4) 50 (29) 48 (78) 43 (107) 38 (140) 35 (124) 30 (72) 29 (49) 25 (12) 36 (18) 6 c 44 (2) 74 (23) 70 (62) 52 (88) 47 (119) 39 (108) 31 (53) 30 (36) 26 (10) 30 (9) 9 d 114 (6) 222 (23) 137 (55) 139 (42) 99 (41) 67 (15) 50 (7) 76 (5) ---54 (4) 11 e 241 ( 6) 261 (31) 150 (40) 127 (30) 110 (20) 66 (12) 43 ( 2) 73 ( 4) ---38 ( 4) Note. Mean grade-level enrollment is rounded to the nearest integer; free/reduced lunch groups are not based on equal-interval units; two-tailed tests of significance for correlations given below.a The N of schools in each group given in parenth eses, beneath mean enrollment for each group.b r of free/reduced lunch rate and grade 3 enroll ment = -.25, p < .0001) c r of free/reduced lunch rate and grade 6 enroll ment = -.30,(p < .0001) d r of free/reduced lunch rate and grade 9 enroll ment = -.46,(p < .0001) e r of free/reduced lunch rate and grade 11 enrol lment = -.51,(p < .0001) The zero-order correlations of size and SES given i n the note are all negative and highly significant. Moreover, this negative association of socioeconomic status and size increases in magnitude with grade level. The general tendency is revealed by the data in the columns and rows of Table 7, which also give an impression of t he numbers of impoverished and affluent students served by schools of various size. In gene ral, the smaller schools in West Virginia in 1990 tended to enroll students from impoverished ba ckgrounds. This tendency increases with grade level; small high schools are more likely to serve impoverished students than are small elementary schools. The overall picture that emerges from the school-le vel regression analyses is that in West Virginia, among regulareducation students in 1990 (1) the indirect (interactive) effect of school size on achievement is a better predictor of studen t achievement than either school size or socioeconomic status alone (Table 3); (2) the natur e of the prediction is that increases in school size imply increasingly more severe negative effect s among impoverished children (Table 4); and (3) impoverished children, as compared to more affl uent children, generally attended smaller schools (Tables 5, 6, and 7). Finally, the data rep orted in Tables 5 and 6 strongly suggest what other studies (e.g., Huang & Howley, 1993) have fou nd, namely that small school size tends to disrupt the negative influence of socioeconomic sta tus on the achievement of impoverished students. County-level results. The basic analysis here is ex actly parallel to the school-level analysis,
16 of 25 with the exception, as previously noted, that an al ternative measure of socioeconomic status is available to join the analysis (percentage of popul ation aged 20 and older with educational attainment less than grade 12). Because of the char acter of free and reduced lunch rates, in which it was suspected that observed rates underestimated the actual prevalence of eligibility at the secondary level, the district-level free and reduce d-price lunch rates were not computed as weighted averages of the all schoollevel rates. I nstead, the district rates were based on weighted averages of the rates for grades 3, 6, and 9 only. The results derived from the socioeconomic status variable constructed in this way and for the alternative measure (the census attainment variable) were quite similar, in any case. As in the school-level analyses all three variables in the Friedkin and Necochea model were entered first, followed by backward stepwise r emoval of the least significant variable. Removal was stopped when all remaining variables we re significant at p < .05. All analyses exclude the state's largest county, which was ident ified in preparatory analysis as a statistical outlier. Tables 8 and 9 report the results of the regression analyses with (1) free and reduced-price lunch rates and with (2) the alternative measure of socioeconomic status. Since there are four grade levels and two equations for each, Tables 8 a nd 9 provide data relevant to eight equations. Table 8 gives statistics about the final regression equations and Table 9 reports the related effect sizes on achievement. Table 8 Summary of Hierarchical Regression Analyses for the Friedkin and Necochea Model: 54 West Virginia School Districts With Two Measures of Socioeconomic Status (Eight Equations) Grade Level Variables in the Equation BSE 3 a,1 Grade 3 EnrollmentInteraction Term +.018044 -.000519 .008636.000254 + .65 .64 3 b,2 Grade 3 EnrollmentInteraction Term +.023862 -.000745 .008743.000273 + .86 .86 6 a,3 Grade 6 EnrollmentInteraction Term +.017703 -.000532 .006257.000177 + .90 .96 6 b,4 Grade 6 EnrollmentInteraction Term +.020666 -.000653 .006135.000182 +1.05 1.12 9 a,5 Grade 9 EnrollmentInteraction Term +.031034 -.001079 .007436.000209 +1.16 1.44 9 b,6 Grade 9 EnrollmentInteraction Term +.032255 -.001155 .007445.000217 +1.21 1.49 11 a,7 Grade 11 EnrollmentInteraction Term +.029209 -.000960 .007644.000221 +1.10 -1.24
17 of 25 11 b,8 Grade 11 EnrollmentInteraction Term +.038620 -.001327 .006825.000210 +1.45 -1.62 Note. All regression coefficients significant at p < .05. Residuals normally distributed for all equations (N = 54). a Free and reduced-price lunch as SES variable.b Educational attainment < grade 12 as SES variab le. 1 F nonsignifi. (p = .1160); R2=.08; Durbin-Watso n = 1.7 2 F significant (p = .0266); R2=.13; Durbin-Watso n = 1.8 3 F significant (p = .0154); R2=.15; Durbin-Watso n = 2.2 4 F significant (p = .0032); R2=.20; Durbin-Watso n = 2.1 5 F significant (p = .0000); R2=.36; Durbin-Watso n = 2.0 6 F significant (p = .0000); R2=.37; Durbin-Watso n = 2.0 7 F significant (p = .0003); R2=.27; Durbin-Watso n = 1.9 8 F significant (p = .0000); R2=.44; Durbin-Watso n = 2.0 Table 9 Effect Sizes of District Enrollment on CTBS Composi te Basic Skills Achievement Test Scores for Varying Levels of Socioeconomic Status (Two Mea sures of Socioeconomic Status) Grade Level a Rate of Socioeconomic Status Variable a,b 5% 15% 25% 35% 45% 55% 65% 75% 85% 95% 3 a,1 3 b,2 +.56+.73 +.37+.46 +.18+.20 .00 -.08 -.19 .35 .38.62 .57.89 -.76 -1.16 -.94 -1.43 -1.13 1.70 6 a,3 6 b,4 +.77+.89 +.50+.55 +.22+.22 -.05-.11 -.32 .44 .59.78 .86 -1.11 -1.13-1.44 -1.40-1.78 -1.67 2.11 9 a,5 9 b,6 +.96+1.0 +.56+.56 +.15+.13 -.25-.31 -.66 .74 -1.07-1.18 1.47 -1.61 -1.88-2.05 2.28 -2.48 -2.69 2.92 11 a,7 11 b,8 +.92 +1.20 +.56+.70 +.20+.20 -.17-.29 -.52 .79 .89 -1.29 1.25 -1.79 -1.61-2.29 -1.97-2.79 -2.33 3.29 Note. Values calculated from the partial differenti al of the applicable regression equations The first grade 3 equation, as noted in t ext, is nonsignificant. Effect size equations, which follow in footnotes 1-8, are based on equation 3, as given in Appendix A; that is, effect size =(a + cy)(s(x)/s(z )), where x denotes the size variable and z denotes the achievement variable.a Free and reduced-price lunch rates as SES varia ble. b Educational attainment < grade 12 as SES variab le. 1 es = (.01804 .000519y)(264.29/7.30)2 es = (.02386 .000745y)(264.29/7.30)
18 of 253 es = (.01770 .000532y)(264.39/5.19)4 es = (.02066 .000653y)(264.39/5.19)5 es = (.03103 .001079y)(263.46/7.00)6 es = (.03225 .001155y)(263.46/7.00)7 es = (.02921 .000960y)(234.27/6.23)8 es = (.03862 .001327y)(234.27/6.23) All variables in the equations reported in Table 8 are significant at p < .05 and each equation (except for the first, where free lunch ra te serves as the measure of socioeconomic status) is significant at p < .05. The data present ed in Table 8 indicate a consistent pattern in both sets of equations (i.e., those for both measures of socioeconomic status) and at all four grade levels (3, 6, 9, and 11). Each equation shows a dir ect positive effect of size and an indirect (interactive) effect of size and socioeconomic stat us. These effects, moreover, generally increase with grade level, as does the amount of variance ex plained. The pattern is clearest with equations using the alternative measure of socioeconomic stat us (see footnotes 2, 4, 6, and 8 in Tables 8 and 9). The null hypothesis given in hypothesis 4, therefore, is rejected. R2 values in these analyses are comparable to those reported by Friedkin and Necochea (1988, p. 246). The R2 values in their district-lev el analyses vary from .27 to .46. The district-level R2 values in this study vary between .13 and .44. In general, Friedkin and Necochea employed a more comprehensive measure of socioecono mic status, which was readily available to them. This difference not only accounts for high er R2 values but also for the fact that in the present analysis socioeconomic status does not rema in significant in the final equations. One can also observe in Table 8 that the values at each grade level for the main and for the interactive effects of size are nearly equal; the p ositive direct effects are canceled by the nearly equal negative indirect effect. For the entire popu lation, and across all grade levels, this result means that the overall effect of size is nearly zer o. Only when the effects are assessed at differing levels of socioeconomic status do the strong dispar ities in the effects of size manifest themselves. Table 9 demonstrates this fact. The increasing magnitude of the direct effect of th e size term ( increasing from a low of +.65 in the grade 3 equations to a high of +1.45 in the grade 11 equations) and of the indirect effect of size via the interaction term ( increasin g from a low of -.64 in the grade 3 equations to a high of -1.62 in the grade 11 equations) accounts f or the pattern of effect sizes given in Table 9. The combined effects of these two independent varia bles produced the increasing severity of differential effects of size calculated for differi ng levels of socioeconomic status. As shown in Table 9, at grade 3 there are moderate positive effects of size (+.56 and +.73) on district-wide achievement in districts in the hi ghest category of average socioeconomic status, whereas in the lowest category of socioeconomic sta tus the negative effects are at least twice as strong (-1.13 and 1.70). This pattern--substantia l negative effect sizes in the districts with lowest socioeconomic status and moderate positive e ffect sizes in the districts with the lowest socioeconomic status (i.e., high rates of free and reduced-price lunch qualifiers)--is consistent across all equations. It is moreover, consistent wi th the pattern of results reported by Friedkin and Necochea (1988). The effects in this study also bec ome more extreme as grade level increases. At the 11th grade level the largest positive effect si zes are +.92 and +1.20 versus the largest negative effect sizes of -2.33 and -3.29. In general the neg ative extremes are about two or three times the magnitude of the positive effects.Conclusions The correlational and regression analyses disclose preponderant support for all four hypotheses. The direct association of size and achi evement is neither practically nor statistically
19 of 25significant, but, instead socioeconomic status gove rns the relationship. As in the California study (Friedkin & Necochea, 1988), large size benefits af fluent students, but afflicts impoverished students and vice versa. And, as in the California study, the negative effects of size on the achievement of impoverished students are much stron ger than the positive effects of size on affluent students. However, small schools and distr icts in West Virginia were shown in the analysis to disrupt the negative relationship of si ze and student achievement, whereas the reverse seems to have been true in California. At least in 1990, the smaller schools in West Virginia tended to serve impoverished communities, an associ ation that was strongest at the high school level (see the note to Table 7). Since that time, h owever, West Virginia has facilitated the closure of many small schools. The findings developed in this study provide strong evidence that small school size benefits the achievement of impoverished West Virgi nia students. The evidence suggests, as well, that increasing school size may produce effec ts that are the opposite of those that policymakers claim they intend in closing smaller s chools. The Case for the Model As noted previously, one may complain that the Frie dkin & Necochea model avoids variables supposedly known (via the various school effectiveness literatures) to influence student achievement. Some species of production function, o n this view, would serve educators much better than an analysis of two structural variables over which educators have little influence. But critics like Steven Hodas (1993) charge that th e notion of an abstract production function substantially misrepresents the nature of teaching and learning. In the view of such critics, teaching and learning are complex, idiosyn cratic, and even chaotic. For them, "re-inventing the wheel" is not only unavoidable, i t is absolutely necessary. Education production functions, on this view, oversimplify the complexit y of schooling as an organic system. In short, the presumption that we can manipulate features of reality in the same way that we can manipulate features of a regression equation is unw arranted. Far more useful would be an analysis that suggests how educators might leverage the features of existing reality to good effect. Call i t the "Zen approach" if you like. The present study and its predecessor provide an example of suc h an approach, with useful findings. The parsimony of the model might even commend it as app ropriate theorizing. Small schools effectively disrupt a dangerous cycle in education. Small scale-schooling seems to accomplish this miracle without extensive staff development budgets, without widespread dissemination of innovative materials an d methods, and without vast systemic aspirations for reform that implicate everything fr om teacher education to American culture itself in the name of enhanced student performance. In Wes t Virginia in 1990, at least, small schools were going about their good work unremarked and una ppreciated; indeed, they were under attack as an embarrassment. The method employed by this study and by Friedkin a nd Necochea is not a production-function, nor is it a truncated form the reof. It is, instead, an analysis of the interactin g effects of key structures per se. In the present st udy, neither the need to manipulate key process variables nor the ability to do so is assumed. The assumption of such a model, rather, is that social structures are significant, meaningful, and perhaps--if we design wise analyses--may reveal themselves as determining. These assumptions, of co urse, are more sociological than educational. The point is that wise policy would work with, rath er than against, such structures, when possible, as it appears to be in the present case: the state of West Virginia could seek to retain rather than to eleiminate its small schools. These assumptions and insights owe more of a debt to the sociological than to the pedagogical imaginatio n. In fact, it is very difficult to believe (on the ba sis of personal experience) that small
20 of 25 schools, serving impoverished students in rural are as of a very rural state are systematically implementing innovative practices that produce the results reported in this study. Instead, it is more likely that the school effectiveness literatur es merely point to an assortment of virtues that somehow persist better in small-scale than in large scale schooling. Most teachers in West Virginia's small rural schools are locals, not cosm opolitans, in Alvin Gouldner's sense of that term; they are generally more committed to the comm unities where they teach than to pursuit of a brilliant career. For the most part, they are skept ical of state and national reform efforts (Seal & Harmon, 1995), perhaps with justification. Moreover these teachers have been shown to cherish a range of values that would subvert any concertedl y systematic (or "systemic") scheme of reform (Howley, Ferrell, Bickel, & Leary, 1994). In short, there is probably considerable resistance in West Virginia schools to the grand plans of reforme rs. Mostly, such plans do not sell well from the courthouse steps: "Rural residents distrust out siders with big plans for making 'deprived' people want to be 'middle-class'" (Seal & Harmon, 1 995, p. 119). There is one additional, important observation that should be made. It is clear that without a structural model such as that used in this study, the contradictory role that socioeconomic status plays in determining the effects of size will conti nue to be overlooked, especially given the prevalent commitment to equilibrium theories versus conflict theories. It is, for instance, a bit strange that the original 1988 study, though so oft en cited in the literatures on school size and rural education, has not been replicated until now. Whatever the motive, a likely result of such an ove rsight is that a bogus conventional wisdom (the nostrum that bigger is better) will be replaced with another, but equally suspect, conventional wisdom (e.g., small is always best). C learly, small is not always best. In West Virginia schools in 1990 seemed well sized to the n eeds of the population. Sadly, the situation may be changing in that state. In other states, however, the danger, of the view t hat small-is-always-best is that it enacts a Malthusian compromise, the greatest good for the gr eatest number. This utilitarian choice, however, is still preferable to a compromise that i mposes sanctions on a great number for the benefit of a privileged minority. Appendix A In calculus, the derivative (often written as "dy/d x") gives the value of the change in a dependent variable, y, associated with change in a single independent variable, x. The derivative is a ratio, often construed as a generalized form o f slope familiar in algebra as the ratio of rise over run, or y 2 y 1 x2 x1The regression coefficient, of course, provides an estimate of slope in regression analysis, with regression coefficients appearing as constants in r egression equations. It is important to note well this fact, since during the process of taking a der ivative one distinguishes which values are constant ("constants") from those that which vary ( "variables"). Unlike the derivative, the partial derivative (like the partial regression coefficient) is particularly useful in working with equations with two or more independent variables (e.g., as in regression equations). The partial derivative merel y gives the rate of change in the function (i.e., the value of the dependent variable) with respect t o one independent variable as another is held constant. This resembles the way in which partial r egression coefficients give the influence of one variable on another when the influence of a thi rd variable is eliminated. To calculate the partial derivative, one variable ( either x or y) is held constant while differentiation proceeds with respect to the other; the variable "held constant" is treated during
21 of 25differentiation as if it were a constant. Afterward s, in the resulting differentiated equation, values of the variable held constant (socioeconomic status in this case) can be substituted in order to calculate actual values of the partial derivative f unction (i.e., to determine the influence of the variable not held constant on the dependent variabl e with respect to differing values of the variable that was held constant during partial diff erentiation). In this proposed study, the effects of size on achievement are hypothesized to vary by socioeconomic status, and the partial derivative hypothetically provides a mechanism to e valuate the differences. For this study, the applicable partial derivative w ill give the effect of change in size (defined as cohort-level enrollment) on achievement (CTBS composite scores) as socioeconomic status (free and reduced-price lunch rates and, alt ernatively, in county-level analyses, percent of the general population with less than a twelfth-gra de education) is held constant. The general form of the mathematical model proposed is given by the following equation: f(z) = ax + by + cxy (equation 1) where:a, b, and c are the unstandardized regression coeff icients (of size, socioeconomic status, and the interaction term, respectively);z represents values of achievement (the dependent v ariable); x represents values of size (one independent variab le); andy represents the values of the socioeconomic status variable (the second independent variable). Holding y constant, and differentiating z with resp ect to x, the relevant partial derivative is given by the equation: fx(z) = a + cy (equation 2) This equation can be used to calculate the effect o n the dependent variable (z, achievement) for differing values of the variable h eld during partial differentiation (y, socioeconomic status). Standardizing the partial derivative renders it as an "effect size," which is more easily interpretable than the unstandardized form. The tot al, standardized effect of school size on achievement (in standard deviation units) is given by the following formula: s(x)effect size = (a + cy)----s(z) (equation 3) This is the final form of the regression equations, and it represents the change in achievement (in standard deviation units) expected with change in s ize (also in standard deviation units) among
22 of 25cases with a particular SES. References Anderson, J. (1991, April). Using the norm-referenc ed model to evaluate Chapter 1. Paper presented at the annual meeting of the American Edu cational Research Association, Chicago. (ERIC Document Reproduction Service No. ED 350 315)Appalachian Land Ownership Task Force. (1981). Land ownership patterns and their impacts on Appalachian communities: A survey of 80 counties. B oone, NC and New Market, TN: Appalachian State University and Highlander Researc h and Education Center. (ERIC Document Reproduction Service No. ED 325 280)Appalachian Land Ownership Task Force. (1983). Who owns Appalachia? Landownership and its impact. Lexington, KY: University of Kentucky P ress. Boyer, E. (1995). Basic school: A community for lea rning (Advance Copy). Princeton, NJ: Carnegie Foundation for the Advancement of Teaching (ERIC Document Reproduction Service No. ED 381 284)Bryk, A. (1994). More good news that school organiz ation matters. [from newsletter of the Center on Organization and Restructuring of Schools ]. Cannell, J. (1987). The Lake Wobegone effect revisi ted. Educational Measurement: Issues and Practice, 7(4), 12-15.Conant, J. (1959). The American high school today: A first report to citizens. New York: McGraw-Hill."Consolidation, Yes". (1992, February 22). Charlest on Gazette, p. 4A. "Consolidation (The state Isn't Wrong About That)". (1993, August 11). Charleston Daily Mail, p. 4A.Cubberley, E. (1922). Rural life and education: A s tudy of the rural-school problem as a phase of the rural-life problem. NY: Houghton-Mifflin.DeYoung, A. (1995). The death and life of a rural A merican high school: Farewell, Little Kanawha. New York: Garland.Fowler, W., & Walberg, H. (1991). School size, char acteristics, and outcomes. Educational Evaluation and Policy Analysis, 13(2), 189-202.Friedkin, N., & Necochea, J. (1988). School system size and performance: A contingency perspective. Educational Evaluation and Policy Anal ysis, 10(3), 237-249. Gaventa, J. (1980). Power and powerlessness: Quiesc ence and rebellion in an Appalachian valley. Chicago: University of Illinois Press.Goodlad, J. (1984). A place called school. New York : McGrawHill. Governor's Committee on Education. (1990). Educatio n first: Our future depends on it.
23 of 25Charleston, WV: Author.Hodas, S. (1993). Is water input to a fish? Problem s with the production-function model in education. Education Policy Analysis Archives [On-l ine serial], 1(12). Available E-mail: email@example.com Message: get hodas v1n12 f= mail Howley, A., Ferrell, S., Bickel, R., & Leary, P. (1994). Teachers' values and the prospect for instructional reform. Planning and Changing, 25(1/2), 56-74.The Holy Bible (King James Version, 1611). New York : American Bible Society. Jencks, C., Bartlett, S., Corcoran, M., Crouse, J., Eaglesfield, D., Jackson, G., McClelland, K., Mueser, P., Olneck, M., Schwarz, J., Ward, S., & Wi lliams, J. (1979). Who gets ahead? The determinants of economic success in America. New Yo rk: Basic Books. Johnson, F. (1989). Assigning type of locale codes to the 1987-88 CCD public school universe. Paper presented to the annual meeting of the Americ an Educational Research Association, San Francisco, CA. (ERIC Document Reproduction Service No. ED 312 113) Katz, M. (1968). The irony of early school reform. Cambridge, MA: Harvard University Press. Marsh, D. (1992, February 14). The seductive myth o f tiny schools. Charleston Gazette, p. 6A. Ornstein, A. (1993). Norm-referenced and criterionreferenced tests: An overview. NASSP Bulletin, 77(555), 28-39.Ornstein, A., & Gilman, D. (1991). The striking con trasts between norm-referenced and criterion-referenced tests. Contemporary Education, 62(4), 287-293. Linn, R. (1990). Comparing state and district resul ts to national norms: The validity of the claims that "everyone is above average." Educational Measu rement: Issues and Practice, 9(3), 5-14. Plecki, M. (1991, April). The relationship between elementary school size and student achievement. Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL."Probation recommended for some schools." (1990, Oc tober 10). Sunday Gazette-Mail. National Center for Education Statistics. (1995). S chool district databook [CD-ROM]. Washington, DC: Author."Scores concern officials." (1995, February 12). Ch arleston Gazette, p. 24. Seal, K., & Harmon, H. (1995). Realities of rural s chool reform. Phi Delta Kappan, 77(2), 119-124.Sergiovanni, T. (1993, April). Organizations or com munities? Changing the metaphor changes the theory. Paper presented at the annual meeting o f the American Educational Research Association, Atlanta, GA.Stemnock, S. (1974). Summary of research on size of schools and school districts. Arlington, VA: Educational Research Service, Inc. (ERIC Docume nt Reproduction Service No. ED 140 459)
24 of 25 "Student test scores beat average, Marockie says." (1994, June 29). Charleston Gazette, p. 3A. Walberg, H., & Fowler, W. (1987). Expenditure and s ize efficiencies of public school districts. Educational Researcher, 16(7), 5-13."West Virginia student test scores improve at least eight percent from last year." (1993, June 11). Charleston Gazette, p. 5A.Whisnant, D. (1980). Modernizing the mountaineer. B oone, NC: Appalachian Consortium Press. Worthen, B., & Spandel, V. (1991). Putting the stan dardized test debate in perspective. Educational Leadership, 48(5), 6569.About the AuthorCraig Howley Director ERIC Clearinghouse on Rural Education and Small Sch ools Appalachia Educational LaboratoryPhone: 800-624-9120 email: firstname.lastname@example.org http://www.ael.org/erichp.htm I've written about, studied, and lived in rural pla ces. (It's debatable whether or not I still live in a rural place, but the local chamber of commerce says I do, given that our house sits 2 miles north of I-64).Culture, politics, economics, and history concern m e. I wish schools were better at promoting 'the life of the mind' (whatever that is; finding out is part of the adventure) among everyone. And I think there are reasons they don't, but these reaso ns constitute more than just inattention or foolishness. Culture, politics, economics, and hist ory suggest reasons. Literature (fiction) may be a much better guide to true education in rural places than the sorts of poor studies we educationists sponsor. Check out We ndell Berry's Second Growth (circa 1950) or Annie Proulx's The Shipping News (circa 1990) and even E.M. Forster's Howards End (circa 1920). These folks have preserved something we have tried desperately to abandon, but can't actually escape. The wonder is that, though these b ooks (and many more) treat the dilemmas of rural life, they also deal with the idea of a true education more universally. Now, that's fun because it's not easy. In particular, novels don't lend themselves to translations as cookbooks. Teaching well is the most difficult work in the wor ld. We make a great mistake with attempts to make it easy or happy. Happiness is not a worthy ai m for education, nor is getting and holding a good job.Copyright 1995 by the Education Policy Analysis ArchivesEPAA can be accessed either by visiting one of its seve ral archived forms or by subscribing to the LISTSERV known as EPAA at LISTSERV@asu.edu. (To sub scribe, send an email letter to LISTSERV@asu.edu whose sole contents are SUB EPAA y our-name.) As articles are published by the Archives they are sent immediately to the EPAA subscribers and simultaneously archived in three forms. Articles ar e archived on EPAA as individual files under the name of
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