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Educational policy analysis archives.
n Vol. 12, no. 58 (October 22, 2004).
Tempe, Ariz. :
b Arizona State University ;
Tampa, Fla. :
University of South Florida.
c October 22, 2004
School size, achievement, and achievement gaps / Bradley J. McMillen.
Arizona State University.
University of South Florida.
t Education Policy Analysis Archives (EPAA)
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EDUCATION POLICY ANALYSIS ARCHIVES A peer-reviewed scholarly journal Editor: Gene V Glass College of Education Arizona State University Copyright is retained by the first or sole author, who grants right of first publication to the Education Policy Analysis Archives EPAA is a project of the Education Policy Studies Laboratory. Articles are in dexed in the Directory of Open Access Journals (www.doaj.org). Volume 12 Number 58 October 22, 2004 ISSN 1068-2341 School Size, Achievement, and Achievement Gaps Bradley J. McMillen North Carolina Department of Public Instruction Citation: McMillen, B. J. (2004, Octobe r 22). School size, achievement, and achievement gaps. Education Policy Analysis Archives, 12 (58). Retrieved [date] from http://epaa.asu.edu/epaa/v12n58/. Abstract In order to examine the relationship between school size and achievement, a study was conducted using longitudinal achievement data from North Carolina for three separate cohorts of public school students (one elementary, one middle and one high school). Results revealed several interactions between size and student characteristics, all of which indicated that the achievement gaps typically existing between certain subgroups (i.e., more versus less-advantaged, lower ver sus higher-achieving) were larger in larger schools. Results varied across the grade level cohorts and across subjects, but in general effects were more common in mathematics than in reading, and were more pronounced at the high school level. Study results are discussed in the context of educational equity and cost-effectiveness. Introduction Concerns about school size in the educationa l research literature tend to center on high schools. The most common sentiment expressed is th at high schools are too large, and that they are getting larger. The U.S. Department of Education (2000a) reports that the number of public schools serving the secondary grades in the U. S. has la rgely held steady between 23,000 and 26,000 since 1930. During that same time, however, the number of public high school students in the U. S. nearly tripled, from approximately 4.4 million to over 13 million. As consolidation trends have created larger sc hools, the issue of school size has become of great interest to educators and policymakers alike. As the demand for safer schools, the need to help all students reach high achievement standards, and the proliferation of school-level monitoring
School size, achievement, and achievement gaps 2 and accountability systems have increased, so has interest in the contribution of many school-level variables including school size to student outcomes. Intuitively, school size would appear to have considerable impact on both student achievemen t and discipline in the school. Smaller size is often associated with more personal attention, more opportunities for involvement, less anonymity for students, and a more caring environment. Thes e factors are then hypothesized to lead to more positive student outcomes (Finn, 1989; Holland & Andre, 1987). Larger schools, however, are said to offer a broader and deeper curriculum along with economies of scale that often appeal to policymakers. Studies of student behavior indicate that sma ller schools are generally associated with more positive behavioral outcomes for students. Larger schools are reported to have higher dropout and expulsion rates than smaller schools (Fetler, 1989; Fowler & Walberg, 1991; Pittman & Haughwout, 1987; Schoggen & Schoggen, 1988). Larger schools also have been shown to have more problems with most major behavioral issues including truancy, disorderliness, physical conflicts among students, robbery, vandalism, alcohol use, drug use sale of drugs on school grounds, tobacco use, trespassing, verbal abuse of teachers, teacher ab senteeism, and gangs (Haller, 1992; Heaviside, Rowand, Williams, & Farris, 1998; Lindsay, 1982; Page 1991). There is also a substantial body of research that indicates that students in sma ller schools are more likely to be involved in extracurricular activities (Baird, 1969; Barker & Gump, 1964; Grabe, 1981; Lindsay, 1982; Morgan & Alwin, 1980; Schoggen & Schoggen, 1998). School size has also been studied in relati onship to student achievement, at both the elementary and high school levels. The majority of studies at the elementary level point toward an inverse relationship, i.e., smaller elementary schools tend to have higher achievement. For example, a study in New York found that reading and math tes t scores were higher in elementary schools with smaller enrollments, even after controlling for so cioeconomic factors (Kiesling, 1968). Caldas (1993) found a small negative relationship between school size and general achievement among elementary schools in Louisiana. Wendling and Cohen (1981) also found that third graders from smaller schools demonstrated higher achievement in reading and math than their counterparts in larger schools. In that study, the average enro llment in the lower-achieving schools was 776, while the average enrollment of the higher-achieving schools was 447. Fowler (1995) reviewed a number of studies of school size and achievement in el ementary schools, all of which again suggested a negative relationship. Several of the studies Fowler reviewed, however, were not widely published or were not published at all. Even so, there is little contrary evidence in the educational research literature to refute the conclusion that smaller elementary schools are associated with higher achievement. Although the findings for elementary schools wo uld appear fairly consistent, the research on high school size and achievement is less conclusi ve. Using state achievement test data from 293 public high schools in New Jersey, Fowler and Walberg (1991) found that school size was inversely related to test scores in mathematics and writ ing. They also found that smaller schools were associated with higher passing rates on the readin g portion of the states Minimum Basic Skills Test as well as on the mathematics and writing portions of the states High School Proficiency Test. These effects were statistically significant even afte r controlling for students fa mily income level, but the actual size of the effects was not clearly re ported. The schools in this study had enrollments ranging from 147 to 4,018, with an average enrollment of 1,070. Other studies have demonstrated similar results. Fetler (1989), in a study of all public high schools in California, found that schools with smaller enrollments tended to have higher achievement scores, although the relationship was not strong and the analysis did not take into account any student background factors. Walber g and Walberg (1994) used data from the 1990 National Assessment of Educational Progress (N AEP) mathematics assessment to examine
Education Policy Analysis Archives Vol. 12 No. 58 3 relationships among size, expenditures and achievem ent. Their analyses demonstrated that states with larger schools tended to score lower on the NAEP mathematics assessment, even after controlling for per-pupil expenditures and percen tage of non-Caucasian students in the state. Other studies, however, have failed to demonstrate higher levels of achievement for smaller high schools. Lindsay (1984), analyzing data from a nationally representative sample of almost 14,000 high school students, found no meaningful relationship between school size and academic ability. Academic ability in this study was measured by a standardized composite score based on four tests (vocabulary, reading, inductive reasoning and mathematics) that were used in the National Longitudinal Study conducted by the U. S. Depart ment of Education. A study by Jewell (1989) reached similar conclusions. In examining the rela tionship between school size and college entrance exam scores across all 50 states and the District of Columbia, he found no significant relationship between high school size and either ACT scores or Scholastic Achievement Test (SAT) scores after controlling for poverty. In another earlier study, Baird (1969) analyzed data from over 21,000 high school students who took the American College Te st (ACT) and found that students from smaller schools actually had lower ACT scores. Haller, M onk, and Tien (1993) also found no relationship between high school size and higher-order thinking skills using data from a nationally-representative sample of 10th graders from the Longitudinal Study of American Youth. Compared to the results for elementary schools, the evidence for the size-a chievement relationship at the high school level appears to be more mixed. One of the more sophisticated studies of si ze and achievement found that students from medium-sized high schools actually demonstrated high er achievement than students in either smaller or larger schools (Lee & Smith, 1997). Using long itudinal data from a nationwide sample of over 9,000 students, the authors studied the relationship between size and achievement gains between 8th grade and 12th grade. The results indicated that afte r controlling for various student-level and school-level demographic characteristics, students in moderate-sized high schools tended to have higher gains in both reading and mathematics, wi th the effects for mathematics being somewhat stronger than for reading. Specifically, they foun d that the highest gains in achievement were found in high schools with enrollments between 600 and 90 0 students. In addition, the finding of lower mathematics gains in larger schools was especi ally pronounced for non-Caucasian students and students from lower socioeconomic backgrounds. The Lee and Smith study is also one of the few studies in this area to control for prior achievement. A recent reanalysis of this same datas et, however, by Howley and Howley (2004) has questioned Lee and Smiths conclusions regarding op timal size, contending in particular that the effects of very small schools were not adequately a ddressed in the analysis. They concluded that the relationship between size and achievement is in fact more linear and that smaller size (less than the 600-student cutpoint posited by Lee and Smith) does in fact benefit students from lower socioeconomic backgrounds (see Lee (2004) for a crit ique of this reanalysis and its conclusions). The interaction between poverty and size was also echoed in a report by the Rural School and Community Trust (Howley & Bickel, 1999) using data from 13,600 public schools in 2,290 districts in Georgia, Montana, Ohio, and Texas. Specifically, schools in less affluent communities in each state demonstrated higher achievement if they were smaller, while the opposite relationship was found in more affluent communities. Howley and colleagues have labeled this phenomenon the excellence effect of small school size, and have al so demonstrated this result across grade levels using data from other states including West Virginia (Howley, 1995) and Arkansas (Johnson, Howley, & Howley, 2002). This line of research ha s also forwarded the notion of an equity effect of size, showing that the ubiquitous poverty-achiev ement correlation is much stronger in larger schools and districts than in smaller schools and di stricts (e.g., Bickel & Howley, 2000; Friedkin & Necochea, 1998).
School size, achievement, and achievement gaps 4 Overall, school size appears to be related to a host of behavioral and academic outcomes for students, with smaller schools being associated with more positive outcomes in most cases. The research on size and achievement at the high school level appears to be somewhat of an exception, however, with multiple studies reaching different conclusions. In addition, both the Lee and Smith (1997) study and the series of studies by Howley and colleagues point toward the possibility that school size may be associated with different outco mes for students from different backgrounds. Many prior studies, however, have failed to cont rol for prior achievement, have not explored the possibility of differential effects fo r subgroups of students, and/or have not been able to analyze student-level variables in conjunction with school-l evel effects. These issues, in conjunction with the federally-driven focus on disaggregated achievement results and progress monitoring, call for further investigation of how the size-achievement relationship may operate among specific types of students. In an effort to better understand how school size relates to achievement among different subgroups of students across various grade levels, a study was undertaken to examine these relationships using data from the North Carolina public schools. North Carolina provides a particularly interesting venue to study this issue du e to the wide ranges in the size of schools across the state, a relatively high average school size (Figure 1), and the availability of longitudinal achievement test data for individual students from the states testing and accountability program. Two primary research questions were formulated to guide the overall study: 1. What are the relationships between school size and achievement at the elementary, middle and high school levels? 2. Do size-achievement relationships vary among students with differing levels of prior achievement, students of different ethnicities and students whose parents have different levels of education? 549 981 484 779 0 200 400 600 800 1000 1200 Elementary/MiddleSecondary1996-97 Enrollment North Carolina U. S. Average Figure 1. Average Enrollment in North Carolina and U. S. Public Schools, 1997-1998 Note. Elementary/middle schools are defined as a school in which the lowest grade is no higher than 6 and the highest grade is 8 or lower. Secondary schools are defined as schools in which the lowest grade is no lower than 7. Vocational schools, alternative schools, special education schools, and other schools not reported by grade level are excluded. From U.S. Dept. of Education, National Center for Education Statistics, Common Core of Data.
Education Policy Analysis Archives Vol. 12 No. 58 5 Method Student Achievement Data Data for the study were gathered from severa l databases maintained by the North Carolina Department of Public Instruction, including read ing and mathematics end-of-grade (EOG) testing databases and databases for the states High Sc hool Comprehensive Test (HSCT). EOG tests in reading and mathematics are administered each spring to most North Carolina public school students in grades 3-81. The HSCT is a test of reading and mathematics administered to 10th graders in the North Carolina public schools each spring These databases also contain a variety of demographic information as well as codes identifying the school attended by each tested student. Using these databases, three separate cohort samples were constructed. The elementary cohort consisted of all tested 3rd graders from the 1996-97 school year, the middle school cohort consisted of all tested 6th graders in 1997-98, and the high school cohort consisted of all tested students in grade 8 in 1997-98. Each students achievement data for that school year was then linked to their achievement test scores in the same subject areas two years later (Table 1). Students were included in the final samples only if they a) had available test data in at least one of the two subjects for both the baseline year and two years later; b) had made regular progress from grade to grade between the baseline year and two years later (i.e., were not retained and did not skip any grades); and c) attended the same school for both years following the baseline year. Table 1 Reading and mathematics test data used for elementary, middle, and high school cohorts Grade 3 4 5 6 7 8 9 10 Elementary Pre Post Middle Pre Post High Pre Post School-Level Data Data on school size was obtained from state student membership databases. The average daily membership for each school was used as an indicator of school size. Average daily membership is calculated as the number of studen ts officially listed on the daily roster of each school averaged across the entire school year. Sin ce the study covered a two-year span, the size data for each school was averaged across the two-year pe riod to produce a two-year average school size estimate. These data were then appended to the re cords in the cohort sample datafiles. In cases where schools either closed or where schools gained or lost large numbers of students between the two years due to consolidations, closings, or redistricting, data for those students were filtered out 1 During the years from which the data were drawn for this study, some students who were exempt from being tested based on limited English proficiency status and some special education students who were exempt based on recommendations in their Individualized Education Plans may not have been tested in one or more subject areas.
School size, achievement, and achievement gaps 6 prior to conducting the analysis. This resulted in the elimination of a very small number of cases (less than 2%) in each of the three analysis datasets. Data on the percentage of students eligible for free or reduced price lunch was also obtained for each school from extant state databases. Preliminary analyses prior to the calculation of the actual results included screening for univariate and various bivariate outliers and othe r unusual conditions in the data that may have adversely impacted results (e.g., test scores beyond the range of possible sc ores, duplicate testing records for the same students in the same year, etc.). These screening analyses resulted in the deletion of a very small number of individual record s due to anomalies that could not be reconciled. Characteristics of the students and schools in the final analysis samples are presented in Tables 2 and 3, respectively. Preliminary regression models also indicated some collinearity problems involving the continuous school size predictor. These condit ions were ameliorated by creating a four-level categorical school size variable (divided at the qua rtiles within each cohort grade level configuration) and using it as the indicator of size. Results gene rated using the four-level school size predictor were not substantively different from those using the c ontinuous predictor; therefore the results reported here are those using the four-level categorical school size variable. Table 2 Sample characteristics Students Elementary Middle High n % n % n % Gender Male 27,235 50 26,12849 30,161 49 Female 27,380 50 27,17851 28,625 51 Ethnicity Caucasian 37,808 69 37,12970 40,861 70 African-American 14,022 26 13,42125 14,874 25 Other 2,785 5 2,756 5 3,051 5 Parent(s) Highest Less than HS 4,609 8 3,642 7 3,349 6 Education Level HS Graduate 21,129 39 19,35036 15,469 26 Some College 4,719 9 4,886 9 4,787 8 2-Year Degree 7,955 15 8,81317 13,294 23 4-Year Degree 12,913 24 12,93824 15,141 26 Graduate Degree 3,290 6 3,677 7 6,746 11 Total 54,615 53,306 58,786 Note Student demographic data are from the samples in the Reading analysis. Because a small number of students took only one test or the other due to variou s exemptions, there are negligible differences between the reading analysis and mathematics analysis samples at each level. Percentages may not add to 100 due to rounding.
Education Policy Analysis Archives Vol. 12 No. 58 7 Table 3 Sample characteristics Schools Elementary (n = 1,053) Middle (n = 508) High (n = 333) Mean Range Mean Range Mean Range Number of students (2-year average) 506 26 1,392 570 21 1,508 859 27 2,352 % of students eligible for free/reduced price lunch 48 0 99.7 44 0 97.5 30 0 94.5 Covariates In all three sets of analyses, both student-level and school-level variables known to be correlate with student achievement were included as covariates in order to get a more precise estimate of the relationship between school size and achievement. The student-level covariates included gender, ethnicity, the highest level of education for the parent(s) in the home, and the students prior achievement status (at/above grade level or below) in the same subject2. The percentage of students in the school who were elig ible for free or reduced price lunch was used as a school-level covariate. In the high school mathemat ics analyses, indicators of whether each student had taken Algebra and Geometry were also used as covariates to help control for differential coursetaking experiences. Analysis Procedures Analyses of each of the th ree cohort datasets involved the estimation of two-level hierarchical linear models. This approach allows for the proper estimation of effects when units of analysis (e.g., students) are nested within a larger contextual unit (e.g., schools). Adjusting for this nesting allows for proper error estimation as well as the inclusion of both student-level and schoollevel predictor variables and their interactions in the models. Traditional least-squares regression methods in a multilevel context require either aggreg ating data to the school level prior to analysis, which results in a loss of statistical power and precisi on, or disaggregating school-level data down to the individual student level, which often results in spuriously significant results that show relationships between variables which may not truly exist (Hox, 1995). Hierarchical linear modeling methods avoid both of these problems by properly incorporating both school-level and student-level factors in the same analysis (Bryk & Raudenbush, 1992; Singer, 1998). In each case, initial null models were generated as a baseline, with predictors added one by one and level by level, to check for and ameliorate any unusual or problematic conditions in the data that may have hindered interpretation of the results. With respect to th e overall models, prior achievement and the other student-level covariates accounted for a notable portion of the explainable between-school and 2 Dichotomized versions of the ethnicity and parent education level variables were employed in lieu of their original forms because of the uneven dist ribution of the ethnicity variable in the sample and because of reliability concerns with parent education level data that are reported by the classroom teacher.
School size, achievement, and achievement gaps 8 within-school variation in achievement in many of the models. These indicators of explained variance are not analogus to a squared multiple corre lation in linear regression, however, and should not be interpreted as such. They are merely rela tive indicators of the proportion of school-level variation and student-level variation that is expl ained by the variables in the model (Snijders & Bosker, 1994). Given prior studies in this area which have found a curvilinear relationship between size and achievement (e.g., Lee & Smith, 1997), all mode ls were initially estimated with both linear and nonlinear terms for school size. However, in every case, better model fit was achieved using only the linear term, indicating that these data did not support a curvilinear relationship between size and achievement. The nonlinear terms were th erefore omitted from the final models. Results Elementary Cohort For the elementary cohort, the reading achievement analyses yielded no statistically significant relationship between school size and ac hievement after controlling for school and student demographic characteristics (Table 4). As was expected, higher 3rd grade achievement scores, female gender, White ethnicity, and higher levels of parent education were all associated with higher EOG scores at the end of grade 5. Attending a school with a lower percentage of students eligible for free or reduced price lunch was also associated with higher achievement (Table 4). The size-prior achievement interaction implied that there was a nega tive size-achievement relationship, but that it was predominantly seen among students who were below grade level in 3rd grade (Figure 2). The size of the effect, however, was rather small (.06 SD3). No statistically significant interactions were found between size and ethnicity or size and pare nt education level; therefore, those terms were dropped from the final model. In the mathematics analyses, there was no significant main effect for size. There was, however, another significant size-prior achievement interaction (Table 5). This interaction indicated that students who were below grade level in mathematics based on their 3rd grade scores scored better in grade 5 if they attended smaller schools, whereas the pattern for students who scored above grade level in grade 3 was more uneven (Figure 3). The actual magnitude of this interaction, as in the reading model, was rather small (.09 SD). The pattern of relationships for the student and school-level covariates in the mathematics model mirrored that of the reading model, with the exception of the gender variable, which was not si gnificant and subsequently dropped from the final model (Table 5). 3 The expression of effect sizes in standard deviati on units in these analyses represents the difference in scale score gaps between students in the smalle st school size quartile and the largest. The standard deviation estimate used for these calculations is the statewide standard deviation for each test (North Carolina Department of Public Instruction, 2000; 2001).
Education Policy Analysis Archives Vol. 12 No. 58 9 Table 4 Elementary cohort 2-level HLM regression model reading Level Variable b SE t F Student-level Prior Reading Achievement 3rd Grade10.13.11 89.41* Gendera -.49.05 -8.99* Ethnicityb 2.33.07 34.10* Parent Education Levelc 2.83.06 48.91* School-level School Sized 2.18 Less than 400 students .05.17 .26 400-549 students .12.16 .74 550-699 students -.05.16 -.28 Free/Reduced Price Lunch (% eligible)-.02.003-5.89* Interactions Size x Prior Achievement 4.70* Summary % between-student variation explained 44.8% % between-school variation explained 66.2% a 0 = female, 1 = male. b 0 = non-White, 1 = White. c 0 = high school diploma or less, 1 = at least some post-secondary education. d Reference group for school size is 700+ students. p < .05. 148.1 147.9147.8 147.5 157.7 157.7 157.6157.6130 140 150 160 170 Less than 400400-549550-699700 or more School SizeAdjusted 5th Grade EOG Reading Scale Score Below grade level in Grade 3 At/above grade level in Grade 3
School size, achievement, and achievement gaps 10 Figure 2. Elementary cohort reading model interaction between prior achievement and school size. Table 5 Elementary cohort 2-level HLM regression model mathematics Level Variable b SE t F Student-level Prior Mathematics Achievement 3rd Grade11.70.18 64.65* Ethnicitya 2.91.09 32.61* Parent Education Levelb 3.63.07 48.92* School-level School Sizec .51 Less than 400 students -.26.27 -.96 400-549 students -.09.25 -.36 550-699 students .18.26 .70 Free/Reduced Price Lunch (% eligible) -.01.004-3.44* Interactions Size x Prior Achievement 5.60* Summary % between-student variation explained 41.5% % between-school variation explained 44.2% a 0 = non-White, 1 = White. b 0 = high school diploma or less, 1 = at least some post-secondary education. c Reference group for school size is 700+ students. p < .05. 152.1 152.1 151.9 151.5 162.9 163.1 163.4 163.2130 140 150 160 170 Less than 400400-549550-699700 or more School SizeAdjusted 5th Grade EOG Mathematics Scale Score Below grade level in Grade 3 At/above grade level in Grade 3
Education Policy Analysis Archives Vol. 12 No. 58 11 Figure 3. Elementary cohort mathematics mo del interaction between prior achievement and school size. Middle School Cohort Similar to the elementary results, the reading achievement analyses for middle school students also yielded no overall relationship b etween school size and achievement after controlling for various school and student demographic characteristics (Tables 6 & 7)4. The pattern of relationships found for the covariates was also iden tical to the elementary cohort analyses with one exception: Male students in the middle school cohort demonstrated slightly higher achievement in mathematics than their female counterparts, which was not the case in the elementary mathematics analysis. As was true at the elementary level, the middle school models also yielded significant interactions between prior achievement and school size for both reading and mathematics. Students who were scoring on grade level in 6th grade tended to do slightly better in larger middle schools over the next two years, whereas students who were below grade level in 6th grade did slightly better in smaller schools (Figures 4 & 5). Although larg er in comparison to the elementary results, the interactions again were not overwh elming (.12 SD for reading and .13 SD for mathematics). Interactions between school size and ethnicity and school size and parent education level were nonsignificant in both the reading and mathem atics models and those terms were therefore dropped. Table 6 Middle school cohort 2-level HL M regression model reading Level Variable b SE t F Student-level Prior Reading Achievement 6th Grade10.04.18 57.01* Gendera -.75.05 -14.65* Ethnicityb 2.22.07 30.33* Parent Education Levelc 2.72.07 41.72* School-level School Sized 2.25 Less than 400 students -.45.21 -2.14* 400-549 students -.35.20 -1.79 550-699 students -.46.20 -2.30* Free/Reduced Price Lunch (% eligible) -.01.003-4.28* Interactions Size x Prior Achievement 5.68* Summary % between-student variation explained 44.8% 4 Although the coefficients associated with the dummy-cod ed size variable did indi cate that achievement was slightly higher in the largest size category compared to some smaller categories using a standard p level of .05, the overall F test was non-significan t. Also, since these t tests largely amount to non-orthogonal a posteori contrasts, a familywise .05 error rate per model is a pr eferable standard (Kirk, 1995) Using this standard of .0167 (.05 divided by 3), none of the specific size contrasts in the middl e school models would have reached significance.
School size, achievement, and achievement gaps 12 % between-school variation explained 63.1% a 0 = female, 1 = male. b 0 = non-White, 1 = White. c 0 = high school diploma or less, 1 = at least some post-secondary education. d Reference group for school size is 700+ students. *p < .05. 156.7 156.4 156.0 156.2 165.8 165.9 165.8 166.3140 150 160 170 180 Less than 400400-549550-699700 or more School SizeAdjusted 8th Grade EOG Reading Scale Score Below grade level in Grade 6 At/above grade level in Grade 6 Figure 4. Middle school cohort reading mode l interaction between prior achievement and school size. Table 7 Middle school cohort 2-level HLM regression model mathematics Level Variable b SE t F Student-level Prior Mathematics Achievement 6th Grade15.09.30 -50.75* Gendera .15.08 1.98* Ethnicityb 4.16.12 35.15* Parent Education Levelc 4.77.12 38.87* School-level School Sized 1.01 Less than 400 students -.91.39 -2.34* 400-549 students -.69.37 1.86 550-699 students -.89.38 -2.33 Free/Reduced Price Lunch (% eligible) -.02.0063.27* Interactions Size x Prior Achievement 6.58* Summary % between-student variation explained 36.7%
Education Policy Analysis Archives Vol. 12 No. 58 13 % between-school variation explained 50.1% a 0 = female, 1 = male. b 0 = non-White, 1 = White. c 0 = high school diploma or less, 1 = at least some post-secondary education. d Reference group for school size is 700+ students. p < .05.
School size, achievement, and achievement gaps 14 165.2 165.4 164.6 164.5 178.7 178.9 178.7 179.6150 160 170 180 190 Less than 400400-549550-699700 or more School SizeAdjusted 8th Grade EOG Mathematics Scale Score Below grade level in Grade 6 At/above grade level in Grade 6 Figure 5. Middle school cohort mathematics model interaction between prior achievement and school size. High School Cohort The high school analyses yielded the largest number of relationships between school size and achievement as well as the largest relationships in terms of effect size. In the reading model, there was a significant and positive main effect for size, along with statistically significant interactions involving size and ethnicity and size and parent ed ucation level (Table 8). Taken together, these relationships implied that while students overall pe rformed better in Reading in larger high schools, the benefits accrued more strongly to White studen ts and students whose parents had at least some post-secondary education. Non-White students and students whose parents had a high school education or less showed a more U-shaped patte rn of performance, with scores being roughly equal in the smallest and largest schools (Figure 6) The size of these interaction effects were as large or larger than any of the interactions found at the middle and elementary levels (.12 SD for the size-parent education level interaction, and .20 SD for the size-ethnicity interaction). The pattern of relationships for the other student and school-level covariates mirrored that of the reading models in the elementary and middle school analyses. As in the high school reading model, the high school mathematics model also yielded a positive main effect for school size. Significant in teractions were also found between size and prior achievement, size and ethnicity, and size and parent education level (Table 9). These interactions, as in previous analyses, again indicated that the benefit of larger school size again accrued disproportionately to students whose prior ac hievement was higher (.28 SD; Figure 8), White students (.10 SD; Figure 9), and students whose parents had at least some education beyond high school (.11 SD; Figure 10). The size-prior achi evement interaction in the high school mathematics model was the largest found in the study. Relationships for other student and school-level covariates were similar to those found in the elementary and middle school models. It should also be noted that indicators representing students exposure to algebra and geometry courses through
Education Policy Analysis Archives Vol. 12 No. 58 15 grade 10 were also available in the extant database to be used as covariates in the high school mathematics model, thereby controlling for course-t aking factors that were not measurable at the elementary and middle school levels. Students who had taken these courses, as expected, demonstrated higher achievement. Table 8 High school cohort 2-level HLM regression model reading Level Variable b SE t F Student-level Prior Reading Achievement 8th Grade11.35.12 92.49* Gendera -1.89.08 -23.62* Ethnicityb 4.23.30 14.17* Parent Education Levelc 3.90.27 14.33 School-level School Sized 5.78* Less than 700 students -2.58.37 -6.96* 700-1,199 students -1.90.33 -5.72* 1,200-1,699 students -.91.36 -2.52* Free/Reduced Price Lunch (% eligible) -.01.005-2.09* Interactions Size x Ethnicity 10.18* Size x Parent Education Level 7.04* Summary % between-student variation explained 32.5% % between-school variation explained 66.7% a 0 = female, 1 = male. b 0 = non-White, 1 = White. c 0 = high school diploma or less, 1 = at least some post-secondary education. d Reference group for school size is 1,700+ students. *p < .05. 162.8 162.5 162.7 163.2 165.5 165.7 166.4 167.1140 150 160 170 180 Less than 700700-11991200-16991700 or more School SizeAdjusted 10th Grade Reading Scale Score HS or less At least some college Figure 6. High school cohort reading mode l interaction between parent education level and school size.
School size, achievement, and achievement gaps 16 163.0 162.5 162.6 163.0 165.3 165.7 166.5 167.3140 150 160 170 180 Less than 700700-11991200-16991700 or more School SizeAdjusted 10th Grade Reading Scale Score Non-White White Figure 7. High school cohort reading model graphic representation of interaction between ethnicity and school size. (Note. School size groups are divided based on quartile cutoffs for purpose of illustration. Grade 8 achievement grou ps are based on a dichotomization of the scale score variable used in the HLM model that corresponds to the official cut point used by the state to determine whether a student is performing at or above grade level.) Table 9 High school cohort 2-level HLM regression model mathematics Level Variable b SE t F Student-level Prior Mathematics Achievement 8th Grade12.55.56 -22.31* Ethnicitya 4.99.41 12.27* Parent Education Levelb 4.09.37 11.06* Completed Algebra I Course 1.27.11 11.50* Completed Geometry Co urse 8.59.19 45.66* School-level School Sizec 5.90* Less than 700 students -5.14.63 -8.16* 700-1,199 students -3.93.57 -6.90* 1,200-1,699 students -2.64.62 -4.28* Free/Reduced Price Lunch (% eligible) -.03.009-3.89* Interactions Size x Prior Achievement 10.66* Size x Ethnicity 2.72* Size x Parent Education Level 3.75* Summary % between-student variation explained 39.6% % between-school variation explained 39.6% a 0 = non-White, 1 = White. b 0 = high school diploma or less, 1 = at least some post-secondary education. c Reference group for school size is 1,700+ students. p < .05.
Education Policy Analysis Archives Vol. 12 No. 58 17 168.9 168.2 168.5 168.7 177.6 178.2 179.1 181.3150 160 170 180 190 Less than 700700-11991200-16991700 or more School SizeAdjusted 10th Grade Mathematics Scale Score Below grade level in Grade 8 At/above grade level in Grade 8 Figure 8. High school cohort mathematics model interaction between prior achievement and school size. 171.4 171.1 171.5 172.5 175.0 175.3 176.2 177.5150 160 170 180 190 Less than 700700-11991200-16991700 or more School SizeAdjusted 10th Grade Mathematics Scale Score Non-White White Figure 9. High school cohort mathematics model interaction between ethnicity and school size. 171.9 171.6 172.1 173.0 174.5 174.8 175.5 177.1150 160 170 180 190 Less than 700700-11991200-16991700 or more School SizeAdjusted 10th Grade Mathematics Scale Score HS or Less At Least Some College Figure 10. High school cohort mathematic s model interaction between parent education level and school size.
School size, achievement, and achievement gaps 18 Discussion According to prior research on school size and its relationship to student achievement and behavior, the majority of studies indicate that smaller is better. There are some inconsistencies with respect to high school size and achievement, but stud ies of school size in general have demonstrated that smaller schools are associated with better behavi oral outcomes, higher rates of participation in extracurricular activities, and higher achievement. In addition, many of these studies have been conducted with large, nationally representative sa mples of students and schools, which would imply that those results should be fairly robust and app licable to a wide range of educational situations. Analyses of North Carolina data, however, show a more complex pattern of results. At the elementary and middle school levels, school size was related to achievement but only through interactions with students prior level of achievem ent. Students who were scoring on grade level in reading and mathematics in the baseline year tended to score higher two years later if they attended larger schools, whereas students who were sc oring below grade level in the baseline year demonstrated slightly lower performance two years later if they attended larger schools. These effects were somewhat stronger in middle school th an in elementary school. At the high school level, size was positively related to both reading and mathematics achievement in the overall sample. The benefits of size at the high school level, ho wever, appeared to accrue disproportionately (or in some cases entirely) to higher-achieving studen ts, White students, and students whose parents had more education, especially in mathematics. Eff ects seen in the high sc hool cohort were the largest in the study. Although the nature of the inter actions involving school size in the current study differed by grade level and in some cases were small in magnitude, in each case the interaction implied that learning was less equitable in larger schools. The results of this study provide interesting parallels to previous studies suggest ing that student and community characteristics interact with size (e.g., Howley & Bickel, 1999; Lee & Smith, 1997). While the Lee and Smith study attributed greater achievement disparities in larger schools to the relatively low performance of lessadvantaged students in those environments, the current study raises the possibility that these disparities may in some cases be due to the relati vely high performance of more-advantaged students in larger schools. Correspondingly, while the lin e of research by Howley and colleagues posits that the poverty-achievement relationship is larger among larger schools, the current study suggests that oft-documented achievement gaps between student sub groups may also larger within larger schools. This suggests that the school-level equity effects of size identified by Howley and colleagues may also translate down to student subgroups within schools. Thus, although the same basic achievement ga ps are identified across different studies of school size, the possible underlying explanations of these results and their implications could be very different. For example, the observed results may be a function of higher-achieving students in larger schools taking disproportionate advantage of broa der and deeper curriculum offerings. The stratification and tracking arrangements that this explanation would suggest may be more easily fostered in larger schools (Haller, Monk, Spotted Bear, Griffith, & Moss, 1990; Monk, 1987). The likelihood of this explanation is further bolstered by the fact that the largest effect in the current study was seen in high school mathematics, wher e stratification and tracking are particularly prevalent (Gamoran & Hannigan, 2000; Haller et al., 1990; Oakes, Gamoran & Page, 1992). If so, interventions that attempt to raise the level of ri gor and breadth of curricula in smaller schools may be warranted (Barker, 1985), or perhaps interve ntions targeted at promoting greater access to accelerated curricula for historically under-represented groups. Technology applications that allow higher-level offerings such as Advanced Placement courses to be taken via the internet in smaller, more remote schools (or for that matter by larg er numbers of students in any school) might be
Education Policy Analysis Archives Vol. 12 No. 58 19 beneficial in this respect, as would programs targ eted at better identifying and serving gifted and talented students from more diverse background s (Darity, Castellino, Tyson, Cobb, & McMillen, 2001). It is also possible that academically-challenged students perform better in smaller schools because of factors related to the school culture an d environment. If so, large schools might take advantage of organizational structures such as those discussed by Cawelti (1993) and Goodlad (1984) in order to create a small-school atmos phere within a large school. These may include vertical house plans (i.e. schools-within-schools) whic h essentially divide a large school into multiple smaller schools on the same campus, each of which operates with its own group of students and with relative autonomy, or special focused curriculu m tracks within high schools that could serve as within-school magnet programs to circumvent th e enormity of a large school. These approaches assume factors such as the social climate, the pe rsonal relationships between students and teachers, and the extent to which students can become engaged and invested in the schooling experience are the true catalysts of positive outcomes in sma ll schools. A recent study by Darling-Hammond, Ancess, & Ort (2002) in New York documents a case where this kind of reorganization strategy was applied to a large urban high school and resulted in improved student outcomes. A five-year evaluation of the Bill and Melinda Gates Foundation program which funds the creation of smaller high schools is also currently underway (American In stitutes for Research, 2003) and should help to inform these issues as well. It would seem, then that the implications of the findings from quantitative studies of the size-achievement relati onship may depend on which interpretations of the identified interactions are found to be most plausible. Given the current movement toward closing achievement gaps and getting student subgroups to meet criterion-based academic standards, which has been reinforced by the recent passage of the federal No Child Left Behind Act, proper delineation of the mechanisms underlying these relationships is critical for designing effective interventions for students who are at risk of not meeting those standards. Further studies of how school size is related to the day-to-day activi ties of students and teachers may provide greater insight into this issue. In any organization, structural factors such as size tend to have their effects on outcomes indi rectly by altering the day-to-day processes and interactions that occur within the organization. Therefore, studies looking for a direct link between school size and student outcomes that fail to includ e these process factors in the analysis may reach different conclusions about the true role of school size in students growth and development. Some studies have suggested that factors such as more personal social relations (e.g., teacher-to-student, student-to-student, etc.), stronger internal acco untability, and opportunities for more varied approaches to instruction and asssessment may play a mediating role (Darling-Hammond et al., 2002; Lee, Smerdon, Alfeld-Liro, & Brown, 2000; Wasley et al., 2000) Further examination of how these relationships may play out differently for di fferent subject areas or across grade levels would also be important, as the current study as well as some prior investigations both imply that school size effects may be more common and relatively la rger in mathematics and at higher grade levels (Howley, 1995; Lee and Smith, 1997; Johns on, Howley, & Howley, 2002). Whether these differential relationships are a function of cumulative developmental effects that are most easily seen in later grades or perhaps qualitative differences in the size-achievement dynamic across levels and subjects is largely unknown at this point. The findings reported here, along with those of prior school size-achievement studies should also lead local boards of education and other policymakers to at least consider whether efforts to consolidate smaller schools into larger ones might be achieving desired efficiencies at some cost to at-risk student groups. When considering only the financial ramifications, larger schools tend to be less expensive to operate, on a per-pupil basis, oth er things being equal (McGuire, 1993, p. 171). Unfortunately, these other things are rarely e qual, and financial savings from consolidation will
School size, achievement, and achievement gaps 20 probably not apply equally across all expenditure ar eas. For example, the consolidation of two schools may save personnel expenses by eliminating a principals position, but it may simultaneously result in an increase in pupil transportation costs. The consolidation of smaller schools into larger units also may or may not result in cost savi ngs depending on how one defines the outputs of schooling (Lawrence et al., 2002). For example, St iefel, Berne, Iatarola, & Fruchter (2000) have shown that smaller high schools, although they may not enroll as many students per dollar as larger schools, may be producing more graduates per dollar Therefore, smaller schools may also be more economically efficient if the output is defined as graduates instead of enrollees while also possibly providing more supportive environments for at-risk students. In attempting to interpret the results of the current and previous studies of the sizeachievement relationship, it has to be acknowledged that size is inextricably intertwined with many other factors that are often associated with acad emic and behavioral outcomes for students. These complexities are further underscored by the findin g in this study that the size-prior achievement interactions were larger and more prevalent than the size-parent education level or size-ethnicity interactions, despite the fact that prior achievement is typically highly correlated with these variables. This overlap, coupled with the fact that school size is typically not manipulated experimentally for research purposes, makes it very difficult to identify which specific factors or combinations thereof might be the most salient. Although including some of these confounding variables in the analysis for control purposes is helpful, it is not a subst itute for random assignment. The 2,200-plus public schools in North Carolina show great variability on a number of these potential factors (e.g., urban/rural location, family/community characteristi cs, student demographics, poverty, etc.), with a good number of schools identified at each point on those spectra. Compared to other states, North Carolina consistently falls at or near the national median on the vast majority of these types of school characteristics (U. S. Department of Educ ation, 2003). The exten t to which the results obtained here could be applied to other geographi c settings, however, may be influenced by the extent to which those other settings mimic that profile. Analyses of similar data from other states or locales that are more homogeneous on some of these factors may in fact yield different results. Howley and Howley (2004) would imply that the generalizability of the results of the current study may also be limited by the fact that there ar e only a handful of truly small high schools (i.e., less than 100 students) in North Carolina compared to states that have been examined in prior research (e.g., Howley & Bickel, 1999), a phenomenon th ey refer to as size bias. The analyses here do not speak to this issue; however, Lee (2004) argues that it is unclear as to whether or how a small number of small schools would result in actual bias of parameter estimates. With respect to the practical significance of the findings, the elemen tary and middle school effects, while statistically significant, are small and therefore should be interpreted cautiously. It is unclear whether the effects at these levels are smaller than at the high school le vel because size is actually less important at those grades. This discrepancy in size of effects may likely be due to statistical factors such as less actual variance in school size at the elementary and middle levels compared to the high school level. Given the findings here, along with previous studies indicating that the achievement gap exists prior to children entering the K-12 system (e.g., Phillips, Brooks-Gunn, Duncan, Klebanov, & Crane, 1998; U. S Department of Education, 2000b), it is unlikely that school size is the primary force behind the well-documented achievement disparities between various student subgroups. Its relationship to achievement and achievement gaps may, however, be mediated by other processrelated factors that are either encouraged or stifled by school size. Delineating the specific mechanisms through which school size affects student outcomes and the ways in which those effects might be selectively experienced by differe nt student subgroups or in different subject areas is a potentially rich area of investigation that may help us to better understand how schools need to be structured so that all students can reach high standards.
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School size, achievement, and achievement gaps 24 Phillips, M., Brooks-Gunn, J., Duncan, G. J., Klebanov P., & Crane, J. (199 8). Family background, parenting practices, and the black-white test scor e gap. In C. Jencks & M. Phillips (Eds.), The black-white test score gap (pp. 103-145). Washington, DC: Brookings. Schoggen, P., & Schoggen, M. (1988). Student voluntary participation and high school size. Journal of Educational Research, 81 288-293. Snijders, T. A., & Bosker, R. J. (1994) Modeled variance in two-level models. Sociological Methods and Research, 22, 342-363. Stiefel, L., Berne, R., Iatarola, P. & Fruchter, N. (2000). High school size: Effects on budgets and performance in New York City. Educational Evaluation and Policy Analysis, 22, 27-39. U. S. Department of Education. (2000a). Digest of education statistics: 2000. Washington, DC: U. S. Government Printing Office. U. S. Department of Education. (2000b). Americas kindergarteners (NCES #2000-070). Washington, DC: Author. U. S. Department of Education. (2003). Overview of public elementary and secondary schools and districts: School year 2001-02 (NCES # 2003-411). Washington, DC: Author. Walberg, H. J., & Walberg III, H. J. (1994). Losing local control. Educational Researcher, 23 (5), 19-26. Wasley, P. A., Fine, M., Gladden, M., Holland, N. E., King, S. P., Mosak, E., & Powell, L. C. (2000). Small schools, great strides. New York: Bank Street College of Education. Retrieved on February 12, 2001 from http://www.bankstreet.edu/gems/publications/smallschoollow.pdf. About the Author Bradley J. McMillen Senior Evaluation Consultant North Carolina Department of Public Instruction Division of Accountabilty Services 6314 Mail Service Center Raleigh, NC 27699-6314 Phone: (919) 807-3808 Fax: (919) 807-3772 Email: firstname.lastname@example.org Brad McMillen conducts policy and program evaluation studies for the North Carolina Department of Public Instruction. His research interests include school organizational characteristics and students at risk for academic failure.
Education Policy Analysis Archives http:// epaa.asu.edu Editor: Gene V Glass, Arizona State University Production Assistant: Chris Mu rrell, Arizona State University General questions about appropriateness of topics or particular articles may be addressed to the Editor, Gene V Glass, gl email@example.com or reach him at College of Education, Arizona State University, Tempe, AZ 85287-2411. The Commentary Editor is Casey D. Cobb: firstname.lastname@example.org. EPAA Editorial Board Michael W. Apple University of Wisconsin David C. Berliner Arizona State University Greg Camilli Rutgers University Linda Darling-Hammond Stanford University Sherman Dorn University of South Florida Mark E. Fetler California Commission on Teacher Credentialing Gustavo E. Fischman Arizona State Univeristy Richard Garlikov Birmingham, Alabama Thomas F. Green Syracuse University Aimee Howley Ohio University Craig B. Howley Appalachia Educational Laboratory William Hunter University of Ontario Institute of Technology Patricia Fey Jarvis Seattle, Washington Daniel Kalls Ume University Benjamin Levin University of Manitoba Thomas Mauhs-Pugh Green Mountain College Les McLean University of Toronto Heinrich Mintrop University of California, Berkeley Michele Moses Arizona State University Gary Orfield Harvard University Anthony G. Rud Jr. Purdue University Jay Paredes Scribner University of Missouri Michael Scriven University of Auckland Lorrie A. Shepard University of Colorado, Boulder Robert E. Stake University of IllinoisUC Kevin Welner University of Colorado, Boulder Terrence G. Wiley Arizona State University John Willinsky University of British Columbia
Archivos Analticos de Polticas Educativas Associate Editors Gustavo E. Fischman & Pablo Gentili Arizona State University & Universidade do Estado do Rio de Janeiro Founding Associate Editor for Spanish Language (1998) Roberto Rodrguez Gmez Editorial Board Hugo Aboites Universidad Autnoma Metropolitana-Xochimilco Adrin Acosta Universidad de Guadalajara Mxico Claudio Almonacid Avila Universidad Metropolitana de Ciencias de la Educacin, Chile Dalila Andrade de Oliveira Universidade Federal de Minas Gerais, Belo Horizonte, Brasil Alejandra Birgin Ministerio de Educacin, Argentina Teresa Bracho Centro de Investigacin y Docencia Econmica-CIDE Alejandro Canales Universidad Nacional Autnoma de Mxico Ursula Casanova Arizona State University, Tempe, Arizona Sigfredo Chiroque Instituto de Pedagoga Popular, Per Erwin Epstein Loyola University, Chicago, Illinois Mariano Fernndez Enguita Universidad de Salamanca. Espaa Gaudncio Frigotto Universidade Estadual do Rio de Janeiro, Brasil Rollin Kent Universidad Autnoma de Puebla. Puebla, Mxico Walter Kohan Universidade Estadual do Rio de Janeiro, Brasil Roberto Leher Universidade Estadual do Rio de Janeiro, Brasil Daniel C. Levy University at Albany, SUNY, Albany, New York Nilma Limo Gomes Universidade Federal de Minas Gerais, Belo Horizonte Pia Lindquist Wong California State University, Sacramento, California Mara Loreto Egaa Programa Interdisciplinario de Investigacin en Educacin, Chile Mariano Narodowski Universidad Torcuato Di Tella, Argentina Iolanda de Oliveira Universidade Federal Fluminense, Brasil Grover Pango Foro Latinoamericano de Polticas Educativas, Per Vanilda Paiva Universidade Estadual do Rio de Janeiro, Brasil Miguel Pereira Catedratico Universidad de Granada, Espaa Angel Ignacio Prez Gmez Universidad de Mlaga Mnica Pini Universidad Nacional de San Martin, Argentina Romualdo Portella do Oliveira Universidade de So Paulo Diana Rhoten Social Science Research Council, New York, New York Jos Gimeno Sacristn Universidad de Valencia, Espaa Daniel Schugurensky Ontario Institute for Studies in Education, Canada Susan Street Centro de Investigaciones y Estudios Superiores en Antropologia Social Occidente, Guadalajara, Mxico Nelly P. Stromquist University of Southern California, Los Angeles, California Daniel Suarez Laboratorio de Politicas Publicas-Universidad de Buenos Aires, Argentina Antonio Teodoro Universidade Lusfona Lisboa, Carlos A. Torres University of California, Los Angeles Jurjo Torres Santom Universidad de la Corua, Espaa Lilian do Valle Universidade Estadual do Rio de Janeiro, Brasil