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PAGE 1 Readers are free to copy display, and distribute this article, as long as the work is attributed to the author(s) and Education Policy Analysis Archives, it is distributed for noncommercial purposes only, and no alte ration or transformation is made in the work. More details of this Creative Commons license are available at http://creativecommons.org/licen ses/byncnd/2.5/. All other uses must be approved by the author(s) or EPAA EPAA is published jointly by the Mary Lou Fulton College of Education at Arizona State Universi ty and the College of Educ ation at the University of South Florida. Articles are indexed by H.W. Wilson & Co. Please contribute commentary at http://epaa.info/wordpress/ and send errata notes to Sherman Dorn (epaaeditor@shermandorn.com). EDUCATION POLICY ANALYSIS ARCHIVES A peerreviewed sc holarly journal Editor: Sherman Dorn College of Education University of South Florida Volume 14 Number 28 Novemb er 3, 2006 I SSN 1068 School Size, Student Achievement, and the Power Rating of Poverty: Substantive Finding or Statistical Artifact? Theodore Coladarci University of Maine Citation: Coladarci, T. (2006). School size, st udent achievement, and the power rating of poverty: Substantive finding or statistical artifact?. Education Policy Analysis Archives, 14 (28). Retrieved [date] from h ttp://epaa.asu.edu/epaa/v14n28/. Abstract The proportion of variance in student ac hievement that is explained by student SESpovertys power rating, as some ca ll ittends to be lower among smaller schools than among larger schools. Smaller schools, many claim, are able to somehow disrupt the seemingly axiomatic association between SES and student achievement. Using eighthgrade data for 216 public schools in Maine, I explored the hypothesis that this in part is a statisti cal artifact of the greater volatility (lower reliability) of schoolaggregated studen t achievement in smaller schools. This hypothesis received no support when readin g achievement served as the dependent variable. In contrast, the hypothesis wa s supported when the dependent variable was mathematics achievement. For reason s considered in the discussion, however, I ultimately concluded that the latter result s are insufficient to a ffirm the statisticalartifact hypothesis here as well. Implications for subsequent research are discussed. Keywords: small schools; student achievement; socioeconomic status;, rural education. PAGE 2 Education Policy Analysis Archives Vol. 14 No. 28 2 Tamao de la escuela, logro acadmico de los estudiantes, y la potencia nominal de la pobreza: Hallazgos sustantivos o artificio estadstico? Resumen La proporcin de varianza en el logro acad mico de los estudiantes que se atribuye al estatus socio econmico (SES)llamado por algunos la potencia nominal de la pobrezatiende a ser menor entre escuelas pequeas que entre escuelas grandes. Las escuelas ms pequeas, segn dicen muchos, logran de alguna manera, romper la aparente asociacin ax iomtica entre el SES y el logro acadmico de los estudiantes. Usando datos de octavo grad o de 216 escuelas pblicas del estado de Maine, en este artculo exploro la hiptes is que esto es, en parte, un artificio estadstico que se da debido a la mayor volatilidad (m enor confianza) del logro acadmico de los estudiantes cuando se toma en cuenta el valor agregado de toda la escuela en las escuelas pequeas. Esta hi ptesis no se sustenta cuando el logro acadmico en lectura es la va riable dependiente. Por el contrario, la hiptesis s se sustenta cuando la variable dependiente es el logro acadmico en matemticas. Sin embargo, por las razones argidas en la di scusin, se concluye qu e los resultados en matemticas tampoco son suficientes para confirmar la hip tesis de artefacto estadstico. Al final se incluyen sugerenc ias para continuar la investigacin sobre este tema. Introduction As every student of education research knows, the relationship between student achievement and socioeconomic status (SES) is wellestablished in the empirical literature: All things equal, as student SES increases, so does student achievem ent (e.g., Sirin, 2005; White, 1982). Further, this holds regardless of the unit of analysis employed (e.g., student, school, multilevel). The seemingly axiomatic nature of this relationship notwithstanding a recurring finding in rural education research is that SES and school size interact in affecting student achievement (e.g., Howley, 1996; Howley & Bickel, 1999; Huang & Howley, 1993; Johnson, Howl ey, & Howley, 2002; McMillen, 2004; also see Friedkin & Necochea, 1988; Lee & Smith, 1997). In other words, the magnitude of the relationship between SES and achievement depends on the size of the school, or, equivalently, that the magnitude of the relationship between school size and achievement depends on the SES makeup of the school. How is such an interaction demonstrated? With th e school as the unit of statistical analysis, for example, interaction is shown by regressi ng achievement on SES, school size, and the mathematical product of SES and school size, and then testing the product term for statistical significance. If the slope associated with this term is statistically significantwhich researchers have been reporting with remarkable consistencythere is an interaction between SES and school size. A common way to illustrate such an interaction is to show that the schoollevel correlation between SES and achievement is weaker among smaller sc hools than among larger schools. That is, SES explains less of the variance in school achievement among smaller sc hools than it does among larger schools. As Huang and Howley (1993) put it, smaller schools mitigate the effect that SES has on student achievement. PAGE 3 School Size, Student Achievement, and the Power Rating of Poverty 3 The mitigatingeffect finding enjoys considerable fanfare by researchers, advocacy groups, and practitioners alike. Johnson, Howley, and Howley (2002), for example, declared this finding to be among the most consistent ever to be reporte d in educational research (pp. 36). The Rural School and Community Trust, a strong advocate fo r rural schools and communities, crafted the phrase povertys power rating to refer to the pe rcentage of variance in achievement that is explained by SES (i.e., the familiar coefficient of determination). In newsletters and press releases, the Rural Trust celebrates the recurring finding that the power rating of poverty is markedly lower sometimes negligibleamong smaller schools than among larger schools. In study after study, the organizations president recently announced, s mall schools have been shown to cut povertys power over student achievement (Tompkins, 2006). And in an oped published in my local newspaper, a school superintendent and his colleagues summed it up this way: Small schools are an antidote to the impact of poverty on school achi evement (Butler et al., 2005, p. A9). Despite my affinity to rural schools and communities, I have always been uneasy with the mitigatingeffect finding and, in particular, the markedly lower power rating of poverty in smaller schools. As much as I am attracted to the notion th at smaller schools, by virtue of their smallness, are somehow able to disrupt the achievement disadvantage of lowerSES students, and as much as I can imagine the many ways in which smaller schools might be able to pull this off (although hard data would be helpful), my immediate suspicion wa s that the diluted SESachievement correlation among smaller schools may have little to do with the educational experience characterizing such schools. Rather, I suspected a statistical artifact at play. Loosely defined, a statistical artifact is where a research result is misleading because of an artificial or extraneous effect due to stat istical considerations. For example, if X has modest variance and, further, r = 0 between X and Y the absence of relationship between X and Y very well could be due to restricted range in X (a statistical artifact) rather than to an absence of relationship between the two constructs underlying X and Y In the present context, the putatively ameliorative role of smaller schools in the SESachievement rela tionship would be a statistical artifact if, say, there were much less variability in either student SES or student achievement among smaller schools than among larger schools. This in fact was my immediate suspicion, both because it is so obvious as a plausible rival hypothesis (when subgroup correlations are comparatively small) and because I saw no acknowledgment of this possibility by those who were doing (or celebrating) the research. However, I was unable to find evidence of restri cted variance in the statistics reported by the researchers, nor did such evidence surface in my qu ick reanalysis of Maine data that had been featured in a newsletter of the Rural Trust (Maines small schools cut povertys power, 2005). My interest in the challenges that small school s face related to the adequate yearly progress requirement of No Child Left Behind suggested another possible statistical artifact: the greater volatility of schoollevel student achievement among smaller schools (Coladarci, 2003). School achievement can differ widely from one year to the next for smaller schools, whereas larger schools enjoy considerably greater stability in this regard (e.g., Hill & DePascale, 2003; Kane & Staiger, 2002; Linn & Haug, 2002). Consider Figure 1, for example, which shows th e relationship between the size of the fourthgrade cohort tested in a Maine school and the oneyea r change in the proportion of students in that school who are proficient on the Maine Educationa l Assessment reading test. Although the average change from one year to the next hovers around zero for all schools (dashed line), there is considerably greater variability among smaller schools in the amount of this change. For schools having 15 or fewer fourth graders, fo r instance, this change ranges from .47 (declining from 60% proficient to 13% proficient) to +.83 (increasin g from 17% proficient to 100% proficient). In PAGE 4 Education Policy Analysis Archives Vol. 14 No. 28 4 contrast, the corresponding figures are only .07 and +.09, respectively, among schools having 150 or more fourth graders. 275 250 225 200 175 150 125 100 75 50 25 0 4th grade enrollment (number tested) 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 oneyear change: proportion proficient in reading Figure 1. The relationship between the number of fourthgrade students tested in a school and the oneyear change in the proportion of studen ts who are proficient. (S ource: Coladarci, 2003, Figure 4) At issue here is the differential reliability of schoolaggregated student achievement for smaller versus larger schools. A schools achievem ent result at any given point in time can be thought of as an estimate of the schools tru e score (Hill, 2002, p. 2). Insofar as school achievement tends to be less stable from one year to next for smaller schools than for larger schools (much as shorter tests tend to have lower testretest reliability than longer tests), there thus is a greater likelihood that the reported level of achievement for a smaller school will be at variance with the schools true level of achievementthe school s effectiveness as an institution (e.g., see Cronbach, Linn, Brennan, & Haertel, 1997, p. 393) Because reliability places an upper limit on a variables ability to correlate with other variables (e.g., Thorndike, 1982, p. 222), a plausible conjecture is that the lower SESachievement correlation among smaller schools is an artifact of the lower reliability of school achievement for such schools. In short, this is the conjecture I investigated in the present study. In pursuing the statisticalartifact hypothesis I was not motivated by a desire to debunk popular opinion regarding the virtues of small sc hools. Rather, I simply wished to determine whether a celebrated proposition in the rural educ ation literature could withstand a sincere attempt to falsify it. If such an attempt were to fail, then we all are entitled to a greater confidence in this PAGE 5 School Size, Student Achievement, and the Power Rating of Poverty 5 propositiongreater warranted confidencethan we presently can claim (e.g., see Phillips, 2000, pp. 137). Method Context, Data Source, and Variables Maine, the context of this study, is a pred ominantly rural state in northern New England. With roughly 1.3 million people spread over 33,215 square miles (a geographical area comparable to the remaining New England states combined), the vast majority (96%) of Maine residents are White, the median household income is $39,212 (versus the U.S. average of $43,318), and 10.7% of Mainers live below the poverty level (versus 12.5% in the U.S.). The number of elementary and secondary students is about one fifth that of the U.S. av erage (198,820 vs. 956,762) with approximately one third (32.34%) of these students qualifying for fr ee or reduced lunch (compared to 37.40% across the nation). 1 My focus is on eighthgrade achievement in public schools, using reading and mathematics data from the Maine Educational Assessment (MEA) for the 2002 and 2003 school years. (The MEA scale range was 501 at that ti me.) For each public school having an eighth grade, I created a weighted twoyear mean for both reading achievement ( reading ) and mathematics achievement ( math ). Similarly, I determined for each school the weighted twoyear percentage of students receiving subsidized lunch ( poverty). As for operationally defining school size, I immediately faced the distinction between a schools total enrollment across all grades and a schools mean enrollment per grade. Howley (2002, pp. 52) argues that the latter is the appr opriate measure of school size because pergrade enrollment takes into account a schools grade configurationthat, say, a K school with 270 students (30 per grade) is arguably smaller than a 6 school with 270 students (90 per grade). I have yet been able to appreciate the logic of this position, which inevitably must fall on how one conceptualizes school and its effects on students. But because most mitigatingeffect studies employed the enrollmentpergrade measure of school size, I followed suit in the analyses reported below. Specifically, I determined the mean enrollment per grade for each school, averaged across 2002 and 2003 ( school size ). (I confess that I also ran all analyses using a totalenrollment measure of school size, which yielded results similar to those based on enrollment per grade.) To estimate a schools volatility in eighthgrade achievement, I determined the difference in mean achievement from 2003004 to 2002 for reading and mathematics separately. I then recoded the absolute value of these differences to obtain a volatility rating for each school. There were separate volatility ratings for reading and math ( volatility ), and both were constructed as shown in Table 1. 1 These statistics are available from the Web sites of the State of Maine, http://www.state.me.us ; the U.S. Census Bureau, http://www.census.gov ; and the National Center for Education Statistics, http://nces.ed.gov PAGE 6 Education Policy Analysis Archives Vol. 14 No. 28 6 Table 1 Volatility definitions Volatility rating Change in school achievement a mean 1 0 to 2.50 points 2 2.51 to 5.00 points 3 5.01 to 7.50 points 4 7.51 to 10.00 points 5 10.01 to 12.50 points 6 12.51 to 15.00 points 7 15.01 to 17.50 points 8 17.51 to 20.00 points a The scale of the Maine Educational A ssessment ranges from 501 to 580. Analyses I restricted my analyses to public schools in Maine that had (a) an eighth grade in 2002 and 2003, (b) data on all variables fo r both 2002 and 2003, and (c) neither changed their grade span from one year to the next nor absorbed in 2003 students from a school that had closed at the end of 2002. Finally, I eliminated schools that did not have at least two eighthgrade students in each of the two school years. These restrictions resulted in a final sample of 216 schools from a universe of 233 public schools having an eighth grade in 2003. The school served as the unit of analysis. After conducting preliminary analyses to establish the trustworthiness of the data, which had been downloaded from state websites, I began by demonstrating the aforementioned interaction between socioeconomic status and school size. I did so using ordinary leastsquares regression, wh ere, in the present case, the equation is (e.g., Aiken & West, 1991). Here, is the predicted value of the dependent variable (either reading or math); a is the intercept; X 1122312 YabXbXbXX Y 1 and X 2 are poverty and school size, respectively; and X 1 X 2 is their mathematical product. Pr ior to creating the product term and consistent with common practice, I centered pover ty and school size at their respective means to reduce the inevitable collinearity engendered by multiplicative terms. All analyses were conducted using SPSS 14.0 for Windows The statistical significance of b 3 the slope of the product term, indicates the presence of interaction between X 1 and X 2 that the magnitude of b 1 varies with X 2 or, symmetrically, that the magnitude of b 2 varies with X 1 In the present context, this means that the degree of association between poverty and achievement ( b 1 ) depends on school size ( X 2 ), or, equivalently, that the degree of association between school size and achievement ( b 2 ) depends on the socioeconomic status of the school ( X 1 ). By entering the product term on a separate step, I obtained the increment in explained variance ( R 2 ) that is associated with the povertysize in teraction, the statistical significance of which is identical to that of b 3 To further illustrate the degree of interacti on between poverty and school size, and, in particular, to recast this interaction in terms of povertys power rating, I fit separate achievementonpoverty regression lines for schools falling above an d below the median pergrade enrollment. The magnitude of interaction is shown by the degree to which the two withingroup regression lines are nonparallel. From this analysis, I also obtained the withingroup correlations between each achievement measure and poverty, which, when squar ed, represents the power rating of poverty. PAGE 7 School Size, Student Achievement, and the Power Rating of Poverty 7 To explore my statisticalartifact hypothesisthat povertys reduced power rating, when examined among smaller schools, reflects the lowe r reliability of schoollevel achievement in such schoolsI repeated these analyses on successively lessvolatile (scorewise) collections of schools. The first set of analyses included all 216 schools (i.e., volatility = 1 through 8); the second set included schools for which volatility = 1 through 7; and so on to the final set of analyses involving the 104 least volatile schools (i.e., volatility = 1) (Again, there were separate volatility ratings for math and reading.) If, in fact, the povertysize in teraction is a statistical artifact due to the lower reliability of schoollevel achievement among smalle r schools, then this interaction should attenuate with successively lessvolatile collections of schoolsand be negligible for schools having the least volatility. Results I begin by portraying the achievement volat ility among these schools and, in turn, the relationship between this volatility and school size. To investigate the statisticalartifact hypothesis, I then report the results of the regression analyses on successively lessvolatile collections of schools. The Volatility of SchoolLevel Achievement As described above, I estimated a schools volatility in eighthgrade achievement by first calculating the difference in mean achievement from 20034 to 2002 for reading and for mathematics. Among these 216 schools, the change in achievement from one year to the next ranges from roughly to +17 MEA points in reading (M = .56, SD = 4.61) and, for math, to +16 MEA points (M = +1.14, SD = 4.79). The wellestablished relationship between school size and achievement volatility is clearly evident in the present data (Figur e 2). Again, there simply is greater volatilitylower reliabilityof schoollevel achievement among sma ller schools than among larger schools. This also can be seen in the correlations between school size and the absolute value of a schools change in achievement from one year to the next: r = .31 and r = .29, respectively, for reading and math. In short, Figure 2 and these two correlations underscore the re levance of the statisticalartifact hypothesis that frames the present study. Table 2 Frequency distribution of vo latility ratings (n = 216). Volatility rating Reading Math 1 104 104 2 62 60 3 22 29 4 16 11 5 4 4 6 6 4 7 2 3 8 0 1 PAGE 8 Education Policy Analysis Archives Vol. 14 No. 28 8 400375350325300275 250 225200 175 150125 100 755025 0 pergrade enrollment 20 15 10 5 0 5 10 15 20 oneyear change in mean achievement: reading 400375350325300275250225200175150125 100 75 50 25 0 pergrade enrollment 20 15 10 5 0 5 10 15 20 oneyear change in mean achievement: math Figure 2. School size and the volatility of achievement in reading (top) and mathematics (bottom). PAGE 9 School Size, Student Achievement, and the Power Rating of Poverty 9 Regression Analyses: All Schools The first set of regression analyses is based on all schools, irrespective of volatility. Table 3 presents descriptive statistics for reading, math poverty, and school size. Not surprisingly, schools vary considerably with respect to both poverty and size: Some schools have as few as 3 students per grade and 3% of their students receiving subsidize d meals, whereas other schools have as many as 358 students per grade and 84% of their students receiving subsidized meals. Reading and math correlate highly ( r = .74), as one would expect, and each correlates with poverty in the customary fashion (Sirin, 2005; White, 1982). There is some tendency for smaller schools to be located in more impoverished communities ( r = .34). School size, however, is unrelated to achievement ( r = .07, p = .16). Table 3 Descriptive statistics: All schools (n = 216). Intercorrelations Variable M SD Range Reading Math Poverty Reading 535.96 3.94 522.72, 547.69 Math 528.16 4.36 514.51, 542.17 .74* Poverty 39.52 16.63 2.68, 83.86 .48* .37* School size 72.78 77.31 2.94, 358.00 .07 .07 .34* Note. For the purpose of this table, poverty and school size are in their original uncentered form (which affects only the mean and range). p < .01. Reading Table 4 shows the regression results fo r reading. Poverty significantly and independently predicts reading at Step 1, wherea s the corresponding effect of school size falls short of statistical significance. An additional 2.21% of the variance in reading is explained by the introduction of the product term at Step 2, which, consistent with prior research, shows a statistically significant interacti on between poverty and school size ( p = .01). Table 4 Regressing reading on poverty, school size, and their prod uct: All schools (n = 216) Variable b s.e. t p R 2 Without interaction (constant) 535.96 Poverty 0.1216 0.0151 0.51 8.07 < .01 School size 0.0055 0.0032 0.11 1.71 .09 With interaction (constant) 535.74 Poverty 0.1273 0.0151 0.54 8.45 < .01 School size 0.0080 0.0034 0.16 2.40 .02 Poverty x size 0.0005 0.0002 2.52 .01 .0221 Note. Poverty and school size were centered for this analysis. At Step 2, the unadjusted and adjusted values of R 2 are .260 and .249, respectively. PAGE 10 Education Policy Analysis Archives Vol. 14 No. 28 10 Because the povertysize interaction presentl y enjoys so much attention in the rural education literature, elaboration on the meaning of the various coefficients reported at Step 2 may be helpful. As we saw above, Step 2 estimates the effects for the full equation, where the last term, reflects the interaction of poverty and school size. As Aiken and West (1991) explain, b 1122312 YabXbXbXX 312bXX 1 is the readingonpoverty slope for schools having a pergrade enrollment equal to the mean (i.e., centered X 2 = 0). For schools of average size, then, reading achievement decreases about .13 MEA points ( b 1 = .1273) with every onepercentagepoint increase in the students receiving subsidized meals. In standardized terms, this corresponds to a decline in reading achievement of roughly half a standard deviation ( 1 = .54) for each standard deviation increase in poverty (again, for schools of average si ze). One interprets b 2 analogously: For schools at the mean for poverty, reading ac hievement decreases about .01 MEA points ( b 2 = 0.0080) for each onestudent increase in sc hool sizean achievement decline of 16% of a standard deviation ( 2 = .16) for each standard deviat ion increase in school size. The statistical significance of b 3 signals the presence of interaction between poverty and school size. Specifically, the negative coefficient for the product term X 1 X 2 coupled with the negative coefficient for poverty, means that the simple slope for povertyi.e., the readingonpoverty slope at a specified value of school sizeis steeper (more negative) for larger schools than it is for smaller schools. The concept of simple slope is central to interpreting a statis tically significant interaction. The simple slope for poverty derives from the full equation, which, when recast as the Y onX 1122312 YabXbXbXX 1 regression at a specified value of X 2 looks like this: The critical term here is 221321 ()( YabXbbXX )) 132( bbX which is the Y onX 1 slope for the specified value of X 2 (expressed as a deviation from the centered mean of zero). Select a deviation score to represent X 2 plug this value into the expression 132( bbX ) and you have the simple slope for poverty at a particular school size. PAGE 11 School Size, Student Achievement, and the Power Rating of Poverty 11 100908070605040302010 0 poverty 550 545 540 535 530 525 520 515 510 reading r = .39 r 2 = .15 median split: fewer than 42 students per grade (solid line) 42 students or more per grade (broken line) r = .64 r 2 = .41 Figure 3. The interaction of poverty and school size ( p =.013), reading: All schools (n = 216). For example, consider a school having 16 stud ents per gradethe 25th percentile in school size and roughly 57 fewer students than the mean ( 2 X = 72.78). The simple slope for schools of this size is b 57 = .0982, which corresponds to a standardized regression coefficient of 57 = .41. 2 Thus, with each standard deviation increa se in poverty, reading achievement in these smaller schools decreases approximately 40% of a standard deviation. The simple slope is slightly steeper for schools having 42 students per grade (the median school size, or 50th percentile): b 31 = .1115 or, in standardized terms, 31 = .47. Now consider a school falling at the 75th percentile in school size, or 105 students per gr ade. Here, the unstandardized and standardized simple slopes are b +32 = .1437 and +32 = .61, respectively. For these larger schools, then, reading decreases approximately 60% of a standard deviation with each standard deviation increase in poverty. Consistent with the statistically signific ant interaction of poverty and school size, simple slopes estimated at various levels of school size illustrate that reading achievement is increasingly 2 I introduced the subscript to ma ke explicit the particular value of X 2 at which the Y onX 1 slope is estimated. The specified value of X 2 is expressed as a deviation score: X 2 2 X = 16 72.68 = 6.58 (rounded to ). PAGE 12 Education Policy Analysis Archives Vol. 14 No. 28 12 related to poverty as school size increases, and decreasingly related to poverty as school size decreases. Figure 3 shows the withingroup regression lin es for belowand abovemedian schools in per grade enrollment. As described above, I obtai ned these by splitting the schoolsize distribution at the median (42 students per grade) and, for ea ch group of schools, fitting a readingonpoverty regression line. These withingroup regression lin es further illustrate the interaction reported in Table 4: There is a flatter slopea weaker re lationship between reading achievement and poverty for smaller schools than for larger schools. Indeed, the correlation for the former is r = .39 versus r = .64 for the latter, which, when squared, yi eld power ratings of 15% and 41%, respectively. Although there is considerable withingroup va riability evident in Figure 3 and, further, the nonparallel displacement of one regression line relative to the other is not great (particularly where most of the data are), there is some tendency for smaller higherpoverty schools to have reading achievement superior to that of larger higherpoverty schools. Math Table 5 shows the regression results for math, based on all schools. The pattern of results is similar to those obtained for reading. At Step 1, poverty is significantly related to math whereas school size is not. And at Step 2, the interaction of poverty and school size explains an additional 5% of variance in mathematics achievement ( R 2 = .0479, p < .01): As with reading achievement, mathematics achievem ent is increasingly related to poverty as school size increases, and decreasingly related to poverty as school size decreases. For example, the mathonpoverty slope for mediansize schools is b 31 = .0860 ( 31 = .33). In contrast, the simple slope for schools at the 25th percentile in school size is b 57 = .0643 ( 57 = .25) and, for schools at the 75th percentile, b +32 = .1385 ( +32 = .53). Table 5 Regressing math on poverty, school size, and their produc t: All schools (n = 216). Variables b s.e. t p R 2 Without interaction (Constant) 528.16 Poverty 0.1025 0.0177 0.39 5.78 < .01 School size 0.0039 0.0038 0.07 1.02 .31 With interaction (Constant) 527.80 Poverty 0.1118 0.0175 0.43 6.40 < .01 School size 0.0080 0.0039 0.14 2.05 .04 Poverty x size 0.0008 0.0002 3.53 < .01 .0479 Note. Poverty and school size were centered for this analysis. At Step 2, the unadjusted and adjusted values of R 2 are .187 and .176, respectively. The withingroup regression lines are presen ted in Figure 4, which shows the nonparallel displacement indicative of interaction. The math onpoverty slope is flattersignifying a weaker relationshipfor smaller schools than for larger schools. The corresponding power ratings are, respectively, 4% for smaller schools ( r = .19) and 46% for larger schools ( r = .68). PAGE 13 School Size, Student Achievement, and the Power Rating of Poverty 13 100908070605040302010 0 poverty 550 545 540 535 530 525 520 515 510 math r = .19 r 2 = .04 median split: fewer than 42 students per grade (solid line) 42 students or more per grade (broken line) r = .68 r 2 = .46 Figure 4. The interaction of poverty and school size ( p =.001), math: All schools ( n = 216). The symmetry of b 3 As noted above, the statistical significance of b 3 indicates that the magnitude of the achievementonpoverty slope ( b 1 ) is a function of school size ( X 2 ) and, symmetrically, the magnitude of the achievementonsize slope ( b 2 ) is a function of poverty ( X 1 ). My emphasis thus far has been decidedly on the former given its direct relevance to the concept of povertys power rating which frames the present study. But many writers blur the distinction between the two interpretations, referring to one and then to the other as their argument develops. I (briefly) will follow suit. Just as the simple slope for poverty ( b 1 ) at a specified value of school size (X 2 ) is equal to the simple slope for school size ( b 13bbX 2 1 2 ) at specified value of poverty ( X 1 ) is equal to At Step 2 of Tables 4 and 5, we see that school size has a negligible, if statistically significant, negative effect on both reading and math for schools of average poverty (i.e., X 23bbX 1 = 0). But when the simple slope is calculated for a sc hool where 23% of its students receive subsidized mealsapproximately one standard deviation, or 17 percentage points, below the mean ( 1 X = 39.52)school size is unrelated to achievement in either reading or math. Specifically, b 17 = 0.0007 and 17 = 0.01 ( p = .91) for reading; for math, b 17 = 0.0062 and 17 = 0.11 ( p = .20). Now consider a comparatively highpoverty school in which 73% of students receive subsidized meals (roughly PAGE 14 Education Policy Analysis Archives Vol. 14 No. 28 14 two standard deviations, or 33 percentage points, above the mean). 3 Here, the effect of school size on reading is statistically significant and large: b +33 = .0249 and +33 = .49 ( p < .01). For math, the effect is larger still: b +33 = .0355 and +33 = .63 ( p < .01). Thus, with a standard deviation decrease in school size, reading achievement in these highpoverty schoolsunlike their lowerpoverty counterpartincreases by half a standard de viation, and math achievement increases almost twothirds of a standard deviation. This findin g, of course, merely restates the povertysize interaction by focusing on the conditional effect of school size rather than the conditional effect of poverty. Regression Analyses: Successively Le ssVolatile Collections of Schools To explore the possible operation of a statisti cal artifact due to the greater volatility in achievement among smaller schools, I repeated the regression analyses reported above for successively lessvolatile collections of schools. Rather than exhaustively delineate these results for each value of the volatility measure, I report in Ta ble 6 the primary statistic for each analysis: the increment in R 2 at Step 2 when the product term, X 1 X 2 is introduced. I then provide additional details for the results based on the 104 leastvolatile schools in reading achievement and the 104 leastvolatile schools in math achievement. 4 Table 6 Volatility in school achiev ement and the magnitude of R 2 Reading Math Volatility n R 2 p Volatility n R 2 p 8 8 216 .0479 < .01 7 216 .0221 .01 7 215 .0470 < .01 6 214 .0216 .01 6 212 .0419 < .01 5 208 .0291 < .01 5 208 .0392 < .01 4 204 .0292 < .01 4 204 .0383 < .01 3 188 .0300 < .01 3 193 .0261 .01 2 166 .0422 < .01 2 164 .0274 .02 1 104 .0305 .03 1 104 .0139 .19 Note. R 2 is associated with the introduction of the pr oduct term (poverty x size) at Step 2 of each regression analysis. Reading As Table 6 shows, the interaction between poverty and school size is unrelated to the volatility of schoollevel achievement in readin g: For each successive analysis, the increment in explained variance associated with the introduction of the product term at Step 2 is statistically significant. Further, there is no evidence that R 2 statistical significance notwithstanding, is systematically smaller when based on successively less volatile schools. 3 One should interpret these deri ved slopes cautiously, of course, given the few data points at the upper end of the poverty scale. 4 Roughly half (51%) of these 104 schools were l east volatile in both reading and mathematics achievement. PAGE 15 School Size, Student Achievement, and the Power Rating of Poverty 15 Table 7 Descriptive statistics: Least volatile sc hools, reading achievement (n = 104) Intercorrelations Variable M SD Range Reading Poverty Reading 535.95 3.76 527.99, 545.95 Poverty 38.78 15.98 2.68, 78.52 .59* School size 89.19 79.67 2.94, 358.00 .09 .35* Note. For the purpose of this table, poverty and school size are in their original uncentered form (which affects only the mean and range). p < .01. Tables 7 and 8 show descriptive statistics and regression results, respectively, based on the leastvolatile schools in reading achievement ( n = 104). Again, these are the schools for which mean achievement on the reading measure did not vary more than 2.5 points across the two years examined. The pattern of results here is similar to that reported earlier for all 216 schools, as are the withingroup regression lines show n in Figure 5. Indeed, regardin g the latter, povertys power rating differential% for smaller schools vs. 42% for larger schoolsis almost indistinguishable from the differential based on all schools (15% and 41 %, respectively). With respect to reading achievement, then, the statisticalartifact hy pothesis is not consistent with the data. Table 8 Regressing reading on poverty, school size, and their product: Schools having minimal volatility in achievement (n = 104) Variables b s.e. t p R 2 Without interaction (Constant) 535.94 Poverty 0.1492 0.0200 0.63 7.45 < .01 School size 0.0065 0.0040 0.14 1.61 .11 With interaction (Constant) 535.72 Poverty 0.1411 0.0200 0.60 7.067 < .01 School size 0.0074 0.0040 0.16 1.875 .06 Poverty x size 0.0005 0.0002 2.237 .03 .0305 Note. Poverty and school size were centered for this analysis. At Step 2, the unadjusted and adjusted values of R 2 are .390 and .372, respectively. PAGE 16 Education Policy Analysis Archives Vol. 14 No. 28 16 100908070605040302010 0 poverty 550 545 540 535 530 525 520 515 510 reading r = .40 r 2 = .16 median split: fewer than 42 students per grade (solid line) 42 students or more per grade (broken line) r = .64 r 2 = .42 Figure 5. Interaction of poverty and school size ( p = .001), reading: Schools having minimal volatility in achievement ( n = 104). Math A different picture emerges with mathemat ics achievement, where we see a gradual decline in R 2 with successively lessvolatile collections of schools (Table 6)to the point of statistical nonsignificance when base d on the 104 leastvolatile schools ( R 2 = .0139, p = .19). Tables 9 and 10 present the relevant statistics for the latter analysis, where, at Step 2 of Table 10, we see the statistically nonsignificant slope for the product term. PAGE 17 School Size, Student Achievement, and the Power Rating of Poverty 17 Table 9 Descriptive statistics: Least volatile schools, math achi evement (n = 104) Intercorrelations Variable M SD Range Math Poverty Math 527.60 4.26 514.51, 542.17 Poverty 38.25 14.71 7.99, 73.89 .41* School size 82.28 81.72 3.39, 327.50 .06 .30* Note. For the purpose of this table, poverty and school size are in their original uncentered form (which affects only the mean and range). The school s in Tables 9 and 10 are the not same 104 schools represented in Tables 7 and 8. (See footnote 4.) p < .01. The withingroup regression lines are shown in Figure 6. While the power ratings of poverty show some differential between smaller and larger schools, it derives from a povertysize interaction that failed to reach statistical significance and, therefore, reflects chance variation. Between the general decline in R 2 values (Table 6) and the absence of a statistically significant povertysize interaction when based on the least volatile schools (Table 10), the statisticalartifact hypothesis is consistent with the data in the case of mathematics achievement. Table 10 Regressing math on poverty, school size, and their product: School s having minimal volatility in achievement (n = 104) Variables b s.e. t p R 2 Without interaction (Constant) 527.47 Poverty 0.1249 0.0278 0.43 4.54 < .01 School size 0.0038 0.0050 0.07 0.76 .45 With interaction (Constant) 527.31 Poverty 0.1309 0.0278 0.45 4.71 < .01 School size 0.0070 0.0055 0.13 1.26 .21 Poverty x size 0.0005 0.0004 1.31 .19 .0139 Note. Poverty and school size were centered for this analysis. At Step 2, the unadjusted and adjusted values of R 2 are .186 and .162, respectively. The schools in Tables 9 and 10 are the not same 104 schools represented in Tables 7 and 8. (See footnote 4.) PAGE 18 Education Policy Analysis Archives Vol. 14 No. 28 18 100908070605040302010 0 poverty 550 545 540 535 530 525 520 515 510 math r = .35 r 2 = .12 median split: fewer than 42 students per grade (solid line) 42 students or more per grade (broken line) r = .57 r 2 = .32 Figure 6. No interaction of poverty and school size ( p = .193), math: Schools having minimal volatility in achievement ( n = 104) Discussion The question posed in the subtitle of this articlesubstantive finding or statistical artifact? does not permit an unequivocal answer. When the dependent variable is reading achievement, I find no support for my hypothesis that povertys power rati ng is lower in smaller schools because of their greater volatility (lower reliability) in schoollevel achievement. Thus, the celebrated interaction of socioeconomic status and school size clearly stands with respect to eighthgrade reading achievement in these Maine schools. But for math ematics achievement, the statisticalartifact hypothesis is supported: Among these smaller sc hools, the lower reliability of schoollevel achievement appears to be a plausible explanation of the reduced power rating of poverty for these schools. Unfortunately, the latter conclusion is complica ted by plausible rival hypotheses of its own. Two problems immediately come to mind. First, my achievementvolatility measure does not distinguish between random variation and variation due to educational practice. Some of the highdiscrepancy schools in Figure 2, as reflected in th eir alignment on the vertical axis, doubtless are revealing realnot randomimprovement or decline in achievement. By treating all variation as PAGE 19 School Size, Student Achievement, and the Power Rating of Poverty 19 random variation, I inevitably exclude some schools from the analysis that should have been included (were it possible to make this distin ction in practice). That said, the results are not systematically biased as a consequence, insofar as the absence of real improvement schools is offset by the absence of real decline schools. The second problem is of greater concern. By conducting the regression analyses on successively lessvolatile collections of schools, and because achievement volatility is more pronounced among smaller schools (Figure 2), I succe ssively compromise the full representation of smaller schools as well. For example, 26 schools in the full sample (12.0%) had fewer than 10 students per grade. Among the least volatile sc hools in mathematics achievement, however, only 6 schools (5.8%) were this small. Would the results of the final analyses, where achievement volatility is minimum, likely differ had the smallest schools been fully represented? It is difficult to say. In short, I arguably exclude some of the very school s required for an adequate test of my statisticalartifact hypothesis (and, in doing so, introduce a certain irony into the present study). Yet this second problemthe successive un derrepresentation of smaller schoolshad no effect on the viability of the povertysize interacti on for reading achievement. This inconsistency presents an interesting challenge: how to explain it If one is inclined to dismiss my findings for mathematics achievement because of this underrepresentation, then the challenge is to explain why a similar outcome was not obtained for readin g achievement. After all, smallerschool underrepresentation operates there as well. So, what is it about reading achievement (or related instruction) that makes the povertysize interaction immune to the successive underrepresentation of smaller schools in these analyses? Or, if one prefers, what is it about mathematics achievement (or related instruction) that makes the povertysize interaction particularly vulnerable in this regard? On the other hand, for those readers whose confidence in the statisticalartifact results for mathematics achievement is unshaken by this underrepresentation, the corresponding challenge is to explain why the statisticalartifact hypothesis did not prevail for reading achievement. After all, reading achievement is not appreciably less volatile than mathematics achievement. So, what is it about reading achievement (or related instruction) that expl ains this apparent invincibilitya greater robustnessof the povertysize interaction? Unfortunately, I cannot answer these questions. But insofar as I cannot explain, even with the benefit of hindsight, a statisticalartifact finding that would surface only for mathematics achievement, I am inclined to attach greater im port to the successive underrepresentation of smaller schools in these analyses than I had at the outset. Although I cannot explain why this underrepresentation has no concomitant effect on the povertysize interaction with respect to reading achievement, this anomaly presently perp lexes me less than does a mathematicsspecific statistical artifact. Furthermore, it is only in the final, most restrictive analysiswhere a sizeable number of the smallest schools are losttha t the povertysize interaction for mathematics achievement fails to reach statistical significance. In view of these considerations, then, I conclude that my results are insufficient to support the statisticalartifact hypothesis with respect to ma thematics achievement. Although this conclusion is not as unequivocal as that for reading achiev ement, I nevertheless believe it is the reasonable conclusion given the considerations above. In shor t, the celebrated interaction of poverty and school size has survived a sincere attempt to empirically cast doubt on it. Consequently, we can have greater confidence in this interaction than I believe was warranted before. Further tests of the statisticalartifact hypothesi s would be informative, if only to show that my somewhat equivocal results for mathematics achievement are a mere anomaly. In this spirit, I encourage other researchers who have explored the mitigatingeffect phenomenon to conduct, where possible, (re)analyses of their own with th e inclusion of an achievementvolatility measure. PAGE 20 Education Policy Analysis Archives Vol. 14 No. 28 20 If the interaction of socioeconomic status and scho ol size is accepted as an established (if modest) phenomenon, we nonetheless are left wanting for a credible explanation of it. Such an explanation seemingly would dr aw on the mechanisms through which smaller schools facilitate student achievement and related outcomes, but, un fortunately, we are wanting there as well. As Fowler and Walberg (1991) said in reference to the thenextant research, [a]lthough these studies show a positive relationship between small school size and student outcomes, they do not suggest why this may occur. In other studies, which only peripherally included school size, researchers have suggested reasons for the beneficial effect that small sc hool size has upon st udent outcomes (p. 191; emphasis added). The situation has changed little in the ensuing 15 years. As an influence on student achievement, school si ze clearly is a proxy rather than a causal force in and of itself. To offer credible explanat ions for the povertysize interaction, then, we first need stronger evidence regarding the mechanismsthe mediating variablesthrough which school size putatively influences student achievement (M cMillen, 2004, p. 20). Howley (2002, p. 62) offers care, attention, and respect as possible mechanisms; Lee and Smith (1997, p. 219) refer to the academic and social organization and functioning of schools. Doub tless there are other contextand processrelated forces at play as well. Whatever the focus, a warranted claim about its relationship to both school size and student achievement must be based on careful empirical investigation, not on casual observation, anecdot al reports, reasonable (but untested) hypotheses, popular opinion, or the will to believe. We need more descriptive research like that conducted by Howley and Howley (2006) and Lee, Smerdon, Alfe ldLiro, and Brown (2000), which should be followed up by analyses that exercise the stat istical control necessary to test hypotheses that fundamentally get at causeandeffect relationships. Equipped with empirically established mediatin g variables regarding the relationship between school size and student achievement, we can then craft defensible conjectures regarding the povertysize interaction. In this regard, of course ones central obligation will be to argue why a mediating variable would be expected to differentially affect student achievement as a function of student SES. For example, if the accumulation of evidence from sound empirical research were to show that smaller schools are characterized by more personalized social relations and, in turn, that these more personalized social relations improve student outc omes, our obligation is to cogently argue why lowerSES students would benefit from such social relations more than higherSES students would. These conjectures should then be subjected to empirical tests of their own. One could introduce a set of socialrelations variables into the full regression equation (in the tradition above) to see whether the povertysize interaction disappearsas it would if the povertysize interaction is in fact due to social relations. In any case, wellcrafted arguments followed by equally wellcrafted investigationsboth premised on warranted claims regarding the mechanisms through which school size influences student achievementshould be the direction of fu ture research on the povertysize interaction. PAGE 21 School Size, Student Achievement, and the Power Rating of Poverty 21 References Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions Newbury Park, MA: Sage. Butler, L., Carr, R., Cook, K., Go uld, R., Keenan, F., MacArthur, S., Power, L., & Ritchie, K. (2005, June 8). Consolidation cant save money. [Letter to the editor.] Bangor Daily News p. A9. Coladarci, T. (2003). Gallup goes to school: Th e importance of confidence intervals fo r evaluating Adequate Yearly Progress in small schools. Policy Brief. Washington, D.C.: The Rural School and Community Trust. Re trieved October 26, 2006, from http://files.ruraledu.org/docs/nclb/coladarci.pdf Cronbach, L. J., Linn, R. L., Brennan, R. L., & Haertel, E. H. (1997). Generalizability analysis for performance assessments of student achievement or school effectiveness. Educational and Psychological Measurement 57, 373. Fowler, W. J., & Walberg, H. J. (1991). Sc hool size, characteri stics, and outcomes. Educational Evaluation and Policy Analysis 13 189. Friedkin, N. E., & Necochea, J. (1988). School system size and perf ormance: A contingency perspective. Educational Evaluation and Policy Analysis 10 237. Hill, R. K. (2002, April). Examining the reliability of accountability systems Paper presented at the 2002 meeting of the American Educational Research Association, New Orleans. Retrieved September 25, 2003, from http://www.nciea.org/publications/NCME_RHCD03.pdf Hill, R. K., & DePascale, C. A. (2003). Reliability of No Ch ild Left Behind Accountability Designs. Educational Measurement : Issues and Practices 22 (3), 12. Howley, C. B. (1996). Compound ing disadvantage: The effects of school and district size on student achievement in West Virginia. Journal of Research in Rural Education 12 (1), 25 32. Howley, C. B. (2002). Small sc hools. In A. Molnar (Ed.), School reform proposals: The research evidence (pp. 49). Greenwich, CT: Information Age Publishing. Howley, C. B., & Bickel, R. (1999). The Matthew Project: National Report Athens, OH: Ohio University, Educational Studies Department. (ERIC Docu ment Reproduction Service No. ED 433 174). Howley, A., & Howley, C. B. (2006). Small schools and the pressure to consolidate. Education Policy Analysis Archives, 14 (10). Retrieved March 29, 2006, from http://epaa.asu.edu/epaa/v14n10/ PAGE 22 Education Policy Analysis Archives Vol. 14 No. 28 22 Huang, G., Howley, C. B. (1993). Mitigating di sadvantage: Effects of sm allscale schooling on student achievement in Alaska. Journal of Research in Rural Education, 9 (3), 137. Johnson, J. D., Howley, C. B., & Howley, A. A. (2002). Size, excellence, and equity: A report on Arkansas schools and districts Athens, OH: Ohio Univer sity, Educational Studies Department. (ERIC Document Reprod uction Service No. ED 459 987). Kane, T.J., & Staiger, D.O. (2002). Volatility in school test scores: Implications for testbased accountability systems. In D. Ravitch (Ed.), Brookings Papers on Education Policy 2002 (pp. 23583). Washington, DC: Brookings Institution. Lee, V. E., Smerdon, B. A., AlfeldLiro, C., & Brown, S. L. (2000). Inside large and small high schools: Curriculum and social relations. Educational Evaluation and Policy Analysis 22, 147. Lee, V. E., & Smith, J. B. (1997). High school size: Which work s best and for whom? Educational Evaluation and Policy Analysis 19 205. Linn, R. L., & Haug, C. (2002). Stability of schoolbuilding ac countability scores and gains. Educational Evaluation and Policy Analysis 24 (1), 29. McMillen, B. J. (2004, Octo ber 22). School size, achiev ement, and achievement gaps. Educational Policy Analysis Archives 12(58). Retrieved Octo ber 22, 2004, from http://epaa.asu. edu/epaa/v12n58/ Phillips, D. C. (2000). The expanded social scientists bestiary Lanham, MD: Rowman and Littlefield. Maines small schools cut povertys power over student achievement. (2005, February). Rural Policy Matters 7 (2), pp. 3, 5. Retrieve d October 30, 2006, from http://www.ruraledu.org/atf/c f/%7BF4BE47E7FA2747A8B6628DE8A6FC0577%7D/rpm7_2.pdf Sirin, S. R. (2005). Socioeconomic status and academic achi evement: A metaanalytic review of research. Review of Educational Research 75 417. Thorndike, R. L. (1982). Applied psychometrics Boston: Houghton Mifflin. Tompkins, R. (2006). Small scho ols, small districts: Good for rural kids, economies, and democracy. Rural Americans (Issue 14). Retrieved June 19, 2006, from http://www.demos.org/pubs/KitchenTable014.pdf White, K. (1982). The relation between socioe conomic status and academic achievement. Psychological Bulletin 91, 461. PAGE 23 School Size, Student Achievement, and the Power Rating of Poverty 23 About the Author Theodore Coladarci University of Maine Email: coladarci@umit.maine.edu Theodore Coladarci is Professor of Educational Psychol ogy at the University of Maine. Since 1992, he has se rved as editor of Journal of Research in Rural Education ( http://www.umaine.edu/jrre/ ). An earlier version of this work was presented at the 2006 meeting of the American Educational Research Association, and the author wishes to thank the discussant, Aimee Howley, for her thoughtful co mments and suggestions. The author also is grateful for the feedback prov ided by Deb Allen, Sandy Ervi n, Ed Kameenui and the four anonymous reviewers. PAGE 24 Education Policy Analysis Archives Vol. 14 No. 28 24 EDUCATION POLICY ANALYSIS ARCHIVES http://epaa.asu.edu Editor: Sherman Dorn, University of South Florida Production Assistant: Chris Murre ll, Arizona State University General questions about ap propriateness of topics or particular articles may be addressed to the Editor, Sherman Dorn, epaaeditor@shermandorn.com. Editorial Board Michael W. Apple University of Wisconsin David C. Berliner Arizona State University Robert Bickel Marshall University Gregory Camilli Rutgers University Casey Cobb University of Connecticut Linda DarlingHammond Stanford University Gunapala Edirisooriya Youngstown State University Mark E. Fetler California Commission on Teacher Credentialing Gustavo E. Fischman Arizona State Univeristy Richard Garlikov Birmingham, Alabama Gene V Glass Arizona State University Thomas F. Green Syracuse University Aimee Howley Ohio University Craig B. Howley Ohio University William Hunter University of Ontario Institute of Technology Daniel Kalls Ume University Benjamin Levin University of Manitoba Thomas MauhsPugh Green Mountain College Les McLean University of Toronto Heinrich Mintrop University of California, Berkeley Michele Moses Arizona State University A. G. Rud Purdue University Michael Scriven Western Michigan University Terrence G. Wiley Arizona State University John Willinsky University of British Columbia PAGE 25 School Size, Student Achievement, and the Power Rating of Poverty 25 EDUCATION POLICY ANALYSIS ARCHIVES Englishlanguage New Scholar Editorial Board Noga Admon New York University Jessica Allen University of Colorado Cheryl Aman University of British Columbia Anne Black University of Connecticut Marisa Cannata Michigan State University Chad d'Entremont Teachers College Columbia University Carol Da Silva Harvard University Tara Donahue Michigan State University Camille Farrington University of Illinois Chicago Chris Frey Indiana University Amy Garrett Dikkers College of St. Scholastica Misty Ginicola Yale University Jake Gross Indiana University Hee Kyung Hong Loyola University Chicago Jennifer Lloyd University of British Columbia Heather Lord Yale University Shereeza Mohammed Florida Atlantic University Ben Superfine University of Michigan John Weathers University of Pennsylvania Kyo Yamashiro University of California Los Angeles PAGE 26 Education Policy Analysis Archives Vol. 14 No. 28 26 Archivos Analticos de Polticas Educativas Associate Editors Gustavo E. Fischman & Pablo Gentili Arizona State University & Universidade do Estado do Rio de Janeiro Asistentes editoriales: Rafael O. Serrano (ASUUCA) & Lucia Terra (UBC) Hugo Aboites UAMXochimilco, Mxico Armando Alcnt ara Santuario CESU, Mxico Claudio Almonacid Avila UMCE, Chile Dalila Andrade de Oliveira UFMG, Brasil Alejandra Birgin FLACSOUBA, Argentina Sigfredo Chiroque IPP, Per Mariano Fernndez Enguita Universidad de Salamanca. Espaa Gaudncio Frigotto UERJ, Brasil Roberto Leher UFRJ, Brasil Nilma Lino Gomes UFMG, Brasil Pia Lindquist Wong CSUS, USA Mara Loreto Egaa PIIE, Chile Alma Maldonado University of Arizona, USA Jos Felipe Martnez Fernndez UCLA, USA Imanol Ordorika IIEUNAM, Mxico Vanilda Paiva UERJ, Brasil Miguel A. 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Stromquist USC, USA Daniel Surez LPPUBA, Argentina Antonio Teodoro Universidade Lusfona, Lisboa Jurjo Torres Santom Universidad de la Corua, Espaa Llian do Valle UERJ, Brasil xml version 1.0 encoding UTF8 standalone no mods:mods xmlns:mods http:www.loc.govmodsv3 xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.loc.govmodsv3mods31.xsd mods:relatedItem type host mods:identifier issn 10682341mods:part mods:detail volume mods:number 14issue 28series Year mods:caption 20062006Month November11Day 33mods:originInfo mods:dateIssued iso8601 20061103 xml version 1.0 encoding UTF8 standalone no record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd leader nam a22 u 4500 controlfield tag 008 c20069999azu 000 0 eng d datafield ind1 8 ind2 024 subfield code a E1100501 0 245 Educational policy analysis archives. n Vol. 14, no. 28 (November 03, 2006). 260 Tempe, Ariz. : b Arizona State University ; Tampa, Fla. : University of South Florida. c November 03, 2006 505 School size student achievement and the power rating of poverty: substantive finding or statistical artifact? / Theodore Coladarci. 650 Education x Research v Periodicals. 2 710 Arizona State University. 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