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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 Colleges of Education at Arizona State University and the Universi ty of South Florida. Articles are indexed by H.W. Wilson & Co. Send commentary to Casey Cobb (c asey.cobb@uconn.edu) and errata notes to Sherman Dorn (epaaeditor@s hermandorn.com). EDUCATION POLICY ANALYSIS ARCHIVES A peerreviewed scholarly journal Editor: Sherman Dorn College of Education University of South Florida Volume 14 Number 2 Januar y 20, 2006 ISSN 10682341 Successive Student Co horts and Longitudina l Growth Models: An Investigation of Elementary School Mathem atics Performance Keith Zvoch University of Nevada, Las Vegas Joseph J. Stevens University of Oregon Citation: Zvoch, K., & Stevens, J. J. (2006) Successive student cohorts and longitudinal growth models: An investigation of elem entary school mathem atics performance. Education Policy Analysis Archives, 14 (2). Retrieved [date] from http://epaa.asu.edu/epaa/v14n2/. Abstract Mathematics achievement data from three longitudinally matched student cohorts were analyzed with multilevel growth mode ls to investigate the viability of using status and growthbased indi ces of student achievement to examine the multiyear performance of schools. Elem entary schools in a large southwestern school district were evaluated in terms of the mean achi evement status and growth of students across cohorts as well as changes in the ac hievement status and growth of students between student cohorts. Results indica ted that the cross and betweencohort performance of schools di ffered depending on whether the mean achievement status or growth of studen ts was considered. Results also indicated that the crosscohort indicators of school performance were more reliably estimated than their betweencohort counterparts Further examination of the performance indices revealed that crosscohort achievement st atus estimates were closely related to student demographics while betweencohort estimates were associated with cohort enrollment size and cohort initial performance stat us. Of the four school performance indices studied, only student growth in achievemen t (averaged across PAGE 2 Education Policy Analysis Archives Vol. 14 No. 2 2 cohorts) provided a relatively reliabl e and unbiased indication of school performance. Implications for the No Ch ild Left Behind sch ool accountability framework are discussed. Keywords: school accountability, longit udinal growth models, No Child Left Behind Act. Over the past several years, states have deve loped educational accountability systems as a means for improving the achievement outcomes fo r students (see Fuhrman & Elmore, 2004; Ladd, 1996). Educational accountability systems have been built on an implicit theory of action that assumes a public airing of student achievement results and a structured program of rewards and sanctions is requisite to motivate school personne l to constructively respond to evidence of substandard student outcomes (ForteFast & He bbler, 2004; Furhman & Elmore, 2004; Marion, et al., 2002). For state policy makers, the substandard outcome most in need of redress by system stakeholders is student performance on standa rdized achievement tests. As reflected in the weighting of accountability outco mes, achievement test scores ha ve been utilized as the key evidential component for determining the relative efficacy of schools in ea ch state accountability system (Goertz & Duffy, 2001; Stevens, Parkes, & Es trada, 2000). Although widespread, the use of standardized test data as the primary or sole mean s for evaluating school performance is not without controversy. Questions regarding measurement prec ision, alignment with instructional content, and fairness in use for special student populations ma ke the reliance on achievement tests a concern for many (e.g., AERA, APA, & NCME, 1999; Baker & Linn, 2004; Barton, 2004 ; Linn, 2000; Popham, 1999). Nonetheless, with passage of the No Child Left Behind federal legislation (NCLB: No Child Left Behind Act, 2002), testing is now more ubiqui tous and of higher stakes than ever before. Under NCLB, states must revise their accountability systems to include annual testing of students in grades 3 through 8 in mathematics and reading/language arts. Consequences for substandard performance have also become more uniform and more stri ngent. Schools now face the clear prospect of a probationary designation, staff restructuring and/or state takeover if achievement standards are not met (NCLB, 2002). The institutionalization of mandatory testing across content area and grade level and the concomitant performance pressures that schools now face place a special burden on the analytic methods used to measure school performance. Fo r accountability systems to work fairly and effectively, school performance indices need to be reliable and valid (Baker & Linn, 2004; ForteFast & Hebbler, 2004; Marion, et al., 2002). The challenge pr esented by the need for scientifically credible school performance data has led to investigation of the assessment approaches that have been used in state accountability systems. St ate approaches to school assessment can be categorized into those that measure school performance as a function of st udent achievement at one point in time (i.e., status) or those that measure the change in student achievement across two or more occasions. Status approaches (e.g., percent proficient, mean achievement) have been most commonly used by states and have had wide appeal because of the relative ease with which these measures can be calculated and understood by system stakeholders However, status measures tend to be problematic when used for evaluative or accountability purposes As singular snapshots of student achievement, status measures capture both the influence of student background and prior educational experience as well as current school contributions to student performance (Raudenbush, 2004; Raudenbush & Willms, 1995). The confounding of different so urces of achievement performance presents a particular challenge under conditions commonly foun d in public school districts. Student assignment PAGE 3 Successive Student Cohorts and Longitudinal Growth Models 3 to schools is not random, but is instead influenced by social and economicbased selection processes. The nonrandom sorting of families into neighborhoods and students into schools tends to result in a differential accountability burden fo r those schools that happen to serve large numbers of disadvantaged students (Raudenbush, 2004). Rela tive to their more advantaged counterparts, schools situated in impoverished contexts typica lly are required to produce a disproportionate increase in student achievement levels if sta te achievement standards are to be met and low performance sanctions are to be avoided. Perhaps in partial recognition of the challe nge that schools with disadvantaged intakes face when statustype measures are used to evaluate sc hool performance, states have also utilized measures that index the change in student achi evement between testing occasions. Measures of student changes in achievement are seen as an alter native means by which schools, particularly those with challenging intakes, can demonstrate positive effects on students. Several states have measured student changes in achievement by comparing th e grade level performance of successive student cohorts (e.g., the mean performance of 3rd graders in 2004 is compared to the mean performance of 3rd graders in 2005: quasi change) in an attempt to mitigate school differences in student intake (Stevens, et al., 2000). However, measuring school e ffectiveness by the change in successive student cohort performance levels can also be problematic for evaluative and accountability purposes (Hill & DePascale, 2003). Recent investigations of the successive cohort approach demonstrate that estimates of yeartoyear changes in the mean achi evement of students tend to be affected in large part by sampling variation, measurement error, and unique, nonpersistent factors (e.g., construction noise) that affect test scores on only one of the testing occasions (Kane & Staiger, 2002; Linn & Haug, 2002). As a result, the observed change in school mean performance across student cohorts may be due in large part to the yeartoyear fl uctuation in student characteristics and testing conditions rather than actual changes in studen t performance (Carlson, 2002; Linn & Haug, 2002). The observed difficulty of obtaining valid and precise estimates of school performance when school compositions differ nonrandomly and/or when the mean performance of successive student cohorts is compared has led to interest in measurin g the achievement progress of individual students as another alternative for evaluating school perf ormance (Teddlie & Reynolds, 2000; Willms, 1992; Zvoch & Stevens, 2003). In this approach, the test scores of individual students are linked across time. Individual growth trajectories are then esti mated by fitting a regression function to the time series data obtained on each student. A measure of school performance follows from averaging the individual growth trajectories within each school Tracking the achievemen t progress of individual students has certain advantages over the status an d quasichange models that states have used for school accountability purposes. Conceptually, longit udinal models of student achievement growth better represent the timedependent process of academic learning (Bryk & Raudenbush, 1988; Seltzer, Choi, & Thum, 2003; Willett, 1988). Further, unlike status models, indices that capture the yeartoyear changes in student achievement provide a degree of control over the stable background characteristics of students that otherwise complic ate the evaluation of school effectiveness (Ballou, Sanders, & Wright, 2004; Sanders, Saxton, & Hor n, 1997; Stevens, 2005). In addition, school performance measures that follow from estimate s of the achievement progress of individual students tend to be more reliable than school pe rformance measures that are based on the changes in achievement status between successive student c ohorts (e.g., Kane & Staiger, 2002). Indices of student achievement growth may thus offer an alter native for monitoring school performance that avoids some of the inherent difficulties associated with the achievement status and the quasichange approaches to school evaluation. Despite the potential of using individual time series data as a basis for measuring and evaluating school performance, states have a cu rrent disincentive for incorporating indices of student achievement growth into their accountabilit y systems. Under NCLB, states are required to PAGE 4 Education Policy Analysis Archives Vol. 14 No. 2 4 utilize a statustype measure (i.e., the percentage of students proficient or above on one testing occasion) as the primary means for evaluating school performance. Secondarily, states are permitted to evaluate schools that fail to meet standard by th e percent proficient methodology by indexing the changes in proficiency between successive student cohorts (i.e., quasichange). States can also choose to track the achievement progress of individu al students as a third approach for evaluating school performance, but under the provisions of NCLB, this methodology can only serve to further identify schools in need of improvement (Olson, 2004). In other words, schools that meet standards either by the percent proficient or quasicha nge approaches can be identified as needing improvement if a growth target is not met, but demonstrating strong student growth is not sufficient to avoid a low performance sanction if the school does not have an adequate percentage of students proficient by either of the two primary methodologies endorsed by NCLB. The disincentive currently associated with using individual time series data to measure and evaluate school performance has not allowed states to take full advantage of the annual testing of students required under the NCLB legislation. At present, only a couple of states and a handful of school districts have examined school performance as a function of student achievement growth (e.g., Kiplinger, 2004; Sanders, et al., 1997; We bster & Mendro, 1997; Zvoch & Stevens, 2003). Even less common are examinations of the multiyear pe rformance of schools using longitudinal data on successive student cohorts (see Ponisciak, & Bryk 2005; Bryk, Thum, Easton, & Luppescu, 1998; Bryk, Raudenbush, & Ponisciak, 2004, for examples). The limited application of longitudinal growth modeling methods to achievement data collected on students over time has left unanswered questions about the viability of using these technique s in state accountability systems. Although the studies conducted to date suggest that indices of student achievement growth tend to provide a less biased and a potentially more stable estimate of school performance than some NCLBendorsed alternatives, questions about the mechanics of im plementation (e.g., crosscohort or between cohort analyses, estimation of unadjusted or valueadded models) and the feasibility of use remain to be clarified (Bryk, et al., 2004; Flicek, 2004; Raude nbush, 2004). In response, the present study was designed to provide one example of how longitud inal growth models can be used to assess school performance across multiple student cohorts. Of particular interest was ascertaining whether estimates of cohorttocohort changes in the achi evement growth of students provide a sound alternative for measuring school improvement. Note however that the in tent of the current investigation was only to provide a preliminary an d exploratory examination of the behavior and viability of certain growthbased approaches to measuring school performance. As such, school performance estimates were examined in relation to student intake characteristics rather than being adjusted by them. The investigation was facilitated by the analysis of achi evement data from three longitudinally matched elementary school stud ent cohorts from a large school district in the southwestern United States. The following research questions were considered: 1) Does the crosscohort performance of schools differ based on an ex amination of school mean achievement vs. an examination of school average rates of growth in achievement? 2) Are the crosscohort school performance estimates related to selected school ch aracteristics? 3) To what degree do estimates of the mean achievement status and achievement gro wth of schools change with each successive student cohort? 4) Are estimates of the cohortto cohort changes in school performance related to selected school characteristics? and, 5) How reliabl e, on average, are each of the school performance estimates? PAGE 5 Successive Student Cohorts and Longitudinal Growth Models 5 Method Participants The multiyear performance of elementary schools was investigated by examining the mathematics achievement of students from three longitudinally matched cohorts. The school district that provided the test score data has 79 kindergar ten through grade 5 elementary schools that serve over 30,000 students each year. The district serves a significant number of students from special populations. At the elementary school level, English Language Learne rs, students eligible for a free or reduced price lunch, and students from ethni c minority groups consti tute approximately 20%, 50%, and 55% of the student body, respectively. Be ginning in the 19992000 school year, all third, fourth, and fifth grade students were assessed annually on the TerraNova/CTBS5 Survey Plus, a normreferenced achievement test (CTB/McGrawHill, 1997). Between 6,000 and 6,500 students in each grade were assessed each spring. Achievemen t data from the three most recent longitudinal cohorts were analyzed in the present study. Table 1 diagrams the data structure associated with the current investigation. In Table 1, it can be seen that third to fifth grade longitudinal matches were available for students who entered the third grad e in 19992000 (cohort 1), 200001 (cohort 2), and 200102 (cohort 3). Cohort 1 thus consisted of stud ents who were third graders in 19992000, fourth graders in 200001, and fifth graders in 200102. The second and third cohorts consisted of the two following elementary school third to fifth grade student cohorts (i.e., cohort 2 from 200001 to 200203, and cohort 3 from 200102 to 200304). Table 1 Cohort Data Structure Year Grade 19992000 200001 200102 200203 200304 3 C1 C2 C3 4 C1 C2 C3 5 C1 C2 C3 Cohort 1 ( N = 3,325), Cohort 2 ( N = 3,347), Cohort 3 ( N = 3,322); School N = 79 Within cohort matches were accomplished by the following set of procedures. For each cohort, students who participated in accountabili ty testing in all three study years were selected ( N ~ 5,000). To facilitate the study of school eff ects, students who attended the same elementary school in all three years were then identified. In each cohort, approximately 900 students transferred schools at least once during the respective threeye ar period studied. Next, students who did not have a mathematics score in any of the three study years ( N ~ 100) were dropped from their cohorts. Finally, students who received one or more modified test administrations were eliminated from the working data files ( N ~ 600). The sample exclusions resulted in the following within cohort sample sizes; cohort 1 ( N = 3,325), cohort 2 ( N = 3,347), cohort 3 ( N = 3,322). The three cohorts were comprised of relatively equal numbers of students from special populations. The percentage of English Language Learners ranged between 1113% per cohort while the percentage of students from economically disadvantaged ba ckgrounds comprised 45 to 46% of the cohorts. The percentage of students from ethnic minority gr oups was also relatively constant at 5455% PAGE 6 Education Policy Analysis Archives Vol. 14 No. 2 6 across cohorts. Note however that th e exclusion of students who did not participate in all three test administrations, students who transferred schools, and students who received at least one modified test administration lowered the percentage of st udents from special populations below district averages. Implications associated with the di sproportionate exclusion of students from special populations will be addressed in the discussion. Measures Outcome data analyzed in the current study were student scale scores on the mathematics subtest of the TerraNova/CTBS5 Survey Plus. The Surv ey Plus is a standardized, vertically equated, norm referenced achievement test. All items are se lectedresponse. According to the publisher, the mathematics subtest measures a students ability to apply grade appropriate mathematical concepts and procedures to a range of problemsolving situ ations. The publisher reports KR20 estimates of reliability of .87 in grade 3, .89 in grade 4, and .87 in grade 5 (CTB/McGrawHill, 1997). Other measures utilized in the study were the fiveyear sc hool average (i.e., 19992000 to 200304) of the percentage of students eligible for a free or reduced lunch ( M = .58, SD = .28) and cohort enrollment size, averaged across the three student cohorts by school ( M = 42.27, SD = 18.81). Analytic Procedures Threelevel longitudinal models were estimated using the Hierarchical Linear Modeling (HLM) program, version 6.0 (Raudenbush, Br yk, Cheong, & Congdon, 2004). Models were estimated using student and school records that were collected in three data files. The first file (level1) contained student and school identifiers, mathematics scale sc ores from students in each of the three cohorts, and a field for grade level. This file contained 30,051 records (i.e., three records for each of 10,017 students). The level2 data file contained student and school identifiers and a field that designated cohort membership ( N = 10,017). The level3 data file contained only school identifiers ( N = 79). After preparing the data for analysis, an unconditional threelevel model was first used to estimate a mathematics growth trajectory for each elementary school student, to partition the observed parameter variance into its within and between school components, and to estimate the average achievement score and average growth ra te for each elementary school across the three student cohorts. The level1 model was composed of a longitudinal growth model that fitted a linear regression function to each individual students grade 3, 4, and 5 achievement scores. Equation 1 specifies the level1 model, Ytij = 0ij + 1ij(Grade 3)+ etij (1) where Ytij is the outcome (i.e., math ematics achievement) at time t for student i in school j, 0ij is the initial st atus of student ij (i.e., 3rd grade performance),1 1ij is the linear growth rate across grades 35 for student ij and etij is a residual term re presenting unexplained variation from the 1 By subtracting a value of 3 from GRADE initial status is defined as the expected achievement of student i in school j at the end of grade 3 [ 0ij + 1ij( 3 3 ) = 0ij]. PAGE 7 Successive Student Cohorts and Longitudinal Growth Models 7 latent growth trajectory. Levels 2 and 3 in the HLM model estimate mean growth trajectories in terms of initial status and growth rate across al l students (equations 2a and 2b) and across all schools (equations 3a and 3b). 0ij = 00j + r0ij (2a) 1ij = 10j + r1ij (2b) 00j = 000 + u00j (3a) 10j = 100 + u10j (3b) In equations 2a and 2b, it can be seen that the initial achievement status and growth of students is conceived as a functi on of school average achievement ( 00j) or school average growth ( 10j) and corresponding residuals ( r0ij, r1ij). Similarly, the initial status and growth by school in equations 3a and 3b is conceived as a function of the grand mean achievement ( 000) or the grand mean slope ( 100) and corresponding residuals ( u00j, u10j). Equations 3a and 3b were used to calculate the pooled estimates of school mean achiev ement (i.e., the mean performance of 3rd graders across the three cohorts) and school mean growth (i.e., the average 3rd to 5th growth rate of students across the three cohorts). The second model estimated included a term to represent changes over time in the performance of successive cohorts. As with the un conditional model, student growth trajectories were estimated at level 1 (see equation 1), but in th is model the achievement and growth of students was conceived to also vary at level 2 as a function of the temporal span from one cohort to another (coded with a value of 0 for the first cohort, a 1 fo r the second cohort, and a 2 for the third cohort). The linear cohort term represents the federal expectat ion, outlined in the NCLB legislation, that regular, annual progress in student proficiency be made from one cohort of students to the next.2 Equations 4a and 4b specify the level2 model. 0ij = 00j + 01j(Cohort) + r0ij (4a) 1ij = 10j + 11j(Cohort) + r1ij (4b) Using the above coding scheme for cohort membership, the intercept status parameter, school average achievement ( 00j) becomes the expected mean performance of 3rd graders in cohort 1 (200002) whereas the intercept growth parameter, school mean growth ( 10j) becomes the expected growth in achievement across grades 3 to 5 for the first cohort (200002). In addition, the cohort term ( 01j) can be interpreted as the expected change in the 3rd grade mean achievement of schools across the three cohorts and the cohort term ( 11j) can be interpreted as the expected change in school mean growth rates across cohorts. At level3, betweenschool variation in the initial achievement status and growth rate of schools and the schooltoschool differences in the cohort changes in achievement and growth were first modeled either in terms of the grand mean achievement ( 000) or the grand mean slope ( 100) of schools and corresponding residuals ( u00j, u10j) or the grand mean achievement change ( 010) or the 2 The expectation of regular annual progress most often assumes a linear increase in school performance over succeeding studen t cohorts. This assumption may not always hold. The performance of schools could, for example, change across student cohorts in a nonlinear fashion. In the present study, the time trend was modeled with a linear function as the time series was relatively short (three data points). When the time series is of longer duration, it may be nece ssary to represent the data with a more complex function. PAGE 8 Education Policy Analysis Archives Vol. 14 No. 2 8 grand mean growth change ( 110) of schools (across cohorts) and corresponding residuals ( u01j, u11; see equations 5a through 5d). Note that estimation of the residual variances enables assessment of the degree to which schools vary in the 3rd grade mean achievement in the first cohort (200002), u00j; the changes in 3rd grade mean achievement between the three cohorts, u01j; the achievement growth of elementary school stud ents in the first cohort (200002), u10j; and the changes in the achievement growth of elementary sch ool students between the three cohorts, u11j. Equations 5a through 5d were used to calculate the within and betweencohort school performance estimates. 00j = 000 + u00j (5a) 01j = 010 + u01j (5b) 10j = 100 + u10j (5c) 11j = 110 + u11j (5d) Results Mathematics Achievement across Cohorts Table 2 presents the results of model 1, the pooled HLM model. In the upper panel of Table 2, the results of the fixed effects regression model are presented. The first estimate shown, the grand mean ( 000), is the average 3rd grade mathematics scale score across all students. The second estimate, the grand slope ( 100), is the average yearly growth rate for those students. Across the three student cohorts, the average 3rd grade mathematics scale score was estimated as 616.97 while the average yearly growth rate across grades 3 to 5 was estimated to increase by 16.74 scale score units per year. In the next panel of Table 2, estima tes of the studenttostudent and schooltoschool variation in achievement and growth rates are pr esented. Chisquare tests of the models variance components indicated that students and schools differ ed significantly in achievement levels and the rate of achievement growth. The other estimates presented in the middle of Table 1 are the parameter reliabilities associated with each outcome measure. As can be seen in the table, most of the observed variability in the crosscohort paramete r estimates was true parameter variance (school mean achievement = .95, school mean growth = .84). The proportion of variation in student outcomes attributable to schools is presented in th e bottom panel of Table 2. Twentyone percent of the variation in student achievement level and 38% of the variation in student achievement growth was due to schooltoschool differences. To illustrate the schooltoschool differences in mathematics achievement averaged across the three cohorts, empirical Bayes (EB) estimates of the 79 elementary school mathematics mean achievement and mean growth rates are presented in the scatterplot in Figure 1. The horizontal line in the interior of the figure represents the cro sscohort grand mean achievement in mathematics. The vertical line in the interior of the figure represents the crosscohort grand mean growth in mathematics. The two grand mean reference lines classify schools into four quadrants of school performance. The upper right quadrant contains schools with above aver age crosscohort mean achievement in grade 3 and above average crosscohort growth from grades 3 to 5. The lower right quadrant contains schools with below average cro sscohort mean scores bu t above average growth. The two quadrants on the left side of the figure contain schools with be low average crosscohort growth and either high or low mean achievement. The spread of points in Figure 1 demonstrates that schools with low mean scores were not alwa ys low performing schools in terms of student PAGE 9 Successive Student Cohorts and Longitudinal Growth Models 9 growth in achievement. Similarly, above average school mean achievement at grade 3 did not always translate into above average growth across grades 3 to 5. Schools with low grade 3 mean scores had above or below average growth as did schools with re latively high grade 3 mean scores. The lack of a consistent relationship between the mean achievemen t and growth of schools is reflected in the correlation between the models level3 residual terms ( = .16). In these data, knowing a schools initial achievement status offered little insight into the subsequent achievement progress of students. Table 2 ThreeLevel CrossCohort Mode l for Mathematics Achievement Variable Parameter estimates Fixed Effects Coefficient SE t School Mean Achievement, 000 616.97 1.69365.81* School Mean Growth, 100 16.74 0.4041.50* Random Effects Variance Component df 2 Individual Achievement, r0ij 790.85 9938 24535.66* Individual Growth, r1ij 17.95 9938 10826.72* Level1 Error, etij 408.11 School Mean Achievement, u00j 214.19 78 2087.91* School Mean Growth, u10j 10.82 78 542.96* Reliability Estimates School Mean Achievement .95 School Mean Growth .84 Level1 Coefficient Percentage of Variation Between Schools Individual Achievement, 0ij 21.3 Individual Growth, 1ij 37.6 Results based on data from 10,017 students distributed across 79 elementary schools. p < .001 To assess the degree to which the estimates of school mean achievement and school mean growth were associated with schools social context (a measure of bias), corr elations between the EB estimates of school performance an d schools percentage free lunch rate were calculated. Percentage free lunch was strongly related to the average performance level of schools, r (77) = .81, p < .001. Schools with a larger percentage of students elig ible for free or reduced price lunch had student achievement levels that were lower than schools wi th smaller rates of free or reduced price lunch eligibility. However, knowing the percentage of the student body eligible for a free or reduced price lunch provided little insight into the average rate at which students learned mathematics across the three cohorts. A systematic relationship between percent free lunch and school mean growth was not observed, r (77) = .17, p > .05. PAGE 10 Education Policy Analysis Archives Vol. 14 No. 2 10 Figure 1. Crosscohort relation ship between school mean achievement and school mean growth in mathematics Mathematics Achievement by Cohort Table 3 presents the results of the second model that examined changes over time in the performance of successive cohorts. Estimates of th e models fixed effects are presented in the top panel of Table 3. The first estimate presented ( 000) is the average 3rd grade mathematics scale score for the first student cohort (200 002). The second estimate ( 010) is the average cohorttocohort change in 3rd grade mean scale scores. Thes e estimates indicate that the 3rd grade mean achievement of the first student cohort was 619.39 and that the 3rd grade mean achievement of schools decreased by 2.43 scale score points on average with each successive student cohort. The next estimates presented are the average growth rate across grades 3 through 5 for the first student cohort ( 100) and the average cohorttocohort change in longitudinal growth rates ( 110). These estimates indicate that the first student cohort grew an average of 15 .75 scale score points per year and that the mean growth rate of schools across grades 3 through 5 wa s increasing by an average of 1.03 scale score units with each successive cohort. Variance estimates are presented next in Table 3. Chisquare tests demonstrated that in the first cohort of students students and schools differed significantly with respect to achievement levels and rates of growth. Further, these tests also indicated that schools differed with respect to the changes in successive cohort performance. Statistically significant schooltoschool variation was observed in the changes in 3rd grade mean achievement and the grade 3 to 5 changes in achievement growth between co horts. Parameter reliability estimates are presented PAGE 11 Successive Student Cohorts and Longitudinal Growth Models 11 in the bottom panel of Table 3. As with model 1, the mean achievement (.92) and mean growth (.78) of schools were estimated with relatively high pa rameter reliability, but note that these estimates were somewhat lower than their counterparts fr om the previous model that estimated the mean achievement and growth of schools across three st udent cohorts. In addition, the between cohort estimates of school performance were noticeably le ss reliable than within cohort mean achievement and growth estimates. Only half of the observed variability in the cohorttocohort changes in mean achievement status (.51) and twothirds of the obs erved variability in the cohorttocohort changes in mean achievement growth (.68) was true parameter variance. Table 3 ThreeLevel BetweenCohort Mode l for Mathematics Achievement Variable Parameter Estimates Fixed Effects Coefficient SE t School Mean Achievement, 000 616.97 1.69365.81* Mean Achievement Change, 010 2.43 0.613.96* School Mean Growth, 100 16.74 0.4041.50* Mean Growth Change, 110 1.03 0.343.02* Random Effects Variance Component df 2 Individual Achievement, r0ij 775.02 9859 23855.67* Individual Growth, r1ij 13.29 9859 10576.21 Level1 Error, etij 408.11 School Mean Achievement, u00j 292.95 78 1212.08 Mean Achievement Change, u01j 15.43 78 167.80 School Mean Growth, u10j 10.82 78 542.96* Mean Growth Change, u11j 6.23 78 259.49* Reliability Estimates School Mean Achievement .92 Mean Achievement Change .51 School Mean Growth .78 Mean Growth Change .68 Results based on data from 10,017 students distributed across 79 elementary schools. p < .001 The betweencohort change in school performan ce is illustrated in Figures 2 and 3. Fitted trajectories representing cohorttocohort changes in the 3rd grade mean achievement of schools are presented in Figure 2.3 In Figure 2, it can be seen that schools differed in terms of the mean achievement of the first student cohort and in ter ms of the change in mean achievement of 3rd graders over time. It can also been seen that while mean achievement of schools was generally decreasing over time, the c ohorttocohort changes in the mathematics achievement of 3rd graders 3 To better demonstrate the directional change in school performance over successive cohorts, fitted trajectories are presented in Figures 2 and 3. The fitted trajectories mask the yeartoyear fluctuations in cohort performance. PAGE 12 Education Policy Analysis Archives Vol. 14 No. 2 12 were relatively modest. The systematic association between the 3rd grade mean achievement of the first cohort and subsequent changes in cohort grade 3 mean performance is also evident in Figure 2. In Figure 2, it can be seen that the mean ac hievement of schools tended to regress toward the district mean and thus become more homogenous with each succeeding cohort. In other words, schools with a highachieving 200002 cohort tended to demonstrate lower 3rd grade mean performance in subsequent student cohorts an d schools with a low achieving 200002 cohort tended to demonstrate higher 3rd grade mean performance over subsequent cohorts. The correlation between the mathematics performance of 3rd graders in the 200002 cohort and the estimated change in the average mathematics performance of 3rd graders in subsequent cohorts was negative and relatively strong ( 00, 01 = .70). 580 590 600 610 620 630 640 650 660 cohort 1cohort 2cohort 3 Student CohortSchool Mean AchievementFigure 2. School mean achiev ement in mathematic s as a function of student cohort A similar picture emerged when cohort growth rates were examined. Figure 3 presents the cohort growth trajectories by school. In Figure 3, schooltoschool differences in the growth rate of cohort 1 and the changes in cohort growth over time can be seen. School changes in cohort growth rates tended to be positive and somewhat more variable than the changes in school mean achievement displayed in Figure 2. However, the sa me overall pattern of relationship between initial status and subsequent change was again evident. Schools with a high performing 200002 student cohort (in terms of growth) had relatively less successful succeeding cohorts while schools that had an initially low performing cohort had higher grow th rates with the following student cohorts. The relationship between the initial and subsequent gr owth of cohorts was negative but smaller in magnitude than the relationship between the me an achievement status and mean achievement change of cohorts ( 10, 11 = .59). PAGE 13 Successive Student Cohorts and Longitudinal Growth Models 13 0 5 10 15 20 25 30 cohort 1cohort 2cohort 3 Student CohortSchool Mean GrowthFigure 3. School mean growth in math ematics as a functi on of student cohort To assess whether the cohortto cohort change scores were al so associated with cohort enrollment size, school change estimates were pl otted against the threeyear cohort enrollment averages. Figure 4 presents the relationship between cohorttocohort changes in school mean growth and cohort enrollment size. In Figure 4, it can be seen that schools with small enrollments were more likely than schools with large enrollments to have above or below average changes in mean growth between cohorts.4 With the exception of one outlyi ng school (school 23), large cohorttocohort changes in school mean growth tended to be concentrated in schools with relatively small enrollments. A similar pattern emerged when changes in school mean achievement were plotted against cohort enrollment size. The greater succe ssive cohort change estimates for smaller schools suggest that relative to their larger counterpa rts, schools with small enrollments have greater potential for changes in the achievement outcomes of students. However, the differential impact (both positive and negative) of small enrollments is likely attributable to the heightened potential for differences in the composition of student cohorts, rather than any systematic differences in school policy or practice. 4 School changes in mean growth (between cohorts) were averaged across two change cycles, thereby reducing some of the variab ility in the change estimates. PAGE 14 Education Policy Analysis Archives Vol. 14 No. 2 14 Figure 4. Relationship between cohorttocohort changes in sc hool mean growth and cohort enrollment size by school Discussion With passage of NCLB, states are required to restructure their accountability systems to comply with a uniform set of federal guidelines. These guidelines outline the content areas (i.e., mathematics, reading/language arts) and students to be tested, the frequency of testing, the methodology to be used for evaluating school perfor mance, and the set of consequences that befall schools failing to demonstrate adequate student achievement outcomes (NCLB, 2002). Of the changes NCLB has introduced to state accountability systems, one of the most far reaching stems from the manner in which school performance is to be evaluated. Under NCLB, states are required to annually evaluate schools in terms of the percenta ge of students who are at or above a particular cutpoint or proficiency standard (i.e., status) and/or by the cohorttocohort change in the percentage of students who reach proficiency (i.e ., quasichange). The proficiency standard used to evaluate school performance is allowed to vary by state, but NCLB requires adequate yearly progress (AYP) toward the goal of having 100% of students reach the stateadopted standard in each content area by the year 201314. PAGE 15 Successive Student Cohorts and Longitudinal Growth Models 15 Although NCLB is laudable in its aim to push schools toward providing an effective and equitable education for all students, concerns about the methodology used to evaluate school performance have been raised (Linn, 2003; Linn, Baker, & Betebenner, 2002; Ponsciak & Bryk, 2005; Raudenbush, 2004; Stevens, 2005). Of pa rticular concern is whether NCLB methods for measuring school progress reliably capture the im pact that schools have on students and whether these methods are biased against schools that se rve students from disadvantaged backgrounds. Validity concerns about the linkage between treatmen t (i.e., instruction) and outcome (i.e., student achievement) and the differential accountability bu rden placed on schools with challenging intakes stem directly from the manner in which school perf ormance is assessed. The achievement status and quasichange approaches endorsed by NCLB m onitor school effectiveness as a function of the absolute level of student performance and/or with respect to the change in student status across successive student cohorts. The use of measures that index the achievement st atus of a single cohort (relative to a proficiency target) or the change in status between two successive cohorts present a challenge for states as schools can be identified as in need of improvement on the basis of factors (e.g., student demographics) that are outside of the schools control (L inn & Haug, 2002; Kane & Staiger, 2002; Raudenbush, 2004). The potential for factors exogenous to the school to confound the measures of school performance endorsed by NC LB has led to calls for a reexamination of the school performance indices that are used to eval uate schools under NCLB (National Conference of State Legislatures, 2005; Olson, 2004). Of particular interest to system stakeholders is the potential for growthbased measures of school performance to enhance the fairness and equity of the federal accountability framework. In the present study, the viability of using stud ent growth rates as a means for evaluating the achievement progress of schools was in vestigated to ascertain whether indices of students growth in achievement provide a reliable and valid alternati ve to the status and quasichange approaches to school evaluation endorsed by NCLB. The investigation was based on the analysis of achievement data from three longitudinally matched elemen tary school student cohorts from a large school district in the southwestern United States. Results indicated that the cro sscohort performance of schools differed depending on whether the mean ac hievement status or growth of students was considered. Across the three cohorts studied, the relationship between the initial achievement status of students and students subsequent achievem ent progress was quite weak as schools with high initial achievement (averaged across cohorts) were generally as likely as schools with low initial achievement to have low, average, or high mean achievement growth. The same was generally true of the relationship between school demographics and school mean rates of achievement growth. Knowing the percentage of the student body eligible for a free or reduced price lunch provided little insight into the average rate at which students learned mathematics across the three cohorts. However, the freelunch percentage was strongly related to the average performance level of schools. Schools with a larger percentage of studen ts eligible for free or reduced price lunch had student achievement levels that were lower than sc hools with smaller rates of free or reduced lunch eligibility. Betweencohort estimates of school improvement (i.e., cohorttocohort changes in student achievement) provided an additional perspective on the effectiveness of schools by indexing the degree to which school performance changed with each succeeding student cohort. Over the study period, schools tended to have lower mean achiev ement scores but increased rates of student growth. On average, the mean achievement of thir d graders decreased by close to two and a half scale score points per cohort while the growth in mathematics achievement across grades 3 to 5 increased by slightly more than one scale scor e point per year with each succeeding cohort. The overall cohorttocohort changes in school perf ormance were thus relatively modest and equalizing across cohorts. In other words, the d ecreases in third grade mean achievement were PAGE 16 Education Policy Analysis Archives Vol. 14 No. 2 16 small in magnitude and offset by increases in student achievement growth so that the grand mean performance of 5th graders remained relatively constant across the study period. The negative relationship between the mean achievement and gr owth of schools was reflected in the correlation between the models residual change estimates (01, 11 = .61). Schools that had increases in the third grade achievement status of subsequent student co horts were less likely to have an increase in cohort growth rates and vice versa. In fact, onl y a handful of schools had either simultaneous increases or decreases in the achievem ent and growth of student cohorts. The difficulty schools face in delivering continual increases in student achievement outcomes was also reflected in the coefficients rela ting the achievement and growth status of cohort 1 to the changes in student achievement and growth between cohorts. In both instances, schools with high initial performance (either in terms of mean achievement or mean growth) were less able than schools with low initial performance to demo nstrate positive changes in the achievement and growth of subsequent student cohorts. Clear eviden ce of the regression effect is displayed in Figures 2 and 3. In these figures, it can be seen that while the mean achievement of third graders was slightly decreasing and the growth in achievement across gr ades 3 to 5 was slightly increasing from cohorttocohort, school performance estimates were becoming more similar over time. For the majority of schools then, student performance changed very little from cohorttocohort on either outcome. However, for those schools with relatively extrem e initial status performance estimates, the observed cohorttocohort changes in student achievement served to homogenize school performance as student achievement and growth tended to regre ss toward the districts achievement status and growth averages. The relationship between the performance status of cohort 1 and the subsequent changes in achievement between cohorts is an indication that a schools ability to increase student achievement outcomes may be contingent upon how well students initially perform.5 However, it is worth noting that the changes in school performance were related to student cohort size as well. Schools with smaller student cohorts had greater changes in st udent outcomes than schools with larger cohorts. The greater volatility of the successive cohort chan ge estimates for smalle r schools follow in part from the heightened potential for differences in the makeup of student cohorts to occur when schools serve relatively small numbers of studen ts (Kane & Staiger, 20 02; Linn & Haug, 2002). The volatility of the cohorttocohort school improvement estimates was also reflected to some degree in the consistency with which these parameters were es timated. Relative to the consistency with which the crosscohort mean achievement and growth of schools was estimated, cohorttocohort changes in school mean achievement and school mean gr owth were noticeably less reliable indicators of school performance. In many respects, results of the current study were consistent with other recent investigations of the reliability and validity of various school performance indicators. As with findings from other recent studies, the level at wh ich students in a school achieved (i.e., school mean achievement) was estimated with high reliability but was closely tied to the level of economic hardship experienced by students (Raudenbush, 2004). In addition, the modest changes in school mean achievement, the negative relationship b etween initial cohort mean achievement status and successive cohort mean change, the greater volatility of the mean change estimates for small schools, 5 The relationship between initial status and school changes in performance could also be due to district policies, including those aimed at school im provement, that are sufficiently uniform to draw achievement scores together. In other districts or in national samples, regressi on effects may not be as pronounced. PAGE 17 Successive Student Cohorts and Longitudinal Growth Models 17 and the overall less reliable estimates of cohor ttocohort changes in mean achievement also mirrored other recent findings (Hill & DePascale, 2003; Kane & Staiger, 2002; Linn & Haug, 2002; Schwarz, Yen, & Schafer, 2001). The current stud y was unique, however, in the focus on evaluating school performance from the perspective of change s in individual student achievement across and between cohorts. Estimates of the growth in student achievement across cohorts tended to provide a relatively reliable and unbiased measure of sc hool performance. However, estimates of the cohorttocohort changes in student achievement growth sh ared similar properties with estimates of the successive cohort mean change score. For ex ample, cohorttocohort changes in student achievement growth were estimated with less reliability than the crosscohort student growth estimates. The average cohorttocohort changes in student achievement growth were also generally small in magnitude and tied closely to cohort enro llment size and the first cohorts initial growth status. These results further highlight the difficult ies associated with comparing successive student cohorts, even those that are longitudinally ma tched over time. Changes in school performance between student cohorts tend to be quite modest wh en averaged across schools while the changes in cohort performance for any one school can resul t from idiosyncrasies associated with the composition of the cohort being evaluated rather th an with any real change in the effectiveness of instruction at a school. Results of the current study provide some indication of the strengths and weaknesses associated with four distinct measures of school performance. However, consideration of sample and data limitations is necessary for contextualizin g the current findings. Specifically, it should be noted that the study was based on the normrefere nced mathematics achievement of students. The patterns seen in normreferenced math achievement ma y not be the same in other subject areas or if scores were taken from a criterionreferenced in strument. Results were also based on achievement data from a select, nontransient student sample. Th e sample analyzed differed (in terms of student demographics) from the districts general student population and may have produced an upward bias on estimates of student and school achievement outc omes (Zvoch & Stevens, 2005). The study also focused on the analysis of entire student cohor ts. The focus on the achievement performance of entire student cohorts differs from the NCLB requirement that achievement outcomes also be disaggregated by student subgroups. The achiev ement outcomes associated with disaggregated groups may or may not mirror the results reported here although it is likely that due to the smaller size of student subgroups, estimates of yeartoyear changes in school performance would be more volatile. Generalizability concerns also follow from the analysis of data from a single southwestern school district. As with other school districts lo cated in the same geographic region, the district studied serves large numbers of Hispanic students and large numbers of Englishlanguage learners. The high percentage of students from these demo graphic groups distinguishes this district from many others in the United States and may limit the generalizability of results. The study also may have been limited to some degree by constraints associated with the data structure. Of particular concern is that achievem ent data were not available until students were in grade 3. Not having data on students kindergar ten entry status and achievement growth from kindergarten to grade 2 makes it difficult to know the true school effect on students. For example, schools that appeared average in terms of student gr owth across grades 3 to 5 may have been either more or less effective for students across kindergarten to grade 2. In the former scenario (i.e., high kindergarten to grade 2 growth, average grade 3 to 5 growth), the school would be judged as less effective than warranted. A related concern follo ws from the number of cohorts available for analysis. In the current study, estimates of school improvement were based on the changes in performance between three student cohorts. The small number of cohorts available for estimating school trends in achievement along with the obser ved fluctuations in cohort performance led to relatively unreliable estimates of school improvement. Although not inconsistent with findings from PAGE 18 Education Policy Analysis Archives Vol. 14 No. 2 18 previous investigation of school achievement trends (Hill & DePascale, 2003; Kane & Staiger, 2002; Linn & Haug, 2002), current school improvement estima tes may not have been as indicative of the true change in school achievement outcomes as would be required for high stakes decisionmaking (see Bryk, et al., 2004; Raudenbush, 2004). Although the aforementioned limitations suggest a need for additional research on the multiyear performance of schools, both within and between cohorts and across different sampling conditions, the current study does provide a glimpse into the potential usefulness of various indicators of school performance. Of the four mea sures examined in the current study, estimates of the growth of students within cohorts (or aver aged across cohorts) offered the most favorable combination of attributes for assessing the effecti veness of schools. Although slightly less reliable than estimates of school mean achievement, esti mates of school mean growth were more reliable than either of the cohort change measures. School mean growth estimates were also less confounded by student demographics than their sc hool mean achievement counterparts. In addition, by capturing the gains that students achieve over time instead of student performance on a particular testing occasion, school mean growth tends to be a more conceptually defensible indicator of school performance. The combination of attributes afford ed by the achievement growth estimates coupled with the difficulties associated with the mean achievement and successive cohort change measures suggest that consideration should be given to in corporating growth measures (either within or averaged across cohorts) into state accountability sy stems. One approach to utilizing growth data for school accountability purposes would be to eval uate schools on the basis of the percentage of students meeting an annual growth target. Assessi ng school performance with respect to the percent of students meeting expected growth instead of the percent of students proficient at any one time would potentially enable schools serving disadvan taged student populations to demonstrate positive instructional impacts on students and simultaneously keep schools with advantaged intakes honest. Utilizing the growth of students as a measure of school performance would also enable states to avoid evaluating schools on the basis of inhe rently volatile shortterm successive cohort comparisons. A change in accountability focus from statusbased measures to student growth indices would not be without difficulty however. I ssues surrounding student mobility, test alignment and equating, the setting of growth targets, demo graphic change, and incomplete time series data lead to a different set of challenges for the design of state accountability systems (Bryk, et al., 2004; Gong, 2004). Nevertheless, if the effectiveness of schools is to be determined on the basis of student performance on standardized tests, it seems reasonable to construct an accountability framework that enables schools to be evaluated on an outcome measure that more closely taps the school contribution to student learning. PAGE 19 Successive Student Cohorts and Longitudinal Growth Models 19 References American Educational Research Association, Am erican Psychological Association, & National Council on Measurement in Education (1999). Standards for educational and psychological testing Washington, DC: Authors. Baker, E. L., & Linn, R. L. (2004). Validity issue s for accountability systems. In S. H. Fuhrman & R.F. 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J., & Mendro, R. L. (1997). The Da llas valueadded accoun tability system. In J. Millman (Ed.), Grading teachers, grading sc hools: Is student achiev ement a valid evaluation measure? (pp. 8199). Thousand Oaks, CA: Corwin Press. Willett, J. B. (1988). Questions and answers in the measurement of change. In E. Rothkopf (Ed.), Review of research in education 198889 (pp. 345422). Wa shington: American Educational Research Association. Willms, J. D. (1992). Monitoring school performa nce: A guide for educators. Washington, DC: Falmer Press. Zvoch, K., & Stevens, J. J. (2005). Sample exclusion and st udent attrition effects in the longitudinal study of middle school mathematic s performance Educational Assessment, 10 (2) 105123. PAGE 22 Education Policy Analysis Archives Vol. 14 No. 2 22 Zvoch, K., & Stevens, J. J. ( 2003). A multilevel, longitudinal model of middle school math and language achievement. Educational Policy Analysis Archives, 11 (20). Retrieved July 8, 2003, from http://ep aa.asu.edu/epaa/v11n20. About the Authors Keith Zvoch University of Nevada, Las Vegas Joseph J. Stevens Affiliation information Email: zvochk@unlv.nevada.edu Keith Zvoch is an Assistant Professo r in the Department of Educational Psychology at the University of Nevada, Las Vegas. His re search interests include program evaluation, educational assessment, an d school acco untability. Joseph J. Stevens is an Associate Professor in the Department of Educational Leadership at the University of Oregon. His rese arch interests are assessm ent, validity, policy and accountability issues. PAGE 23 Successive Student Cohorts and Longitudinal Growth Models 23 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 Greg 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 Appalachia Educational Laboratory 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 Anthony G. Rud Jr. Purdue University Michael Scriven Western Michigan University Terrence G. Wiley Arizona State University John Willinsky University of British Columbia PAGE 24 Education Policy Analysis Archives Vol. 14 No. 2 24 EDUCATION POLICY ANALYSIS ARCHIVES Englishlanguage Graduate Student 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 University of Minnesota 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 25 Successive Student Cohorts and Longitudinal Growth Models 25 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 (19982003) Roberto Rodrguez Gmez Editorial Board Hugo Aboites Universidad Autnoma MetropolitanaXochimilco 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 EconmicaCIDE 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 Mariano Narodowski Universidad To rcuato 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 Un iversidad 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 PublicasUniversidad de Buenos Aires, Argentina Antonio Teodoro Universidade Lusfona Lisboa, Carlos A. Torres UCLA Jurjo Torres Santom Universidad de la Corua, Espaa 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 E1100475 0 245 Educational policy analysis archives. n Vol. 14, no. 2 (January 20, 2006). 260 Tempe, Ariz. : b Arizona State University ; Tampa, Fla. : University of South Florida. c January 20, 2006 505 Successive student cohorts and longitudinal growth models : an investigation of elementary school mathematics performance / Keith Zvoch [and] Joseph J. 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