Class size, class composition, and the distribution of student achievement

Abstract

Using richly detailed data on fourth‐ and fifth‐grade students in the North Carolina public school system, I find evidence that students are assigned to classrooms in a non‐random manner based on observable characteristics for a substantial portion of classrooms. Moreover, I find that this non‐random assignment is statistically related to class size for a number of student characteristics and that failure to control for classroom composition can severely bias traditionally estimated class size effects. Teacher‐fixed effects and classroom composition controls appear to be effective at addressing selection related to classroom composition. I find heterogeneity in class size effects by student characteristics – students who struggle in school appear to benefit more from class size reductions than students in the top of the achievement distribution. I find that smaller classes have smaller achievement gaps on average and that class size reductions may be relatively more effective at closing achievement gaps than raising average achievement; however, class size effects on both average achievement and achievement gaps are small.

Keywords:

• class size
• education
• achievement gap

JEL Classifications:

• I20
• I21
• I28

Acknowledgements

I am grateful for helpful comments from Roger von Haefen, Laura Taylor, Trudy Cameron, two anonymous referees, and seminar participants at North Carolina State University and Utah State University. Any errors are mine.

Notes

1. Plausible mechanisms for this sorting include (1) administrators and teachers purposefully assigning students that struggle to smaller classes in an effort to help close achievement gaps; (2) administrators may assign the best teachers to larger classes and/or a different mix of student types in an effort to maximize the sphere of influence of these teachers (Burns and Mason 1998, 2002; Gamoran 1989); and (3) parents who highly value education may pressure administrators to have their children assigned to the best teachers (Clotfelter, Ladd, and Vigdor 2006b; Hui 2003; Lareau 1987; Sieber 1982).The first two explanations would be consistent with the incentives provided by the No Child Left Behind Act. Clotfelter, Ladd, and Vigdor (2006b) offer persuasive empirical evidence that ‘more highly qualified teachers are matched with more advantaged students.’ (See also Clotfelter, Ladd, and Vigdor 2005, 2006a). Cohen‐Zada and Reuven (2008) and Urquiola and Verhoogen (2009) show how non‐random assignment can arise from the institutional features of the school. See also Burns and Mason (1998, 2002).

2. NCERDC data are not available to the general public, but academic researchers can apply for access. Detailed information on the data available from the NCERDC can be found on their website: http://www.pubpol.duke.edu/centers/child/ep/nceddatacenter/index.html.

3. For this study, I restrict the analysis to classrooms with at least 15 students. Classrooms smaller than 15 students generally contain high proportions of students categorized as having one or more learning disabilities.

4. I calculate these tests for 13 different student characteristics whereas Clotfelter, Ladd, and Vigdor (2006b) calculate them for six characteristics. I have also performed this analysis using (less‐familiar) G‐tests. Sokal and Rohlf (1994) suggest that G‐tests may perform better in small samples, although the results I obtain using G‐tests are very similar to those obtained using χ² tests. See also Clotfelter, Ladd, and Vigdor (2006b).

5. Formally, χ² tests are computed as: where O_i is the observed count in a classroom in category i and E_i is the count expected under the null hypothesis.

6. It is worth noting that ‘academically gifted’ and ‘learning disabled’ are qualitative categories and the standards used to judge whether or not a student belongs in these categories are not necessarily uniform across, or even within, schools.

7. The estimation of education production functions has a very long history. See Todd and Wolpin (2003) for an in‐depth discussion of specification issues. See also, for example, Brown and Saks (1975), Babcock and Betts (2009), Belfield and Fielding (2001), and Foreman‐Peck and Foreman‐Peck (2006). Barrow and Rouse (2005) provide an overview of methodologies in this area.

8. The exceptionality heterogeneity index for classroom j (EH_j) is calculated as follows: The race/gender index is calculated analogously using the following groups: Black females, Black males, Hispanic females, Hispanic males, White females, White males, other race/ethnicity females, and other race/ethnicity males.

9. These sub‐samples are more restrictive than those used by Clotfelter, Ladd, and Vigdor (2006b) in their study of teacher effectiveness because I test for evidence of non‐random assignment using 13 student characteristics rather than six.

10. School‐fixed effects models suggest that classrooms with large concentrations of students categorized as ‘other ethnicity’ are larger, but teacher‐fixed effects models suggest they are smaller. One possible explanation for this result may be the existence of several schools in North Carolina with large populations of Native American students.

11. The dependent variable is standardized by subtracting the mean EOG test score for that year and grade and dividing by the standard deviation of EOG test scores for that year and grade.

12. I have also estimated these models with standard errors clustered at the classroom level. These results are similar to those reported. Full results are available upon request.

13. Only the coefficients on the class size variable and the associated t‐statistics from these models are reported in Table 5. Full results are available from the author upon request.

14. Although the inclusion of the previous year’s test score as a control variable is appealing on theoretical grounds, the inclusion of a lagged dependent variable may result in serial correlation in the errors. Although ordinary least squares (OLS) parameter estimates remain unbiased in the presence of serial correlation, the standard errors of those estimates may be calculated incorrectly. This bootstrapping procedure randomly draws, with replacement, individual student observations from the original dataset. When in a sample the size of the original dataset is obtained, OLS estimates of class size effects are obtained and recorded. By repeating this procedure, the characteristics of the resulting distribution of class size estimates can be used to construct standard errors. Given the large amount of computation time required to process the very large dataset and large number of fixed effects I use only 100 replications. Bootstrapping relies on the assumption that the estimation sample is representative of the population. This assumption is very defensible here because the estimation sample is the population of fourth‐ and fifth‐grade students in North Carolina public schools.

15. Sims (2008) finds that schools may also respond strategically to class size reductions by changing classroom composition. Bosworth and Caliendo (2007) provide theoretical evidence.

16. The unit of observation is the classroom standard deviation. These data are standardized by subtracting the mean classroom standard deviation for that year and grade and dividing by the standard deviation of classroom standard deviations for that year and grade.