Class size effects on student achievement in Norway: patterns and explanations.

• Author: Bonesronning, Hans

1. Introduction

   Most recent studies of class size effects are occupied with the problem of simultaneity. Parents who care about the education of their children may seek schools with small classes, which confound the true relationship between class size and student achievement. Alternatively, students may be allocated to small classes on the basis of "unfavorable" student characteristics that are not fully observable to the researcher, which makes class size negatively correlated with the error term in ordinary least squares estimations. While recognizing these problems, many prior studies are based on little variation in class size, and what variation there is may be endogenous. The present study takes advantage of two features of the Norwegian schooling system. First, there is a rule limiting maximum class size to 30 students, which causes exogenous class size variation. Second, class size variation is substantial because there is a large number of small schools. The approach is very similar to the one outlined by Angrist a nd Lavy (1999), who use Maimonides' rule of 40 in Israel to construct instrumental variable estimates of class size effects on student achievement.

   The maximum class size rule of 30 students in lower secondary schools in Norway is used to isolate the exogenous portion of class size variation and estimate its effect on achievement. This instrument solves the potential endogeneity problems that emerge from nonrandom assignments of students to classes by teachers and school administrators, but a crucial identifying assumption is that parents do not selectively choose schools on the basis of school quality. Angrist and Lavy, following Campbell (1969), argue that if the up-and-down pattern in the class size/enrollment relationship is reflected in the achievement growth pattern, parents selectively choosing schools are likely to be less of a problem for the analysis. The present paper seeks to take advantage of these discontinuities in a similar way to Angrist and Lavy.

   The initial empirical analyses show that a class size reduction induces a small increase in student achievement in the lower secondary school. The class size effect is significantly different from zero only for some student subgroups. Several recent studies report small and significant class size effects, at least for some student subgroups (Akerhielm 1995; Hanushek, Rivkin, and Kain 1998; Angrist and Lavy 1999), but Hoxby (2000) reports no class size effects. The studies referred to apply data samples that come from different educational systems and involve students of different ages. Class size may matter somewhere--for specific class size ranges, for specific students, for specific subjects, for specific incentives, and so on. Little is known about these circumstances, and this motivates the second section of the paper.

   The issue to be investigated is why are (some of) the students in the Norwegian lower secondary school positively affected by class size reductions? The theoretical class size literature provides the point of departure. This literature starts out from the notion of teaching as a public good, but the issue is more complex. Some element of teaching may have the character of a private good. Brown and Saks (1975, 1980) assume that the teachers choose between lecturing, which is like a public good, and tutoring, which is like a private good. While lecturing implies that class size does not matter, tutoring implies that it might because tutoring time per student increases when class size decreases. Brown and Saks (1987) provide empirical evidence that teachers respond to a decrease in class size by reallocating resources toward low-achieving students. Thus, there may be winners and losers from class size changes.

   Lazear (2001) views teaching as a congested public good. Students who are disruptive in class generate congestion. The probability that one student is not impeding his or her own or any others' learning at any moment of time is given by p. Learning time is consequently [p.sup.n], where n is the number of students in the class. Given p

   An investigation of whether the revealed class size effects are larger in environments that generate many disruptive students is carried out to evaluate Lazear's theory. Surprisingly, the estimation results seem to indicate that the class size effects are larger in "nice" environments, that is, in environments that generate few disruptive students. Notably, class size effects for female students are revealed only in such environments. Thus, if congestion effects are present, they seem to be dominated by other effects.

   One such effect might be that individual students make more effort in nice environments and that student effort and teaching are complementary inputs in education production. Then even a small increase in teaching time (induced by a class size reduction in a nice environment) may generate much improvement in student performance. A measure of student effort in class is used to evaluate this hypothesis. The investigations show that female students respond to nice environments by providing more effort, which can potentially explain why class size effects are revealed only for females in such environments.

   The class size theories presented previously emphasize that the achievement effects of class size changes emerge from changes in teaching time (per student). Neither theory is occupied with the potential substitution problems that occur in the classroom. Effort substitution is a major issue in the teacher-student interaction model established by Correa and Gruver (1987). In this model, student achievement is portrayed as a function of student effort and teacher effort, and it is shown that an increase in teacher effort toward individual students may not raise student achievement if individual students respond by decreasing their own efforts. Correa and Gruver show that the teachers' grading practices may mitigate the substitution problems. Inspired by this model, the analysis presented here establishes a measure of the teachers' grading practices and provides empirical evidence that the class size effects depend on the grading practices chosen. Moreover, the effects differ across student subgroups. An interpr etation suggested by the findings is that when the proper incentives are established, some student subgroups exploit the opportunities generated by smaller classes.

2. Sampling Procedures and Data

   From a researcher's point of view, the exogenous variation in class size generated by interaction between enrollment and the maximum class size rule is an attractive feature of the Norwegian schooling system. Unfortunately, investigations of class size effects cannot easily be carried out because no national test programs are established. The test scores applied in the present paper come from tests initiated as part of the current research program. The Appendix provides further description of this project.

   Testing all Norwegian students within a cohort was beyond the capacity of the project. A sample had to be chosen. I wanted to maximize the advantages of having substantial variation in the explanatory variables without generating problems for the estimation strategy. A nonrandom sample was generated according to the following guidelines. Norway was separated into five regions (north, mid, west, south, and east). From each region, the county (from among a group of three to five counties) that revealed the largest variation in expenditure per student among its local governments was chosen. In the resulting sample, the average expenditure per student is 59,800 Norwegian kroner (NOK) (approximately $7000) with a standard deviation of 15,900 NOK among the local governments. This should be compared with a national average (standard deviation) of 57,500 NOK (13,000 NOK). Thus, the sample average expenditures are 4% higher than the population average, and the standard deviation in the sample is 22% higher than in the population. Both sampling criteria contribute to this pattern. First, the requirement that all five regions are represented with one county implies that relatively small schools located in rural areas are oversampled. For instance, the northern region (average expenditure 65,300 NOK) has a very scattered population and relatively few schools, most of which are small, while the eastern part (average expenditure 52,600 NOK) has a highly concentrated population with many large schools. Second, the requirement that the chosen county has the highest within-county variation in expenditures separates out the most heterogeneous counties. The resulting sample consists of counties that broadly are characterized by a few cities surrounded by sparsely populated areas. The city schools are fairly large with high average class sizes, and the rural schools are smaller with smaller average class sizes. Since the variation in enrollment among the schools represented in the sample is higher than in a representative national s ample, the sampling procedure fulfills the requirement of substantial variation in the explanatory variables. It is difficult to see that this procedure introduces additional endogeneity problems. This issue is discussed further here.

   The lower secondary school covers the 8th through the 10th grade, with students from 13 to 16 years old. Typically, students from a number of small elementary schools are enrolled in one larger lower secondary school. The cumulative class size effects up to the age of 13 have to be separated out to reveal the class size effects experienced in the lower secondary school. Hence, a longitudinal design was chosen for the sampling of individual student data. One cohort of lower secondary school students in the five counties was tested in mathematics on four occasions during two years starting in the fall of 1998 (with tests at the beginning and end of both 9th and 10th grades). A longitudinal design is vulnerable to...