School Crowding, Year-Round Schooling, and Mobile Classroom Use: Evidence

from North Carolina

Steven C. McMullena,*

, Kathryn E. Rouseb

a Calvin College, Department of Economics, North Hall #177, Grand Rapids, MI 49546, United States

b Elon University, Department of Economics, 2075 Campus Box, Elon, NC 27244, United States

First draft: Feb 2011

This Draft: May 1, 2012

* Corresponding author. Tel.: 616 526 6460; Fax: 616 526 8410

Email address: scm9@calvin.edu

Highlights:

• Severely crowded schools have a negative impact on reading achievement.

• Severely crowded schools have no discernable impact on math achievement.

• Mobile classrooms moderately hurt achievement.

• Year-round calendars moderately hurt achievement.

• Mobile classrooms and year-round calendars partially offset the impact of crowding.

1

Abstract

This study exploits a unique policy environment and a large panel dataset to evaluate the impact

of school crowding on student achievement in Wake County, NC. We also estimate the effects of

two education policy initiatives that are often used to address crowding: multi-track year-round

calendars and mobile classrooms. We estimate a multi-level fixed effects model to identify

effects that are not confounded by other school, family, and individual characteristics. Results

suggest that severely crowded schools have a negative impact on reading achievement but have

no discernable impact on math achievement. Both mobile classrooms and year-round calendars

are found to have a small negative impact on achievement in the absence of crowding, but a

positive impact in crowded schools, though these policies are only able to partially offset the

negative impact of crowding.

JEL classification: H75; I21; I28

Keywords: human capital, educational economics

2

1. Introduction

Faced with budget shortfalls and increased enrollments, many U.S. students now find

themselves in a severely crowded school (NCES, 2000; Kolodner, 2008; Graves, 2010). In this

paper, we investigate this little studied, yet important education policy concern by estimating the

impact of school crowding on academic achievement, as measured through standardized reading

and math test scores. We also estimate the effectiveness of two common policies often used to

address overcrowding: the use of multi-track year-round calendars and the use of mobile

classrooms. We study the case of Wake County, North Carolina, where school construction has

been unable to keep up with rapid population growth, leading to high levels of crowding in its

public schools. In fact, in 2007, 92 of the 124 (80.7%) Wake County Public School System

(WCPSS) schools were above 106 percent of the school’s capacity.1 The troubling trend is not

unique to the WCPSS. For instance, 38% of New York public school students attended an

overcrowded school in 2006-072 and more than 60,000 Chicago public school students attended

a school deemed overcrowded during the 2008-09 school year. 3 School overcrowding is an

important policy problem for several reasons. Most directly, it frequently leads to large class

sizes, which, though the evidence is mixed, have been found to negatively impact student

achievement (Angrist & Lavy, 1999; Krueger & Whitmore, 2000; Mishel, Rothstein, Krueger,

Hanushek, & Rice, 2002). Teachers report crowding affects their ability to facilitate classroom

activities, alters their instructional techniques, and leads to burnout (Rivera-Batiz & Marti,

1995). Crowded schools have also been associated with poor facility conditions, such as sub-

1 106% of school capacity is the NCES (2000) definition of an “overcrowded school.”

2 Kolodner, Meredith. “New York City schools suffer massive overcrowding, statistics show.” October 2, 2008.

http://www.nydailynews.com/fdcp?1285873431282. 3 ‘School Overcrowding: Limiting Hispanic Potential.” United Neighborhood Organization Report, April 2009.

3

standard electrical and lighting, heating/ventilation and air conditioning systems, among others

(Ready, Lee & Welner, 2004).

Because new school construction is often prohibitively expensive, schools often address

overcrowding concerns with alternative policies, which are themselves controversial and often

viewed as highly inadequate solutions (Ready, Lee & Weiner, 2004; National Center for

Education Statistics 2000). Mobile classrooms are perhaps the most common response to

overcrowding (Philipp and Wang, 2008; NCES, 2007). They have been criticized for health and

safety concerns (Coles, 1999; Heise and Bottoms, 1990; Kennedy, 2000; Callahan et. al., 1999),

their unappealing façade (Chan, 2009), and poor construction quality (Fickes, 1998; Chan,

2009). The use of multi-track year-round calendars is an additional solution that does not require

increased physical space. In contrast to the traditional 180-day calendar where nine months of

instruction are followed by a three month summer break, multi-track year-round calendars spread

the same 180 instructional days across an entire calendar year. Students are placed into a

particular track that comes with its own schedule, where at any point in time at least one track

(and their teachers) is on break. This attribute allows the school to accommodate a larger number

of students. According to the WCPSS, for instance, depending on enrollment, a multi-track

school can hold 20 to 33 percent more students than a traditional calendar school.4

Despite widespread public concern over the perceived negative impacts of school

crowding, few studies have examined the effects of crowding or these education policy responses

on student outcomes. While there is a substantial body of academic literature on the effects of

school size (Andrews, Duncombe & Yinger, 2002; Foreman-Peck & Foreman-Peck, 2006;

Kuziemko, 2006; Jones, Toma & Zimmer, 2007) and class size (Angrist, 1999; Krueger &

Whitmore, 2001, Mishel et al., 2002), the literature on school crowding and mobile classroom

4 http://www.wcpss.net/year-round/year-round-overview.html. (accessed February 10, 2011).

4

use is sparse and is primarily limited to public policy research briefs. Consequently, most of the

evidence is drawn from studies that are largely descriptive in nature. There is more recent work

on year-round calendars (Graves 2010, 2011; AUTHORS 2012), however, to date these studies

have not examined the impact of YRS as a solution for school crowding with individual level

data.

This study brings new empirical evidence to an area of education policy discussion

which, to date, has received little attention in the research literature. Wake County, North

Carolina, makes for an ideal study because fast population growth and a diverse policy response

has lead to a large amount of variation in school crowding, mobile classroom use and school

calendar models. By taking advantage of these policy changes, we are able to disentangle the

effects of school crowding from the policy responses. To estimate the academic effects of

crowding, we merge a large restricted-use panel dataset from the North Carolina Education

Research Data Center (NCERDC) with publically available data from the WCPSS. The unique

policy environment in the WCPSS combined with this rich panel dataset enables us to exploit

within-student and within-school variation using a multi-level fixed effects model with student,

school and grade-by-year fixed effects. Using this estimation method, our study improves upon

a literature that, with few exceptions, has been unable to address selection bias. In doing so, this

paper provides the first empirical estimates of these important policy concerns which are free of

the biases inherent in the limited existing literature.

The remainder of the paper is organized as follows. Background on is provided in

Section 2. Section 3 describes the data. Our empirical methodology is discussed in Section 4. In

Section 5, we present our results, in Section 6 we discuss issues raised by the study, and Section

7 concludes.

5

2. Background

2.1 Related literature

As mentioned above, despite it being an important and prevalent education policy

concern, there is little empirical evidence on school crowding or policy responses. Perhaps the

most sophisticated study is Rivera-Batiz and Marti (1995), which uses school-level regression

analysis to evaluate the impact of overcrowding in New York City public schools. Looking at

low-income schools, their findings suggest two to nine percent fewer students pass reading and

math proficiency tests in crowded schools. While the study is able to control for some observed

school-level covariates, it does not control for unobserved student or school characteristics,

which are likely correlated with both school crowding and achievement. This is problematic

because students are not randomly allocated into crowded schools. For instance, high achieving

students may be attracted to a particular school, increasing demand and causing crowding in

these schools. Alternatively, schools that have a high proportion of low-performing students

may have fewer resources and be less able to address growth with new school construction,

leading to crowding in existing schools. The failure of this and other existing studies to

appropriately control for unobserved heterogeneity makes it difficult to draw any causal

inferences on the impacts of crowding.

Similar to the crowding literature, most of the evidence on mobile classrooms, which is

summarized by Chan (2009), comes from qualitative case-study analyses or simple descriptive

quantitative methods. The general consensus reported in that paper is that portable classrooms

have little impact on student achievement, but again, it is difficult to draw any strong conclusions

without using more sophisticated methods.

6

Though still small, there is more robust empirical evidence on the effects of multi-track

year-round calendars.5 Graves (2010, 2011) studies the impact of multi-track year-round

calendars using school-level longitudinal data from California. In contrast to prior research, her

data allows her to include both school fixed effects and school time trends. Once she includes

these controls, estimates imply schools on multi-track calendars score slightly lower than their

traditional calendar counterparts. Graves also finds that this effect is larger in schools deemed

“critically overcrowded.” Using the same WCPSS and North Carolina data used in this study

AUTHORS (2012) use a model with both student and school fixed effects. We find multi-track

year-round schools have no discernable impact on student achievement.

This study builds on these previous studies in three ways. First, because of our rich panel

data, we can control for both school and student-level permanent unobserved heterogeneity. To

date, no previous literature on school crowding or mobile classrooms has addressed these

concerns. Second, our previous paper on year-round schooling does not address whether these

calendars have differential impacts in overcrowded schools, the schools for which they are most

often implemented. Though this question is explored in Graves (2010), it does not include

controls for student-level selection. Finally, and most importantly, no previous study is able to

explore crowding in a setting with this level of variation and with plausibly exogenous variations

in policy responses. We show in this paper that the effect of crowding can only be understood

when considered in conjunction with policies like year-round schooling and the use of mobile

classrooms.

5 For further year-round schooling research that does not include the attention to experimental design of these more

recent studies, see Cooper et. al.’s (2003) review, and McMillan’s 2001 study of schools in North Carolina.

Moreover, there is a large and closely related literature on school calendar inputs to student achievement, including

Goodman (2012), Marcotte (2007), Marcotte and Hemelt (2008), and Sims (2008).

7

2.2 Wake County Public Schools

Over the past decade, Wake County has experienced steady population growth, which, in

turn, has led to a rapidly growing school system, now the 18th

largest in the nation.6 During the

2003-2004 school year enrollment in the WCPSS was 108,970 and by 2008-2009 it had grown

by more than 26 percent to 137,706 students.7 This growth has resulted in excessive school

crowding. In 2007, 92 of the 124 (80.7%) Wake County schools were above 106 percent of the

school’s capacity, the threshold over which the NCES deems overcrowded (NCES 2000). This

large percentage of crowded schools is well-above the national average of roughly 25% as

reported by the U.S. Department of Education (2000).

The WCPSS has addressed this unprecedented growth in several ways. For one, from

2003 to 2009, the WCPSS built 33 new schools. While the number of new schools helped to

address growing demand, the excessive population growth often led to situations where new

schools were already too small when they opened or were needed before they were completed.

As a result, the school system has had to rely on additional policy changes. Many new WCPSS

schools were opened with mobile units and three schools were opened early as temporary

modular campuses.8 Mobile units were also added to existing schools to help ease

overcrowding. In 2003, the WCPSS had 649 mobile classrooms in its 126 schools. By 2009, the

system had more than 1,000 mobile units in its 159 schools.9 As an additional approach, a

growing number of multi-track year-round calendars were implemented in the WCPSS. These

calendars ease overcrowding by using the building for the entire calendar year. Students are

allocated into four distinct tracks, at least one of which is “tracked out” (not in school) at any

6 Wake County Public School System. 2011. http://www.wcpss.net/demographics/. (accessed February 7, 2011).

7 http://www.wcpss.net/demographics/quickfacts/index_qf.html (accessed February 16, 2011).

8 http://webarchive.wcpss.net/growth/big-picture/options_for_providing_classrooms.html (accessed May 20, 2011).

9 http://www.wcpss.net/demographics/quickfacts/index_qfreports.html (accessed March 3, 2011).

8

point in time. This allows a school to accommodate a larger number of students in the same

physical space. Use of these calendars in the WCPSS began in the early 1990s. Over the next

10 years, the WCPSS added 12 more year-round schools and by 2006, nearly 17,000 WCPSS

students attended a year-round school. The largest policy change, however, occurred in 2007-

2008, when the school system converted 22 elementary and middle schools from traditional to

year-round academic calendars.10

While these policies have been the source of concern and

controversy for the school system, from an econometric perspective, these attributes of the

WCPSS are desirable because they provide a relatively large amount of across-school and

within-school variation needed to identify the effects of interest.

In addition to the large variation in the variables of interest provided by the policy

changes in the WCPSS, the WCPSS also has a unique student assignment policy. In contrast to

most neighborhood-based school systems across the nation, students are assigned to schools

based on a goal of maintaining within-school socioeconomic diversity. Specifically, the goal of

the policy is to keep each school with fewer than 40% of its students receiving reduced-price or

free lunch. The school system does this through a controversial busing policy. Under this

policy, students may be switched to a different school if they have not moved or graded-out of a

school. As a result of this system, students and their parents are given little choice over their

school assignment, which helps ease some concerns with self-selection bias. Taken as a whole,

the rapid population growth, the policy experimentation, and busing policy make the WCPSS a

desirable location to study the effects of crowding and these education policy responses.

3. Data

10

http://www.wcpss.net/year-round/year-round_factsheet.html (accessed March 3, 2011).

9

This paper uses a restricted-use dataset from the NCERDC, which was created through a

partnership with the North Carolina Department of Instruction in 2000. The NCERDC, housed in

the Center for Child and Family Policy at Duke University, holds and manages student, school,

teacher, and district level data for all of N.C.’s public schools. This study uses student data

coming primarily from the individual-level EOG (end-of-grade) test and school level information

from the School Report Card files. In addition, we merge our main data with WCPSS school

capacity and mobile classroom use data that are publicly available on the WCPSS website, which

is available beginning in the 2004-2005 school year. While we observe the NCERDC student-

level data prior to 2005, the WPCSS data limits our analysis to WCPSS students in years 2004-

2005 through 2008-2009. The primary identification strategy relies on repeated observations of

students over time. The sample is therefore restricted to those cohorts for which we can

calculate at least two test score gains. The sample construction is illustrated in Table 1.11

The

final analysis sample includes 69,353 unique students, in 125 schools and results in a total of

186,711 student-year observations.

The primary crowding variables of interest include two discrete measures of crowding,

which we construct from the WCPSS data: moderately crowded and severely crowded. Schools

that are above 106% capacity are considered moderately crowded, while schools that are above

130% capacity are considered severely crowded.12

These crowding measures capture the relative

enrollment-to-building capacity ratio. In order to evaluate the use of mobile classrooms as a

response to crowding, we define the crowding level relative to the permanent building space by

11

Note, testing data from the prior year is used to calculate the test score gain for the first observation of each

cohort. For example, to calculate the 2004-2005 test score gain for cohort 1, we use testing data from the 2003-2004

files for sixth graders. 12

These definitions closely follow those used by NCES (2000), except that we increase the threshold for “severe

crowding” from 125 to 130% in order to better capture the distribution of crowding in our data, since we have an

unusually large number of highly crowded schools in our sample. This change in cutoff has almost no impact on the

reported results (results available upon request).

10

eliminating mobile classroom space from the denominator.13

Additionally, the building capacity

measure provided in the data by the WCPSS is adjusted for year-round schools, assuming that,

due to the alternate calendar, these schools have a higher capacity. To test whether year-round

calendars are better than traditional calendars for a given number of students, the margin of

interest for policy making, we reverse this adjustment. For those schools that we observe under

multiple calendars, we assume that the building capacity remains constant after a switch to a

year-round calendar, and re-compute the crowding measure. For those schools that are always

year-round, we impute the traditional-calendar building capacity with a linear prediction rule,

using parameters estimated with those schools that we observe under both calendars.14

The two primary policy response variables of interest include the use of mobile

classrooms and the use of multi-track year-round calendars. Unfortunately, we do not observe

whether a particular student is in a mobile classroom. However, we do observe the number of

mobile classrooms in each school and, with capacity information, are able to construct a variable

for the percent of a school’s campus capacity that comes from mobile classroom use.15

In this

study, we use the percent mobile classroom use, because it better captures the extent to which a

school relies on mobile classrooms compared to the raw number of mobile units. We determine

whether a student attended a year-round school by using a test date indicator taken from the EOG

files.16

13

If we use the common “campus capacity” measure of crowding, the relevant counterfactual is different and less

useful, but the results are similar, though the magnitude of the effects diminishes. 14

Unfortunately, we do not have access to the formula used by the WCPSS to adjust capacity for the year-round

schedule. Instead, using only the schools that we observe under both calendars, we use their year-round capacity to

predict their traditional calendar capacity, yielding these results: ���������� � 97.3 � 0.65 ����������. 15

Our percent mobiles variable is defined as ������� � 22 �!��"�#$ �������� � ������� � 22%⁄ where 22 is the

average class size. 16

For students who attended a year-round school, this variable is coded as “YROxx”, where xx indicates the year of

the test. In cases where this indicator variable is not constant across all students within the same school in a

particular year, we refer to publically available information on the WCPSS website regarding school type and

11

Table 2 reports yearly descriptive statistics for the crowding and policy indicators.

Looking first at the crowding data, it is apparent that the WCPSS has experienced excessive

levels of crowding over the last several years. In 2004-2005, nearly 80% of sampled students are

in a school that is deemed moderately or severely crowded. This percentage remains above 70%

across the entire time period and reaches a peak of 88.2% in 2007-2008. In fact, in this school

year, more than half of the county’s students are attending a school deemed severely

overcrowded. The high level of severe crowding in this school year reflects the high number of

schools that underwent calendar conversions, from the traditional to the multi-track year-round

model. While daily crowding likely decreased (because one track is out of the school at any

point in time), the adjusted measure used in this paper, (which captures the overall student-to-

building ratio) increases as more students are attending the same physical building.

The increased use of year-round calendars is reflected in the summary statistics in the

lower panel of Table 2. Whereas roughly 18% of the sampled schools are on the multi-track

year-round calendar from the 2004-2005 school year to the 2006-2007 school year, this number

jumps dramatically with the large policy change in the 2007-2008 school year (to 34.4%). We

also observe a relatively high percent of mobile classroom use in the sample. The average

mobile classroom in 2004-2005 is 14.3%; it reaches a peak of 18.4% in 2005-2006, and by 2008-

2009, drops to roughly 15%. Recall, our 2008-2009 sample includes 5th

to 8th

graders. Thus, if

mobile classroom use is more common in elementary schools, the slight drop in mobile

classroom use observed in the analysis sample may simply be a function of this sample

restriction.

manually change the indicator, if necessary. In our study, the four modified calendar schools are not considered

year-round. All YRSs in our sample, therefore, follow the school-wide, multi-track model of YRS.

12

The primary outcomes of interest are a student’s EOG math and reading test score gains.

These tests, typically administered in the spring, are given to all NC students in the third through

the eighth grade. Test dates are adjusted so that all students are tested at roughly the same point

in their school year. For example, in 2010-2011, traditional calendar students began testing on

May 11, day 156, while YRS students on track 4 began testing on June 1, or day 158.17

These

test date adjustments ensure that student outcomes are comparable across calendar type. The

tests are multiple-choice format and are designed to measure growth in competency. The scores

are therefore expected to increase as a student progresses through school. Similar to Bifulco and

Ladd (2006) and AUTHORS (2012), we address this issue and ensure that test score gains are

comparable by normalizing raw scores such that grade-by-year test score means are equal to zero

with a standard deviation of one. The variable of interest, the test score gain, is defined as the

change in test score from year 1−t to year t , or 11 −−− stigigst YY . In addition to test scores, the

EOG files also have demographic information on the students. These characteristics include a

student’s gender, ethnicity, and parents’ education level. Unfortunately, the parents’ education

data is often missing, particularly in the later years. However, examining student demographics

by the level of school crowding can provide an idea of the extent to which there appears to be

non-random sorting of students into schools.

Student descriptive statistics for these observable demographics and for achievement

gains are reported in Table 3. The observations are classified into the following three groups

based on the level of school crowding: (1) not crowded, (2) moderately crowded, and (3)

severely crowded. First looking at the math and reading test score gains, the raw means suggest

that there is no clear pattern in the mathematics achievement between students in crowded

17

Testing schedules are available at http://www.wcpss.net/evluation-research/reports/calendars/t_calendars.html

(accessed June 9, 2011).

13

schools and non-crowded schools. Compared to non-crowded schools, the raw mean in math

achievement is higher in moderately crowded schools, but is smaller in severely crowded

schools. However, the raw means suggest a clearer pattern for students’ reading achievement

growth, indicating lower achievement in crowded schools. Turning to the demographics, the

summary statistics reported in Table 3 indicate that students in crowded schools are, on average,

more likely to be white, Asian, and/or female, than students in non-crowded schools. The racial

differences between these students indicate that it is possible that the sample of students in

crowded schools is different than that of non-crowded schools. This suggests simple empirical

models that fail to control for the non-random selection of students may result in biased

estimates.

Table 4 reports school descriptive statistics by level of crowding. Similar to the student

level descriptive statistics, schools are classified according to their level of crowding (not

crowded, moderately crowded and severely crowded), which could vary within-school over time.

Not surprisingly, the more crowded schools have higher student-to-teacher ratios, class sizes, and

more students than their less crowded counterparts. It is worth noting, though, that the impact of

crowding on class size appears to be quite small. The means of school characteristics also

suggest that crowded schools tend to have students of higher socioeconomic status, a higher

percentage of fully licensed and experienced teachers, a lower teacher turnover rate, and a

slightly higher average daily attendance rate. These simple differences imply highly crowded

schools are non-randomly crowded and suggest school crowding may be associated with higher

quality students and higher quality school inputs. If crowding has a negative impact on student

achievement, empirical evaluations which fail to account for the student selection bias will lead

to estimates that understate the true impact of school crowding.

14

The potential selection bias suggested by the descriptive evidence presented in Tables 3

and 4 is addressed in this paper using a multi-level fixed effects approach. While this empirical

approach (described in detail in the following section) should give us estimates of the impact of

these policies that are free from many of the biases that could plague the cross-sectional studies

in the literature, there is the possibility that our panel data techniques will result in a non-

representative sample. As mentioned above, our analysis sample is restricted to those students

for whom we can calculate multiple test score gains.18

Moreover, our study identifies the effects

of interest from those students whom we observe multiple times in the same school under

different crowding levels, numbers of mobile classrooms, or calendar type. This means that

students who switch schools often are less likely to appear in our identifying sample. If the

impact of crowding or one of the policy responses is different for these more mobile students,

then our estimates will not reflect the true effects for the larger student population.

Such systematic differences in the samples could occur if students are intentionally

moving to avoid, or to take advantage of, one of the policies evaluated in this study. If students

move for this sort of reason and are left out of the sample as a result, our results could be biased

in favor or against the policies, depending on the type of movement. This scenario is unlikely to

be a problem in our case for a couple of reasons. First, because the district actively assigns many

students to schools to maintain a degree of economic diversity, student choice of schools is

somewhat more limited in Wake County than in other districts. Second, because many of the

schools are so crowded they likely would not accept students from outside their geographic

zones, giving students relatively limited mobility options.

18

A comparison of the full sample of WCPSS students in the relevant grades during the time period of interest

shows that the characteristics of the two samples are very similar, though our analysis sample is slightly more likely

to be white and female.

15

If the school system selects these movers in a non-random way, the results may still not

accurately reflect the impact on the larger student population. To explore this issue, Table 5

compares descriptive statistics for non-structural school switchers19

with the non-switchers in our

analysis sample. Results suggest that there is basically no difference in achievement across the

two groups, but the two groups do differ by race. Students in the switcher group are more likely

to be African-American and Hispanic. This kind of movement is consistent with the WCPSS’s

bussing for diversity policy and suggests the school system is more likely to bus minority

students, many of which are low income. Importantly, the similarities in test scores imply the

school system is not systematically choosing these minority students based on their achievement.

Moreover, when we estimate the effects by race, we find no evidence that minority groups have a

clear incentive to move in a particular direction or that the differential busing policy will have a

noticeable impact on our results.20

Given these observations along with the relatively small

proportion of non-structural switchers (<10%) in the analysis sample, the average estimated

effects on the larger Wake County student population would likely be very similar to the results

we report in this paper.

Finally, despite some assurances regarding the within-district movers in the analysis

sample, some students leave the district entirely over the time period. For similar reasons as

outlined above, this sort of exit will be problematic if it is a result of crowding or the policy

responses. We observe the number of students who leave the district over the years studied.

Despite a relatively high amount of variation in crowding and policy responses, the number of

19

Non-structural switchers are defined as those observations for which a student is observed in two subsequent years

in either two different elementary schools or two different middle schools. 20

In general, results from our preferred specifications are qualitatively similar across race. African American

students have slightly lower magnitude effects, with the exception of mobile classroom use and reading scores.

Hispanic students have higher magnitude impacts, though given the sample sizes it is difficult to draw firm

conclusions. Results by race are available upon request.

16

exits as a percentage of the school district population stays relatively constant over the period we

study.21

Furthermore, if students are constrained in their choice of public schools and students

want to avoid a year-round calendar, this should show up in the number of students leaving the

Wake County system when the large policy change occurred in 2007-2008. Yet, the exit rate

stays relatively stable even around the time of this large year-round schooling policy change. The

small changes in this exit rate that we do observe are not large enough to dramatically alter the

sample we work with, and thus the results.

4. Empirical Approach

The empirical challenge in estimating the causal impact of school crowding (or the policy

responses) on achievement is that students are not randomly allocated into overcrowded schools,

and policy responses might not be randomly allocated across schools. For instance, it may be

that high achieving students are attracted to a particular school, increasing demand and causing

high levels of crowding in these schools. Alternatively, schools that have a relatively high

proportion of low-performing students may also have fewer resources and be less able to address

growth with new school construction, leading to high levels crowding in existing schools. The

busing policy used by the WCPSS helps to partially offset this problem of reverse causation by

assigning students to schools across the county to balance socioeconomic diversity. Yet, even

with this unique policy, there are areas of the county that grow faster than others and, as

indicated by the summary statistics reported in Table 4, quality and crowding differences across

schools still persist. These attributes likely result in variation in school crowding levels that is not

totally random.

21

In the 2005-2006 school year, 4.36% of the population left the district for other schools, compared to 4.58% in the

following year, 4.68% in the first year after the policy change, and 4.26% in 2008-2009.

17

Similarly, in addition to student selection problems, the policy responses that we examine

might not be randomly assigned to schools. Schools might differ in unobserved characteristics

that help determine which policy response is chosen to alleviate the school crowding. If this is

the case, and if these unobserved characteristics also influence student achievement, then our

estimation strategy must deal with these factors in order to accurately measure the policy effects.

By including observed demographics and school characteristics, we can eliminate a

portion of the selection bias. However, it is still likely that there exist other, unobserved

attributes that are correlated with school crowding and achievement. To deal with this problem

of unobserved heterogeneity, we exploit the panel design of the dataset using a multi-level fixed

effects model. We estimate the impact of crowding on achievement using a variation of the

“value-added” approach common in the economics of education literature (Todd and Wolpin

2003). Specifically, the dependent variable is defined as the change in test scores between

periods and the model is written as follows:

igstgtsistststststigigst scrowdmcrowdYY εγµϕδβαα +++++++=− −− SP2111 , (1)

where igstY is the test score for student i in grade g at school s at time t. The variables

stmcrowd and stscrowd are indicator variables that are set equal to one if school s is

moderately or severely crowded at time t, respectively. stP is a vector containing the two policy

variables ( percent mobile classroom and year-round calendar use) for school s at time t, stS is a

vector of other school-level characteristics of school s at time t, iϕ is a student specific fixed

effect, sµ is a school fixed effect, gtγ is a set of grade-by-year fixed effects, and igstε is an error

18

term.22

Using this approach, the α parameters capture the impact of crowding on the change in

achievement over the course of the year, the β variables capture the effects of the policy

responses on achievement, the student and school fixed effects capture average rates of change in

achievement within student and school, and the grade-by-year effects capture any unobserved

effects that may differ within a grade in a particular year (e.g. change in tests).

This technique identifies the impact of crowding using the exogenous variation in

crowding that remains after controlling for the two policy responses, other observed covariates

and the multi-level fixed effects. By using this estimation technique, the parameters of interest

are identified from within-student and within-school variation. In other words, we have to

observe a particular student multiple times in a particular school under different policy regimes

or crowding levels in order to identify these effects. This strategy, which is common in the

economics of education literature (Rivkin, Hanushek, and Kain, 2005; Bifulco and Ladd, 2006;

Hanushek and Rivkin, 2009; AUTHORS, 2011),23

is plausible with this data because of the large

changes in school crowding, mobile classrooms, and school calendars that we observe in this

school district over this timeframe.

We estimate several specifications based on equation (1) using, in turn, mathematics and

then reading test scores as the outcome of interest. We begin with a baseline model that does not

include the policy variables, student or school fixed effects. Then, we add the policy variables in

the second specification. In the third specification, we add school fixed effects. The fourth

specification includes both student and school fixed effects. Since crowding, year-round school

22

Models of this type generally include a vector of time-varying student characteristics. We do not have any time

varying student-level variables in our dataset. Thus, this vector is eliminated from the model. 23

For a detailed discussion of the conceptual and empirical concerns regarding estimation of the education

production function, see Hanushek (1979) and Todd and Wolpin (2003).

19

and mobile classroom use are school-level variables, all standard errors are clustered at the

school-level.

While controlling for the policy variables is important because it disentangles the direct

impact of crowding from the possible indirect effects of the policy response, it may be the case

that mobile classrooms and year-round calendars have a differential impact based on the extent

to which a school is crowded. For example, in a school that is not crowded, the use of mobile

classrooms might unnecessarily provide a learning environment that is inferior to a permanent

classroom. However, in the context of a crowded school, a mobile classroom might be far better

than the alternative, if the alternative is to hold class in a space not designed for that purpose. If

this is the case, then our estimated impact of crowding might average different crowding effects

based on different policy combinations. We therefore also examine whether the mobile

classrooms and year-round calendars impact student outcomes differently at alternative levels of

crowding by adding interaction terms to equation (1). Specifically, we add the following four

interaction terms: (1) year-round*moderately crowded, (2) year-round*severely crowded, (3)

percent mobiles*moderately crowded, and (4) percent mobiles*severely crowded.

5. Results

5.1 School crowding and achievement

The first goal of this research is to estimate the impact of school crowding on student

achievement gains. Table 6 shows the empirical results. Math results are given in panel A, while

reading results are in panel B. The first specification (column 1) shows the results of a simple

pooled cross-section specification, with observed school-level covariates and grade by year

indicator variables included as controls. These results follow the pattern observed in Table 3:

20

there is no significant difference in the mathematics achievement between students in crowded

schools and non-crowded schools, but the students’ reading achievement growth is significantly

lower in crowded schools. Because the achievement test scores are normalized to have a

standard deviation of one, these baseline estimates indicate that the achievement loss due to

crowding amounts to about 3% of a standard deviation in achievement, or slightly more than one

percentile point.24

This first specification, however, does not control for other important school differences

that might be correlated with both school crowding and achievement. The most obvious

omission of this type is the two important policy variables that are related to crowding: the use of

mobile classrooms and year-round school calendars. Column 2 includes these two policy

variables as controls. The inclusion of these policy indicators has little impact on the estimates

of the impact of crowding. Importantly, however, there are likely other unobserved school

characteristics that, when not accounted for, will bias the estimated impact of school crowding.

To control for these characteristics, we include school fixed effects in the third column. In the

math achievement specification, the addition of school fixed effects results in a slightly higher

estimated impact of moderate crowding, which is significant at the 10% level, but the estimated

impact of high levels of crowding is close to zero. In the reading specification, the estimated

impacts of moderate crowding and severe crowding remain virtually unchanged, implying failure

to control for unobserved school heterogeneity results in little bias of these estimates.

While school fixed effects will effectively control for time-invariant unobserved school

characteristics, there is also evidence for student level selection into schools, a phenomenon

which is likely related to school crowding. For example, the summary statistics in Table 3

24

At the 50th

percentile, an improvement of 1 percentile point in achievement is equal to about 0.025 standard

deviations, relative to a student’s peers.

21

indicate that students in crowded schools are, on average, more likely to be white, Asian, and/or

female, than students in non-crowded schools. In order to control for student selection, the

fourth column shows the results of specifications that include student fixed effects in addition to

the school fixed effects. This specification identifies the effect of crowding using only the

within-school and within-student variation, which will control for any school or student

characteristic that does not change over time. The addition of the student fixed effects does not

qualitatively change the estimates of the impact on math achievement gains, but the impact of

moderate and severe crowding on reading achievement increases noticeably. This evidence

implies failure to control for the non-random selection of students into schools leads to

understated impacts of crowding on reading achievement. This result is consistent with the

descriptive evidence provided in Tables 3 and 4, which reflects differences in both student and

school characteristics by school crowding levels.

While controlling for the policy variables and these unobserved student characteristics is

important, it may be that mobile classrooms and year-round calendars have a differential impact

based on the extent to which a school is crowded. If this is the case, then our estimated impact of

crowding might average different crowding effects based on different policy combinations. The

fifth specification in Table 6 interacts the crowding variables with the percentage of mobile

classrooms and the year-round calendar indicator variable. In this specification the estimated

impact of crowding on mathematics achievement is not statistically significant from zero. The

impact of attending a severely crowded school on reading achievement is negative, however, and

has a much larger impact than those effects estimated in other specifications. This result implies

that relative to a school that is not crowded, in those schools where there is no attempt to

mitigate school crowding with mobile classrooms or a multi-track year-round school calendar,

22

growth in reading achievement will be 0.054 standard deviations or 2 percentile points lower for

moderately crowded schools, and 0.11 standard deviations, or 4 percentile points, lower for high

levels of crowding.

5.2 Mobile classroom use and achievement

The first policy we consider here is the most common solution to school crowding: the

use of mobile classrooms. It is important to note that our definition of crowding used for these

analyses considers only the “building capacity” or the capacity of the permanent classroom

space. Thus, when we estimate the impact of increasing the percentage of classrooms that are

mobile, the counterfactual scenario is one in which more students are crowded into the crowded

building. In our preferred specification (column 5), we find a large negative impact of mobile

classrooms of around 0.20 standard deviations, though this effect is only statistically significant

for the reading specification. This estimate indicates that, in the absence of school crowding, a

mobile classroom is inferior to permanent classroom space, perhaps especially for reading and

language classes.

In the context of a crowded school, we find that the estimated impact of mobile

classrooms is likely net positive. For moderately crowded schools, we estimate that this negative

impact of mobile classrooms is entirely offset for mathematics achievement, and partially offset

for reading achievement. Similarly, for highly crowded schools, in both math and reading

achievement, the impact of mobile classrooms is estimated to be positive. It is worth noting that

the level of precision is higher for all of these estimates in the reading specification than the

mathematics specification, though qualitatively, the results are similar.

23

Since we measure mobiles as a percentage of the total classrooms, it is worth further

extrapolating how many mobile classrooms a school would need to invest in to offset the

negative impact of school crowding. For example, taking the case of reading achievement in a

highly crowded school, the estimated net impact of making 10% of total classroom space mobile

(which would be a net increase in classrooms, not a conversion of existing classroom space) is

only a 0.011 standard deviation improvement in achievement, which is only 1/10th

of the

negative impact of having a crowded school. Thus these estimates indicate that, unrealistically, a

school would have to increase mobile classrooms until they amounted to 100% of classroom

space to totally offset the crowding effect. In practice it is likely not realistic to increase the

percentage of mobile classrooms above 33% of total classroom space, which would offset only a

third of the negative impact of school crowding.

5.3. Multi-track year-round calendars and achievement

The multi-track year-round calendar is often used as a remedy for school crowding, since

it allows schools to keep the school facilities in constant use throughout the year. Our estimates

of the impact of year-round schooling in these specifications are, not surprisingly, similar to

those found previously with these data (AUTHORS, 2011). The estimates are also similar to

estimates found using other datasets (Graves, 2010). Small positive significant impacts of year-

round schooling are estimated in specifications without school fixed effects, but when

unobserved school characteristics are controlled for, this positive effect goes away, indicating

that the primary bias in the year-round schooling estimates is due to unobserved school-level

characteristics.

24

Similar to Graves’ (2010) analysis at the school level, we are also able to estimate the

effect of year-round schooling separately for overcrowded schools and those that are not

crowded. Moreover, we are able to take advantage of the “natural experiment” created by the

conversion of 22 schools to the year-round calendar and control for student selection. In our

preferred specification (column 5) for math achievement, the base estimate of the impact of year

round schooling is negative. Conversely, the estimate on the interaction term is positive for those

schools that are crowded, though none of these estimates are statistically significant at

conventional levels. A similar result is found for reading achievement: year round schooling

results in 0.088 standard deviations lower achievement in the schools that are not crowded.

However, in crowded schools this negative effect is offset and it is more than offset in highly

crowded schools. These results indicate that year round calendars, while not helpful on their

own, may be partially effective as a solution to school crowding. Our estimates indicate that in a

highly crowded school reading achievement will be only 0.062 standard deviations lower for

those in a multi-track year-round school, rather than the 0.11 standard deviation penalty realized

by those in similarly crowded schools without a policy response.

5.4. Class Size

Any discussion of school crowding must consider class size, because the easiest way for

a school to deal with an influx of students is to increase the number of students in each class. If

this were the case, the negative impact of crowding might operate through the increase in class

size. Our descriptive data indicates that these schools have a strong commitment to holding class

size constant even if a school becomes more crowded. Table 4 compares class size across

25

crowding levels, and while the average increases only slightly due to crowding, the variance in

class size falls as schools get more crowded.

In order to explore whether these class size changes drive our results, our preferred

specification is estimated three times. Column 5 shows the estimates with no control for class

size. This, then, gives the “full effect” of crowding. Column 6 includes a class size control, and

column 7 includes class size and the student-teacher ratio. Interestingly, in this case, the changes

in class size that we observe seem to have no impact on achievement. The estimated impact of

class size and student teacher ratio is consistently small (and sometimes positive) and never

statistically significant. Moreover, the estimated impact of crowding stays the same regardless

of how we control for class size. This means that the mechanism at work here is not one of

crowding within classrooms, but instead it is crowding at the school level.

6. Discussion

These results raise a series of questions for researchers and policymakers considering the

issues discussed in this project. First, our results indicate that the issue of school crowding

cannot be separated from a consideration of student mobility. Because, in our sample25

,

crowding appears to be accompanied by the selection of students with favorable characteristics

and higher achievement, cross sectional studies of the impact of crowding, as well as school

level analyses that do not deal with student-level behavior, will find estimates of the impact of

crowding that are biased downward. Similarly, studies of student mobility that do not consider

the negative impacts of crowding may also be missing an important consideration.

25

This might not be the norm in other contexts, since there are varied causes of school crowding and the Wake

County school system has a policy of minimizing socio-economic disparities across schools. For example, unlike in

our sample, NCES (2000) found that over-crowded schools were more likely to have high proportions of minority

students than less crowded schools.

26

The second question raised concerns schools’ response to crowding. Our results indicate

that the crowding effect is not operating through changes in class size. Given this, it is worth

asking how schools actually respond to crowding if mobile classrooms are not used, a year-round

calendar is not implemented, and class size is held constant. This is an important consideration

for the interpretation of our estimates of the impact of crowding since we control for these other

policies. The mechanism at work is not immediately apparent. Fortunately, other studies can

help answer this question. A NCES report (2000) indicates that crowded schools use space that

is not intended for instruction as classroom space. Moreover, if classroom space is used more

efficiently, teachers may have to teach in many classrooms throughout the day, and lose a degree

of control over the classroom space that they are in. If teachers in less crowded schools can use

the classroom space to complement their curriculum or pedagogy, then the loss of this control

could lead to lower achievement. Finally, the NCES documents a number of correlations

between school crowding and infrastructure concerns which indicate that a crowded school may

also be a lower-quality school setting.

The last significant issue raised by our results is the noticeable difference between the

impacts on mathematics and reading. This difference does not seem to be driven by issues of

language acquisition by English language learner (ELL) students.26

This difference may also be

an artifact of the way schools prioritize subjects within the school. If mathematics classes

always received prime classroom space while reading classes were placed in mobile classrooms,

this could account for the observed effects. This pattern seems unlikely, however, given the high

status of both math and reading relative to other disciplines in the context of a high-stakes testing

26

We estimate these specifications separately for different racial groups and find that those groups that are more

likely, in the Wake county context, to be non-native English speakers do not have statistically different results.

Moreover, for a part of our sample we observe English as a second language (ESL) status. Again, for this subgroup

the results are not significantly different.

27

regime. A final, and likely, explanation is that there exist pedagogical differences between the

two subjects that make crowding more detrimental for reading and language classes than for

math.

7. Conclusion

In this study we are able to take advantage of a unique policy environment in Wake

County, NC, where fast population growth and policy experimentation has lead to a large amount

of variation in school crowding, mobile classroom use, and school calendar models. We exploit

these policy changes to disentangle the effects of school crowding from the policy responses.

Our results indicate that the inclusion of both student and school-level fixed effects are important

for controlling for unobserved school characteristics and student selection into schools.

We find that school crowding has a negative impact on students’ reading achievement

growth, while student’s mathematics achievement growth does not seem to suffer. Additionally,

we find some evidence that year-round schools have a small negative impact on achievement,

but that they do offset some of the negative impact of crowding. Similarly, the use of mobile

classrooms has a small adverse impact on student achievement growth. In a crowded school,

however, the effect of mobile classrooms is net positive, but only large enough to partially offset

the negative impact of crowding.

Despite the obvious policy-usefulness of these estimates, we should note three cautions

about applying these results outside of Wake County. First, the rapid population growth in this

district over this time period is unusual, and thus the magnitude of the crowding concerns might

differ here than in other places. In such cases, the results might differ as well, since a different

mix of policies might be used to address the crowding. Second, the cause of the school crowding

28

is likely important for understanding the effect. In our data, crowded schools tended to have a

positively selected group of students, whereas other crowded schools might be crowded because

of limited resources, and also have student body that is more economically disadvantaged.

Crowding of this type might have a different effect, as might the same policy response on a

student population that is demographically different. Our third note of caution deals with our

experimental design. Because we only observe a short time frame of data before and after the

large year-round schooling change, we don’t have a long enough time period to control well for

possible time-varying unobserved influences on student performance. Our fixed-effects strategy

should control for any changes in the student demographic composition, and we try to include

the relevant policy changes going on in this time period, but in times of rapid population growth,

it is possible that there are omitted variables that we cannot address. These concerns are

common to many studies of this type that try to focus attention on a “natural experiment,” but

should not be ignored. Hopefully other similar studies will lend more evidence to the effects in

question.

Given the large amount of resources at stake for school districts when making choices

regarding school facilities, it is important that policy makers have accurate information. These

results should at the very least reassure interested parties that while school crowding is a

problem, these common solutions will partially remedy the crowding impact. Moreover, in the

cases in which one of these policy responses is significantly cheaper than additional construction,

these estimates will help schools weigh the costs and benefits of various short and long term

policy responses.

29

Table 1. Sample Construction

Cohort 2004-2005 2005-2006 2006-2007 2007-2008 2008-2009

1 Grade 7 Grade 8

2 Grade 6 Grade 7 Grade 8

3 Grade 5 Grade 6 Grade 7 Grade 8

4 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8

5 Grade 4 Grade 5 Grade 6 Grade 7

6 Grade 4 Grade 5 Grade 6

7 Grade 4 Grade 5

Table 2. Yearly descriptive statistics for crowding, year-round and mobile classroom use

2005 2006 2007 2008 2009

Crowding indicators:

Moderately crowded school 0.410 0.431 0.409 0.374 0.355

(0.492) (0.495) (0.492) (0.484) (0.478)

Severely crowded school 0.361 0.395 0.447 0.508 0.455

(0.480) (0.489) (0.497) (0.500) (0.498)

Policy indicators:

Year-Round 0.187 0.186 0.180 0.344 0.343

(0.390) (0.389) (0.384) (0.475) (0.475)

Mobile Classroom Use 0.143 0.167 0.184 0.176 0.149

(0.117) (0.151) (0.161) (0.147) (0.129)

Number of Observations 29,823 39,459 40,997 42,218 34,214

Notes:

a. Standard deviation in parentheses.

30

Table 3. Student descriptive statistics by school crowding

Mean Std. Dev. Mean Std. Dev. Mean Std. Dev.

Achievement:

Math gain 0.015 (0.529) 0.029 (0.526) 0.005 (0.533)

Reading gain 0.028 (0.028) 0.011 (0.600) -0.001 (0.601)

Demographics:

Male 0.513 (0.500) 0.509 (0.500) 0.504 (0.500)

White 0.516 (0.500) 0.540 (0.498) 0.598 (0.490)

Black 0.300 (0.458) 0.277 (0.448) 0.225 (0.418)

Hispanic 0.104 (0.305) 0.092 (0.289) 0.080 (0.271)

Indian 0.003 (0.055) 0.003 (0.052) 0.003 (0.051)

Asian 0.035 (0.183) 0.050 (0.219) 0.057 (0.231)

Mixed 0.041 (0.199) 0.037 (0.189) 0.038 (0.190)

Number of Observations

Moderately crowded Severely crowdedNot crowded

31,039 73,946 81,726

Table 4. School descriptive statistics by school crowding

Mean Std. Dev. Mean Std. Dev. Mean Std. Dev.

% Free or reduced price lunch 0.406 (0.141) 0.329 (0.145) 0.281 (0.143)

Student-teacher ratio 12.754 (1.750) 13.728 (1.565) 14.392 (1.517)

Class size 21.767 (3.603) 22.227 (3.121) 23.396 (2.644)

# Students 597.271 (185.581) 742.528 (226.851) 858.539 (223.695)

% Teachers fully licensed 0.946 (3.603) 0.941 (0.058) 0.959 (0.048)

% Teachers with 0-3 yrs 0.230 (1.750) 0.227 (0.088) 0.249 (0.083)

% Teachers with 4-10 yrs 0.342 (0.080) 0.322 (0.116) 0.314 (0.072)

% Teacher turnover 0.176 (0.097) 0.173 0.000 0.163 (0.080)

% Average daily attendance 0.954 (0.009) 0.955 (0.931) 0.957 (0.006)

Not crowded Moderately Crowded Severely Crowded

N = 118 N = 212 N = 247

31

Table 5. Comparison of Students who Switch Schools to Non-Switchers

Mean Std. Dev. Mean Std. Dev.

Math Gain 0.016 (0.555) 0.023 (0.527)

Reading Gain 0.008 (0.633) 0.009 (0.599)

Male 0.507 (0.500) 0.512 (0.500)

White 0.583 (0.475) 0.344 (0.493)

Black 0.242 (0.494) 0.425 (0.428)

Hispanic 0.084 (0.343) 0.136 (0.278)

Indian 0.003 (0.058) 0.003 (0.051)

Asian 0.051 (0.209) 0.046 (0.220)

Mixed 0.037 (0.208) 0.045 (0.190)

N

Notes:

a. Switchers are defined as non-structural switchers: those observations for which

students are observed in two subsequent years in either two different elementary

schools or two different middle schools.

Non-Switchers Switchersa

170,174 16,537

32

Table 6. Crowding, multi-track year-round schools, mobile classrooms and academic achievement

(1) (2) (3) (4) (5) (6) (7)

A. Math Achievement

Moderately Crowded 0.007 0.007 0.0238** 0.0339** 0.010 0.006 0.006

(0.012) (0.012) (0.012) (0.017) (0.019) (0.020) (0.020)

Severely Crowded -0.019 -0.015 0.004 -0.010 -0.028 -0.030 -0.030

(0.016) (0.017) (0.018) (0.027) (0.039) (0.038) (0.038)

Year-Round 0.000 0.0226** -0.025 -0.024 -0.095 -0.093 -0.098

(0.011) (0.016) (0.038) (0.061) (0.060) (0.062)

Percent Mobile -0.022 0.007 0.037 -0.199 -0.199 -0.193

(0.042) (0.053) (0.079) (0.140) (0.140) (0.142)

Year-round*Moderately crowded 0.063 0.062 0.067

(0.058) (0.056) (0.059)

Year-Round*Severely Crowded 0.075 0.073 0.080

(0.054) (0.053) (0.057)

Percent Mobiles*Moderately crowded 0.365** 0.375** 0.367**

(0.168) (0.169) (0.169)

Percent Mobiles*Severely crowded 0.274 0.272 0.270

(0.179) (0.180) (0.180)

Class Size -0.002 -0.003

(0.002) (0.003)

Student-Teacher Ratio 0.009

(0.010)

B. Reading Achievement

Moderately Crowded -0.0328***-0.0326*** -0.0328*** -0.0511*** -0.0540*** -0.0542*** -0.0542***

(0.010) (0.010) (0.011) (0.015) (0.017) (0.017) (0.017)

Severely Crowded -0.0362*** -0.0340** -0.0307** -0.0514** -0.110*** -0.110*** -0.110***

(0.011) (0.013) (0.015) (0.020) (0.022) (0.022) (0.022)

Year-Round 0.0178*** -0.004 0.011 -0.0894* -0.0893* -0.0887*

(0.007) (0.011) (0.020) (0.053) (0.053) (0.053)

Percent Mobile -0.011 0.004 -0.015 -0.197*** -0.197*** -0.198***

(0.027) (0.030) (0.047) (0.056) (0.056) (0.056)

Year-round*Moderately Crowded 0.073 0.073 0.072

(0.050) (0.050) (0.050)

Year-Round*Severely Crowded 0.137** 0.137** 0.136**

(0.059) (0.059) (0.058)

Percent Mobiles*Moderately Crowded 0.070 0.070 0.071

(0.098) (0.099) (0.099)

Percent Mobiles*Severely Crowded 0.308*** 0.307*** 0.308***

(0.080) (0.079) (0.080)

Class Size 0.000 0.000

(0.002) (0.002)

Student-Teacher Ratio -0.001

(0.007)

Number of observations = 186,711

Number of students = 69,253

Student Fixed Effects No No No Yes Yes Yes Yes

School Fixed Effects No No Yes Yes Yes Yes Yes

Notes:

a. All models include grade/year fixed effects and school characteristics (elementary, middle, number of students, % teachers fully licensed,

% teachers with 0-3 yrs experience, % teachers with 4-10 yrs experience, and % teacher turnover).

b. Robust standard errors, clustered at the school level, are in parenthesis. * ,** , *** denote statistical significance at the

10, 5, 1 percent levels, respectively.

33

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