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Technical Report #3 – November 2002




 The Influence of District Size, School Size and
Socioeconomic Status on Student Achievement
   in Washington: A Replication Study Using
          Hierarchical Linear Modeling




              Martin L. Abbott, Ph.D.

               Jeff Joireman, Ph.D.

              Heather R. Stroh, M.ED.
The Washington School Research Center (WSRC) is an independent research and data analysis
center within Seattle Pacific University. The Center began in July 2000, funded through a gift from
the Bill and Melinda Gates Foundation. Our mission is to conduct sound and objective research on
student learning in the public schools, and to make the research findings available for educators,
policy makers, and the general public for use in the improvement of schools. We believe that sound
data and appropriate data analysis are vital components for the identification of school and
classroom practices related to increased student academic achievement.



                               Washington School Research Center
                                 3500 188th St. S.W., Suite 328
                                     Lynnwood, WA 98037
                                     Phone: 425.744.0992
                                      Fax: 425.744-0821
                                   Web: www.spu.edu/wsrc




     Jeffrey T. Fouts, Ed.D.          Martin L. Abbott, Ph.D.           Duane B. Baker, Ed.D.
       Executive Director               Senior Researcher                      Director -
     Professor of Education           Professor of Sociology          School Information Services




Copyright  2002 by the Washington School Research Center, Seattle Pacific University. All rights
reserved. Additional copies of this report may be downloaded in pdf format free of charge at
www.spu.edu/wsrc.
The Influence of District Size, School Size and
      Socioeconomic Status on Student
 Achievement in Washington: A Replication
  Study Using Hierarchical Linear Modeling




              A Technical Report For
       The Washington School Research Center
Foreword



In recent years there has been a growing interest in the role that school size
plays in creating effective learning environments for students. Serious questions
have been raised about the “bigger is better” approach to schools, and policy
makers are asking researchers if there are research findings on this important
topic. In fact, there have been numerous studies, both quantitative and
qualitative, strongly suggesting students generally do better in smaller schools
than larger schools.

Such a study published in the year 2000 on education in the state of Georgia
caught the interest of the Urban Issues Committee of the Washington State
School Directors’ Association (WSSDA). Recognizing that such research
findings have direct policy implications, the Association approached the
Washington School Research Center (WSRC) about replicating the study in the
state of Washington. Through the joint sponsorship of WSSDA and WSRC, this
technical report is a replication of that study using achievement, poverty, school
and district size data from Washington State.

The research findings on school size show that the question is a complex one,
and that there are numerous factors that might interact with school size to
account for variation in student and school performance. Using a statistical
procedure called Hierarchical Linear Modeling, WSRC researchers Abbott,
Joireman, and Stroh attempt to identify the ways in which district size, school
size, and family income level interact to effect student achievement.

Replication research across states is difficult because of differing state tests,
grade structures, data bases, and other factors. And, in fact, while this study
replicates the general approach used in the Georgia study, it does differ in some
significant ways. First, WASL scores were used for this study rather than the
ITBS; second, because of questions about the reliability of the high school
poverty data, 4th and 7th grade data were analyzed, while the Georgia study
analyzed 8th and 11th grade data. Still, the results presented here complement
the original findings and add to the body of research that strongly suggests that
school size and district size do matter.

The WSRC researchers conclude: “We found that large district size is
detrimental to achievement in Washington 4th and 7th grades in that it strengthens
the negative relationship between school poverty and student achievement.”
Further, they state, “the negative relationship between school poverty and
achievement is stronger in larger districts,” and “small schools appear to have the
greatest equity effects.” In other words, when school poverty is high, children



                                         i
perform better in small districts, and the effect of school level poverty on
achievement is smallest when both the district and school are small.

Jeffrey T. Fouts
Executive Director
Washington School Research Center




                                          ii
Table of Contents



Introduction ........................................................................................................... 1

Literature Review .................................................................................................. 2

A Replication Study ................................................................................................ 4

Nature of the Data ................................................................................................. 4

Joint Effects of School and District Variables Using Hierarchical Linear Modeling . 5

Fourth and Seventh Grade Equity Effects............................................................... 14

Conclusions ........................................................................................................... 16

References ........................................................................................................... 18




                                                          iii
The Influence of District Size, School Size and Socioeconomic Status
 on Student Achievement in Washington: A Replication Study using
                    Hierarchical Linear Modeling

                                    Introduction

New interest in the effects of school size on academic achievement has grown in
recent years as nationwide school reform efforts have gained momentum. Policy
makers and practitioners have suggested new models of schools based on the
idea that smaller is better. As these models have gained currency, there is also
a call for research to investigate the likely effects of creating smaller schools.
Are smaller schools needed as a strategy to improve student academic
performance?

Several studies have examined this question by employing single-level
regression models to examine the influence of size (district and school) and low-
income on academic achievement. A recent article by Robert Bickel and Craig
Howley (2000), however, advanced the research in this area by introducing a
multi-level approach to their analysis examining the joint influence of district and
school size on academic performance in Georgia. Their research focuses on the
“cross-level interaction” of district and school socioeconomic status and size
through a single-equation relative-effects model.

The results of the Bickel and Howley study particularly highlight the sizeable
influences of two such cross-level (i.e., district and school level) interactions: the
product of district size and school socioeconomic status (SES), and the product
of school size and district SES (especially at the 8th grade level). School-level
academic performance is negatively affected both by the district poverty-school
size cross-level interaction, and the school poverty-district size cross level
interaction. (These findings are not as consistent at the 11th grade level.) Single
level regression models cannot identify these important cross-level dynamics.

Another important contribution of the Bickel and Howley study is their “equity”
analyses. According to the authors, “equity” refers to non-equivalence in the
socioeconomic status-academic performance relationship as a function of size.
In effect, this asks whether “the amount of variance in achievement associated
with SES is substantially reduced in smaller units” (p. 5). The analysis is
accomplished by examining the variance in school performance associated with
low-income in four different configurations of school and district size (i.e., large
school-large district, large school-small district, small school-large district, and
small school-small district).




                                          1
Literature Review
The issue of school and district size effects on student achievement is not new.
Beginning in the 1920s schools and school districts grew larger as districts
attempted to consolidate both administration and curriculum and instruction.
Ellwood Cubberly, a former urban superintendent, advocated for the creation of
large schools as immigrant populations in major cities grew. Meanwhile, Joseph
Kennedy, dean of the school of education at North Dakota State University, was
concerned about losing the participation and identity of local communities as
rural schools consolidated (Robertson, 2001; Howley, 1996). The press for
larger schools continued through the 1950s as U.S. educators felt the pressure of
the “space race” and the need to provide a wider, more academically rigorous
curriculum for future scientists.

Large schools, however, are not without fault. Bracey (2001) noted:

      …large schools, especially large high schools, produce their own set of
      problems, which a growing number of researchers and policy makers think
      can be solved by returning to small schools. Advocates for small schools
      have argued that they can
      • raise student achievement, especially for minority and low-income
         students;
      • reduce incidents of violence and disruptive behavior;
      • combat anonymity and isolation and, conversely, increase the sense of
         belonging;
      • increase attendance and graduation rates;
      • elevate teacher satisfaction;
      • improve school climate;
      • operate most cost-effectively;
      • increase parents and community involvement; and
      • reduce the amount of graffiti on school buildings. (p. 413)

Such arguments by advocates for small schools bear some merit, as research
indicates that school (Lee & Loeb, 2000) and district (Bickel & Howley, 2000;
Johnson, Howley, & Howley, 2002) size can impact student achievement. Lee
and Loeb examined 264 Chicago elementary schools and found that school size
influenced student achievement both directly and indirectly. They reported that
teachers in small schools (less than 400 students) take more responsibility for
students’ academic and social development, and that this in turn enhances
student achievement. They noted that small schools facilitate more intimate and
personal relationships among both teachers and students, and that it is these
relationships that impact student learning. Another study of Chicago’s public
schools found that small schools increase attendance, student persistence,


                                        2
performance, graduation rates, grades, course completion, and parent, teacher,
student, and community satisfaction (Walsey, Fine, Gladden, Holland, King,
Mosak, & Powell, 2000).

Other research on school size has found socioeconomic status to be the
confounding variable in the size/achievement equation, noting that as school size
increases, achievement levels for schools with less advantaged students
decreases (Bickel, 1999; Howley & Bickel, 2000). Bickel, Howley, Williams, and
Glascock (2000) confirmed these findings while controlling for ethnicity,
language, size, cost, and curricular composition factors. Interestingly, Howley,
Strange, and Bickel (2000) noted that “size exert[s] a negative influence on
achievement in impoverished schools, but a positive influence on achievement in
affluent schools. That is, all else being equal, larger school size benefits
achievement in affluent communities, but it is detrimental in impoverished
communities” (p. 4). Additionally, the authors’ results indicated that “the
relationship between achievement and SES is substantially weaker in the smaller
schools than in the larger schools” (p. 5).

The authors also noted, however, that small size doesn’t necessarily guarantee
student success. “Small size is a necessary but insufficient condition for school
improvement. . . . It is important to avoid seeing small schools as the sole
solution to all that ails education. Rather, we would suggest that it is a key
ingredient in a comprehensive plan to improve education” (p. 66).

District size may also moderate the effects of school size. Bickel and Howley
(2000) found an interesting interaction between district and school size. The
authors explained:

       Larger schools in larger districts seem to propagate inequality of outcomes
       by comparison to smaller schools and smaller districts. In fact, smaller
       schools in larger districts demonstrate a useful equity effect, as well. For
       large schools in smaller districts, however, the improvements in equity
       might be so slight as to be called negligible. (p. 21)

The authors (Bickel & Howley, 2000) found that in communities with high rates of
poverty, small schools in small districts increase student achievement. Overall,
“smaller districts and smaller schools demonstrate greater achievement equity”
(Howley, 2000, p. 7).

Simply stated, current research indicates that the amount of impact
socioeconomic status has on student achievement is dependent upon several
factors, including the size of the school and the size of the district in which the
school functions.




                                          3
A Replication Study
The present study is a replication of the Bickel and Howley (2000) method to the
Washington state academic performance of 4th and 7th grade students. While
Bickel and Howley focused on the 8th grade Iowa Test of Basic Skills (ITBS) and
the 11th grade Georgia High School Graduation Test, this study examines 4th and
7th grade academic performance on the Washington Assessment of Student
Learning (WASL)1 test that is mandated across the state. A great deal of
attention in Washington has focused on the use of the WASL, which has
replaced other tests (e.g., ITBS) for assessing statewide standards-based
learning objectives.

Bickel and Howley’s method is accomplished in this study by the use of
Hierarchical Linear Modeling through the HLM software program (Raudenbush,
S., Bryk, A., and Congdon, R., 2000). The two approaches attempt to specify the
joint relationships and cross-level interactions of two structural levels (district and
school) on school academic performance.

                                   Nature of the Data
The data used for this replication study were provided by Washington State’s
Office of the Superintendent of Public Instruction, and are from the testing year
2001. The data consist of all 4th and 7th grade student WASL scale scores in
reading and mathematics, aggregated to the school level.2 Schools with less than
ten students were excluded from the study, and, because there was a concern
over the unique characteristics of some “types” of schools, those labeled
“Alternative,” “Institutional,” and other “Unclassified” types were not included.

Table 1 indicates the descriptive data for the variables used in this study.
Percent Free and Reduced (F/R) Lunch is used on both school and district levels
as a measure of low socioeconomic status3. Spansize is the number of students
per grade level and is used as our measure of school size, as it was by Bickel
and Howley (2000). Enrollment is the total number of students per district.




1
 For more information on administration of the WASL and ITBS in Washington, visit
www.k12.wa.us/assessment/WASLintro.asp. For more technical information on the WASL, visit
www.k12.wa.us/assessment/qawasl.asp. For more technical information on the ITBS, visit
www.riverpub.com/products/group/itbs.htm.
2
  Grade 10 WASL scores were not used since, in our experience, the % Free/Reduced Lunch
data are less reliable at that level.
3
  Bickel and Howley (2000) refer to free or reduced price meals as SES.


                                            4
Table 1

Descriptive Statistics for 4th and 7th Grades


                      4th Grade

     School-Level           Mean     SD       N
Percent F/R Lunch           37.86   23.62    1035
Spansize                    69.67   28.29    1035
Math Scale Score           392.83   14.79    1035
Reading Scale Score        405.45   7.07     1035

     District-Level
Percent F/R Lunch           37.92   18.84  251
Enrollment                 3903.34 6089.11 250

                      7th Grade

     School-Level           Mean      SD      N
Percent F/R Lunch           33.27    21.33   417
Spansize                   177.87   112.74   417
Math Scale Score           366.29    17.81   417
Reading Scale Score        393.79    6.81    417

     District-Level
Percent F/R Lunch           37.92   18.75  255
Enrollment                 3924.40 6277.40 255




    Joint Effects of School and District Variables Using Hierarchical Linear
                                   Modeling
Major Goal

As noted earlier, our major goal was to determine whether the cross-level
interactions (school x district level) between size and socioeconomic status
reported by Bickel and Howley (2000) would replicate within Washington. More
specifically, we were interested in whether the data in Washington would
replicate two major patterns reported by Bickel and Howley: (1) larger schools
are beneficial within affluent communities, whereas smaller schools are more
beneficial within less affluent districts; and (2) the “achievement cost” associated
with less affluent schools is greater in large districts (i.e., the negative


                                             5
relationship between school-level poverty and achievement is stronger in larger
districts). The first pattern would require an interaction between school size and
district-level SES; the second pattern would require an interaction school-level
SES and district size.

Hierarchical Linear Modeling (HLM)

To determine whether these two patterns (interactions) were present within the
current Washington State data, we conducted a series of analyses using
hierarchical linear modeling (HLM). HLM is a statistical technique that is
appropriate for analyzing multi-level data (e.g., schools nested within districts).
HLM has a number of advantages over other approaches to multi-level data,
such as ignoring the nested nature of the data (which violates the assumption of
non-independent errors) or pooling the data by, for example, aggregating across
schools within a district (which results in a loss of data, and eliminates the
possibility of examining cross-level interactions).4 In each of the analyses, school
level variables were group centered (district means) and district level variables
were grand mean centered.

Overview of Models 1 and 2

Tables 2 and 3 summarize the results of two different models, respectively.
Model 1 examines the influence of school size and district poverty (on math and
reading at both the 4th and 7th grades), whereas Model 2 examines the influence
of school poverty and district size (on math and reading at both the 4th and 7th
grades).

Model 1 Summary – School Size and District Poverty

We begin by interpreting the results for Model 1, using the 4th grade, shown in
the top half of Table 2. As can be seen, results for math and reading were quite
similar. First, scores in math and reading each showed a highly significant
negative relationship with district poverty (Bs = -47.10 and -24.10, ps < .0001).
Second, math and reading showed no significant relationship with school size (ps
> .35). Third, results did not reveal interactions between school size and district
poverty, findings that were not consistent with Bickel and Howley’s (2000)
findings.

While the interactions were not significant, they were in the expected direction.
To further examine the nature of the interactions, we plotted the relationship
between school size and achievement for districts in the 25th (thin lines) and 75th
(bold lines) percentiles on poverty, as shown in Figure 1. While the interaction
was not significant in either case, there was a tendency for larger schools to be
somewhat more beneficial in more affluent districts (25th percentile, thin line) than
in less affluent districts (75th percentile, bold line).
4
    For more information on Hierarchical Linear Modeling, see Raudenbush and Bryk (2002).


                                                6
We now turn to the results for the 7th grade, summarized in the bottom half of
Table 2. As can be seen, the 7th grade results replicated the 4th grade results.
First, both math and reading showed a significant negative relationship with
district poverty (Bs = -61.05 and -23.41, ps < .0001). Second, math and reading
showed no significant relationship with school size (ps > .28). Last, results at the
7th grade failed to reveal a significant interaction between school size and district
poverty on either math or reading (ps > .27). For comparison with the 4th grade
results, we present graphs of the relationship between school size and
achievement at the 25th and 75th percentile on district poverty for the 7th grade
(Figure 2). As can be seen, while the interaction was not significant in either
case, there was a tendency for larger schools to be somewhat more beneficial in
more affluent districts (25th percentile, thin line) than in less affluent districts (75th
percentile, bold line).

Taken as a set, results for Model 1 do not replicate Bickel and Howley’s (2000)
findings concerning the interaction between school size and district poverty. As
we explain in the next section, results did replicate Bickel and Howley’s finding
concerning the interaction between school poverty and district size.

Model 2 Summary – School Poverty and District Size

We now turn to the results for Model 2, in which we examine the influence of
school poverty and district size on math and reading in the 4th and 7th grade, as
summarized in Table 3. First, math and reading showed significant negative
relationships with school level poverty in both grades (ps < .001). Second, math
and reading showed a significant positive relationship with district size in the 4th
grade (ps < .004), but showed no significant relationship with district size in the
7th grade (ps > .09). Finally, in each case, school poverty and district size
showed a significant interaction (ps < .001). As we will explain, the nature of this
interaction was consistent with Bickel and Howley’s (2000) findings (for their 8th
grade only – comparable 11th grade results were not significant).

To further examine the nature of this interaction, we plotted the relationship
between school poverty and achievement for small districts (25th percentile, thin
line) and large districts (75th percentile, bold line) at both the 4th and 7th grade.
Figure 3 presents the results for 4th grade; Figure 4 presents the results for the
7th grade. As can be seen, in every case, the negative relationship between
school poverty and achievement is stronger in large districts, thus replicating
Bickel and Howley’s (2000) findings (for the 8th grade only).




                                            7
Table 2

Summary of HLM Runs – Model 1 – Effects of School Size and District Poverty
on Math and Reading Achievement in 4th and 7th Grades.


4th Grade (Figure 1)

Math                            B         SE          t     df   p-value
Intercept              389.99776    0.519294   751.015    246      0.000
School Size                  0.03   0.031156     0.925    246      0.355
District Poverty           -47.10   2.843856   -16.562    246      0.000
SS x DP                     -0.16   0.143109    -1.127    246      0.260

Reading                        B          SE         t      df   p-value
Intercept                 404.00    0.242452 1666.317     246      0.000
School Size                 0.01    0.014902    0.574     246      0.566
District Poverty          -24.10    1.377889 -17.492      246      0.000
SS x DP                   -0.087    0.073228   -1.191     246      0.234

7th Grade (Figure 2)

Math                           B          SE          t     df   p-value
Intercept                 364.36    0.708944    513.95    253      0.000
School Size                 0.04    0.038757      0.95    253      0.343
District Poverty          -61.05    4.394758    -13.89    253      0.000
SS x DP                    -0.25    0.229240     -1.09    253      0.275

Reading                        B          SE          t     df   p-value
Intercept                 393.02    0.277042   1418.64    253      0.000
School Size                 0.01    0.011056      1.06    253      0.289
District Poverty          -23.41    1.776674    -13.18    253      0.000
SS x DP                    -0.05    0.068546     -0.76    253      0.445



Note. DP = District Poverty (% of students in district on free or reduced lunch);
SS = School Size (spansize).




                                               8
Table 3

Summary of HLM Runs – Model 2 – Effects of School Poverty and District Size
on Math and Reading Achievement in 4th and 7th Grades.


4th Grade (Figure 3)

Math                            B         SE              t     df   p-value
Intercept                 390.21    0.806115       484.062    247      0.000
School Poverty              -0.24   0.038737        -6.228    247      0.000
District Size            .000287    0.000107         2.682    247      0.008
SP x DS                 -0.00001    0.000003        -3.809    247      0.000

Reading                         B         SE         t          df   p-value
Intercept                404.075    0.397746 1015.914         247      0.000
School Poverty              -0.14   0.017709   -7.719         247      0.000
District Size            .000151    0.000052    2.907         247      0.004
SP x DS                -0.000005    0.000001   -4.497         247      0.000

7th Grade (Figure 4)

Math                            B         SE              t     df   p-value
Intercept                 364.83    1.020967        357.34    253      0.000
School Poverty              -0.40   0.120292         -3.29    253      0.001
District Size            .000250    0.000149          1.68    253      0.093
SP x DS                 -0.00002    0.000006         -3.51    253      0.001

Reading                         B         SE              t     df   p-value
Intercept                 393.24    0.401891        978.48    253      0.000
School Poverty              -0.18   0.051623         -3.42    253      0.001
District Size            .000084    0.000052          1.60    253      0.108
SP x DS                -0.000008    0.000002         -3.51    253      0.001



Note. DS = District Size; SP = School Poverty (% students on free or reduced
lunch).




                                               9
Figure 1

Math and Reading Achievement as a Function of School Size (Spansize) and
District Poverty (25th and 75th Percentiles) – 4th Grade.




                          4th WASL
    409.7
                                                                DISTFR = -0.137913
                                                                DISTFR = 0.113489
M   407.1
a
t   404.6
h
    402.0
A
    399.4
c
h
    396.9
i
e   394.3
v
e   391.7
m
e   389.1
n
t   386.6

    384.0

        -60.94   2.94   66.83   130.71   194.60        258.48

                         School Size




                          4th WASL
    412.6
R                                                               DISTFR = -0.137913
e   411.5
                                                                DISTFR = 0.113489
a
d   410.3
i
n   409.1
g
    408.0
A
c   406.8
h
i   405.6
e
v   404.4
e
m   403.3
e
    402.1
n
t
    400.9

        -60.94   2.94   66.83   130.71   194.60        258.48

                         School Size




                                                  10
Figure 2

Math and Reading Achievement as a Function of School Size (Spansize) and
District Poverty (25th and 75th Percentiles) -- 7th Grade.


                                         7th Grade
               398.3
                                                                          DISTFR = -0.146881
                                                                          DISTFR = 0.114901
           M   394.1
           a
           t   389.9
           h
               385.6
           A
               381.4
           c
           h
               377.2
           i
           e   372.9
           v
           e   368.7
           m
           e   364.5
           n
           t   360.2

               356.0

                   -168.95   -67.15   34.65    136.45   238.24   340.04

                                        School Size



                                         7th Grade
               403.2
           R                                                              DISTFR = -0.146881
           e   401.8
                                                                          DISTFR = 0.114901
           a
           d   400.4
           i
           n   399.0
           g
               397.6
           A
           c   396.3
           h
           i   394.9
           e
           v   393.5
           e
           m   392.1
           e
               390.7
           n
           t
               389.4

                  -168.95    -66.13   36.68    139.50   242.32   345.13

                                       School Size




                                                11
Figure 3

Math and Reading Achievement as a Function of School Poverty and District
Size (25th and 75th Percentile) – 4th Grade




                             4th WASL
    399.7
                                                                 DISTENRL = -3357.289063
                                                                 DISTENRL = 558.710938
M   397.3
a
t   394.9
h
    392.4
A
    390.0
c
h
    387.6
i
e   385.1
v
e   382.7
m
e   380.3
n
t   377.8

    375.4

        -37.93   -18.20   1.52       21.24    40.96    60.69

                          School Poverty



                            4th WASL
    409.4
R                                                              DISTENRL = -3357.289063
e   408.1
                                                               DISTENRL = 558.710938
a
d   406.7
i
n   405.3
g
    403.9
A
c   402.6
h
i   401.2
e
v   399.8
e
m   398.5
e
    397.1
n
t
    395.7

        -37.93   -18.20   1.52      21.24    40.96    60.69

                          School Poverty




                                                 12
Figure 4

Reading Achievement as a Function of School Poverty and District Size (25th and
75th Percentile) for 4th and 7th Grade.


                                       7th Grade
               378.4
                                                                        DISTENR = -3406.395996
                                                                        DISTENR = 443.604004
           M   374.7
           a
           t   370.9
           h
               367.2
           A
               363.4
           c
           h
               359.7
           i
           e   355.9
           v
           e   352.2
           m
           e   348.4
           n
           t   344.7

               340.9

                   -33.27   -14.76   3.76      22.27    40.79   59.30

                                     School Poverty




                                       7th Grade
               399.3
           R                                                            DISTENR = -3406.395996
           e   397.6
                                                                        DISTENR = 443.604004
           a
           d   395.9
           i
           n   394.2
           g
               392.5
           A
           c   390.9
           h
           i   389.2
           e
           v   387.5
           e
           m   385.8
           e
               384.1
           n
           t
               382.5

                   -33.27   -14.57   4.13      22.83    41.53   60.23

                                     School Poverty




                                                   13
Fourth and Seventh Grade Equity Effects

In order to measure equity effects, Bickel and Howley (2000) used squared zero-
order correlation values (r2) between SES and achievement within four
categories of district and school size (i.e., large school-large district, large school-
small district, small school-large district, and small school-small district). They
found that,

       the predicted equity effect of reducing district size but not school
       size would be practically significant; the predicted equity effect of
       reducing school size but not district size would also be practically
       significant and perhaps somewhat larger; and the combined
       strategy of reducing both school and district size would be
       predicted to yield substantial equity and excellence effects . . .
       (p. 20)

This study replicates Bickel and Howley’s method by creating four categories of
district and school size based on median split values: large school-large district,
large school-small district, small school-large district, and small school-small
district. Within each of these categories are the r2 values between F/R lunch %
and achievement (math and reading WASL scores) for 4th and 7th grades. Table
4 lists the results of these analyses.

Both 4th and 7th grade results reflect Bickel and Howley’s (2000) results in that
the data in this study show the small district-small school category shows the
smallest proportion of variance in achievement associated with school poverty.
Apart from this however, the results are different from Bickel and Howley in at
least one major instance: the large district-large school category did not result in
the greatest amount of variance explained across grade levels and subjects.

The overall results of both 4th and 7th grades reflect the school poverty – district
size interaction results reported in Table 3: that the negative relationship
between school poverty and achievement is stronger in larger districts. Small
schools in small districts explain the least amount of variance (13% to 24% of the
variance in achievement associated with poverty), but the largest amount of
variance explained is in large districts irrespective of school size (41% to 54%).

This finding is consistent across grade levels and subjects despite large
differences between 4th and 7th grade spansizes. The 4th grade spansize is
approximately 2 ½ times smaller than the 7th grade spansize while the mean
district enrollments are approximately equal. Therefore, small schools appear to
have the greatest equity effects, while large districts are the most detrimental.




                                          14
Table 4
Variance in Achievement Explained by School Poverty as a Function of
District and School Size: Washington Multi-Level Equity Effects1
Large and Small Districts and Schools—4th, 7th WASL Scores

                   4th Grade Math Scale Scores
                                             Districts2
                                     Large                    Small
                                 2                        2
                                r             n           r            n

Grade Span      Large         0.41           487      0.27            30
   Size3        Small         0.45           416      0.17            101


                 4th Grade Reading Scale Scores
                                             Districts2
                                     Large                    Small
                                 2                        2
                                r             n           r            n

Grade Span      Large         0.47           487      0.46            30
   Size 3       Small         0.54           416      0.13            101


                   7th Grade Math Scale Scores
                                             Districts4
                                     Large                    Small
                                 2                        2
                                r             n           r            n

Grade Span      Large         0.48           208          -            0
   Size 5       Small         0.46           80       0.24            129


                 7th Grade Reading Scale Scores
                                             Districts4
                                     Large                    Small
                                 2                        2
                                r             n           r            n
Grade Span      Large         0.53           208          -            0
   Size 5       Small         0.51           80       0.20            129


1
  Variance (r2) in Scale Scores attributable to % Free/Reduced Lunch
2
  Median Split = 1,514.5
3
  Median Split = 68.9
4
  Median Split = 1,411
5
  Median Split = 186



                                                    15
Conclusions
Our study replicated the method of Bickel and Howley (2000) for understanding
the influence of district size, school size and socioeconomic status on student
achievement in Washington. We found that large district size is detrimental to
achievement in Washington 4th and 7th grades in that it strengthens the negative
relationship between school poverty and student achievement. This finding
replicated that of the Bickel and Howley (2000) study.

We did not replicate another of Bickel and Howley’s (2000) findings, however. In
our study, district affluence did not have a significant impact over the school size
– student achievement relationship. The tendency for larger schools to be
somewhat more beneficial in more affluent districts (and, equivalently, for smaller
schools to be more beneficial in less affluent districts) is shown in the analyses,
but was not found to be statistically significant.

The nature and configuration of Washington schools may partially explain the
discrepancy between the findings of the two studies with respect to district
affluence. First, the majority of districts (for both 4th and 7th grades) in
Washington are single-school districts that tend to be smaller, poorer, and more
rural than multi-school districts.5 Second, our study used the WASL as the
measure of student achievement (both 4th and 7th grades) in contrast to Bickel
and Howley’s use of the ITBS (8th grade) and the Georgia High School
Graduation Test (11th grade). Preliminary analyses in Washington indicate that
the WASL and the ITBS have different correlations with school poverty,
especially in math.6 Third, there are a number of other variables not addressed
in either study that may exert important influences on student achievement.

Taken together, the method used by Bickel and Howley (2000) and that of this
study were useful for explaining relationships in multi-level data that could not be
explained by more traditional (single-level) analyses. School and district size are
often assumed to be primary, independent influences on student achievement.
In fact, this is a commonly expressed sentiment even among practitioners and
policy makers. However, based on this study, it appears that size is a more
complex matter, and needs to be viewed in the context of other influences in
order to determine its contribution to school-level achievement.

Certainly, the multi-level findings of our study argue against the simplistic
conclusion that reducing school and/or district size will automatically improve
student achievement, or be more equitable. We are in complete agreement with
Bickel and Howley (2000) in their comment that, “the conclusions of this study
would seem to require rather wide debate and reconsideration of the size issue,

5
  We use “single-school districts” to indicate districts with a school containing one (4th and 7th)
grade level.
6
  For a comparison of the WASL and ITBS in Washington, see Joireman and Abbott (2001)


                                                  16
across the spectrum of poverty and wealth, and not just in the case of
impoverished communities” (p. 21).




                                       17
References

Bickel, R. (1999). School size, socioeconomic status, and achievement: A
       Georgia replication of inquiry in education. Huntington, WV: Marshall
       University.

Bickel, R., & Howley, C. (2000). The influence of scale on student performance:
       A multi-level extension of the Matthew Principle. Education Policy Analysis
       Archives, 8(22).

Bickel, R., Howley, C., Williams, T., & Glascock, C. (2000). High school size,
       achievement equity, and cost: Robust interaction effects and tentative
       results. Washington, D.C.: Rural School and Community Trust.

Bracey, G. (2001). Small schools, great strides. Phi Delta Kappan, 82(5), 413-
      414.

Howley, C. (1996). Ongoing dilemmas of school size: A short story. Retrieved
     June 4, 2002 from
     http://www.aasa.org/issues_and_insights/district_organization/howley_dile
     mmas.htm.

Howley, C. (2000). School district size and school performance. Rural Education
     Issue Digest.

Howley, C., & Bickel, R. (2000). When it comes to schooling…small works:
     School size, poverty, and student achievement. Randolph, VT: Rural
     School and Community Trust.

Howley, C., Strange, M., & Bickel, R. (2000). Research about school size and
     school performance in impoverished communities [ERIC Digest].
     Washington, DC: Office of Educational Research and Improvement.

Joireman, J., & Abbott, M. (2001) The relationships between the Iowa Test of
      Basic Skills and the Washington Assessment of Student Learning in the
      State of Washingon [ERIC Document TM033 308]. Lynwood, WA:
      Washington School Research Center, Seattle Pacific University.

Johnson, J. D., Howley, C. B., Howley, A. A. (2002). Size, excellence, and equity:
      A report on Arkansas schools and districts. Athens, OH: Ohio University.

Lee, V., & Loeb, S. (2000). School size in Chicago elementary schools: Effects
      on teachers’ attitudes and students’ achievement. American Educational
      Research Journal, 37(1), 3-31.




                                        18
Raudenbush, S., & Bryk, A. (2002). Hierarchical linear models: Applications and
     data analysis methods (2nd edition). Thousand Oaks: Sage Publications.

Raudenbush, S., Bryk, A., and Congdon, R. (2000) HLM (Version 5) [Computer
     Software]. Lincolnwood, IL: Scientific Software International.

Robertson, S. (2001). The great size debate [A CEFPI Brief on Educational
      Facility Issues]. Scottsdale, AZ: Council of Educational Facility Planners,
      International.

Walsey, P. A., Fine, M., Gladden, M., Holland, N. E., King, S. P., Mosak, E., &
     Powell, L. C. (2000). Small schools: Great strides. A study of new small
     schools in Chicago. New York: Bank Street College of Education.




                                        19

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Wsrc hlm district size final 10 2-02

  • 1. Technical Report #3 – November 2002 The Influence of District Size, School Size and Socioeconomic Status on Student Achievement in Washington: A Replication Study Using Hierarchical Linear Modeling Martin L. Abbott, Ph.D. Jeff Joireman, Ph.D. Heather R. Stroh, M.ED.
  • 2. The Washington School Research Center (WSRC) is an independent research and data analysis center within Seattle Pacific University. The Center began in July 2000, funded through a gift from the Bill and Melinda Gates Foundation. Our mission is to conduct sound and objective research on student learning in the public schools, and to make the research findings available for educators, policy makers, and the general public for use in the improvement of schools. We believe that sound data and appropriate data analysis are vital components for the identification of school and classroom practices related to increased student academic achievement. Washington School Research Center 3500 188th St. S.W., Suite 328 Lynnwood, WA 98037 Phone: 425.744.0992 Fax: 425.744-0821 Web: www.spu.edu/wsrc Jeffrey T. Fouts, Ed.D. Martin L. Abbott, Ph.D. Duane B. Baker, Ed.D. Executive Director Senior Researcher Director - Professor of Education Professor of Sociology School Information Services Copyright  2002 by the Washington School Research Center, Seattle Pacific University. All rights reserved. Additional copies of this report may be downloaded in pdf format free of charge at www.spu.edu/wsrc.
  • 3. The Influence of District Size, School Size and Socioeconomic Status on Student Achievement in Washington: A Replication Study Using Hierarchical Linear Modeling A Technical Report For The Washington School Research Center
  • 4. Foreword In recent years there has been a growing interest in the role that school size plays in creating effective learning environments for students. Serious questions have been raised about the “bigger is better” approach to schools, and policy makers are asking researchers if there are research findings on this important topic. In fact, there have been numerous studies, both quantitative and qualitative, strongly suggesting students generally do better in smaller schools than larger schools. Such a study published in the year 2000 on education in the state of Georgia caught the interest of the Urban Issues Committee of the Washington State School Directors’ Association (WSSDA). Recognizing that such research findings have direct policy implications, the Association approached the Washington School Research Center (WSRC) about replicating the study in the state of Washington. Through the joint sponsorship of WSSDA and WSRC, this technical report is a replication of that study using achievement, poverty, school and district size data from Washington State. The research findings on school size show that the question is a complex one, and that there are numerous factors that might interact with school size to account for variation in student and school performance. Using a statistical procedure called Hierarchical Linear Modeling, WSRC researchers Abbott, Joireman, and Stroh attempt to identify the ways in which district size, school size, and family income level interact to effect student achievement. Replication research across states is difficult because of differing state tests, grade structures, data bases, and other factors. And, in fact, while this study replicates the general approach used in the Georgia study, it does differ in some significant ways. First, WASL scores were used for this study rather than the ITBS; second, because of questions about the reliability of the high school poverty data, 4th and 7th grade data were analyzed, while the Georgia study analyzed 8th and 11th grade data. Still, the results presented here complement the original findings and add to the body of research that strongly suggests that school size and district size do matter. The WSRC researchers conclude: “We found that large district size is detrimental to achievement in Washington 4th and 7th grades in that it strengthens the negative relationship between school poverty and student achievement.” Further, they state, “the negative relationship between school poverty and achievement is stronger in larger districts,” and “small schools appear to have the greatest equity effects.” In other words, when school poverty is high, children i
  • 5. perform better in small districts, and the effect of school level poverty on achievement is smallest when both the district and school are small. Jeffrey T. Fouts Executive Director Washington School Research Center ii
  • 6. Table of Contents Introduction ........................................................................................................... 1 Literature Review .................................................................................................. 2 A Replication Study ................................................................................................ 4 Nature of the Data ................................................................................................. 4 Joint Effects of School and District Variables Using Hierarchical Linear Modeling . 5 Fourth and Seventh Grade Equity Effects............................................................... 14 Conclusions ........................................................................................................... 16 References ........................................................................................................... 18 iii
  • 7. The Influence of District Size, School Size and Socioeconomic Status on Student Achievement in Washington: A Replication Study using Hierarchical Linear Modeling Introduction New interest in the effects of school size on academic achievement has grown in recent years as nationwide school reform efforts have gained momentum. Policy makers and practitioners have suggested new models of schools based on the idea that smaller is better. As these models have gained currency, there is also a call for research to investigate the likely effects of creating smaller schools. Are smaller schools needed as a strategy to improve student academic performance? Several studies have examined this question by employing single-level regression models to examine the influence of size (district and school) and low- income on academic achievement. A recent article by Robert Bickel and Craig Howley (2000), however, advanced the research in this area by introducing a multi-level approach to their analysis examining the joint influence of district and school size on academic performance in Georgia. Their research focuses on the “cross-level interaction” of district and school socioeconomic status and size through a single-equation relative-effects model. The results of the Bickel and Howley study particularly highlight the sizeable influences of two such cross-level (i.e., district and school level) interactions: the product of district size and school socioeconomic status (SES), and the product of school size and district SES (especially at the 8th grade level). School-level academic performance is negatively affected both by the district poverty-school size cross-level interaction, and the school poverty-district size cross level interaction. (These findings are not as consistent at the 11th grade level.) Single level regression models cannot identify these important cross-level dynamics. Another important contribution of the Bickel and Howley study is their “equity” analyses. According to the authors, “equity” refers to non-equivalence in the socioeconomic status-academic performance relationship as a function of size. In effect, this asks whether “the amount of variance in achievement associated with SES is substantially reduced in smaller units” (p. 5). The analysis is accomplished by examining the variance in school performance associated with low-income in four different configurations of school and district size (i.e., large school-large district, large school-small district, small school-large district, and small school-small district). 1
  • 8. Literature Review The issue of school and district size effects on student achievement is not new. Beginning in the 1920s schools and school districts grew larger as districts attempted to consolidate both administration and curriculum and instruction. Ellwood Cubberly, a former urban superintendent, advocated for the creation of large schools as immigrant populations in major cities grew. Meanwhile, Joseph Kennedy, dean of the school of education at North Dakota State University, was concerned about losing the participation and identity of local communities as rural schools consolidated (Robertson, 2001; Howley, 1996). The press for larger schools continued through the 1950s as U.S. educators felt the pressure of the “space race” and the need to provide a wider, more academically rigorous curriculum for future scientists. Large schools, however, are not without fault. Bracey (2001) noted: …large schools, especially large high schools, produce their own set of problems, which a growing number of researchers and policy makers think can be solved by returning to small schools. Advocates for small schools have argued that they can • raise student achievement, especially for minority and low-income students; • reduce incidents of violence and disruptive behavior; • combat anonymity and isolation and, conversely, increase the sense of belonging; • increase attendance and graduation rates; • elevate teacher satisfaction; • improve school climate; • operate most cost-effectively; • increase parents and community involvement; and • reduce the amount of graffiti on school buildings. (p. 413) Such arguments by advocates for small schools bear some merit, as research indicates that school (Lee & Loeb, 2000) and district (Bickel & Howley, 2000; Johnson, Howley, & Howley, 2002) size can impact student achievement. Lee and Loeb examined 264 Chicago elementary schools and found that school size influenced student achievement both directly and indirectly. They reported that teachers in small schools (less than 400 students) take more responsibility for students’ academic and social development, and that this in turn enhances student achievement. They noted that small schools facilitate more intimate and personal relationships among both teachers and students, and that it is these relationships that impact student learning. Another study of Chicago’s public schools found that small schools increase attendance, student persistence, 2
  • 9. performance, graduation rates, grades, course completion, and parent, teacher, student, and community satisfaction (Walsey, Fine, Gladden, Holland, King, Mosak, & Powell, 2000). Other research on school size has found socioeconomic status to be the confounding variable in the size/achievement equation, noting that as school size increases, achievement levels for schools with less advantaged students decreases (Bickel, 1999; Howley & Bickel, 2000). Bickel, Howley, Williams, and Glascock (2000) confirmed these findings while controlling for ethnicity, language, size, cost, and curricular composition factors. Interestingly, Howley, Strange, and Bickel (2000) noted that “size exert[s] a negative influence on achievement in impoverished schools, but a positive influence on achievement in affluent schools. That is, all else being equal, larger school size benefits achievement in affluent communities, but it is detrimental in impoverished communities” (p. 4). Additionally, the authors’ results indicated that “the relationship between achievement and SES is substantially weaker in the smaller schools than in the larger schools” (p. 5). The authors also noted, however, that small size doesn’t necessarily guarantee student success. “Small size is a necessary but insufficient condition for school improvement. . . . It is important to avoid seeing small schools as the sole solution to all that ails education. Rather, we would suggest that it is a key ingredient in a comprehensive plan to improve education” (p. 66). District size may also moderate the effects of school size. Bickel and Howley (2000) found an interesting interaction between district and school size. The authors explained: Larger schools in larger districts seem to propagate inequality of outcomes by comparison to smaller schools and smaller districts. In fact, smaller schools in larger districts demonstrate a useful equity effect, as well. For large schools in smaller districts, however, the improvements in equity might be so slight as to be called negligible. (p. 21) The authors (Bickel & Howley, 2000) found that in communities with high rates of poverty, small schools in small districts increase student achievement. Overall, “smaller districts and smaller schools demonstrate greater achievement equity” (Howley, 2000, p. 7). Simply stated, current research indicates that the amount of impact socioeconomic status has on student achievement is dependent upon several factors, including the size of the school and the size of the district in which the school functions. 3
  • 10. A Replication Study The present study is a replication of the Bickel and Howley (2000) method to the Washington state academic performance of 4th and 7th grade students. While Bickel and Howley focused on the 8th grade Iowa Test of Basic Skills (ITBS) and the 11th grade Georgia High School Graduation Test, this study examines 4th and 7th grade academic performance on the Washington Assessment of Student Learning (WASL)1 test that is mandated across the state. A great deal of attention in Washington has focused on the use of the WASL, which has replaced other tests (e.g., ITBS) for assessing statewide standards-based learning objectives. Bickel and Howley’s method is accomplished in this study by the use of Hierarchical Linear Modeling through the HLM software program (Raudenbush, S., Bryk, A., and Congdon, R., 2000). The two approaches attempt to specify the joint relationships and cross-level interactions of two structural levels (district and school) on school academic performance. Nature of the Data The data used for this replication study were provided by Washington State’s Office of the Superintendent of Public Instruction, and are from the testing year 2001. The data consist of all 4th and 7th grade student WASL scale scores in reading and mathematics, aggregated to the school level.2 Schools with less than ten students were excluded from the study, and, because there was a concern over the unique characteristics of some “types” of schools, those labeled “Alternative,” “Institutional,” and other “Unclassified” types were not included. Table 1 indicates the descriptive data for the variables used in this study. Percent Free and Reduced (F/R) Lunch is used on both school and district levels as a measure of low socioeconomic status3. Spansize is the number of students per grade level and is used as our measure of school size, as it was by Bickel and Howley (2000). Enrollment is the total number of students per district. 1 For more information on administration of the WASL and ITBS in Washington, visit www.k12.wa.us/assessment/WASLintro.asp. For more technical information on the WASL, visit www.k12.wa.us/assessment/qawasl.asp. For more technical information on the ITBS, visit www.riverpub.com/products/group/itbs.htm. 2 Grade 10 WASL scores were not used since, in our experience, the % Free/Reduced Lunch data are less reliable at that level. 3 Bickel and Howley (2000) refer to free or reduced price meals as SES. 4
  • 11. Table 1 Descriptive Statistics for 4th and 7th Grades 4th Grade School-Level Mean SD N Percent F/R Lunch 37.86 23.62 1035 Spansize 69.67 28.29 1035 Math Scale Score 392.83 14.79 1035 Reading Scale Score 405.45 7.07 1035 District-Level Percent F/R Lunch 37.92 18.84 251 Enrollment 3903.34 6089.11 250 7th Grade School-Level Mean SD N Percent F/R Lunch 33.27 21.33 417 Spansize 177.87 112.74 417 Math Scale Score 366.29 17.81 417 Reading Scale Score 393.79 6.81 417 District-Level Percent F/R Lunch 37.92 18.75 255 Enrollment 3924.40 6277.40 255 Joint Effects of School and District Variables Using Hierarchical Linear Modeling Major Goal As noted earlier, our major goal was to determine whether the cross-level interactions (school x district level) between size and socioeconomic status reported by Bickel and Howley (2000) would replicate within Washington. More specifically, we were interested in whether the data in Washington would replicate two major patterns reported by Bickel and Howley: (1) larger schools are beneficial within affluent communities, whereas smaller schools are more beneficial within less affluent districts; and (2) the “achievement cost” associated with less affluent schools is greater in large districts (i.e., the negative 5
  • 12. relationship between school-level poverty and achievement is stronger in larger districts). The first pattern would require an interaction between school size and district-level SES; the second pattern would require an interaction school-level SES and district size. Hierarchical Linear Modeling (HLM) To determine whether these two patterns (interactions) were present within the current Washington State data, we conducted a series of analyses using hierarchical linear modeling (HLM). HLM is a statistical technique that is appropriate for analyzing multi-level data (e.g., schools nested within districts). HLM has a number of advantages over other approaches to multi-level data, such as ignoring the nested nature of the data (which violates the assumption of non-independent errors) or pooling the data by, for example, aggregating across schools within a district (which results in a loss of data, and eliminates the possibility of examining cross-level interactions).4 In each of the analyses, school level variables were group centered (district means) and district level variables were grand mean centered. Overview of Models 1 and 2 Tables 2 and 3 summarize the results of two different models, respectively. Model 1 examines the influence of school size and district poverty (on math and reading at both the 4th and 7th grades), whereas Model 2 examines the influence of school poverty and district size (on math and reading at both the 4th and 7th grades). Model 1 Summary – School Size and District Poverty We begin by interpreting the results for Model 1, using the 4th grade, shown in the top half of Table 2. As can be seen, results for math and reading were quite similar. First, scores in math and reading each showed a highly significant negative relationship with district poverty (Bs = -47.10 and -24.10, ps < .0001). Second, math and reading showed no significant relationship with school size (ps > .35). Third, results did not reveal interactions between school size and district poverty, findings that were not consistent with Bickel and Howley’s (2000) findings. While the interactions were not significant, they were in the expected direction. To further examine the nature of the interactions, we plotted the relationship between school size and achievement for districts in the 25th (thin lines) and 75th (bold lines) percentiles on poverty, as shown in Figure 1. While the interaction was not significant in either case, there was a tendency for larger schools to be somewhat more beneficial in more affluent districts (25th percentile, thin line) than in less affluent districts (75th percentile, bold line). 4 For more information on Hierarchical Linear Modeling, see Raudenbush and Bryk (2002). 6
  • 13. We now turn to the results for the 7th grade, summarized in the bottom half of Table 2. As can be seen, the 7th grade results replicated the 4th grade results. First, both math and reading showed a significant negative relationship with district poverty (Bs = -61.05 and -23.41, ps < .0001). Second, math and reading showed no significant relationship with school size (ps > .28). Last, results at the 7th grade failed to reveal a significant interaction between school size and district poverty on either math or reading (ps > .27). For comparison with the 4th grade results, we present graphs of the relationship between school size and achievement at the 25th and 75th percentile on district poverty for the 7th grade (Figure 2). As can be seen, while the interaction was not significant in either case, there was a tendency for larger schools to be somewhat more beneficial in more affluent districts (25th percentile, thin line) than in less affluent districts (75th percentile, bold line). Taken as a set, results for Model 1 do not replicate Bickel and Howley’s (2000) findings concerning the interaction between school size and district poverty. As we explain in the next section, results did replicate Bickel and Howley’s finding concerning the interaction between school poverty and district size. Model 2 Summary – School Poverty and District Size We now turn to the results for Model 2, in which we examine the influence of school poverty and district size on math and reading in the 4th and 7th grade, as summarized in Table 3. First, math and reading showed significant negative relationships with school level poverty in both grades (ps < .001). Second, math and reading showed a significant positive relationship with district size in the 4th grade (ps < .004), but showed no significant relationship with district size in the 7th grade (ps > .09). Finally, in each case, school poverty and district size showed a significant interaction (ps < .001). As we will explain, the nature of this interaction was consistent with Bickel and Howley’s (2000) findings (for their 8th grade only – comparable 11th grade results were not significant). To further examine the nature of this interaction, we plotted the relationship between school poverty and achievement for small districts (25th percentile, thin line) and large districts (75th percentile, bold line) at both the 4th and 7th grade. Figure 3 presents the results for 4th grade; Figure 4 presents the results for the 7th grade. As can be seen, in every case, the negative relationship between school poverty and achievement is stronger in large districts, thus replicating Bickel and Howley’s (2000) findings (for the 8th grade only). 7
  • 14. Table 2 Summary of HLM Runs – Model 1 – Effects of School Size and District Poverty on Math and Reading Achievement in 4th and 7th Grades. 4th Grade (Figure 1) Math B SE t df p-value Intercept 389.99776 0.519294 751.015 246 0.000 School Size 0.03 0.031156 0.925 246 0.355 District Poverty -47.10 2.843856 -16.562 246 0.000 SS x DP -0.16 0.143109 -1.127 246 0.260 Reading B SE t df p-value Intercept 404.00 0.242452 1666.317 246 0.000 School Size 0.01 0.014902 0.574 246 0.566 District Poverty -24.10 1.377889 -17.492 246 0.000 SS x DP -0.087 0.073228 -1.191 246 0.234 7th Grade (Figure 2) Math B SE t df p-value Intercept 364.36 0.708944 513.95 253 0.000 School Size 0.04 0.038757 0.95 253 0.343 District Poverty -61.05 4.394758 -13.89 253 0.000 SS x DP -0.25 0.229240 -1.09 253 0.275 Reading B SE t df p-value Intercept 393.02 0.277042 1418.64 253 0.000 School Size 0.01 0.011056 1.06 253 0.289 District Poverty -23.41 1.776674 -13.18 253 0.000 SS x DP -0.05 0.068546 -0.76 253 0.445 Note. DP = District Poverty (% of students in district on free or reduced lunch); SS = School Size (spansize). 8
  • 15. Table 3 Summary of HLM Runs – Model 2 – Effects of School Poverty and District Size on Math and Reading Achievement in 4th and 7th Grades. 4th Grade (Figure 3) Math B SE t df p-value Intercept 390.21 0.806115 484.062 247 0.000 School Poverty -0.24 0.038737 -6.228 247 0.000 District Size .000287 0.000107 2.682 247 0.008 SP x DS -0.00001 0.000003 -3.809 247 0.000 Reading B SE t df p-value Intercept 404.075 0.397746 1015.914 247 0.000 School Poverty -0.14 0.017709 -7.719 247 0.000 District Size .000151 0.000052 2.907 247 0.004 SP x DS -0.000005 0.000001 -4.497 247 0.000 7th Grade (Figure 4) Math B SE t df p-value Intercept 364.83 1.020967 357.34 253 0.000 School Poverty -0.40 0.120292 -3.29 253 0.001 District Size .000250 0.000149 1.68 253 0.093 SP x DS -0.00002 0.000006 -3.51 253 0.001 Reading B SE t df p-value Intercept 393.24 0.401891 978.48 253 0.000 School Poverty -0.18 0.051623 -3.42 253 0.001 District Size .000084 0.000052 1.60 253 0.108 SP x DS -0.000008 0.000002 -3.51 253 0.001 Note. DS = District Size; SP = School Poverty (% students on free or reduced lunch). 9
  • 16. Figure 1 Math and Reading Achievement as a Function of School Size (Spansize) and District Poverty (25th and 75th Percentiles) – 4th Grade. 4th WASL 409.7 DISTFR = -0.137913 DISTFR = 0.113489 M 407.1 a t 404.6 h 402.0 A 399.4 c h 396.9 i e 394.3 v e 391.7 m e 389.1 n t 386.6 384.0 -60.94 2.94 66.83 130.71 194.60 258.48 School Size 4th WASL 412.6 R DISTFR = -0.137913 e 411.5 DISTFR = 0.113489 a d 410.3 i n 409.1 g 408.0 A c 406.8 h i 405.6 e v 404.4 e m 403.3 e 402.1 n t 400.9 -60.94 2.94 66.83 130.71 194.60 258.48 School Size 10
  • 17. Figure 2 Math and Reading Achievement as a Function of School Size (Spansize) and District Poverty (25th and 75th Percentiles) -- 7th Grade. 7th Grade 398.3 DISTFR = -0.146881 DISTFR = 0.114901 M 394.1 a t 389.9 h 385.6 A 381.4 c h 377.2 i e 372.9 v e 368.7 m e 364.5 n t 360.2 356.0 -168.95 -67.15 34.65 136.45 238.24 340.04 School Size 7th Grade 403.2 R DISTFR = -0.146881 e 401.8 DISTFR = 0.114901 a d 400.4 i n 399.0 g 397.6 A c 396.3 h i 394.9 e v 393.5 e m 392.1 e 390.7 n t 389.4 -168.95 -66.13 36.68 139.50 242.32 345.13 School Size 11
  • 18. Figure 3 Math and Reading Achievement as a Function of School Poverty and District Size (25th and 75th Percentile) – 4th Grade 4th WASL 399.7 DISTENRL = -3357.289063 DISTENRL = 558.710938 M 397.3 a t 394.9 h 392.4 A 390.0 c h 387.6 i e 385.1 v e 382.7 m e 380.3 n t 377.8 375.4 -37.93 -18.20 1.52 21.24 40.96 60.69 School Poverty 4th WASL 409.4 R DISTENRL = -3357.289063 e 408.1 DISTENRL = 558.710938 a d 406.7 i n 405.3 g 403.9 A c 402.6 h i 401.2 e v 399.8 e m 398.5 e 397.1 n t 395.7 -37.93 -18.20 1.52 21.24 40.96 60.69 School Poverty 12
  • 19. Figure 4 Reading Achievement as a Function of School Poverty and District Size (25th and 75th Percentile) for 4th and 7th Grade. 7th Grade 378.4 DISTENR = -3406.395996 DISTENR = 443.604004 M 374.7 a t 370.9 h 367.2 A 363.4 c h 359.7 i e 355.9 v e 352.2 m e 348.4 n t 344.7 340.9 -33.27 -14.76 3.76 22.27 40.79 59.30 School Poverty 7th Grade 399.3 R DISTENR = -3406.395996 e 397.6 DISTENR = 443.604004 a d 395.9 i n 394.2 g 392.5 A c 390.9 h i 389.2 e v 387.5 e m 385.8 e 384.1 n t 382.5 -33.27 -14.57 4.13 22.83 41.53 60.23 School Poverty 13
  • 20. Fourth and Seventh Grade Equity Effects In order to measure equity effects, Bickel and Howley (2000) used squared zero- order correlation values (r2) between SES and achievement within four categories of district and school size (i.e., large school-large district, large school- small district, small school-large district, and small school-small district). They found that, the predicted equity effect of reducing district size but not school size would be practically significant; the predicted equity effect of reducing school size but not district size would also be practically significant and perhaps somewhat larger; and the combined strategy of reducing both school and district size would be predicted to yield substantial equity and excellence effects . . . (p. 20) This study replicates Bickel and Howley’s method by creating four categories of district and school size based on median split values: large school-large district, large school-small district, small school-large district, and small school-small district. Within each of these categories are the r2 values between F/R lunch % and achievement (math and reading WASL scores) for 4th and 7th grades. Table 4 lists the results of these analyses. Both 4th and 7th grade results reflect Bickel and Howley’s (2000) results in that the data in this study show the small district-small school category shows the smallest proportion of variance in achievement associated with school poverty. Apart from this however, the results are different from Bickel and Howley in at least one major instance: the large district-large school category did not result in the greatest amount of variance explained across grade levels and subjects. The overall results of both 4th and 7th grades reflect the school poverty – district size interaction results reported in Table 3: that the negative relationship between school poverty and achievement is stronger in larger districts. Small schools in small districts explain the least amount of variance (13% to 24% of the variance in achievement associated with poverty), but the largest amount of variance explained is in large districts irrespective of school size (41% to 54%). This finding is consistent across grade levels and subjects despite large differences between 4th and 7th grade spansizes. The 4th grade spansize is approximately 2 ½ times smaller than the 7th grade spansize while the mean district enrollments are approximately equal. Therefore, small schools appear to have the greatest equity effects, while large districts are the most detrimental. 14
  • 21. Table 4 Variance in Achievement Explained by School Poverty as a Function of District and School Size: Washington Multi-Level Equity Effects1 Large and Small Districts and Schools—4th, 7th WASL Scores 4th Grade Math Scale Scores Districts2 Large Small 2 2 r n r n Grade Span Large 0.41 487 0.27 30 Size3 Small 0.45 416 0.17 101 4th Grade Reading Scale Scores Districts2 Large Small 2 2 r n r n Grade Span Large 0.47 487 0.46 30 Size 3 Small 0.54 416 0.13 101 7th Grade Math Scale Scores Districts4 Large Small 2 2 r n r n Grade Span Large 0.48 208 - 0 Size 5 Small 0.46 80 0.24 129 7th Grade Reading Scale Scores Districts4 Large Small 2 2 r n r n Grade Span Large 0.53 208 - 0 Size 5 Small 0.51 80 0.20 129 1 Variance (r2) in Scale Scores attributable to % Free/Reduced Lunch 2 Median Split = 1,514.5 3 Median Split = 68.9 4 Median Split = 1,411 5 Median Split = 186 15
  • 22. Conclusions Our study replicated the method of Bickel and Howley (2000) for understanding the influence of district size, school size and socioeconomic status on student achievement in Washington. We found that large district size is detrimental to achievement in Washington 4th and 7th grades in that it strengthens the negative relationship between school poverty and student achievement. This finding replicated that of the Bickel and Howley (2000) study. We did not replicate another of Bickel and Howley’s (2000) findings, however. In our study, district affluence did not have a significant impact over the school size – student achievement relationship. The tendency for larger schools to be somewhat more beneficial in more affluent districts (and, equivalently, for smaller schools to be more beneficial in less affluent districts) is shown in the analyses, but was not found to be statistically significant. The nature and configuration of Washington schools may partially explain the discrepancy between the findings of the two studies with respect to district affluence. First, the majority of districts (for both 4th and 7th grades) in Washington are single-school districts that tend to be smaller, poorer, and more rural than multi-school districts.5 Second, our study used the WASL as the measure of student achievement (both 4th and 7th grades) in contrast to Bickel and Howley’s use of the ITBS (8th grade) and the Georgia High School Graduation Test (11th grade). Preliminary analyses in Washington indicate that the WASL and the ITBS have different correlations with school poverty, especially in math.6 Third, there are a number of other variables not addressed in either study that may exert important influences on student achievement. Taken together, the method used by Bickel and Howley (2000) and that of this study were useful for explaining relationships in multi-level data that could not be explained by more traditional (single-level) analyses. School and district size are often assumed to be primary, independent influences on student achievement. In fact, this is a commonly expressed sentiment even among practitioners and policy makers. However, based on this study, it appears that size is a more complex matter, and needs to be viewed in the context of other influences in order to determine its contribution to school-level achievement. Certainly, the multi-level findings of our study argue against the simplistic conclusion that reducing school and/or district size will automatically improve student achievement, or be more equitable. We are in complete agreement with Bickel and Howley (2000) in their comment that, “the conclusions of this study would seem to require rather wide debate and reconsideration of the size issue, 5 We use “single-school districts” to indicate districts with a school containing one (4th and 7th) grade level. 6 For a comparison of the WASL and ITBS in Washington, see Joireman and Abbott (2001) 16
  • 23. across the spectrum of poverty and wealth, and not just in the case of impoverished communities” (p. 21). 17
  • 24. References Bickel, R. (1999). School size, socioeconomic status, and achievement: A Georgia replication of inquiry in education. Huntington, WV: Marshall University. Bickel, R., & Howley, C. (2000). The influence of scale on student performance: A multi-level extension of the Matthew Principle. Education Policy Analysis Archives, 8(22). Bickel, R., Howley, C., Williams, T., & Glascock, C. (2000). High school size, achievement equity, and cost: Robust interaction effects and tentative results. Washington, D.C.: Rural School and Community Trust. Bracey, G. (2001). Small schools, great strides. Phi Delta Kappan, 82(5), 413- 414. Howley, C. (1996). Ongoing dilemmas of school size: A short story. Retrieved June 4, 2002 from http://www.aasa.org/issues_and_insights/district_organization/howley_dile mmas.htm. Howley, C. (2000). School district size and school performance. Rural Education Issue Digest. Howley, C., & Bickel, R. (2000). When it comes to schooling…small works: School size, poverty, and student achievement. Randolph, VT: Rural School and Community Trust. Howley, C., Strange, M., & Bickel, R. (2000). Research about school size and school performance in impoverished communities [ERIC Digest]. Washington, DC: Office of Educational Research and Improvement. Joireman, J., & Abbott, M. (2001) The relationships between the Iowa Test of Basic Skills and the Washington Assessment of Student Learning in the State of Washingon [ERIC Document TM033 308]. Lynwood, WA: Washington School Research Center, Seattle Pacific University. Johnson, J. D., Howley, C. B., Howley, A. A. (2002). Size, excellence, and equity: A report on Arkansas schools and districts. Athens, OH: Ohio University. Lee, V., & Loeb, S. (2000). School size in Chicago elementary schools: Effects on teachers’ attitudes and students’ achievement. American Educational Research Journal, 37(1), 3-31. 18
  • 25. Raudenbush, S., & Bryk, A. (2002). Hierarchical linear models: Applications and data analysis methods (2nd edition). Thousand Oaks: Sage Publications. Raudenbush, S., Bryk, A., and Congdon, R. (2000) HLM (Version 5) [Computer Software]. Lincolnwood, IL: Scientific Software International. Robertson, S. (2001). The great size debate [A CEFPI Brief on Educational Facility Issues]. Scottsdale, AZ: Council of Educational Facility Planners, International. Walsey, P. A., Fine, M., Gladden, M., Holland, N. E., King, S. P., Mosak, E., & Powell, L. C. (2000). Small schools: Great strides. A study of new small schools in Chicago. New York: Bank Street College of Education. 19