Issues in Educational Research, 6(1), 1996, 57-78.

School size and academic achievement in the HSC Examination: Is there a relationship?

Magdalena Mok
Macquarie University

Marcellin Flynn
Australian Catholic University

This paper examines two related research questions. First, irrespective of other school and student background variables, does school size make a difference to achievement in the HSC examination? Second, does school size affect HSC achievement after taking into account school and student background characteristics, student academic motivation and the educational component of the school culture? The sample used in this study comprised 4949 year 12 students from 44 New South Wales Catholic High Schools. To the extent that this sample is representative, results from multilevel analysis of the data suggest that students from larger Catholic schools, on the average tend to achieve more highly than their peers from smaller schools, even after controlling for students' background, motivation and school culture variables.

Introduction: Education in New South Wales

Currently school size is a topic of concern in the New South Wales educational arena. Reviewing school size in terms of whether or not some smaller high schools can continue to provide the breadth of curriculum needed today was an item on the reform agenda proposed by Dr Ken Boston, the Director-General of School Education, soon after signing a new five-year contract with the Government (Sydney Morning Herald, 2 January 1997, page 1). This move has political as well as academic implications. Critics would view the motive as purely economic: closing smaller schools in order to save running costs. They could compare the debate about school size with the Wyndham 'scheme' in NSW (1960-1980) which brought about a transition towards large secondary schools of over one thousand students. Teachers, parents and students, however, may be more likely to recall the recent forced closure of about one thousand small schools and the associated loss of jobs in Victoria. However, how does school size relate to academic achievement? Does the relationship change after controlling for other student and school variables? This study aims to provide some insights into these issues.

Research literature on school size and its effects on academic achievement

Over the past three decades much research has been undertaken both in Australia and overseas on the relationship between school size and academic achievement at secondary level. These studies tended to adopt one of two opposing perspectives and yielded contradictory recommendations. Researchers taking the first perspective focussed on economies of scale (Conant, 1959; 1967). Their arguments were based on the premise that, as school enrolments increased, so did the school's budget, and as a result, more efficient use of funds could be realised for smaller schools. Larger schools had more resource opportunities, better market influence and were able to provide students with more varied and diverse curricula, better qualified teachers and more superior school physical environment and facilities (for example, Haller, Monk, Bear, Griffith and Moss, 1990). Such views received wide support among policy makers.

Research in Australia

An alternative to the economies of scale perspective is one which focussed on outcomes. Research adopting this perspective concentrated on the "output" variables which might include various measures of cognitive and affective achievement as well as attainment. Recommendations were made for large or small schools depending on whether school size correlated positively or negatively with these outcome measures.

In general, research pursuing this outcome line of inquiry found school size effects on cognitive achievement to be bordering on the trivial, if they were found at all. For instance, a thorough review of Australian research by Spearritt (1987) convinced him that achievement in language and number skills was not related to school size. Included in Spearrit's review were periodic surveys conducted by State Education Departments (Tasmania, Queensland and Western Australia) and the Australian Council for Educational Research (Victoria), as well as the Australian studies of literacy and numeracy between 1967 and 1985.

Research overseas

Research undertaken in the United States (Caldas, 1993; Lamdin, 1995; Luyten, 1994; Ramiez, 1990), as well as in Europe, Norway (Bonesronning, 1996), the Netherlands, and Sweden (Luyten, 1994), failed to provide compelling evidence one way or the other. Exceptions to this general conclusion arose when other variables such as socio-economic status (SES) (Howley, 1989), or engagement in restructuring practices (Lee and Smith, 1995) were controlled. In these instances, higher academic achievement tended to be associated with smaller schools.

In contrast to these findings, a strong body of international research consistently favouring smaller schools in terms of various staff and student affective outcomes is now available. The Australian scene was well described in Keeves' (1987) and McKenzie's (1995) separate reviews. The reviewers reported a lack of research on school size and achievement in Australia, but both observed that students in smaller schools tended to enjoy a better quality of school life than those in larger schools. These observations were also found in studies undertaken in the United States.

A negative relationship, however, between school size and affective achievement is well documented in the American research literature. Small schools were found to enhance interpersonal relationships (Bryk, 1996), increase student extra-curricular participation (Barker and Gump, 1964; Oxley, 1994; Stevens and Peltier, 1994), provide more opportunities for developing students' leadership potential (Bryk, Lee and Holland, 1993), have more effective school discipline (Haller, 1992; Rowan, Raudenbush, and Kang, 1991), improved school climate (Lindsay, 1984; Oxley, 1994), and have lower dropout rates of students from schools (Fetler, 1989; Kleinfeld, McDiarmid, and Hagstrom, 1989; Toenjes, 1989). The advantages of small schools over large ones were particularly salient in the case of disadvantaged students (Howley, 1994; Klonsky, 1995) and schools located in rural areas (Haller, 1992).

Despite the existing volume of research, various authors (for example, Bryk, Lee and Holland, 1993; Friedkin and Necochea, 1988; Luyten, 1994) advocated the utilisation of Multilevel Modelling methods in future analyses to explore the effect of school size on achievement. This statistical approach is adopted in this study and will be discussed in the following sections.

Research questions

This study seeks to extend the scope of previous research concerning school size and academic achievement by utilising multilevel models in the analysis. Two research questions are addressed:

1. Irrespective of other school and student background variables, does school size make a difference to students' achievement at the HSC examination?
2. Does school size affect academic achievement in the Higher School Certificate (HSC) Examination after controlling for school and student background characteristics, academic motivation and the educational component of the school culture?

Methods used in this study of school size effects on academic achievement

Description of the sample of students from Catholic schools

This study is a component of a larger investigation concerning the culture of Catholic schools (Flynn, 1993). In this study, Year 12 students from 50 of the 102 Catholic High Schools in New South Wales, Australia, were surveyed in May 1990 regarding their experiences of the quality of school life in their schools. In early 1991, after the HSC Examination results were published, Principals of the 50 sampled Catholic schools were invited to provide, in confidence, the TER (Tertiary Entrance Rank) obtained by their students in the HSC Examination. Of the 50 schools involved in the study, 44 schools (88% participation rate) involving 4,949 students provided their TER information (Flynn, 1993). This report is based on responses from these students on their background characteristics, expectations, motivation, perception of their school's educational culture and their HSC Examination results.

Background of the students in the sample

The sampling procedure adopted ensured that Year 12 students in each Diocese were adequately represented. Of the schools in the sample, 13 were boys' schools, with 1,217 male students, 10 were girls' schools, with 1,002 female students and 21 were co-educational schools with 1,405 male and 1,325 female students. The majority, 87 % of the students, were Catholics. Of the non-Catholic students 3% belonged to the Orthodox Church, 6% to other Christian Churches, 1% belonged to non-Christian religions, and 3% reported that they had no religion. Most of the students were born in Australia (88%). The others were born in Asia (4%), another English-speaking country (3%), in Europe or the Middle East (2%), or in other countries (3%).

Family background of students

Students in the sample came from a variety of family backgrounds: about one in ten of the fathers had only attended primary school, about half of the fathers had received some secondary education (12%) or had completed Year 12 (45%). The remainder had either some tertiary education (9%) or a degree or diploma (23%). Generally, the mothers' educational level tended to be lower than that of the fathers, with only 24% having undertaken some tertiary education or having a degree or diploma.

Measurement of achievement and school size

Academic achievement in this study was measured by the students' performance in the Higher School Certificate Examination (HSC) in 1990. Psychometricians (for example, MacCann, 1995) recommend that the original Tertiary Entrance Scale (TES) aggregate marks, instead of the Tertiary Entrance Ranks (TER), should be used when statistical techniques such as t-test, analysis of variance or regression are used. For the purpose of the current study, analyses were therefore based on the TES scores which were obtained from the TER forwarded by the Principal of each participating school.

School size was measured as a continuous variable representing the total student enrolment in the period of the study. The smallest school had 234 students and the largest school had 1,274 students. The mean and median school sizes were respectively, 759 and 769 students.

Methods of analysis

This study attempts to address two research questions:

1. the global effect of school size on academic achievement, and
   
2. school size effect on academic achievement after taking into account student background, school characteristics and education culture of the school.

School level simple regression analysis

To begin with, school size was modelled as the predictor of average Tertiary Entrance Scale (TES) score utilising a simple regression analysis. TES scores were aggregated to the school level for each of the 44 sampled schools. This was augmented by a scatter plot of school mean TES score against school size. While this first analysis suggested an insight into the possible form of the relationship between school size and academic achievement, results from the analysis could also be quite misleading. The analysis does not take into account:

1. Control for intervening variables,
2. Correlation does not imply causality, and
3. Nested data structure.

Control for intervening variables

First, students' achievement in the Higher School Certificate (HSC) examination may be affected by other contributing factors which are related to school size. Without first controlling for these other factors, one is unable to discern the precise relationship between school size and TES score. Contributing factors to school effectiveness identified in the literature include school socio-economic status (Levine, 1992), school type (single-sex/co-education) (Flynn, 1993), leadership of the school principal (Mortimore, 1995), school culture (Flynn, 1993), school processes (Mortimore, 1995), school climate (Levine and Lezotte, 1990), student motivation (Keeves, 1986) and student achievement at intake (Sammons, Nuttall & Cuttance, 1993).

Correlation does not imply causality

Second, correlation does not imply causality. While school size might relate strongly to academic achievement, the former might not be the cause, or the outcome, of the latter, and vice versa. For instance, it could be argued that a group of caring and well-qualified teachers at a school attracted more students to enrol in the school, while at the same time, teaching methods employed by the teachers contributed to high student achievement. If so, excellent teachers influenced both large school size and better HSC results. Academic achievement, however, was neither the cause nor the effect of school size.

Nested data structure which requires multilevel modelling

Third, as a consequence of the sampling design used in this study, data have an inherent nested structure and ordinary least squared regression methods are considered inappropriate (see for example, Aitkin and Longford, 1986; Goldstein, 1987, 1995). In this study, data consisted of responses from intact classes of students nested within randomly selected Catholic schools. One would expect students coming from the same schools to have more commonalities in their social background, academic inclination and educational values than those coming from different schools. Also, students from the same school, being subjected to the same curriculum and influenced by the same school culture, would tend to be more alike than students from different schools in their academic achievement. The similarities within a cluster of students could give rise to what is known as 'intraclass correlation' in the error term of ordinary least square regression models. Numerous studies have provided evidence, both empirically and theoretically, that ignoring a non-zero intraclass correlation will result in the standard errors of the regression coefficients being underestimated. (See for example, Goldstein, 1995; Kish and Frankel, 1974; Mason, Wong, and Entwistle, 1984; Pfefferman, 1985). This leads to erroneous inferences and conclusions. Instead of ordinary least square regression approaches, recent statistical advances have enabled researchers to use the family of multilevel models which explicitly model cluster effects.

The effect of institutions on student achievement

Further, this research concerns the effect of institutions (school size) on student achievement. Thus, it involves a cross-level research question. Analysis based on school aggregates ignores the within-school variations and might lead to false conclusions. For example, a school with a high average achievement level and heterogeneous student performance might not be more effective than another school with a lower average achievement level but homogeneous student performance. Such within-school variations, however, are not reflected in aggregation analyses. Multilevel modelling on the other hand, enables cross-level questions to be addressed directly in one analysis without having to aggregate or to choose the unit of analysis.

Multilevel models

Taking into consideration both conceptual and statistical issues discussed above, a series of four multilevel models were tested empirically in order to answer the research questions:

1. What is the global effect of school size on academic achievement?
2. What is school size effect on academic achievement after taking into account student background, school characteristics and education culture of the school?

The models used were as follows:

• The basic model
• The school size model
• The education culture model, and
• The conditional model

The basic model

The basic model included no explanatory variables in the equation. In this model, the total variance of Tertiary Entrance Scale (TES) score was decomposed into a school-level component and a student-level component. The Basic Model served as the baseline of comparison for all subsequent models in that this model gave the maximum amount of TES variance to be explained at both the school-level and the student-levels. When more and more explanatory variables were included in subsequent models, both school- and student-levels TES variance decreased. An explanatory variable was deemed important if it accounted for a large proportion of TES variance. In addition, in the Basic Model, the intraclass correlation was computed by taking the ratio of the school-level variance to the total TES variance. The intraclass correlation, therefore, represented the proportion of TES variance accounted for by school membership.

The school-size model

The School-size Model included school size as the only explanatory variable for variations in TES. It provided an answer to the first research question: Irrespective of other school and student background variables, does school size make a difference to achievement in the Higher School Certificate (HSC) examination?

The education culture model

The Education Culture Model included school and student background characteristics, students' educational motivation, and the educational dimension of the school culture, but not school size, as explanatory variables of TES scores. Background variables considered in this model comprised: School type (single-sex versus co-educational), school socio-economic status (SES) (high/medium/low), student gender (male/female) and the interaction effect of these variables with the school.

Students' educational motivation was measured by a total of eight scales. These included three scales to measure each student's reason for staying on at school after Year 10 and five scales to measure the student's expectations of Catholic schools with respect to vocational, academic, personal, social and religious developments. Details of the scales were reported in Flynn (1993). The scales were:

1. Inability to Find Work Scale (7 items): A high score in this scale suggests that the students stay on at school after Year 10 because they cannot find the job the student wanted. All items in this scale are response statements to the question: How important were the following reasons in your decision to stay on at school until Year 12? The scale reliability (Cronbach's Alpha) is 0.65. A typical item in the scale is, "I could not find any kind of job".

2. Student's Enjoyment of School Work Scale (7 items): This scale measures the extent to which a students stay on at school after Year 10 because they enjoy school work. All items in this scale are response statements to the question: How important were the following reasons in your decision to stay on at school until Year 12? The reliability (Cronbach's Alpha) is 0.83. A typical item for this scale is, "School work is interesting".

3. HSC Improves Future Job Opportunities Scale (6 items): This scale measures the extent to which students continue at school after Year 10 because they consider a good Higher School Certificate pass is essential to future employment prospects. All items in this scale are response statements to the question: How important were the following reasons in your decision to stay on at school until Year 12? The reliability (Cronbach's Alpha) is 0.66. A typical item for this scale is, "Some jobs require you to have the HSC".

4. Vocational Development Expectation Scale (6 items): A high score in this scale indicates that the student assigns high priority to the vocational or career goals of the school. Reliability (Cronbach's Alpha) of the scale is 0.86. A typical item is, "Catholic schools should help students prepare for employment".

5. Academic Development Expectation Scale (8 items): A high score in this scale indicates that the student assigns high priority to the academic goals of the school. Reliability (Cronbach's Alpha)of the scale is 0.84. A typical item is, "Catholic schools should prepare students for the HSC".

6. Personal Development Expectation Scale (7 items): A high score in this scale indicates that the student assigns high priority to personal development goals of the school. Reliability (Cronbach's Alpha)of the scale is 0.84. A typical item is, "Catholic schools should help students develop their personality and character".

7. Social Development Expectation Scale (8 items): A high score in this scale indicates that the student assigns high priority to the social development goals of the school. Reliability (Cronbach's Alpha)of the scale is 0.86. A typical item is, "Catholic schools should prepare students to become good citizens".

8. Religious Development Expectation Scale (11 items): A high score in this scale indicates that the student assigns high priority to the religious development goals of the school. Reliability (Cronbach's Alpha) of the scale is 0.93. A typical item is, "Catholic schools should provide an environment in which students' faith in God can develop".

School process variables

School processes, as perceived by students, were measured by six Likert-type scales:

1. The Formal Curriculum Scale (6 items): A high score in this scale indicates that the student is satisfied with the curriculum offered by school in Years 11 and 12. Reliability (Cronbach's Alpha) of the scale is 0.83. A typical item is, "Subjects offer useful knowledge/skills".

2. The Out-of-School Curriculum Scale (4 items): A high score in this scale indicates that the student is satisfied with the out-of-school or 'extra-curricular' activities of the school. Reliability (Cronbach's Alpha) of the scale is 0.67. A typical item is, "Out of school activities have variety and scope".

3. The Principal Scale (5 items): A high score in this scale indicates that the student has a warm appreciation of the leadership of the school Principal. Reliability (Cronbach's Alpha) of the scale is 0.80. A typical item is, "The Principal provides good leadership of the school community".

4. The School Discipline Scale (6 items): A high score in this scale indicates that the student has a positive attitude to school discipline policies. Reliability (Cronbach's Alpha) of the scale is 0.65. A typical item is, "There are too many rules which restrict freedom (reversely coded)".

5. The Students' Relationship with Teachers Scale (13 items): A high score in this scale indicates that the student has a warm appreciation of teachers. Reliability (Cronbach's Alpha) of the scale is 0.91. A typical item is, "Most teachers go out of their way to help you".

6. The Student Morale Scale (12 items): A high score in this scale indicates that the student has a high level of morale and is happy at school. Reliability (Cronbach's Alpha) of the scale is 0.91. A typical item is, "My experience of this school has been a happy one".

The conditional model

The conditional model takes into account students' background, school characteristics and education culture. The Conditional Model addressed the main research question, namely,

Does school size affect academic achievement independently of school and student background characteristics, student academic motivation and the education component of the school culture?

The education culture model served as a point of reference for the conditional model, which included, in addition to all aforementioned background and culture variables and school size.

A synthesis

Common to all multilevel models considered here were two components: (a) the fixed component which in this study expresses the overall effect of the school size on TES across schools, and (b) the random component which contains random variations, due to uniqueness of both students and schools, in the relationship between school size and TES between the schools (ie, school-level variation), and within schools (ie, student-level variation). The mathematical and statistical principles underlying multilevel models have been discussed in detail by Goldstein (1995) and others. These mathematical discussions will not be repeated here.

Model comparisons

Model comparisons were based on the difference in a statistical concept called the model "deviance". The model deviance is a measure of how well the model reflects the data. The smaller the deviance, the better the model. If, for example, the deviance of the Conditional Model was significantly smaller (in a statistical sense) than that of the Education Culture Model, then one could conclude that the Conditional Model reflected the data better. In addition, given that the two models differed only with respect to school size being included in the Conditional Model, but not in the Education Culture Model, one could conclude that school size contributed significantly to the explanation of achievement at the Higher School Certificate Examination.

The MLn package utilised in this analysis

Several computer packages are now available for analysing multilevel data. All multilevel analyses undertaken in this study used the MLn computer software package (Goldstein, 1995; Rasbash & Woodhouse, 1995).

Results

School size effects as the sole predictor of HSC achievement: Simple regression analysis

As a first step, a simple regression analysis of school size effects on TES was undertaken when data were aggregated to the school mean level. The analysis identified a significant (p < 0.05) linear positive relationship between school size and school mean TES score. Larger schools tended to gain higher HSC results. School size explained about 19% of school mean TES variance and there was an expected increase of about 4 TES marks for every 100 additional students in the school. These results are presented in Table 1.

Figures 1a and 1b show the scatter plots of TES against school size, controlling for (a) student gender composition of the school, and (b) school SES level. It is evident from these figures that, in general, after controlling for school background characteristics, there was an increasing trend of TES achievement as school size increased. It is also evident that, for this sample at least, the best TES results occurred at schools of 900 to 1,000 students. In addition, all of these very effective schools (School 10, 25 and 30) were single-sex and of high SES. Both boys and girls schools were included in these highly effective schools, but no co-educational schools were included.

Figure 1a: School Average TES by school size, controlling for student gender

School size effects as an independent predictor of HSC achievement: Multilevel analysis

In the next stage of analysis, a series of four multilevel models were examined.

1. Findings from the basic model
The Basic Model, which included no explanatory variables, served as a baseline for subsequent models. In this analysis, school size effects on academic achievement were examined irrespective of other background and process variables. The Basic Model established that the intraclass correlation was 7% implying that 7% of the total variance in HSC achievement can be attributed to between-school differences.

2. Findings from school size model
The School Size Model considered next included school size as the only explanatory variable for academic achievement. It addressed the first research question and provided an estimate of the amount of variance in Tertiary Entrance Scale score accounted for by school size, irrespective of student background, school characteristics and the educational culture of the school. Results from this analysis indicated that about one-fifth of the between-school variance was accounted for by school size. As expected, school size did not account for a significant proportion of within-school variance (See Table 2).

Figure 1b: School average TES by school size, controlling for school SES

3. Findings from education culture model
Next, school and student background variables and education culture variables were included in, and school size excluded from, the Education Culture Model. This approach was chosen because it provided a baseline to determine the contribution of school size to explaining HSC achievement over and above the school's educational culture. With the exception of three variables, all variables considered contributed significantly to the prediction of TES. The three exceptions were: (a) Medium socio-economic status (SES) schools did not score significantly better than low SES schools in the HSC examination, although there was a substantial difference between high and low SES schools; (b) Neither male nor female students had a clear advantage over the other in terms of average TES marks, after taking into account all other variables in the equation, and (c) No significant interaction effect between student gender and school type was detected. The background and education culture variables together accounted for 44% of the school level variance and 12 % of the student level variance of TES.

Predictor	Basic model	School size model	Ed culture model	Condition- al model
FIXED PART
Grand Mean School Size	273.0	244.7 0.037*	273.5	253.6 0.028*
1. School background
Sch SES (High - Low) Sch SES (Med - Low) Sch Type (Coed - Single-sex)			35.80** 11.30 (ns) -21.72**	30.53** 10.34 (ns) -22.09**
2. Student background
Student Gender (Female - Male) Gender * Sch Type			-10.82 (ns) 12.59 (ns)	-10.56 (ns) 12.56 (ns)
3. Student motivation
No job Enjoy Job HSC Vocational expect Academic expect Personal expect Social expect Religious expect			-22.68** 8.59** 7.69** -29.31** 31.93** 12.88** -9.16** 4.38**	-22.68** 8.86** 7.69** -29.31** 31.93** 12.86** -9.17** 4.39**
4. School processes
Curriculum Out-of-school Principal Teacher Discipline Morale			7.50** -5.43** -12.26** 8.75** 11.97** 7.24**	7.50** -5.43** -12.27** 8.75** 11.95** - 7.25**
RANDOM PART	Variance	Proportion of variance explained compared with basic model
School level Student level	406.4 5161.0	20.6% 0.0%	43.9% 11.8%	55.1% 11.8%
Deviance Difference in degrees of freedom compared to Basic Model Difference in deviance compared to Basic Model Model improvement (p)	56449.7	56440 1 9.7 < 0.005	55793.6 19 656.1 < 0.001	55784.3 20 665.4 < 0.001
Notes: The intra-class correlation for TES can be calculated from the Basic Model by: (406.4/(406.4+5161))=7% 1. * indicated that the estimate was significant at 5% 2. ** indicated that the estimate was significant at 1% 3. (ns) indicated that the estimate was not significant at 5% 4. The deviance of the Conditional Model was less than that of the Education Culture Model by 9.3 with a loss of 1 degree of freedom. The gain in model fit was significant at 0.5%

4. Findings from conditional model
The Conditional Model included school size in addition to the previous variables in the Education Culture Model discussed in the previous section. The Conditional Model addressed the second research question and provided an estimate to the amount of variance of Tertiary Entrance Scale score which was accounted for by school size, after taking into account student background, school characteristics and the education culture of the school. Results from this analysis confirmed the contribution of school size to TES marks independently of the background and education culture. School size explained a further 11% over and above the proportion of school level variance already explained by background characteristics, motivation and school processes. Holding other variables constant, for every increase of 100 students more than the average school enrolment, there was a corresponding increase of 3 TES marks. All predictors considered accounted for more than half of the between-school variance in HSC achievement.

Conclusion

This study proposed to identify possible relationships between school size and HSC achievement of students in NSW Catholic schools. The relationships were studied both before and after taking into account other school characteristics as well as student background and school process variables. The analysis suggests that there is a significant school size effect on academic achievement, even after controlling for background variables and processes in Catholic schools. There is empirical evidence to support the hypothesis that larger Catholic schools are, as far as public examination achievement is concerned, more effective than smaller ones. However, some caveats are necessary to avoid an oversimplification of the effects of school size. For example, it is extremely unlikely that school size alone is sufficient for school effectiveness. Perhaps a more dynamic and varied culture is at work in large schools which leads to better academic achievement in these schools. These issues are developed in the discussion below.

Economies of scale

The notion of economies of scale has been used to explain observed effects of school size on academic achievement. It is argued that, since large schools have relatively less fiscal and human resource constraints than small ones, they are more capable of providing a full range of teaching facilities such as computer laboratory, recruiting better teachers and offering a more diverse and comprehensive curriculum. Better teachers may also tend to choose employment in large, well-established high schools. However, the gain in efficiency from increasing school size might be counter-balanced against the effect of the warm and affective social climate achieved within smaller schools.

Academic achievement is important but it is not the whole of school life

It is necessary to acknowledge that academic achievement, despite the emphasis placed on it by the media, is only one facet of Catholic school life. The other important aspects comprise students' experience of school life and the development of personal, social and religious values. It was the advantages offered by smaller schools over bigger schools on these other aspects of schooling that prompted Goodlad (1984) to suggest 500-600 pupils constitute an ideal school size. Previous studies reported that students in smaller schools experienced a richer environment, participated in more varied activities and had more opportunities in taking up leadership roles than their counterparts in larger schools. In addition, research suggested that smaller schools had less centralised decision-making, higher level of staff cooperation, less disruptive classroom behaviour, higher level personalism and social intimacy, as well as a more favourable culture than larger schools. It is easier in a small school for students and teachers to know one another as persons rather than as an enrolment figure or a subject-matter expert. In other words, interpersonal relationships go well beyond classroom doors to other dimensions of students' school and personal life (Bryk, Lee and Holland, 1993). It was through these informal encounters that teachers developed Catholic values and facilitated the realisation of the school's mission. Our data on school culture corroborated these previous research findings. There were incidences of small schools with very low average TES where students reported an excellent quality of school life.

Practical significance of the findings

A second caution against placing over emphasis on results from this study is that we have used a sophisticated method of analysis, and the interpretation of the outcomes of such analyses are usually more complex than the analysis itself. For example, the regression coefficient of school size was statistically significant, that is, for every increase of 100 students above an average-sized school, students' TES scores increased by about 3 marks. Whether or not the findings were significant in practice in the daily life of schools is a matter of debate. In general, Catholic schools are smaller in size than Government schools. One rarely finds Catholic schools with more than 1,300 students. Suppose one is ready to make the strong and perhaps erroneous assumption that increasing school size would not compromise the quality of social interactions and student development, then utilising statistical results from this study, increasing the average-size schools (that is with about 760 students) to very large schools with about 1,300 students would bring about a corresponding change of only 16 TES marks (or about 3 TER) for an average student at the HSC examination. Perhaps the extent to which this social experiment would be worthwhile would depend, in part, on what one believes is the central mission of the school.

Generalising to all schools

Another obvious caution, necessary against misinterpretation of our statistical findings, is one of generalisability. This sample was representative of schools from the Catholic sector, which distinguishes itself from the Government sector in terms of school mission, institutional structure and interpersonal relationship. It might not reflect Government schools. Specifically, the mechanism through which school size affected achievement in Catholic schools might not be mirrored in Government schools. These limitations on generalisability should be taken into account when drawing conclusions from the research.

Future directions for research

In addition to confirming the relationship between school size and achievement, the analysis revealed some interesting information about school size effects. About half of the predictive power (20%) of school size on achievement at the school level was unique to school size and the other half was shared with background and school process variables. This might provide an explanation of the sometimes contradictory research findings concerning school size effects on cognitive achievement. In other words, whether or not school size effect was a significant influence depended on what other background or process variables were already included in the equation. Consequently, research findings might not be directly comparable. For example, in this study, neither students' ability nor their previous academic achievement were available to the researchers. Had these two powerful predictors of achievement been included, school size might have ceased to be significant. Further replications of this research in Australia, with a carefully mapped list of control variables, are needed.

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Acknowledgments

The authors would like to thank Dr John McCormick, University of New South Wales, and Dr Judy Goyen and Mrs Louise Kobler of Macquarie University for their very helpful comments on earlier versions of this paper.

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