Contextual moderation of primary-school achievement gaps: repeated cross-sectional evidence from Luxembourg

Introduction

Background of the study

Educational inequalities, defined in terms of systematic differences in access to educational institutions, educational achievement (e.g., competencies) and educational attainment (e.g., certificates) structured by certain axes of inequality (e.g., social origin, migrant background, gender, dis/ability), remain a distinct social problem in most education systems. Disadvantages in the uptake of education result in lower labor market chances, lower income, higher risk of unemployment, lower political participation, lower health and even lower lifetime expectancy (Breen, 2022).

Particularly from a sociology of education perspective, the causes of educational inequalities relate to different levels and cannot be attributed to individual students or their direct familial environment only. Although the classical study of Coleman et al., 1966 as well as contemporary studies (e.g., OECD, 2018) that took both context and individual characteristics into account concluded that family background seems to be more important than the school context, educational contexts beyond the family, such as school effects, gains increasing attention in educational research. The present study attempts to contribute to this field by analyzing educational inequalities in achievement taking into account the individual level as well as several context levels, including the classroom, the school and the district levels.

The country of Luxembourg is an appropriate case to study contextual factors of educational inequalities as the Luxembourgish education system is prone to perpetuating educational inequalities with its residential segregation and high stratification in secondary schooling (LUCET and SCRIPT, 2018). While the primary schooling system is comprehensive, i.e., there are no distinct school tracks within primary schooling institutions, it is still affected by residential segregation and grouping of students into classrooms that differ among schools. A reform of the primary schooling system in 2009 attempted to increase student success and reduce educational inequalities. One of the accompanying measures was a new resource distribution mode based on a social index calculated for every commune by the Luxembourg Institute of Socio-Economic Research (LISER) in partnership with the Ministry of National Education, Children and Youth (MENJE) and the General Inspectorate of Social Security (IGSS). Based on this reform, socially-disadvantaged communes would receive up to 20% more resources—in terms of additional teaching hours—than socially-advantaged communes.

In the multilingual and multicultural context of the Luxembourgish education system, the national reports highlight gender, social background, and ethnic (language) background as the most influential student background variables shaping educational inequality (e.g., LUCET and SCRIPT, 2018, 2021). Moreover, student compositions concerning these individual characteristics and schools socio-physical conditions encompassing the number of teachers and students per school and school locale are among the primary school and classroom-level determinants associated with achievement disparities, as reported by two comprehensive systematic reviews (Erer et al., 2025a, 2025b). Accordingly, this study aims to investigate the impact of various classroom compositions and school's teacher-student ratio, as well as the social index on shaping educational disparities in primary schools of Luxembourg over a span of six consecutive years.

The decision to focus on the school years 2014/15 to 2019/20 for analysis is driven by the introduction of the social index reform in 2009, which was intended to be implemented gradually over a ten-year period, and the availability of datasets containing the variables of interest during those years. Additionally, the scholar attention is given to an earlier stage, grade-3, in primary schooling to ascertain which classroom and school variables significantly explain variations in academic achievement, to be able to diminish the impact of cumulative advantage mechanism, or the Matthew effect (Merton, 1968), resulting in enduring achievement gaps and therefore unequal educational gains and opportunities (Diprete and Eirich, 2006).

Aim and research questions

The main aim of this study is to examine moderation effects of various classroom compositions, teacher-student ratio and social index on the relationship between different axes of inequality (gender, socioeconomic background, and language) and grade-3 students' academic achievement in Luxembourg, assessing cross-sectional student data gathered between 2014 and 2019 on an annual basis. Thus, the following research questions that relate to individual, classroom, school and commune levels are examined:

General research question: What are the factors amplifying and reducing achievement gaps in Luxembourgish primary schools?

• What are the patterns of advantage and disadvantage on math and German reading comprehension scores among grade-3 students in Luxembourgish primary schools, with respect to certain axes of inequalities (demographic characteristics) such as gender, socioeconomic background, and language of instruction spoken at home (individual level)?
• How do various classroom composition types (classroom level) shape educational inequalities, i.e., how do these moderate the relationships between axes of inequality (student demographics) and subject-specific academic achievement?
• How do teacher-to-student ratio (school level) and social index (commune level) moderate the relationships between different axes of inequality and subject-specific academic achievement?

Conceptual framework and state-of-research

Based on the research questions in this study, the conceptual framework necessitates thorough considerations across three dimensions: general educational inequality, peer effects, and teacher/school effects.

Starting out with the dimension of general educational inequality, the theoretical frameworks of cultural capital and stereotype threat provide a solid starting point. Resources play an important role in shaping social inequalities. As proposed by Bourdieu, 1986, cultural capital, encompassing knowledge, skills, and cultural preferences/signals, is among crucial resources arising from the familial background of students with its potential to influence their educational experiences and outcomes. According to this theory, students from different social backgrounds possess varying levels of cultural capital; thereby, their entry level characteristics differ even prior to their first schooling experience. For instance, children from higher socioeconomic backgrounds, often due to having parents more actively involving in schooling (see “concerted cultivation” by Lareau, 2011), may be more exposed to cultural resources such as additional extracurricular learning opportunities and language skills that align with general educational norms in the schools. As a result, these accumulated advantages can facilitate their fit and success in the educational system, leading to better educational outcomes. In contrast, students from lower socioeconomic backgrounds may face challenges in accessing and utilizing cultural capital. Similarly, students who do not speak the language of instruction might lack either the cultural capital deemed valuable in schools, or the language proficiency and knowledge to communicate the signals of cultural capital they already possess. Therefore, students from these backgrounds lack these advantage-setting attributes, resulting in perpetuating educational disparities in the educational system.

Regarding students' gender and ethnic (language) background, stereotype threat, coined by Steele and Aronson, 1995, provides valuable insights. Examining the impact of stereotypes on educational performance and achievement, this theory highlights that students belonging to stigmatized groups, such as disadvantaged gender group or ethnic minorities, may experience stereotype threat, referring to the fear of confirming negative stereotypes about one's group. This threat can create psychological pressure and anxiety, leading to underperformance in academic settings (Boucher et al., 2012; Rydell and Boucher, 2017). For example, research has shown that females, despite having the same abilities as males, may be affected by the stereotype that they are less competent in math and science (Régner et al., 2014). The same pattern emerges for males in social science subjects (Pansu et al., 2016). Similarly, non-native students who do not speak the language of instruction, mostly ethnic minorities in most societies, might face difficulties in communication within learning and socialization, and therefore might be stigmatized by peers and faculty (Padilla et al., 2022; Wagner et al., 2008). Consequently, this stereotype threat can hinder their performance in schooling subjects, resulting in perpetuating gender- or minority-based educational inequalities.

Research on peer effects in educational inequality has shown that classroom composition plays a crucial role in shaping educational outcomes. The impact of classroom composition has been mostly recognized after the influential studies of the Coleman-Report in the 1960s (Coleman et al., 1966) or of the Plowden Report (Peaker, 1971) in the United Kingdom. Albeit reported as less influential than the family factors in educational inequalities, the impact of peers is underlined as a crucial factor in educational attainment, especially for the students of the working class (Stienstra et al., 2023). One of the most prominent models of classroom composition, or peer effects, is the boutique model, also known as the tracking model (Sacerdote, 2011). The model posits that students benefit from being surrounded by peers possessing similar abilities and characteristics. Yet, there are some counterarguments to the benefit of more homogenous settings. Scholars highlights that low aspiration levels and low motivation among students may prevail in SES deprived or immigrant environments when the student body composition in schools and classrooms are rather homogeneous, as the case in highly-residentially segregated environments (Hadjar and Becker, 2016). When vulnerable student groups stick with their kind, they reinforce themselves in their dislike of school and low learning motivation, thereby perpetuating educational inequalities (Baumert et al., 2006; Becker and Schubert, 2011; Hadjar and Berger, 2010). Nevertheless, when the residentially segregated context of Luxembourg is taken into consideration, the boutique model might be a naturally expected occurrence in the procedure of student body composition of classrooms and schools.

Fortunately, the impact on educational outcomes of students is not solely emanated from peer effects, as teacher effects also play a significant role (Hattie, 2002; Holmlund and Öckert, 2021). In a classroom with a more homogeneous composition (the boutique model), teachers can tailor the curriculum and pace of learning to meet the needs of the specific group of students (Sacerdote, 2011; Thrupp and Lupton, 2006). Moreover, teachers also adapt their expectations and aspiration levels according to student background (Demanet and Van Houtte, 2012). As teachers are the key component of the learning process (Hattie, 2002, 2010), an increase in the number of teachers in schools (higher teacher-to-student ratios) can bring opportunities for each teacher to find more time per student, align their instructional approaches more effectively, and to have more teacher-student reciprocal interaction (Folmer-Annevelink et al., 2010; Schwartz et al., 2012).

Ultimately, the theoretical concepts of stereotype threat and cultural capital already explain why students belonging to disadvantaged gender, social and language groups might be at-risk concerning the educational inequality. Due to the existence of residential segregation in Luxembourg, schools might experience a lack of substantial diversity to compose heterogenous classrooms. Accordingly, the expected emergence of more homogenous compositions (boutique/tracking model) sounds risky (Baumert et al., 2006) as the stereotype threat might be amplified for the composed group. Yet, teacher effects take greater importance here as they are the key to tackle educational inequality (Hattie, 2002, 2010). In a classroom with more homogeneity, teachers might more effectively to meet the needs of their students by tailoring the pace of learning (Sacerdote, 2011). Schools with a higher teacher-to-student ratio might be in a better situation as teachers have more opportunities to align themselves to their students’ needs (Folmer-Annevelink et al., 2010; Schwartz et al., 2012). Specifically, with the policy implementation of Luxembourg, the provision of additional teaching hours based on the social index to the schools in more deprived communes might be an effective mechanism to compensate the lack of cultural capital of students from distinct backgrounds, diminish the stereotype threat by fostering the possible teacher effects, thereby directly impact educational inequalities.

Developed cross-level hypotheses

Driven by the theory and state-of-research outlined above, the following hypotheses were developed:

• Classroom Level:
• School level:
• Commune level:

The context of Luxembourg

Concerning the endurance of educational inequality throughout time and geography, Luxembourg, as one of the most diverse countries in Europe, has its own assets and complications. On the positive sides, its commitment to promoting educational equality (Dunajeva, 2022), its attempts to provide high-quality school environments, and its society accommodating more than 170 nationalities while operating with three official languages (Luxembourg Ministry of Education, Children and Youth, 2021) are some examples of its unique assets. Yet, as also highlighted in international and national educational reports (LUCET and SCRIPT, 2018, 2021; OECD, 2019, 2021), in this diverse and wealthy country, students from distinct backgrounds still face some common struggles to keep up with their peers from advantageous backgrounds when their gender, language proficiencies, and socioeconomic backgrounds are taken into consideration (Hadjar et al., 2015, 2018).

As described, the Luxembourg population is characterized by a high level of cultural diversity, which has increased markedly over the past decades. While in 2001, 63% of the population was of Luxembourg nationality, in 2023, the figure was only 53%. The Portuguese represent the second largest nationality group with a population share of 15% of a total population of 634,730 in 2023 (statistique.lu). As such, more than 60% of students speak a language other than Luxembourgish at home (Eurydice, 2022). In addition, a widening disparity in the income of the population as measured by the Gini coefficient characterizes the picture (ibid.). The Luxembourg education system is organized on a trilingual basis, which imposes high language demands on students. The literacy and instruction language at the primary school level is German in the traditional Luxembourgish school system. Therefore, speaking Luxembourgish or German at home is assumed to provide students with the necessary language resources for literacy acquisition (Hadjar et al., 2018). The track orientation after six years of primary school is a binding decision jointly made by teachers and parents, or an orientation commission in case of disagreement (Hadjar and Backes, 2021). The Luxembourg secondary school system is characterized by a high degree of stratification into parallel tracks with different levels of aspiration and further career options. Since features relevant for school success such as performance and a trajectory without class repetition in primary school are predictors for the transition into the different tracks in secondary education (in addition to non-performance related variables such as social origin, ibid.; Klapproth et al., 2013), the competencies acquired at primary school age have to be considered highly relevant.

After several decades without large-scale educational reforms (since 1912), the 2009 Luxembourg primary school reform introduced a social index for needs-based resource allocation at the municipal level (Mémorial 20, 2009), which can be used to provide a school with up to 20% additional teaching resources beyond the basic contingent (ONQS, 2022). Thus, contrary to the “watering can” principle, a “fair[ly] unequal treatment” (Sendzik, 2018) is intended to be applied in accordance with context-specific student demographics.

Within the framework of a ministerial project aiming to ensure the continuance of providing equal educational opportunities to students in primary schools of Luxembourg, this study taps on longitudinal patterns of classroom and school impacts on educational inequality.

Method

Utilized datasets

To achieve the research aim, a dataset was created by merging different data provided by various stakeholders within Stata 17.0, resembling census data. First, the “Épreuves Standardisées” (ÉpStan) school monitoring dataset provided by Luxembourg Centre for Educational Testing (LUCET) was utilized. The ÉpStan conduct annually standardized achievement tests in the schooling subjects of German, French, and mathematics in all Luxembourg's mainstream schools on primary and secondary level (Martin et al., 2015). For this study, the anonymized ÉpStan dataset contained standardized math and German reading comprehension test scores of grade-3 students, along with the participants' background information for six consecutive years (see Table 1).

Table 1. Descriptive statistics for math score of the cross-sectional study.

Additionally, this dataset enabled the researchers to employ the nested structure of schools with anonymized classroom, school, commune and year identifiers and to merge different datasets (on the basis of an anonymized dataset combining Épstan and ministerial administrative data, provided by the project partners and according to data sharing agreements). Second, the administrative school data administered by the data division of the school development department (SCRIPT, Service de coordination de la recherche et de l'innovation pédagogiques et technologiques) under the Ministry of National Education, Childhood and Youth in Luxembourg was merged over ÉpStan to further enrich the ad-hoc dataset with two variables on the school level (see Tables 1, 2). Third, to enable the researchers to scrutinize possible impacts emerging from the commune level, a dataset containing a proxy variable called Social Index supplied by the Luxembourg Institute of Socio-economic Research (LISER) was conjoined onto the main dataset. Consequently, the final dataset constituted valuable information on grade-3 students, their classrooms, schools and communes in which their schools are governed.

Table 2. Descriptive statistics for German Reading comprehension score of the cross-sectional study.

On the one hand, the different data sources attempted to cover entire student cohorts. As significances/p-values relate to sampling error, for the analysis of a population census, the sampling error is zero, and the interpretation of analyzes would focus on effect sizes and practical significance in terms of real-world importance, meaningfulness, or usefulness of a research finding (Mohajeri et al., 2020) rather than significances. On the other hand, as data was not available for the entire population in both datasets, our matching procedure led to a reduced sample and thus assessing errors and significances becomes meaningful again.

Participants and variables

It is imperative to note that this dataset solely encompasses information from the conventional state-funded schools, thereby excluding data from alternative and international project schools in Luxembourg. Furthermore, the procedure of pseudonymization employed by the project partners, along with both the subsequent data merging process carried out by the researchers and the complete case approach, led to a reduction in the number of available cases. Consequently, this led to the formation of the final datasets, which represent relatively substantial subsamples derived from the initial ÉpStan data. The proportion of participants within these final datasets, when compared to the initial ÉpStan data, ranges from 81% to 88%.

Tables 1, 2 indicate the total number of participants in the final datasets, along with mean, standard deviation, minimum and maximum values of all the variables used in the analysis for each year, separately for math and German reading comprehension scores. The final datasets for six consecutive years from 2014 to 2019 consisted of 4109–4794 students (for Math) and of 4052–4698 students (for German) in grade 3 within 334–392 classrooms operating under 145–157 schools from 94 to 99 communes in Luxembourg. Regarding the outcome variables, math and German reading comprehension performances of students in the ÉpStan were provided as already standardized scores to enable and facilitate the comparison over the years. The standardization process established the mean score for each test as 500 points with a standard deviation of 100 points.

On the individual level, the dataset provided students' gender, socioeconomic status, and the language spoken at home (multiple responses possible). Concerning gender, the percentages are shown in Tables 1, 2. The socioeconomic status (SES), as measured by the International Socio-Economic Index of Occupational Status (ISEI), is based on the parents' occupational status, with higher ISEI values indicating higher socioeconomic status. For students with two parents or legal guardians, only the highest ISEI (HISEI) value was provided in the ÉpStan. For the language of instruction spoken at home, the binary variable was set to 0 for students who speak Luxembourgish or German with at least one parent or legal guardian, and to 1, for students who primarily speak another language at home. Beyond ensuring adequate statistical power across subgroups, this dichotomization captures the reality of primary schooling in Luxembourg, where initial literacy instruction relies heavily on German, advantaging Germanophonic language speakers (Luxembourgish and German speakers).

Classroom compositions were represented by female percentage, average SES, and percentage of non-Luxembourgish-or-German speakers (nLGs) at home. These composition types were created by aggregating the individual-level demographics onto the classroom level. As illustrated in Tables 1, 2, the mean values of these classroom compositions ranged from 0.48 to 0.50, 46.65 to 47.71, and 0.51 to 0.55, respectively. Additionally, the dataset revealed the presence of cases in which classrooms were exclusively composed of a single gender group or solely comprised nLGs, with such occurrences being observed for nearly every year. Furthermore, the average SES of classrooms exhibited a considerable fluctuation, ranging from as low as 20.82 in 2017 to as high as 68.05 in 2019. Despite the seemingly exceptional nature of these values, it is crucial to acknowledge that they are inherent to the Luxembourgish educational system itself, rather than being outliers or unusual occurrences.

The school level predictors were school student population and teacher-to-student ratio at a given year provided by SCRIPT. In the final datasets, the mean values of these school variables ranged from 304.06 to 323.39 and from 0.10 to 0.11, respectively. Moreover, the final datasets revealed considerable variation in school student population, ranging from 57 students in 2015 to 1008 students in 2018. Furthermore, there was a marked difference in teacher-to-student ratios between the most disadvantaged and advantaged schools, with the former having 5 full-time equivalent teachers per 100 students (0.05) in 2016, while the latter had 27 teachers per 100 students (0.27) in 2019. Nevertheless, despite the values appearing extraordinary at first glance, it is crucial again to acknowledge that they are actually a natural part of the educational system itself, rather than being unusual outliers or uncommon situations.

Lastly, the social index in this study corresponds to a proxy for the monetary compensation a commune receives to hire more full-time equivalent teachers, ranging from 100 to 120. To clarify, a social index of 100 for a given commune denotes a lack of supplementary monetary compensation as it is wealthier than others in comparison, whereas a commune with a social index of 120 is awarded the maximum funding as monetary compensation for additional teaching hours as it is more deprived than its counterparts. As shown in Tables 1, 2, the mean values for the social index ranged from 108.93 to 110.07 in the final datasets.

Data analysis

Consisting of four levels, the ad-hoc dataset created for this study had a nested structure: students (Level-1) gathered in classrooms (Level-2) located within schools (Level-3) governed under communes (Level-4). For the analysis of such nested repeated cross-sectional datasets (2014–2019), the utilization of a data analysis method embracing the violation of observation independency is recommended (Garson, 2013). Therefore, multilevel modelling was selected as the statistical modelling technique not only to account for the interdependence and variation within and between levels, but also to precisely inspect the effects of predictors at different layers of the nested dataset. For the analytical approach, the moderation analysis, conducted with cross-level interactions in this study, was utilized to gain insights into the specific conditions under which higher-level factors are more influential in perpetuating or reducing educational inequality. Accordingly, with an aim to examine moderation effects of various factors introduced in different levels on the relationship between different axes of inequality and two subject-specific academic achievements of grade-3 students in Luxembourg over six consecutive years, this study utilized a multilevel moderation modelling approach.

In order to apply the chosen analytical approach, Classroom-ID, School-ID, and Commune-ID were utilized as clustering variables, with Year-ID used to conduct multilevel modeling separately for each year. Each continuous predictor on every level in the dataset was also centered on their grand mean to make the interpretation of intercepts more meaningful and to reduce any possible multicollinearity (Garson, 2013) that might arise from the aggregation of student demographic variables to create the classroom composition types. Additionally, a stepwise approach was preferred to develop the multilevel models, beginning with the unconditional model (Model 0) and incrementally adding complexity to arrive at the final model (Model 4). First, Model 0 was run with only the corresponding outcome variable. Then, student-level variables were added in Model 1. In Model 2, the commune-level variable and its cross-level interactions with each student-level variable were introduced to examine its impact on the relationship between the student-level predictors and the corresponding outcome. Following the same logic, the classroom-level variables and their interactions with the corresponding student-level variables were introduced in Model 3. After, the school-level variables and only the interactions between teacher-to-student ratio and each student-level variable were introduced in Model 4. The reason behind the sole utilization of teacher-to-student ratio in the cross-level interactions in the final model is a hypothesized impact of the social index as it represents the monetary compensation on the commune level to hire more full-time equivalent teachers to schools under the corresponding communes.

Using Stata 17.0, the models were developed with maximum likelihood and complete case analysis, repeatedly in a nested loop for each year in the dataset and for each outcome variable utilized. For each model in the analyses (see Tables 3–14), the intraclass correlations (ICC), were calculated to represent the proportion of variance in the corresponding outcome variable that is explained by the group-level variations. For math scores, they ranged between <1% to 2%, 2.8% to 4.6%, and between 5% to 9%, respectively on the commune, school and classroom levels (see Tables 3–8). For German scores, ICCs ranged between 1% to 3.1%, 3.1% to 6.1%, and between 4.8% to 9.7%, at the commune, school and classroom levels, respectively (see Tables 9–14). Even though some variances attributed to the levels in the final models were computed relatively low, the researchers decided to continue with the same nested structure as it represents the nature of the education system in Luxembourg. Finally, the stepwise model sequence was structured according to theoretical hierarchy, moving from unconditional models to main effects and cross-level interactions.

Table 3. 2014 cross-sectional math models.

Standard errors in parentheses.

+p < 0.10, *p < 0.05, **p < 0.01, ***p < 0.001.

Table 4. 2015 cross-sectional math models.

Standard errors in parentheses.

+p < 0.10, *p < 0.05, **p < 0.01, ***p < 0.001.

Table 5. 2016 cross-sectional math models.

Standard errors in parentheses.

p < 0.10, *p < 0.05, **p < 0.01, ***p < 0.001.

Table 6. 2017 cross-sectional math models.

Standard errors in parentheses.

p < 0.10, *p < 0.05, **p < 0.01, ***p < 0.001.

Table 7. 2018 cross-sectional math models.

Standard errors in parentheses.

+p < 0.10, *p < 0.05, **p < 0.01, ***p < 0.001.

Table 8. 2019 cross-sectional math models.

Standard errors in parentheses.

p < 0.10, *p < 0.05, **p < 0.01, ***p < 0.001.

Table 9. 2014 cross-sectional German Reading comprehension models.

Standard errors in parentheses.

p < 0.10, *p < 0.05, **p < 0.01, ***p < 0.001.

Table 10. 2015 cross-sectional German Reading comprehension models.

Standard errors in parentheses.

+p < 0.10, *p < 0.05, **p < 0.01, ***p < 0.001.

Table 11. 2016 cross-sectional German Reading comprehension models.

Standard errors in parentheses.

p < 0.10, *p < 0.05, **p < 0.01, ***p < 0.001.

Table 12. 2017 cross-sectional German Reading comprehension models.

Standard errors in parentheses.

+p < 0.10, *p < 0.05, **p < 0.01, ***p < 0.001.

Table 13. 2018 cross-sectional German Reading comprehension models.

Standard errors in parentheses.

p < 0.10, *p < 0.05, **p < 0.01, ***p < 0.001.

Table 14. 2019 cross-sectional German Reading comprehension models.