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Examining high achievement in mathematics and science among postprimary students in Ireland: a multilevel binary logistic regression analysis of PISA data
Largescale Assessments in Education volume 10, Article number: 14 (2022)
Abstract
In Ireland, while, on average, students have performed well on national and international assessments of mathematics and science, the low proportions of high achievers in these subjects are noteworthy. Given these patterns and the multifaceted benefits in individual and societal terms that expertise in mathematics and science has been associated with, policymakers in Ireland have begun to place an increasing emphasis on high achievement in these subjects. This emphasis has coincided with ongoing efforts during the last decade to raise interest and improve academic performance within the realm of science, technology, engineering, and mathematics (STEM) education.
Despite this policy attention, research on high achievement in mathematics and science nationally, but also internationally, has been particularly scarce. In an attempt to provide research evidence that could add further impetus to the ongoing efforts, this study examines high achievement in mathematics and science among postprimary students in Ireland using data from the 2012 and 2015 cycles of the Programme for International Student Assessment (PISA). Specifically, the study aimed to evaluate the contribution of various contextual characteristics stemming from students, their families, teachers, and schools in the prediction of high achievement in mathematics and science within a twostage analysis that included a series of bivariate tests and multilevel binary logistic regression modelling.
The results showed that variables related to students’ selfbeliefs, engagement, and socioeconomic background were consistently associated with high achievement in mathematics and science. Overall, the significant role of students’ homes and families in predicting students’ chances of being high achievers in the two subjects was highlighted. In turn, this indicated that further efforts to enhance collaboration between teachers, schools, and parents may be warranted if progress in the area of high achievement in mathematics and science is to be made. The implications of these findings for policy and practice within the Irish context, the limitations of the study, and recommendations for future research are discussed.
Background
In Ireland, as in many other countries, results from national and international largescale assessments are often used as key performance indicators for primary and postprimary students, exerting a significant and everincreasing influence on educational policymaking (Greaney & Kellaghan, 2008; Johansson, 2016). Ireland’s results in several national and international largescale assessments over the last 20 years indicate that while, on average, students perform well in mathematics and science, there are low proportions of highachieving students in these two subjects and students’ scores at the highest national percentiles in these subjects tend to be lower than those of their counterparts in other countries with similar average performance (e.g., McKeown et al., 2019; Perkins & Clerkin, 2020; Pitsia, 2021; Shiel et al., 2016). Such patterns in national and international largescale assessments have been accompanied by patterns of decreases in the percentages of high achievers in mathematics and sciencerelated subjects in the Junior and Leaving Certificate examinations, the major highstakes examinations in Ireland (with the latter constituting the university entry examination), over the last 15 years (e.g., Pitsia, 2021; Shiel & Kelleher, 2017). Against this background, students in Ireland have consistently performed very well in reading in both national and international largescale assessments, with adequate percentages of high achievers in the subject reflective of the respective patterns in average performance.
Within the context of policymakers’ heightened interest in science, technology, engineering, and mathematics (STEM) education, high achievement in mathematics and science and the issues described above have attracted considerable attention by the Irish authorities. This is primarily due to the multifaceted benefits to individuals and society with which expertise in these areas, as fundamental components of STEM within compulsory education, has been associated. The 2011 National Strategy to Improve Literacy and Numeracy was the first governmental policy document to explicitly focus on highachieving students in mathematics (but not in science, due to the focus of the Strategy on literacy and numeracy only). The Strategy set specific targets for increasing the proportions of students performing at the highest proficiency levels in the National Assessments of Mathematics and English Reading (NAMER) (i.e., levels 3 and 4) and the Programme for International Student Assessment (PISA) (i.e., levels 4 to 6) (Department of Education and Skills [DES], 2011).
Following the formulation of the 2011 Strategy, policy attention to high achievement in mathematics and science in Ireland has been everincreasing. Additional policies have been established and other relevant documentation has emerged at a national level, such as Ireland’s National Skills Strategy 2025 (DES, 2016b), the Action Plans for Education 2016–2019 and 2018 (Government of Ireland, 2018; DES, 2016a), the Chief Inspector’s 2018 report (DES, 2018), and the Report of the STEM Education Review Group (The STEM Education Review Group, 2016). These reiterated the already established national targets for high achievement in mathematics and introduced more specific ones, while also setting corresponding targets for science.
After the interim review report of the 2011 National Strategy (DES, 2017a) and the development of the STEM Education Policy Statement in 2017 (DES, 2017b), high achievement in mathematics and science became a national priority in Ireland. National targets focusing on raising the proportions of students performing at the highest levels of proficiency and improving the scores of students performing at the highest national percentiles in mathematics and science in national and international largescale assessments were introduced by the DES (DES, 2016b, 2017a, 2017b; Government of Ireland, 2018). Most educational policy documents on mathematics and/or science education across primary and postprimary levels in Ireland published since then have included these national targets.
While this policy attention to high achievement in mathematics and science might have been expected to prompt empirical research on the topic, research literature at a national level has been particularly limited, making the formulation of specific guidelines tailored to the needs of high achievers, in an effort to achieve the aforementioned targets, difficult. The few international research studies that examined high achievement in mathematics and/or science (see, for example, Kartal & Kutlu, 2017; Kourti, 2019; Tourón et al., 2018; Veas Iniesta et al., 2017), which could either provide some insights into the topic for Ireland or serve as a starting point for further Irishbased research, tended to focus on certain themes, such as sex differences (e.g., Ellison & Swanson, 2010; Stoet & Geary, 2013). It is also noteworthy that most of these studies have not involved indepth analyses of data, such as examining a wide range of factors originating from different aspects of a student’s life (e.g., individual, family, class, school) simultaneously or including domainspecific variables (e.g., mathematics anxiety, motivation for learning science) in the analysis. Also, very few studies examined the relationships of contextual variables with high achievement in a multiple regression context or applied robust statistical techniques (e.g., multilevel modelling) to account for the clustered nature of educational data and the particularities of largescale datasets. Analyses that meet the aforementioned criteria could obtain more accurate and credible predictions of high achievement compared to other more descriptive analyses, while also mitigating potential limitations of the collected data (e.g., clustering).
Gilleece et al.'s (2010) study is the only one that set out to determine the role of a range of student and schoollevel characteristics in high (and low) achievement in mathematics and science among Irish students. Gilleece et al.'s (2010) research made a significant contribution to the area; however, it was limited to one cohort of 15yearold Irish students and employed a limited range of variables in the analysis. In acknowledging the limitations of their study and the need for further indepth research on high achievement in mathematics and science in Ireland, Gilleece et al., (2010) provided a comprehensive list of recommendations for further research. Amongst others, the authors suggested that inclusion of contextual variables pertaining to individual domains (e.g., attitudes toward mathematics, career expectations related to science) in the analysis would provide a unique insight into the factors associated with high (and low) achievement in mathematics and science in Ireland. Although their recommendations had the potential to add further impetus to the investigation of high achievement in mathematics and science nationally, these recommendations have not yet been realised.
The current study
Prompted by the aforementioned policy attention and in an effort to examine the magnitude and consistency of issues related to high achievement in mathematics and science in Ireland, Pitsia et al., (2022) conducted an indepth longitudinal investigation of high achievement across education levels and student cohorts using PISA, Trends in International Mathematics and Science Study (TIMSS), and Progress in International Reading Literacy Study (PIRLS) data for Ireland. The authors found that issues pertaining to high achievement in mathematics and science in Ireland have been more apparent and consistent at postprimary level, with issues at primary level being less clearcut. Specifically, Pitsia et al., (2022) found that at postprimary level, proportions of highachieving students in mathematics and science have not been in line with Ireland’s good mean performance in the two subjects, having fewer highachieving students compared to other similarly performing countries, with high achievers also often underperforming relative to their peers in other countries. In turn, these findings indicate that attention to high achievement at postprimary level should be considered a priority in Ireland.
Taking into account Gilleece et al.'s (2010) recommendations and Pitsia et al.'s (2022) findings, the study described in this paper is one of the first investigations of what may predict high achievement in mathematics and science among postprimary students in Ireland. The research question guiding the study is:
Which student, home, class, and school characteristics predict high achievement in mathematics and science among 15yearold students in Ireland?
To answer this research question, the analysis of the data (i.e., modelbuilding process) and the discussion of the results have been framed along the lines of the socioecological model of human development (Bronfenbrenner, 1994). According to this theoretical framework, students are nested within different environments with which they interact, and the components of these environments may, either directly or indirectly, contribute to their academic outcomes. Focusing on the microsystem of Bronfenbrenner’s framework, which constitutes the most proximal environment experienced by students comprised of people and systems with which they have direct contact, the analysis of the data and the discussion of the results begin with variables stemming from the students themselves (e.g., demographics, selfbeliefs, dispositions, drive, and engagement, mathematics and sciencerelated activities) and their parents (e.g., parental dispositions and support), and they, next, focus on the variables stemming from the students’ classes and schools (e.g., class disciplinary climate, school resources).
Within the context of ongoing efforts to introduce reforms in mathematics and science teaching and learning in Ireland, the results of this study are intended to extend the existing body of knowledge about high achievement, highlight the areas on which existing initiatives in mathematics and science education could focus to better address the needs of high achievers in these subjects, and prompt the formulation of new policies and initiatives that could assist towards this end.
Methods
Data
This paper presents results from a secondary analysis of PISA 2012 and 2015 data (Organisation for Economic Cooperation and Development [OECD], 2014, 2017b). PISA is a collaborative effort among the OECD countries to measure how well 15yearold students are prepared to meet the challenges of the future (OECD, 2017b). The assessment, which is crosssectional in nature, has taken place every three years since 2000, following a cyclical design within which it changes its major domain in every cycle, starting with reading in 2000, mathematics in 2003, and science in 2006. Ireland, which is the focus of this study, has participated in all PISA cycles since 2000. Data from the 2012 and 2015 cycles are used in this study as, at the time of writing, they are the most recent cycles in which mathematics and science were the major assessment domains, respectively.
Sampling and participants
PISA defines its international target population in terms of students’ age, focusing on 15yearold students in school in seventh grade or higher. These students are approaching the end of compulsory schooling in most participating countries and, thus, this age is considered as particularly important for personal and academic decisions, while school enrolment at this level is close to universal in most OECD countries (OECD, 2017b).
PISA selects its nationally representative samples based on a twostage stratified sampling design. At the first stage, individual schools constitute the sampling units and are selected with probability proportional to size. The secondstage sampling units are students within sampled schools. Typically, 35 students or more are sampled with equal probability per school, although this size may vary depending on the total enrolment size of the school (OECD, 2017b). In Ireland, 5,015 15yearolds in 182 schools took part in PISA 2012 and 5,741 15yearolds in 167 schools took part in PISA 2015 (Perkins et al., 2013; Shiel et al., 2016); these constitute the samples involved in the analysis in the current study.
Measures and variables
PISA uses tests to assess, amongst others, 15yearold students’ reading, mathematics, and science achievement, providing an overall domain score and subdomain scores for each assessed student. The reading, mathematics, and science scales had been standardised to have a mean of 500 (on average across OECD countries) and a standard deviation (SD) of 100 when each was a major assessment domain for the first time (2000 for reading, 2003 for mathematics, and 2006 for science), with these figures slightly varying in subsequent PISA cycles. The concept of “literacy” is used in assessing and interpreting students’ performance in the three domains. “Literacy” refers to “students’ capacity to apply knowledge and skills in key subjects, and to analyse, reason and communicate effectively as they identify, interpret and solve problems in a variety of situations” (OECD, 2017a, p. 13).
In PISA, each student is administered a subset of the test items from the total item pool for each domain due to time restrictions. Consequently, different groups of students answer different, although overlapping, sets of items. Hence, student proficiencies are not observed; instead, they are missing data that must be inferred from the observed item responses. Given this design and to generate populationlevel proficiency estimates, PISA uses the imputation methodology of plausible values. Plausible values constitute random numbers drawn from the distribution of scores that could be reasonably assigned to each individual (Wu, 2005). Five plausible values were generated in PISA 2012 and ten plausible values were generated in PISA 2015.
In the current study, students’ performance in each domain was treated as a discrete rather than a continuous outcome, with high achievers and nonhigh achievers as the two categories. This process was facilitated by the fact that PISA reports students’ performance not only using continuous scores but also levels of performance to indicate the proficiency of students in a given domain. Apart from being allocated a specific score along the performance scales, groups of students are also allocated to specific proficiency levels and descriptions of the competencies associated with such levels are generally provided (see OECD, 2014, 2017b). In this study, mirroring OECD’s approach, students scoring at proficiency levels 5 and 6 in each of the domains were identified as high achievers.
Prior to conducting any analysis, the main outcome variables had to be computed. In the PISA databases, binary variables indicating the proficiency level to which students belong do not exist; hence, using the relevant cutoff points for high achievement (i.e., proficiency level 5 in both mathematics and science) and in order to use all plausible value estimates of student performance in the analysis, each student was assigned the value 0 or 1 based on whether each plausible value estimate was below or above the established cutoff point for each domain, respectively. This was done separately for each plausible value estimate. These binary variables constituted the outcome variables of the analysis.
In addition to collecting information about students’ academic literacy, PISA uses questionnaires to collect information about students’ attitudes, interests, motivations, and beliefs as well as students’ family and school backgrounds to facilitate interpretation of achievement outcomes (e.g., OECD, 2017b). In Ireland, student and school questionnaires were administered in 2012, and student, parent, teacher, and school questionnaires were administered in 2015. Information from these questionnaires was also used for the analysis in this study.
Statistical analysis
To examine the role of student and schoollevel variables in students’ high achievement in mathematics and science, PISA 2012 and 2015 data were analysed in two stages: (i) bivariate analysis and (ii) multilevel binary logistic regression analysis.
Bivariate analysis
In the first stage, bivariate analysis was performed to facilitate the decisionmaking about the inclusion of predictor variables into the next stage of the analysis (i.e., multilevel binary logistic regression analysis). The International Association for the Evaluation of Educational Achievement (IEA) International Database Analyzer (IDB Analyzer) (IEA, 2021) was used to calculate all estimates (e.g., percentages, means, standard errors [SEs], SDs) involved in the bivariate analysis. By using the IEA IDB Analyzer, the plausible values and replicate weights were used appropriately, and adjusted SEs were computed. The cutoff points for high achievement for each domain were used in the benchmarks statistic type in the IEA IDB Analyzer to facilitate treatment of student achievement as a binary outcome.
The effect size of each of the relationships between the predictor variables and the binary outcome variable in each subject (i.e., high achievement), rather than the statistical significance level alone, was the criterion for either including or not including variables in the next stage of the analysis. This was because with large sample sizes, such as the ones analysed in this study, very low coefficients are likely to be statistically significant. Guidelines provided by L. Cohen et al. (2017) and Fritz et al. (2012) related to the computation and reporting of effect sizes based on the types of variables and samples involved in the analysis were followed in the selection of the appropriate effect size measures. Based on these guidelines, the phi (φ) and the Cramer’s V (φ_{c}) effect size measures were used for categorical variables with two or more categories, respectively; the etasquared (η^{2}) effect size measure was used for ordinal variables; and the Hedges’ g (g) effect size measure was used for the continuous variables due to the considerably different sample sizes of the two comparison groups (high and nonhigh achievers).
J. Cohen’s (1988) in conjunction with Hattie’s (2009) guidelines were used to identify the thresholds above which predictor variables would progress to the next stage of the analysis. The fact that effect sizes from different effect size families had to be used in this study meant that equivalent thresholds (i.e., thresholds that indicate effects of the same magnitude) were required. Predictor variables that yielded an effect size of 0.10 (φ and φ_{c}), 0.010 (η^{2}) or 0.20 (g) progressed to the multilevel binary logistic regression analysis as effect sizes at or above these thresholds are considered as small, intermediate or large effects (in general) and teacher or desired effects (in education).^{Footnote 1} Statistical significance tests and calculation of effect sizes for the estimates provided by the IEA IDB Analyzer were conducted using the relevant formulas for each estimate as recommended by the literature (see Gonzalez, 2014).
By selecting predictors for the multilevel binary logistic regression analysis based on both their statistical and practical significance, the aim was to build as parsimonious models as possible that examine the strongest predictors of high achievement in mathematics and science, and also signify the extent to which these predictors retain their significance after accounting for other variables.
Multilevel binary logistic regression analysis
Following the bivariate analysis, multilevel binary logistic regression analysis was conducted on Mplus (Muthén & Muthén, 2017) to examine the contribution of a range of factors stemming from the students, their parents, classes, and schools in the prediction of high achievement in mathematics and science. The sampling design that leads to the clustered nature of PISA samples means that students within the same schools may have more characteristics in common than with students from other schools (L. Cohen et al., 2017; OECD, 2017b). A statistic, the intraclass correlation (ICC), represents the proportion of the total variance in the outcome variable that is attributable to the cluster (here, school) (Field, 2018). Clustering constitutes a problem because many statistical models assume that cases are independent of each other, but students who study at the same school are less likely to be independent of each other. This clustering was taken into account in the regression analysis by conducting multilevel analysis, which can estimate the variation in the outcome variable that is attributable to differences within or between the clusters and identify the factors at each level that are associated with this influence, while not underestimating the SEs of the regression coefficients (L. Cohen et al., 2017; Woltman et al., 2012). Specifically, the contribution of student and schoollevel variables in predicting postprimary students’ high achievement in mathematics and science was evaluated through hierarchical twolevel binary logistic regression models. Students were the unit of analysis at level 1 and schools were the unit of analysis at level 2.
The binary outcome variable used in this study can be conceptualised as the discretisation of an underlying continuous latent variable (i.e., high achievement/nonhigh achievement is a binary representation of the underlying continuous latent variable denoting the test score). Therefore, the continuous latent response variable approach for binary outcomes was used for the calculation of the estimates (McKelvey & Zavoina, 1975; Snijders & Bosker, 2012). Also, given that the continuous latent response variable approach for binary outcomes was employed, cluster sampling was performed in the assessments that were employed, and sampling weights were used in the analysis, parameters for the models were estimated using the maximum likelihood estimation with robust SEs and the logit link function (Muthén & Muthén, 2017).
Modelbuilding strategy
The first step taken in the modelbuilding process was the construction of null models that contained no predictor variables at either level of the analysis for each cycle to compute the ICCs and, thus, determine the extent to which high achievement in mathematics and science can be attributed to betweenstudent and betweenschool differences. Given that when modelling binary outcomes, the variability within groups (σ^{2}) is not normally distributed, the ICCs were estimated using the following approximation:
…where \({\tau }_{00}\) is the random intercept variance (i.e., the level2 variance component) and \({\pi }^{2}/3(\approx 3.29)\)refers to the variance of a standard logistic distribution (i.e., the assumed level1 variance component; given that the logistic regression model does not include level1 residual) (Goldstein, 2011; Goldstein et al., 2002; Hox et al., 2018; Sommet & Morselli, 2017).
The null models also served as a reference point against which the final models were compared, to evaluate the extent to which the addition of predictor variables contributed to the prediction of high achievement in mathematics and science.
Next, predictor variables were hierarchically entered into the models in blocks (see Fig. 1). Steps 1–4 involved studentlevel variables (level 1 of the analysis) and step 5 involved schoollevel variables (level 2 of the analysis). For the PISA 2012 data, there were four rather than five steps in the model as parent and teacherlevel variables were not measured in that cycle in Ireland.
Preliminary analysis (whereby variables were retained in all steps of the models regardless of their statistical significance) indicated that as the sample size decreased due to listwise deletion with each step of the model, the percentages of high achievers in the analysis samples increased. This might be linked to the fact that high achievers tend to have fewer missing responses compared to students with lower performance and, thus, more cases among the nonhigh achievers were being excluded from the analysis. Dropping the nonsignificant variables in every subsequent step of each model helped to mitigate this issue, resulting both into fewer missing cases and minor variations in the percentages of high achievers involved in the analysis (in the PISA 2012 model, percentages of high achievers across the steps ranged from 10.8 to 11.3% being very close to the percentage of high achievers in the overall sample [10.7%]; similarly, in the PISA 2015 model, percentages of high achievers across the steps ranged from 7.2 to 7.7% being very close to the percentage of high achievers in the overall sample [7.1%]).
The final iteration of the modelling process explored the possibility of statistically significant interactions between predictor variables in predicting the outcome. Due to the lack of existing research on the significance of certain interactions in the prediction of high achievement in mathematics and science, this process was exploratory in nature and examined interactions that could be meaningful for policy and practice purposes. Thus, the significance of crosslevel interactions of student sex and studentlevel socioeconomic status with the schoollevel variables included in the analysis was explored. The interactions terms were entered one by one into each final model regardless of whether the main effects were statistically significant or not because, as L. Cohen et al., (2017) argue, an interaction effect may occur even when no main effects are present.^{Footnote 2}
Weights, plausible values, and centring
As recommended by the relevant research literature, sampling and replicate weights (where appropriate) were used in the analysis. The appropriate use of weights for the bivariate analysis was secured through generating all necessary estimates using the IEA IDB Analyzer, while in the multilevel binary logistic regression analysis, Asparouhov’s (2009) recommendations for the use of sampling weights at the two levels of the analysis were followed. Specifically, the final student weight was decomposed into a “withinschool weight” and a “betweenschool weight” for the appropriate estimation. This was done by dividing the final student weight by the final school weight. The resulting “withinschool weight” was used as the level1 weight and the original final school weight was used as the level2 weight (see also Withincluster Weights method in Mang et al., 2021).
The use of plausible values in the analysis adhered to the relevant procedures outlined in the PISA technical reports and data analysis manual, but also in von Davier et al.'s (2009) study. Specifically, all plausible values were used for the estimation of student achievement, and the imputation variance was taken into account in the estimation of the SEs. The correct use of the plausible values for the bivariate analysis was secured through generating all necessary estimates using the IEA IDB Analyzer. For the multilevel modelling, imputation techniques were applied to involve all plausible values in the analysis (i.e., type = imputation).
Continuous predictor variables were centred for the multilevel modelling. Given that the goal of the analysis was to examine the absolute (betweenstudent) contribution of variables in the prediction of high achievement in mathematics and science, grandmean centring was used (Sommet & Morselli, 2017).
Results
Bivariate analysis
Bivariate analysis was conducted to examine the relationships of a range of contextual variables stemming from students, their parents, teachers, and schools with students’ high achievement in mathematics and science and facilitate the decisionmaking about the inclusion of predictor variables into the multilevel binary logistic regression analysis. All contextual variables included in the PISA 2012 and 2015 databases with data for Ireland were included in the bivariate analysis. Background variables that (i) had a statistically significant relationship with the outcome variable and (ii) yielded effect sizes of 0.10 (φ and φ_{c}), 0.010 (η^{2}) or 0.20 (g) progressed to the multilevel binary logistic regression analysis. Tables 1, 2, 3 and 4 present the percentages of students in each performance group (high achievers and nonhigh achievers) and the means and SDs of high achievers and nonhigh achievers for each of the background variables that met these criteria and, thus, were included in the multilevel binary logistic regression analysis along with student sex and socioeconomic status (which were always included in the models) at the student and school levels. The tables also include the results of the tests that were conducted to examine the statistical significance of the differences in the distribution of students across each background variable between the two performance groups or the statistical significance of the differences in the means for each of the background variables between high achievers and nonhigh achievers.^{Footnote 3}
In PISA 2012, none of the categorical variables examined yielded large effect sizes. In PISA 2015, however, there was a number of categorical background variables in which the distribution of high and nonhigh achievers across the different variable categories differed (Tables 1 and 2). Specifically, a set of variables related to the students’ family involvement in sciencerelated careers and their parents’ perceptions and expectations in relation to science were found to be significantly related to students’ belonging to either the high or the nonhighachieving science performance group (Table 1). Although parents’ perceptions and expectations yielded the largest differences between the two performance groups among the examined variables, the magnitude of these differences was relatively small (φ = 0.18 and φ = 0.19). In general, there were approximately four times more high achievers in the groups of students whose parents had more positive perceptions of and higher expectations about science compared to the rest of the students. Additionally, there were significantly fewer high achievers in science among the students who reported watching TV/DVD/video, using the internet, chats or social networks, meeting or talking to friends on the phone, and exercising or practising a sport on school days, either before or after going to school.
Among the ordinal variables examined, student educational expectations in the 2015 cycle was the only one that reached the effect size threshold of η^{2} = 0.010 (Table 2), with high achievers in science having significantly higher educational expectations compared to their nonhighachieving peers.
Results presented in Tables 3 and 4 reveal that the patterns for mathematics and science were highly consistent across the two PISA cycles with regards to differences between high and nonhigh achievers in the continuous background variables. Student selfbeliefs and attitudes, including student mathematics and science selfefficacy, selfconcept, anxiety, and enjoyment, their environmental awareness, openness for problemsolving, familiarity with mathematical concepts, epistemological beliefs about science, and interest in broad science topics were among those with the highest effect sizes. High achievers had higher selfefficacy, selfconcept, enjoyment, environmental awareness, openness for problemsolving, familiarity with mathematical concepts, epistemological beliefs about science, and interest in broad science topics and lower levels of mathematics anxiety compared to nonhigh achievers. The magnitude of these differences ranged from moderate to very large. Effect sizes of these differences between the two performance groups were relatively larger for mathematics compared to science, indicating that the two performance groups in mathematics differed in these noncognitive factors to a greater extent compared to the two performance groups in science.
Students’ family socioeconomic background was also significantly different for the two performance groups in both subjects (PISA 2012: g = 0.70; PISA 2015: g = 0.74), with high achievers coming from families of more affluent socioeconomic background compared to their nonhighachieving peers.
Results also indicated that high achievement in mathematics and science is not only dependent on individual characteristics but on class and school characteristics too. Class and school characteristics such as disciplinary climate, school autonomy, school size, teaching time, and school socioeconomic composition were significantly different between the two performance groups in mathematics and science, suggesting that students in the two performance groups tended to attend somewhat different classes and schools.
Multilevel binary logistic regression analysis
Following the bivariate analysis, hierarchical twolevel binary logistic regression models were applied to the PISA 2012 and 2015 data. All variables that reached the effect size thresholds for each PISA cycle were included in the models.
Given the sampling procedures followed by PISA, whereby students are nested within the same schools (i.e., clusters), variation in achievement (and hence, high achievement) in these assessments can be separated into betweenstudent and betweenschool components. The ICCs, which indicate the extent to which schools differ with respect to high achievement, were calculated through a null model, which contained no predictors at either level of the analysis, for each cycle. In 2012 and 2015, 12.2% and 13.8% of the variance in student high achievement in mathematics and science, respectively, was attributed to betweenschool differences. The significant proportion of variance in high achievement in mathematics and science attributed to the cluster warranted the consideration of the hierarchical nature of the data through conducting multilevel regression models.
The tables that summarise the results of the hierarchical twolevel binary logistic regression models below present the proportions of variance (R^{2}; expressed as a percentage of the total variance)^{Footnote 4} in high achievement explained at each level by each step of the models, the threshold (statistic equivalent to the more commonly used intercept, with the two being the same except that they have opposite signs; Muthén & Muthén, 2017), the standardised coefficients (βs) accompanied by their SEs for each predictor variable, the odds ratios (ORs) for the statistically significant variables in the final model, and the fit statistics for each step of the models, including the null models. Although all steps of the models are presented in the tables, only results from the final models and ORs based on the coefficients of the final models are discussed below.^{Footnote 5}
PISA 2012 – mathematics
Table 5 summarises the results of the hierarchical twolevel binary logistic regression model for high achievement in mathematics that was applied to PISA 2012 data. It should be noted here that in PISA 2012, rotated student questionnaires covering attitudinal and other noncognitive constructs were used to increase the content coverage without increasing the response time for individual students (OECD, 2014). Due to this rotated design, there was an increased number of missing cases for these variables (i.e., missing data by design). Preliminary exploratory analysis, whereby attitudinal variables were entered into the model individually, revealed that the missing patterns for the student familiarity with mathematical concepts variable were different from those for other variables, which precluded the coexistence of this variable with the other variables into the model. Therefore, a decision was made to exclude this variable from the analysis. Despite this exclusion, there was still relatively large missingness in the PISA 2012 model, due to the rotated design. However, the percentage of high achievers in the sample involved in the analysis varied from 10.8 to 11.3% across the different steps of the model, being very close to the percentage of high achievers in the overall sample (10.7%), thus, confirming that the analysis sample was representative of the overall sample despite the relatively large missingness.
After accounting for the other predictors, four studentlevel variables were significantly associated with the odds of students being high as opposed to nonhigh achievers in mathematics in the final model. One of the variables related to students’ socioeconomic background and the remaining three variables related to students’ selfbeliefs and engagement. The ORs in the final model (step 4) indicated that with an increase of one unit (i.e., one point) in the students’ family economic, social, and cultural status index, students were 87% more likely to belong to the highachieving compared to the nonhighachieving group in mathematics (OR = 1.87). In other words, students who were otherwise identical with regards to all the other variables included in the model but had a higher economic, social, and cultural status were much more likely to be high achievers in mathematics. Additionally, with an increase of one unit in the student mathematics anxiety index, students who were otherwise identical with regards to all the other variables were 47% less likely to be high achievers in mathematics, while students were 2.2 times more likely to be high achievers in mathematics, with every extra unit increase in the selfefficacy index. Finally, every oneunit increase in the student openness for problemsolving variable was associated with a 55% greater chance of high achievement in mathematics among 15yearolds.
Student sex was a statistically significant predictor of 15yearolds’ high achievement in mathematics, with sex differences favouring boys, when first entered into the model (step 1), and after accounting for students’ family socioeconomic status. However, sex differences were no longer significant when students’ selfbeliefs, dispositions, drive, and engagement were taken into account.
At schoollevel, none of the four variables included in the model significantly predicted students’ odds of belonging to the highachieving group in mathematics in PISA 2012. This indicated that the proportion of schoollevel variance in students’ high achievement in mathematics that was explained by the model (36.4%) might have, in fact, been explained by the studentlevel variables that were already included in the model rather than the schoollevel variables.
The final model accounted for 41.8% of the betweenstudent (level1) and 36.4% of the betweenschool (level2) variance in high achievement in mathematics, indicating that the student and schoollevel variables that were included in the model contributed considerably to the prediction of the odds of 15yearold students belonging to the highachieving as opposed to the nonhighachieving group in mathematics.
PISA 2015 – science
Table 6 summarises the results of the hierarchical twolevel binary logistic regression model for high achievement in science that was applied to PISA 2015 data. In the final model, a range of studentlevel characteristics were significantly associated with the odds of 15yearold students belonging to the highachieving group in science, while none of the variables at the school level retained their statistical significance, after partialling out the variability in the outcome variable due to the other variables included in the model. After accounting for other predictors, students’ educational expectations, their interest in broad science topics, and their engagement in exercising or playing a sport before going to school were the strongest predictors of high achievement in science, yielding the largest βs at the student level.
Specifically, students who reported that they expected to complete tertiary education at NFQ level 8 (i.e., higher diploma, honours bachelor’s degree) were 2.9^{Footnote 6} and 3.46 times more likely to belong to the highachieving group in science compared to students who expected to complete up to lower secondary education and Leaving Certificate and Vocational programmes (i.e., upper secondary education), respectively. Additionally, students’ odds of being high achievers would be increased by 90% with every extra unit in the interest in broad science topics index. Finally, students who reported exercising before going to school were 60% less likely to be high achievers in science compared to their peers who were otherwise identical with regards to the other variables but did not exercise or play a sport before school.
Student sex along with a number of other studentlevel noncognitive and attitudinal factors also remained significant predictors of high achievement in science in the final model. Females were 43% less likely to be high achievers in science compared to their male peers. Also, greater environmental awareness, higher test anxiety, and value of cooperation were linked to decreased odds of a 15yearold student belonging to the highachieving group in science. Higher science selfefficacy and more positive views on scientific approaches, on the other hand, were associated with increased odds of 15yearold students being high achievers in science. Finally, more frequent engagement of students in sciencerelated learning activities at home at the age of 10, according to their parents’ reports, was linked to increased odds of belonging to the highachieving group in science; students were 1.5 times more likely to be high achievers in science, with every extra unit in the past science activities index.
Even though student enjoyment of science was the variable that yielded the strongest effect size (g = 0.84) in the bivariate analysis for PISA 2015, and was a significant predictor of student high achievement in science when initially entered into the model (step 2), it was no longer significant after the introduction of additional variables in step 3. This indicates that variables such as students’ activities and schedule on school days as well as science activities at a younger age, parental expectations and support at home may have accounted for the contribution of student enjoyment of science to the prediction of high achievement in the subject.
With regard to science learning in school, higher availability and usage of ICT and more unfair treatment of students by teachers, according to students’ reports, were linked with students’ decreased odds of belonging to the highachieving group in science after accounting for all the other predictor variables. With every extra unit in both the ICT availability at school and teacher fairness indices, students were 10% less likely to be high achievers in science. While no causal links of ICT availability at school and teacher fairness with high achievement in science can be established here, meaning that decreased levels of ICT availability at school or increased teacher fairness would not necessarily lead to higher chances of high achievement, it is noteworthy that these two variables retained their statistical significance after other important variables, such as student sex and socioeconomic status, were taken into account.
The predictor variables that were included in the model explained a considerable proportion of the variance in high achievement in science at both levels of the analysis. Specifically, the final model (step 5) accounted for 50.6% of the betweenstudent (level1) and 83.6% of the betweenschool (level2) variance in high achievement in science.
Discussion
The analysis of PISA 2012 and 2015 data reported in this paper provided some insights into potential predictors of high achievement in mathematics and science among 15yearold students in Ireland. In light of earlier findings in the area, the most consistent findings of this study and their implications for policy and practice are discussed.
Overall, the findings of this study indicate that students’ families might play a crucial role in enhancing students’ chances of being high achievers in mathematics and science. Hence, it seems reasonable to suggest that further efforts to enhance collaboration between teachers, schools, and parents are needed, while there is also a need for educational policy to focus teaching and public awareness on the value of noncognitive factors and wellbeing as well as of appropriate leisure activities for improving achievement. Along these lines, preschool practitioners, teachers, and school principals could raise parents’ awareness of their important role as learning partners in shaping their children’s learning on an ongoing basis from the early years. Broadly speaking, greater attention could be placed on initial preschool practitioner and teacher education and continuing professional development programmes to preparing these professionals as well as school principals to work in partnership with parents. Specific ways of how these could be materialised with reference to student and familylevel factors that were found to predict high achievement in mathematics and science in this study are discussed below. In the interest of clarity, the discussion of findings mirrors the approach adopted in the modelbuilding process (see Fig. 1).
Student and family demographics
Student sex
In comparison with sex differences noted in overall achievement in mathematics and science in Ireland (e.g., Clerkin et al., 2015; Shiel et al., 2016), sex differences in high achievement in the two subjects at the bivariate level, as identified by this study, were larger. In general terms, this corroborates earlier findings in the area (e.g., Ellison & Swanson 2010; Gilleece et al., 2010; Stoet & Geary, 2013) indicating that the sex gap in mathematics and science among highachieving students favouring males is consistently larger compared to the respective gaps in overall achievement in the two subjects. After taking other variables into account, though, sex differences in high achievement remained relatively small, following the unsystematic patterns found in research studies on overall achievement (e.g., Zhou et al., 2017).
These findings do not necessarily undermine the practical significance of sex differences in the context of high achievement; rather, they point towards potential factors on which the Irish education system could focus to address these differences. In the mathematics model, the gap between males and females in high achievement was no longer significant after student selfbeliefs, dispositions, drive, and engagement were considered. This suggests that these variables may have accounted for sex differences among high achievers, indicating that males and females with equivalent selfbeliefs, dispositions, drive, and engagement tended to be equally likely to be high achievers in mathematics. Based on these findings, it seems that one possible way to deal with sex differences in high achievement in mathematics in Ireland would be to focus on students’ noncognitive attributes, especially those of females.
It is also noteworthy that although sex differences were not large or systematic, they were in favour of males in both mathematics and science, suggesting that female students may be lagging behind in high achievement in the two subjects. Given that high achievement in these areas during schooling is associated with higher participation in STEMrelated courses at third level, a way of bridging this sex gap would be for schools to assure that all students but, especially, females are aware of the range of STEM courses and careers that are available to them from an early age. However, as it was highlighted in the STEM Education Review Group report, “one key barrier in this regard [underrepresentation of females in the STEM workforce in Ireland] arises from the fact that, while parents are the main influencers when it comes to advising their daughters on how to define educational and career paths, they generally lack information about career options.” (The STEM Education Review Group, 2016, p. 8). Hence, schools could cooperate with parents to make sure that the latter are in a position to make STEM careers and their associated benefits more desirable for female students, in particular, without, of course, neglecting male students in these efforts. Making STEM careers more desirable for all students but, especially, females falls under the key priorities of the current STEM Education Policy in Ireland (DES, 2017b). This study highlights the importance of such policies and of closely monitoring sex differences in high and overall achievement in all the subjects, as an important aspect of educational equity.
Socioeconomic status
One factor that yielded large differences between the two performance groups in both subjects was students’ family socioeconomic status. Prior to and even after accounting for a range of variables, students coming from households with higher socioeconomic status were more likely to be high achievers in mathematics and science. This finding suggests that aspects of postprimary students’ lives pertaining to their home possessions (e.g., books at home), and their parents’ occupation and education are particularly important when it comes to their probabilities of belonging to the highachieving groups in mathematics and science.
This finding corroborates those of the few studies that have examined high achievement in mathematics and science. For instance, studies by Gilleece et al. (2010) and Tourón et al. (2018) have shown that higher socioeconomic status is associated with higher chances of high achievement. Further, it echoes findings from a body of research that investigated the relationship between family socioeconomic status and overall achievement in a range of subjects, including mathematics and science (e.g., Reardon, 2011). This is an important finding in the context of educational equity, especially when the relationship is examined while other variables are taken into account, as it indicates that over and above anything else that is going on in students’ lives as well as in their classes and schools, the socioeconomic status of their family is still an important predictor of their academic outcomes even at the highest levels of performance. This is despite significant efforts that have been made by the Department of Education in Ireland over the years to support the more socioeconomically disadvantaged (e.g., Delivering Equality of Opportunity in Schools [DEIS] initiative). Consequently, this finding begs questions for policymakers and teachers alike about what more can be done to deal with this important aspect of equity in education.
Although schools alone might not be able to decrease socioeconomic inequalities per se, their role in preventing such inequalities from shaping students’ learning and outcomes is crucial. Along with existing initiatives that primarily target socioeconomically disadvantaged schools, efforts could also be made at the individual level. Specifically, to support learning among disadvantaged students, schools need to target their efforts to improve communication with parents in the most disadvantaged homes, and help develop home environments conducive to learning in mathematics and science. Parents from socioeconomically disadvantaged families should be made aware of the importance of the availability of adequate educational resources at home (e.g., books and appropriate space to study) for their children’s learning as well as of useful resources that they could use at home. Additionally, they should be supported by teachers and schools in getting involved in their children’s learning, and in encouraging their children to take part in extracurricular activities related to mathematics and science that are organised by the school or other bodies. Such practices could contribute towards mitigating the impact of inequalities on students’ outcomes. Alongside such efforts, there must also be a stronger emphasis on identifying and nurturing talent of all students but especially of those coming from socioeconomically disadvantaged backgrounds. This could involve a stronger role for special education teachers in Irish schools, who currently seem to focus on addressing the needs of atrisk lowachieving students.
Student selfbeliefs, dispositions, drive, and engagement
A range of variables related to students’ selfbeliefs, dispositions, drive, and engagement emerged as important predictors of high achievement in mathematics and science. This is not surprising given cognate research, such as Bandura’s (1997) work on the importance of such noncognitive constructs and, especially, selfbeliefs in the context of the socialcognitive theory as well as Stankov’s (2013) predictability gradient hypothesis. Such noncognitive student attributes are likely malleable and responsive to change through appropriate schooling and interventions and, thus, merit consideration.
Overall, more positive selfbeliefs and dispositions and greater drive and engagement were linked with higher chances of high achievement in mathematics and science. Student selfbeliefs and, more specifically, domainspecific selfbeliefs were among the strongest predictors of high achievement in mathematics and science in both the bivariate and the multilevel binary logistic regression analysis, with selfefficacy yielding the largest differences. The bivariate analysis indicated that selfefficacy yielded larger effect sizes for high achievement in mathematics compared to high achievement in science. Lee and Stankov (2018), who analysed, amongst others, PISA data to investigate the role of noncognitive factors in overall mathematics achievement, rather than high achievement, also indicated that students’ selfefficacy was the strongest predictor of achievement, even after accounting for home possessions and parental education as proxies for students’ socioeconomic status. This study echoed and extended this finding to high achievement in both mathematics and science. This highlights the importance of retaining and, potentially, increasing the current emphasis on initiatives that purport to enhance these students’ selfbeliefs in mathematics and science, such as those established following the Junior Cycle wellbeing guidelines in Ireland (National Council for Curriculum and Assessment [NCCA], 2017).^{Footnote 7}
There were also variables related to students’ selfbeliefs, dispositions, drive, and engagement (e.g., enjoyment of science, expected occupational status) that yielded particularly large differences between the two performance groups in mathematics and science in a bivariate context but, when examined in a multiple regression context, were no longer significant predictors of high achievement. Given that most of the existing studies examined the relationships of these variables with high achievement in mathematics and science in a bivariate context (see, for example, Tourón et al., 2018), the findings of the current study extend existing research by taking into account other important student and schoollevel variables in a multiple and multilevel regression context.
There were also a small number of instances where differences between the two performance groups in each subject in the bivariate analysis were in favour of one group but, when examined in a multiple regression context, were in favour of the other. Tourón et al. (2018) showed that, in a bivariate context, highachieving students had higher levels of environmental awareness compared to their lowachieving peers, which was also the case in this study. However, when students’ environmental awareness was included in the multilevel model of the PISA 2015 data in this study, it was found to be a negative predictor of high achievement in science. This suggests that students who were identical in all of the other examined variables but had a more thorough understanding of environmental issues were less likely to be high achievers in science. It is plausible that students with high knowledge of science have a greater awareness of the difficulties associated with reversing environmental problems. In any case, this is a finding that raises more questions than it answers, prompting further investigation of the underpinning mechanisms behind this relationship.
These findings provide a strong rationale for the current emphasis that the Irish education system has placed on students’ selfbeliefs and other noncognitive attributes, as integral parts of their wellbeing. They also suggest that preschool practitioners, primary and postprimary teachers, and schools along with parents could work together to help students develop and strengthen such selfbeliefs, which, in turn, are expected to work positively towards raising students’ chances of becoming high achievers in mathematics and science.
Research and relevant initiatives in Ireland and elsewhere acknowledge wellbeing as a key issue in the holistic development of children and young people. The promotion of students’ selfefficacy, as an important aspect of their wellbeing, is among the emphasised areas during early childhood education as well as at primary and postprimary levels in an everincreasing number of countries around the world, including Ireland; for example, students’ wellbeing is a central aspect within Aistear, the early childhood curriculum framework in Ireland (NCCA, 2009), the proposed revised primary curriculum (NCCA, 2020), and the curriculum frameworks for the Junior and Senior Cycles (i.e., postprimary education) (NCCA, 2011, 2017). This emphasis, which is justified by the findings of this study but also by data from PISA 2018^{Footnote 8}, acknowledges the important role of teachers, schools, and parents in enhancing students’ interest and engagement in learning, by ensuring that students develop positive perceptions about their abilities that, in turn, can help them improve and sustain their gains in achievement.
In Ireland, this emphasis has also led to the creation of a wide range of guidelines, innovative courses, and other resources (for examples, see the NCCA’s website) to assist preschool practitioners, teachers, schools, and parents in their efforts to enhance students’ wellbeing. Specifically, NCCA’s guidelines include relevant information on how preschool practitioners, teachers, and school principals can work together with parents to promote children’s wellbeing, including their selfbeliefs. For instance, such a collaboration could be facilitated by individuals who serve as connection points between students’ homes and their schools and who can gain insights into individual students’ home circumstances (such as the Home School Community Liaison Officers within the DEIS initiative in Ireland) and, thus, act as coordinators of action. However, a requirement for parents to be able to support their children in developing and improving their selfbeliefs in relation to mathematics and science is that parents themselves have strong selfbeliefs in their mathematical and scientific abilities. This highlights that part of the overall efforts in improving students’ selfbeliefs may also need to be the corresponding improvement of their parents’ selfbeliefs. This is an area, though, that has not been examined by the current study, and one that could be examined by future research.
On the whole, the current emphasis on students’ wellbeing especially within early childhood education in Ireland should be retained and increased, given that these constructs tend to be more malleable during these early years and, thus, positive developments during these early years are likely to have longterm benefits for students.
Student mathematics and sciencerelated activities and schedule, parent dispositions and support
Students who, according to their parents, were more frequently engaged in sciencerelated activities at the age of 10 were significantly more likely to be high achievers in the subject, again, highlighting the importance of early knowledge about and engagement with materials related to school subjects for future academic success. It is noteworthy that this finding, based on a set of PISA questions about parents’ support for science learning in the middle childhood years, highlights not only the importance of parental encouragement, curiosity, and involvement in their children’s learning, but also how these can contribute to subsequent academic outcomes. The review of the existing literature on high achievement in mathematics and science indicated that previous studies in the area had not explored the role of early skills and engagement with relevant materials in high achievement in either of the two subjects, rendering this one of the first studies to address this topic.
This finding highlights the crucial role of early years and parents’ support in promoting students’ high achievement. On this basis, policymakers and schools should aim to raise parents’ awareness about behaviours and practices that stimulate students’ progress and those that do not. For instance, early interventions and programmes targeted at parents of young children should provide resources and supports to assist parents in enhancing their children’s early knowledge and skills (e.g., literacy, numeracy etc.) and engaging their children in mathematics and sciencerelated activities during the early years.
In recognition of the usefulness of the resources for mathematics and science education that have already been developed by various Irish educational agencies, including those charged with supporting teachers to implement Project Maths^{Footnote 9}, it is imperative that teachers and parents are made aware of this range of supplementary programmes available for teaching these two subjects and for supporting their children at home, respectively. This is important as solutions to educational issues may not always require newly developed resources but rather awareness and appropriate use of existing ones. Bringing such resources together and organising them thematically, such as efforts being made by scoilnet.ie, sfi.ie, and smartfutures.ie as well as making them readily available for teachers, parents, and students are likely to assist towards this end.
Class and school characteristics
None of the schoollevel variables that yielded effect sizes large enough to progress to the multilevel binary logistic regression analysis retained their statistical significance in the multilevel models once other variables were accounted for. This does not imply that the role of classes and schools in the context of high achievement should be neglected. In fact, as discussed above, considerable proportions of variance in students’ high achievement in mathematics and science were attributed to betweenschool differences. The fact that consistent findings about predictors of high achievement in the two subjects at the school level were not detected may be linked with the limited availability of information about these contexts. In light of the findings of this study, though, it appears that teachers and schools can act as mediators and facilitators in tackling differences between high and nonhighachieving students based on certain characteristics stemming from the students themselves and their families that were found to contribute to the prediction of high achievement in mathematics and science.
At this point, it should be acknowledged that a singular focus on high achievers is highly likely to result in increased variance in performance and, hence, increased inequalities among students (e.g., Ferreira & Gignoux, 2014). Hence, this study advocates that a stronger emphasis on meeting the needs of high achievers, as discussed above, should occur with continuing attention to the needs of low achievers as well. Not only is such an approach expected to enhance Irish students’ achievement across the performance continuum, but it is also expected to improve equity within the Irish education system.
Limitations
There are some limitations underlying this study that should be acknowledged and taken into account in the interpretation of the findings. Firstly, even though results based on data from international largescale assessments that draw on nationally representative samples have the potential to be scalable and transferable, the nonexperimental nature of these data did not allow for the establishment of causal relationships among the examined variables (L. Cohen et al., 2017). Research that attempted to explore the potential of these data in examining causal effects (see Rutkowski & Delandshere, 2016) has indicated that methodological limitations prompt more cautious interpretations than those of strict causeandeffect. Hence, any inferences about the relationships between the examined variables should consider that the relationships may be reciprocal.
Secondly, most of the measures of the contextual information about students, their parents, classes, and schools administered were based on selfreports. It is acknowledged that this could lead to selfreport response bias, as respondents may, intentionally or unintentionally, have provided distorted responses. This may, in turn, have reduced the validity of the inferences from these measures (R. J. Cohen & Swerdlik, 2009). Relatedly, PISA uses selfreport measures to capture aspects of teaching and learning such as studentteacher interactions and teaching practices. It has been aptly argued, however, that the multifaceted nature of teaching and learning may not be fully captured through a series of responses to selfreport questionnaires (see Kaplan & Kuger, 2016) and that other additional measures such as direct observations could complement such selfreport data. Consequently, despite the wealth of information that PISA collects, it is acknowledged that, as a secondary analysis of data, this study was somewhat limited in terms of the measures and variables available.
Recommendations for future research
This study has contributed to the literature on high achievement in mathematics and science at postprimary level in Ireland by providing a detailed examination of PISA data. Notwithstanding this contribution, the limitations of this study and aspects of the research problem that were not explored lead naturally to recommendations for future research.
The series of bivariate and multilevel binary logistic regression analyses reported in this paper uncovered a range of contextual characteristics stemming from the students themselves, their homes, classes, and schools that are associated with high achievement in mathematics and science. Nevertheless, experimental and/or longitudinal studies would allow for the detection of potential causal links between such contextual characteristics and students’ high achievement and of developments or changes in the characteristics of the target population at both the individual and the group levels. This was not possible in this study due to the nonexperimental and crosssectional nature of the data. While longitudinal studies that track students’ development from early childhood up to postprimary levels and even beyond exist in Ireland (e.g., Growing Up in Ireland), longitudinal studies with a much stronger focus on mathematics and science and the needs of highachieving students than that of the current efforts are required.
Despite the wealth of contextual information collected by PISA, there are many important variables that are not captured. Employing qualitative approaches to compare the distinct characteristics of education systems, curricula, and teaching practices from other countries to the reality in Irish schools could shed some more light on aspects that are not measured by PISA, such as the classroom and school contexts. Such investigations could, also, be complemented with national and international largescale assessment data. Research projects, such as the TIMSS 1999 Video Study, a study of eighthgrade mathematics and science teaching in seven countries that involved recording and analysis of teaching practices in more than one thousand classrooms (Hiebert et al., 2003; Roth et al., 2006) and a study by Lyons et al., (2003) that was based on the TIMSS Video Study and collected videotape evidence from a small number of postprimary classrooms in Ireland, constitute illustrative examples of such research.
Future studies could seek to replicate the findings reported here and include information from the Junior and Leaving Certificate examinations, as the major highstakes examinations in Ireland, by matching these data with information from PISA and other national or international largescale assessments. In this way, patterns of high achievement in the subjects of interest and the relationships of contextual characteristics with high achievement in these subjects within a highstake and a lowstake assessment context could be compared. It should be acknowledged, however, that under the General Data Protection Regulation (GDPR), such endeavours are likely to have their own challenges. At a minimum, for instance, they will need to ensure that students taking the national and international assessments give their permission for their state examination results to be used in this way.
Conclusion
This study focused on high achievement in mathematics and science among 15yearold students in Ireland. This is an area that has attracted considerable policy attention in Ireland in the past decade primarily in light of national and international largescale assessment results indicating that the Irish education system may be lagging behind with regards to high achievement in these subjects. High achievers, especially those in STEMrelated areas, have a unique contribution to make to the future social and economic wellbeing of countries; hence, affording more students the opportunity to perform at the highest levels in these subjects should constitute an important part of all educational agendas. This research is timely given that it provides muchneeded evidence during a period of ongoing efforts to raise interest and improve academic performance within the realm of STEM education and that a comprehensive review of the 2011 Literacy and Numeracy Strategy (to also include Digital Literacy) is among the stated priorities for primary and postprimary education in the current Programme for Government (Department of the Taoiseach, 2020), which also includes a focus on raising achievement standards at the upper end of the performance distribution. It is anticipated that this evidence will provide further impetus towards this end and inform efforts to address the challenges in relation to high achievement facing the Irish education system.
Availability of data and materials
The PISA datasets analysed in this study are publicly available on the PISA website (https://www.oecd.org/pisa/data/).
Notes
Student sex and socioeconomic status at the student and the school level were excluded from this criterion. These variables were used in the multilevel binary logistic regression analysis regardless of the effect sizes they yielded in the bivariate analysis.
In the exploration of the statistical significance of interaction terms, corrections for multiple comparisons were applied (Armstrong, 2014).
Although these results provide insights into the relationships of a range of contextual variables with high achievement in mathematics and science, they do not provide information about the direction of these relationships, which means that any observed relationships may be reciprocal.
R^{2} at both levels of the analysis was computed based on McKelvey and Zavoina’s (1975) and Snijders and Bosker’s (2012) instructions for the estimation of R^{2} in multilevel logistic regression models whereby the binary outcome variable is generated through the dichotomisation of an underlying continuous latent variable. Mplus calculates R^{2} in line with these instructions (Muthén & Muthén, 2017). With the caveat that the R^{2} refers to this continuous latent outcome variable as opposed to the observed binary outcome, its interpretation is similar to the interpretation of the R^{2} for linear regression. For further technical information, see McKelvey and Zavoina (1975) and Snijders and Bosker (2012).
Across both models, none of the crosslevel interactions of student sex and studentlevel socioeconomic status with the schoollevel variables that were examined reached statistical significance.
Calculated based on the inverted OR (1/OR).
Some caution is warranted in the interpretation of the relationship between students’ selfefficacy (and how the construct is measured in PISA) and student achievement, including high achievement. In PISA, selfefficacy in mathematics and science could be described as a proxy for achievement in each of the two subjects, as students are asked to indicate their level of confidence in performing various types of mathematics and science tasks. Hence, its relationship with achievement could be interpreted as achievement explaining achievement, with those who think that they are able to perform complex tasks in the two subjects being the highest achievers.
The PISA data for Ireland in 2018 indicated that students reported lower levels of life satisfaction compared to the OECD average, fewer students felt cheerful, joyful or proud than students, on average, across OECD countries, students often compared themselves to others, and over half of students reported that they tend to worry about what would happen if they fail an exam or test, indicating relatively poor wellbeing among students in Ireland compared to their counterparts in other countries (McKeown et al., 2019).
Initiative developed as a response to a series of identified potential difficulties with mathematics education in Irish postprimary schools, including Ireland’s results in international largescale assessments, involving the development of revised syllabi in both Junior and Leaving Certificate mathematics (DES, 2010).
Abbreviations
 DEIS:

Delivering Equality of Opportunity in Schools
 DES:

Department of Education and Skills
 GDPR:

General Data Protection Regulation
 IDB:

International Database Analyzer
 IEA:

International Association for the Evaluation of Educational Achievement
 NAMER:

National Assessments of Mathematics and English Reading
 NCCA:

National Council for Curriculum and Assessment
 NFQ:

National Framework of Qualifications
 OECD:

Organisation for Economic Cooperation and Development
 PIRLS:

Progress in International Reading Literacy Study
 PISA:

Programme for International Student Assessment
 STEM:

Science, Technology, Engineering, and Mathematics
 TIMSS:

Trends in International Mathematics and Science Study
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Acknowledgements
The author is grateful to Dr Anastasios Karakolidis for his valuable assistance during multiple stages of this research and for proofreading the manuscript.
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This research was supported by the Irish Research Council Government of Ireland Postgraduate Scholarship. The funding body was not responsible for the design of the study, the collection, analysis, and interpretation of data or the writing of the manuscript.
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Pitsia, V. Examining high achievement in mathematics and science among postprimary students in Ireland: a multilevel binary logistic regression analysis of PISA data. Largescale Assess Educ 10, 14 (2022). https://doi.org/10.1186/s4053602200131x
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DOI: https://doi.org/10.1186/s4053602200131x
Keywords
 High achievement
 Mathematics
 Science
 PISA
 Multilevel binary logistic regression modelling
 Ireland