Early mathematical competencies and later achievement: insights from the Longitudinal Study of Australian Children

Importance of early childhood education

The importance of early childhood education is well established. Conclusive international evidence demonstrates that early childhood is a vital period in children’s learning and development (Department of Education, Employment and Workplace Relations [DEEWR] 2009), and that what happens in early childhood affects later development (Council of Australian Governments [COAG] 2009). There is consistent evidence (e.g. Karoly et al. 2006; Sammons et al. 2002) showing that high-quality early childhood education has a positive impact on young children’s cognitive and social outcomes and adjustment to school (Elliott 2006). With approximately 87% of Australian children spending time in early childhood education and care settings before starting school (Baxter 2015), the educational significance of children’s experiences and outcomes increases (Elliott 2006). In 2009, Australia’s first national early childhood curriculum framework, Belonging, Being and Becoming: The Early Years Learning Framework for Australia (EYLF) (DEEWR 2009), was established “to assist educators to provide young children with opportunities to maximise their potential and develop a foundation for future success in learning” (p. 5). Furthermore, the EYLF contributes to realising COAG’s (2009) vision that by 2020 “all children have the best start in life to create a better future for themselves and for the nation” (p. 4). Participation in effective early childhood education has a number of significant benefits, including improved child development, improved school readiness and performance at school and improved educational attainment (Robinson et al. 2011).

Young children’s mathematical competencies

Doig et al. (2003) have suggested that the increasing numbers of children participating in early childhood programmes, and the growing recognition of the importance of mathematics in general, provide compelling reasons for understanding children’s mathematical development in the early childhood years. Indeed, children begin developing mathematical skills from a very young age, with international research showing that babies and toddlers demonstrate competence in regard to a range of mathematical concepts and processes. In a study of 1003 children aged between 30 and 33 months, Reikerås et al. (2012) found that the toddlers showed mathematical competencies in all areas observed (encompassing number and counting, geometry and problem solving). Similarly, Björklund’s (2008) study of children aged between 13 and 45 months demonstrated that toddlers interact with concepts of dimensions or proportions, location, extent, succession and numerosity and use a range of strategies to express their understanding.

Research with preschool-age children has further demonstrated the ways in which children engage with a range of complex mathematical ideas in the early childhood years. Kinnear and Clark (2014) explored data modelling with a group of 5-year-old children and found that children were able to construct data models which provided evidence of their ability to use existing data, knowledge of the data context and probabilistic reasoning to make predictions. Research by MacDonald (2010, 2012) revealed that children aged 4 to 6 years were able to demonstrate knowledge of measurement concepts such as identifying measureable attributes, comparing attributes such as height or area and using units to describe measurements. A number of studies have shown that young children engage with powerful mathematical processes such as problem solving, argumentation and mathematical reasoning (English and Mulligan 2013; Fielding-Wells and Makar 2015; Makar 2014).

Furthermore, several Australian studies have found that much of the content that forms the mathematics curriculum for the first year of school is already understood clearly by many children on arrival at primary school (Clarke et al. 2006; Gervasoni and Perry 2015; MacDonald 2010), a finding echoed in international studies (e.g. Aubrey 1993; Wright 1994). The seminal Australian study, the Early Numeracy Research Project (see, for example, Clarke et al. 2006) investigated the mathematical knowledge of over 1400 children entering primary school and found that most children already understood much of the mathematical content that was to be taught in the first year (Clarke et al. 2006). Gould (2012), reporting on data from a standardised interview schedule from 65,000 children entering the first year of school, found that approximately 16% of the children entering kindergarten (first year at school) showed the facility with number expected of children in Year 1 (second year at school). Similar findings were noted by Gervasoni and Perry (2013, 2015) in their longitudinal evaluation of the Let’s Count early numeracy programme. Approximately 75% of the children were able to demonstrate the Australian Curriculum – Mathematics Foundation Standard “Students make connections between number names, numerals and quantities up to ten” (Australian Curriculum, Assessment, and Reporting Authority [ACARA] 2013) before they started school (Gervasoni and Perry 2015).

Differences in early mathematics achievement

Early mathematical competencies and later achievement

In recent years, studies have emerged which indicate the predictive power of early mathematical knowledge for later achievement, both in mathematics, specifically, and in terms of more general academic achievement. Watts et al. (2014) conducted a tracking study of 1364 children from 4.5 to 15 years and found that there were statistically significant associations between preschool mathematical ability and adolescent mathematics achievement, even after accounting for early reading, cognitive skills and family and child characteristics. Furthermore, gains in mathematical knowledge from preschool through first grade were even more predictive of mathematics achievement at age 15 than preschool knowledge. Geary et al. (2013) conducted a longitudinal study of 180 children from kindergarten (first year of school) through to age 13 and found that early number system knowledge predicted functional numeracy more than 6 years later, controlling for intelligence, working memory, in-class attentive behaviour, mathematical achievement, demographic and other factors. Duncan et al. (2007) examined six longitudinal data sets to identify links between school-entry academic, attention and socio-emotional skills and later school reading and mathematics achievement. Across all six studies, the strongest predictors of later achievement were school-entry mathematics, reading and attention skills. A meta-analysis of the results showed that early mathematics skills have the greatest predictive power. This finding was consistent for boys and girls, and for children from high and low socio-economic backgrounds. Similarly, Claessens et al. (2009) used data from a nationally representative sample of children from the Early Childhood Longitudinal Study – Kindergarten Cohort (ECLS-K) to estimate the predictive power of school-entry academic, attention-related and socio-emotional skills for reading and mathematics achievement in first, third and fifth grades. School-entry mathematics skills were consistently predictive of fifth grade achievement. Early mathematics skills were not only highly predictive of later mathematics achievement, but of later reading achievement as well. Mathematics skills were the single most important set of kindergarten-entry skills emerging from the analyses.

Given the growing body of research which suggests a relationship between early mathematics and later school achievement (Levine et al. 2010), it is important to ascertain the extent to which children’s early mathematical competencies predict later mathematical achievement.

The current study

Most of the studies cited in the last section reporting associations between school-entry mathematics achievement and later achievement are based on US data, and none have specifically analysed this association in an Australian context. National variations, however, exist in school structures and mathematics curricula, suggesting US results may not be replicated in other jurisdictions. As noted by Duncan et al. (2007), for example, a British study that they analysed did not report any association between school-entry level mathematics and later academic achievement. Further, these studies have tended to base their findings on bivariate relationships between early and later mathematics achievement, ignoring growth in mathematics achievement during the intervening period. Such analyses, however, have long been regarded as inadequate for the systematic study of change (e.g. Rogosa and Willett 1985; Bryk and Raudenbush 1987), primarily because they do not fully utilise the available information. The current study seeks to build on the studies reported earlier by analysing Australian longitudinal data from a large representative sample and studying change through the analysis of individual growth trajectories, as recommended by Bryk and Raudenbush (1987). As such, it considers the research question: To what extent, and how, are competencies at age 4/5 related to mathematical achievement in Years 3, 5, 7 and 9?

Sample

The study is based on 2343 students from the LSAC K-cohort for whom mathematical competency assessments were available at age 4/5 and for whom at least three NAPLAN numeracy test results were available when these students were in Years 3, 5, 7 and 9. Details of the design and implementation of LSAC are available in Sanson et al. (2002). The size of this sample is considerably smaller than the original 4983 contained in the LSAC K-cohort. At least three numeracy test results were available for just 3500 of the children in the K-cohort. Many (55%) of the 1483 children without these results, however, were no longer participating in the LSAC study. Further, only 3176 children in the K-cohort had teacher reports of their mathematical competency at the age of 4/5. Some of the children (n = 221) without these data did not attend school at that age, and parents of others (n = 96) declined to give permission for their child’s teacher to be contacted. It can be assumed, however, that for the majority of cases without these data (n = 1490) teachers declined to provide this information about their students.

Variables

Analysis

Analysis of possible bias in the sample

Given the substantial reduction in sample size, we investigated possible bias between students in the included sample (n = 2343) of the K-cohort and those in the excluded sample (n = 2640). There were no significant differences between the mean ages of children included in the sample and those excluded. Similarly, there were no significant differences in the proportions of boys in the included and excluded samples. However, there were significant differences in the mean socio-economic status (SEP) scores of children in these samples, with those included in the study more likely to come from wealthier family backgrounds than those excluded from the study (d = .32, t = 11.3, p < .01). Further, K-cohort children included in the study were more likely to score higher on the PPVT than those excluded from the study (d = .29, t = 9.4, p < .01). Thus, our sample of children was more likely to come from slightly wealthier families and score higher on the PPVT than those excluded. These results are consistent with the NAPLAN means reported in Table 1, which shows that our sample performed slightly better than the population on each of the four tests.

Bivariate analyses

As is also seen in the table, high-competency students, those with four or five competencies, achieved significantly higher than their low-competency peers in Year 3 (d = .6, t = 11.0, p < .01), Year 5 (d = .5, t = 12.0, p < .01), Year 7 (d = .6, t = 13.0, p < .01) and in Year 9 (d = .6, t = 12.0, p < .01). These students achieved on average more than one half a standard deviation higher than their peers and appeared to maintain that advantage throughout their schooling years.

Each of the individual competencies was assessed against later NAPLAN performance. The students who were able to sort and classify (C1) at age 4/5, on average, achieved significantly better than their peers in Year 3 (d = .6, t = 4.2, p < .01), Year 5 (d = .5, t = 3.6, p < .01) and Year 7 (d = .6, t = 4.2, p < .01), but not significantly better in Year 9. Similar results were evident for students who were able to count objects (C2). Interestingly, those students who could count to 20 (C3) at age 4/5 achieved significantly higher than their peers in Year 3 (d = .6, t = 11.3, p < .01) and broadly maintained this advantage in Year 5 (d = .5, t = 12.2, p < .01), Year 7 (d = .6, t = 13.4, p < .01) and Year 9 (d = .6, t = 11.5, p < .01). Students competent with recognising numbers (C4) and doing simple additions (C5) reported similar results.

Students’ SEP was moderately correlated with achievement in Years 3 and 5 (r = 0.3), and more strongly in Years 7 and 9 (r = 0.4). As is seen in Table 2, students’ measures of verbal ability at age 4/5 (PPVT) were correlated with mathematics achievement in each year that it was measured. Also reported in the table is a slight advantage for boys, who achieved 0.13 standard deviations higher than girls in Year 3 and increased this advantage slightly in the remaining years.

Multivariate analyses

The modelling process applies lines of best fit to the three or four NAPLAN outcomes for each of the 2343 students. The intercepts and gradients from each of these 2343 lines, in turn, form two response variables that are predicted by the explanatory and demographic variables. More formally, the model assumes that, at the student level, growth in NAPLAN numeracy is linear and expressed as:

With results from all 2343 students considered, this model predicts that the mean achievement result should be 437 in Year 3 and 614 in Year 9, both figures higher than those reported in Table 1. This null model also provides estimates for each of the variances of the random effects, shown at the bottom of the table.

Following the fitting of Model A, the total number of competencies (Ncomp), together with demographic variables, was entered into the model as predictors of the random intercept. These variables were also tested as predictors of the random slope. Significant predictors are shown as Model B in Table 3. Those interacting with time, in this case SEP and the standardised PPVT score (PPVTz), are also predictors of the random slope.

As shown in Table 3, after controlling for demographic variables, the number of competencies students gained at age 4/5 was a significant predictor of the random intercept but not of the random slope. Thus, the model suggests that mathematical competence at age 4/5 is an important predictor of achievement in Year 3, but not of subsequent growth in achievement. The model also predicts that boys will achieve 17.5 points higher than girls in Year 3, but that gender will not influence subsequent growth.

Interestingly, SEP predicted both the random intercepts and slopes. Students from homes with an SEP one standard deviation higher than average (SEP = 1) are predicted to achieve 15.9 points higher in Year 3 than students from homes with an average SEP and then increase this advantage in Year 9 (t = 3) to 24.0² points. Similarly, the standardised PPVT-III score of students at age 4/5 (PPVTz) predicted their initial score and also their growth. Interestingly, the effect of PPVTz on growth was negative, suggesting a waning effect. Though not reported in Table 3, age differences when data were first collected did not impact later achievement.