Preschool Growth in Mathematics and Long-Run Achievement: An Instrumental Variables Approach

We estimate the causal links between preschool mathematics growth and late elementary school mathematics achievement, using variation in treatment assignment to an early mathematics intervention as an instrument for preschool mathematics growth. Our two-stage least squares (2SLS) estimates indicate that a standard-deviation of intervention-produced growth during preschool is associated with a 0.24 standard deviation gain in achievement in middle and late elementary school. This impact is approximately half the size of the effect produced by OLS models relating later achievement to preschool math growth, and is approximately 30% smaller than the effect reported by highly-controlled OLS regression models (Claessens et al., 2009; Watts et al., 2014) using national datasets. Implications for developmental and educational theory and practice are discussed. Impact of preschool math on later achievement 3 An accumulating body of research suggests that early mathematical skills are critical to developing long-run success in school (Aunola, Leskinen, Lerkkanem, & Nurmi, 2004; Byrnes & Wasik, 2009; Claessens & Engel, 2013; Duncan et al., 2007; Geary, Hoard, Nugent, & Bailey, 2013; Jordan, Kaplan, Ramineni, & Locuniak, 2009; Stevenson & Newman, 1986; Watts, Duncan, Siegler & Davis-Kean, 2014). Among these studies, Duncan and colleagues’ (2007) analysis of six longitudinal datasets provides the most robust evidence for the strong association between early and later mathematics achievement. Their investigation of school readiness skills asked a seemingly straight-forward question: if one examined a broad range of child skills and behaviors at school entry, and controlled for a host of child and family background characteristics, which characteristics would emerge as the strongest predictors of the child’s eventual school achievement? Among the candidates investigated were academic competencies, attention problems, and internal and externalizing problem behaviors. Across the datasets, a consistent pattern emerged: mathematics achievement at school entry was the strongest predictor of later success in mathematics, and in come cases reading, even when all other characteristics tested were controlled. Since the publication of this study, other correlational studies have found similar results (Claessens, Duncan, & Engel, 2009; Claessens & Engel, 2013; Foster, 2010), including one that extended the outcome measurement into high school (Watts et al., 2014). Developmental and cognitive theories would predict that mathematics knowledge is associated with later mathematics achievement because early numerical skills facilitate students’ future mathematical skill acquisition (e.g. Aunola et al., 2004; Entwisle & Alexander, 1990; Gersten et al., 2009; Jordan et al., 2009). This skill-building framework rests on the idea that mathematics is a particularly hierarchical subject, in which mastery of simple concepts and procedures is required for understanding more difficult mathematics. For example, solving even Impact of preschool math on later achievement 4 a simple algebraic equation would be impossible without knowledge of operations such as division and multiplication, and this operational knowledge depends on understanding the basic principals of counting. Relatedly, Siegler, Thompson, and Schneider (2011) describe how students gradually broaden the class of numbers that they understand as they progress through mathematics, with successful students moving from mastery of whole numbers in early grades to fractions in later elementary and middle school. Indeed, a well-developed body of empirical work documents the carefully-sequenced cognitive steps students take as they expand their understanding of numbers and mathematics (e.g. Booth & Siegler, 2006; Gilmore, McCarthy, & Spelke, 2007; Laski & Siegler, 2007; Opfer & Thompson, 2008; Sarama & Clements, 2009). Further, mathematical thinking may be cognitively fundamental. The quantitative, spatial, and logical-reasoning competencies children learn as they progress in mathematics may form a cognitive foundation for thinking and learning across subjects (Clements & Sarama, 2011). As one example, expressive oral language skills are strongly related to later academic success in mathematics and reading (e.g., Pankratz, Plante, Vance, & Insalaco, 2007; Paul, Hernandez, Taylor, & Johnson, 1996; Pearson & Hiebert, 2010; Snow, Barnes, Chandler, Goodman, & Hemphill, 1991) and learning mathematics at an early age may help build these skills (Sarama, Lange, Clements, & Wolfe, 2012). Beyond the cognitive skill-building framework lie other developmental reasons to expect that early success in mathematics would set children on a successful trajectory throughout school. Complex interactions between the child and her environment in the early schooling years are likely to leave long-lasting influences on the child’s developmental trajectory (Bronfenbrenner & Morris, 2006). For example, a high-achieving child in kindergarten is more likely to receive positive feedback regarding their academic proficiency from teachers, parents, Impact of preschool math on later achievement 5 and peers, which in turn may boost a child’s perception of their own math competence (Bong & Skaalvik, 2003; Meisels, 1998). Relatedly, early mathematics achievement could be a gateway to higher-ability tracking in school, which would also support further academic development. Indeed, these pathways from early to later mathematics achievement have received empirical support, as recent evidence suggests that self-concepts and placement into gifted and talented programs both mediate the association between early and later mathematics (Watts et al., 2015). From Level to Growth in Early Mathematics Much of the correlational evidence linking early mathematics ability and later achievement is based on measures of early levels of mathematics achievement. Other studies show strong associations between early gains in mathematical ability and later success in school. For example, using longitudinal data from the NICHD Study of Early Childcare and Youth Development (SECCYD), Watts and colleagues (2014) found that growth in mathematical skills during the first 2 years of school was more predictive of later achievement than were levelmeasures of school-entry skills. Moreover, early-math gains were just as predictive of high school achievement as grade-3 achievement, and this held after controlling for concurrent gains in other cognitive skills, such as working memory and reading achievement. Claessens et al. (2009) employed nationally-representative data and found that kindergarten growth in mathematics achievement was highly predictive of both fifth grade mathematics and reading achievement. Finally, using a growth-curve modeling approach, Jordan and colleagues (2009) found that growth in number competence, measured over six time-points in kindergarten and first grade, strongly predicted third-grade mathematics achievement. Taken together, these studies suggest that the process of learning mathematics during the early-grade years may set students on a higher-achievement trajectory throughout their time in Impact of preschool math on later achievement 6 school. If the associations between early growth and later achievement reported by these correlational studies approximate causal effects, then such long-run impacts could be expected from educational interventions that successfully promote early mathematics learning. Although past studies of early math growth controlled for a host of child characteristics, including initial level of mathematics achievement, it is still unclear whether the regression-adjusted association between early growth in mathematics and later achievement represents a causal effect. Here we ask: Do early mathematics gains produced by random assignment to an intervention predict later math achievement as strongly as the naturally-occurring gains used in past studies? If the associations reported in past studies are driven by unobserved characteristics, such as interest, motivation, parental support for mathematics, or cognitive aptitude, then even highly successful early mathematics interventions may have no detectable impact on later achievement. Indeed, experimental and observational studies suggest that the regression-adjusted associations reported by correlational research overstate the potential long-run impacts of early mathematics intervention. Bailey, Watts, Littlefield and Geary (2014) hypothesized that the stable correlation observed between measures of early mathematical ability and the sequence of later mathematics measures may be due to stable but unobserved factors that heavily influence mathematics achievement throughout development. Using a latent-factor state-trait model, they separated the variance in longitudinal measures of mathematics achievement into time-variant (state) and time-invariant (trait) components. They found that most of the variation in repeated measures of mathematics achievement was trait-like, as variation in individual differences in mathematics achievement were highly stable over time. Conversely, changes in any single measure of mathematics ability had relatively small effects on subsequent achievement scores once the stable variance was partitioned into a single, latent, factor. They concluded that Impact of preschool math on later achievement 7 correlational studies investigating the association between early and later measures of achievement fail to take into account the multitude of stable environmental and individual factors that likely influence achievement over time, and this omission leads to an overstatement of the importance of a single measure of achievement on later measures. Further, experimental evidence from intervention studies also suggests that long-run correlational models may not accurately represent causal impacts. Building Blocks, a preschool mathematics curriculum