Does the Opportunity–Propensity Framework predict the early mathematics skills of low-income pre-kindergarten children?

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Highlights

  • The O–P Framework accounted for differences in the early math skills of low-income, pre-kindergarten children.

  • High achievement in early math was a function of three latent constructs: antecedent, opportunity, and propensity.

  • Low-income children’s school readiness improved when they had more exposure to typical math learning activities.

  • Typical math learning activities included both teacher-initiated basic math and integrated, play-based activities.

  • Early childhood educators should consider spending more time on these easy-to-implement math learning activities.

Abstract

Prior studies have shown that the variables described in the Opportunity–Propensity (O–P) Framework have successfully accounted for the mathematics and science achievement of students in grades 1–3 and 8–12. The two goals of the present study were to (1) determine whether the O–P Framework could also account for individual differences in the early mathematics skills of low-income, pre-kindergarten children and (2) determine whether latent variables constructed from measured variables would account for performance in the manner specified in the O–P model. The O–P Framework assumes that high achievement in mathematics is a function of three categories of factors: (a) antecedent factors, variables that operate early in a child’s life and explain the emergence of opportunities and propensities, (b) opportunity factors, variables that measure a child’s opportunity to learn mathematics content at home and school, and (c) propensity factors, variables that capture a child’s propensity for learning in terms of self-regulation, motivation, and prior cognitive skills. To test the fit of this model for low-income children during the year before they attend kindergarten, the authors conducted a secondary analysis of achievement and background data from the Early Childhood Longitudinal Study-Birth (ECLS-B) Cohort data set. Structural equation modeling indicated significant associations between the antecedent factor, opportunity factor, and propensity factor, and between the opportunity factor and pre-kindergarten mathematics achievement. The results confirmed the fit of the model and identified the kinds of learning experiences that could promote the acquisition of mathematics skills in low-income children and improve their readiness to learn in first grade and beyond.

Introduction

The number of young children from low-income families who attend pre-kindergarten programs (e.g., preschool or daycare) in the United States is on the rise (see National Institute for Early Education Research (NIEER), 2011; U.S. Department of Education (USDOE), National Center for Education Statistics (NCES), 2000), reaching almost 6 million and representing 48% of all children under age three in the United States (see Addy & Wight, 2012). Low-income children are at risk for a host of early and prolonged academic and social disadvantages. This includes risks of lower academic achievement (see for example, Jordan et al., 2009, Lee and Burkam, 2002), academic failure (Duncan, Yeung, Brooks-Gunn, & Smith, 1998), being placed in special education programs (National Research Council, 2002, Zhang and Katsiyannis, 2002), and being under-employed later in life as adults (Baydar et al., 1993, Senechal et al., 1998). Research suggest that very early opportunities to learn mathematics can promote the development of fundamental mathematics skills and knowledge during pre-kindergarten years (National Association for the Education of Young Children, 2002, Ramani and Siegler, 2008, Starkey et al., 2004), and that mathematics achievement is among the strongest predictors of later academic success (Duncan et al., 2007). Despite the increasing focus on early math skills, little is known about the relationship between greater exposure to typical pre-kindergarten mathematics activities and the mathematics achievement of this particularly vulnerable group of children. By understanding whether low-income pre-kindergarten children would benefit from greater exposure to typical pre-kindergarten mathematics activities, policymakers, educators, and parents would learn whether our current pre-kindergarten programs can provide an opportunity to strengthen the mathematics skills of this growing population of children and improve their readiness to learn in first grade and beyond.

Although mathematics achievement has been found to be one of the strongest predictors of academic success (Duncan et al., 2007) and mathematics performance in primary and secondary grades (Aubrey et al., 2006, Aunio and Niemivirta, 2010, Byrnes and Wasik, 2009, Hooper et al., 2010), there is less agreement on the factors that promote the development of early mathematics knowledge and skills prior to kindergarten (see Bowen et al., 2002, Hillemeier et al., 2010, Pintrich, 2000, Ramani and Siegler, 2008). Some evidence suggests that very early opportunities to learn mathematics can promote the development of fundamental mathematics skills and knowledge during pre-kindergarten years (National Association for the Education of Young Children (NAEYC), 2002; Starkey et al., 2004, Ramani and Siegler, 2008), particularly since pre-kindergarten children tend to have fewer opportunities to learn mathematics than literacy at home (Tudge & Doucet, 2004). In response, a number of researchers (see Chard et al., 2008, Clarke et al., 2011, Clements and Sarama, 2007, Clements and Sarama, 2008, Clements et al., 2011) conducted randomized or quasi-experimental studies on comprehensive, structured early mathematics curriculum to see the impact of these programs on the early math knowledge of children from low-income families. Not surprisingly, these studies have found that students who had the structured early math curricula scored significantly higher than students from comparison or control groups with an average effect size of 0.85 as measured by Cohen’s d (see Chard et al., 2008, Clarke et al., 2011, Clements and Sarama, 2007, Clements and Sarama, 2008, Clements et al., 2011).

However, most pre-kindergarten curriculum on mathematics are not structured and consisted mostly of activities that can be roughly grouped into activities relating to counting, shape identification, the identification and completion of repeating patterns, and integrated activities, where mathematics were integrated in everyday activities such as music, creative movement, and cooking (Greenes, Ginsburg, & Balfanz, 2004). While many researchers have criticized that the typical mathematics programs at the pre-kindergarten levels are limited (Clements and Sarama, 2007, Greenes et al., 2004), emerging evidence suggests that children from low-income families may have differential exposure to mathematics learning opportunities during the early school years (Wang, 2010), and more evidence is needed to understand the relationship between differential opportunities to learn mathematics and mathematics learning for pre-kindergarten children from low-income families.

Moreover, another body of research points to children’s self-regulation (Duckworth and Seligman, 2005, Ponitz et al., 2009), motivation (Pintrich, 2000), and propensity for learning (Byrnes & Wasik, 2009) as strong predictors of mathematics skills and knowledge. Additionally, there is another large body of research that suggests there are a number of factors, such as low birth weight (Bowen et al., 2002, Hillemeier et al., 2010, Sajaniemi et al., 2001), poor early cognitive functioning (Cooper & Schleser, 2006), and low parental expectations (Fehrmann et al., 1987, Rutchick et al., 2009), that put low-income children at greater risk of academic failure than their more affluent peers. The purpose of the present study was to test a hypothesized structural model of predictors that includes many of these factors, linking antecedent factors (those that occur early in a child’s life) to opportunity to learn mathematics and propensity for learning, and, in turn, linking opportunity to learn mathematics and propensity for learning to low-income children’s pre-kindergarten mathematics skills and knowledge.

As noted above, researchers in early childhood have identified a number of factors associated with readiness to learn and achievement. And increasingly, researchers have taken a multivariate approach to predict achievement in which they demonstrated the independent effect of an important construct such as behavioral or emotional regulation while controlling for several other key predictors such as parent education and prior achievement (e.g., Howse et al., 2003, McClelland et al., 2007). While much has been learned from these studies, it is important to extend this work in several ways. First, whereas the aforementioned multivariate studies investigated more factors than researchers typically examined in the past (e.g., four to seven variables instead of one or two), the list of unique, significant predictors of achievement extends well beyond the set analyzed in these studies. Second, and more importantly, most of the multivariate studies have not organized the set of significant predictors into a coherent and comprehensive explanatory framework. Instead, their goal was to demonstrate the unique role of one or two of the factors examined (e.g., behavioral regulation). Constructing an explanatory framework is the first step in designing more effective forms of intervention to elevate the performance of low-income children and get them ready for school. In the absence of effective and more stringent controls for powerful and authentic predictors, intervention efforts may be directed toward a variable that is less related to outcomes than it appears. In addition, an accurate comprehensive account that lays out factors in a temporally arranged manner would help interventionists to understand what to target and when. This approach of identifying most of the significant predictors and arranging them into a proposed explanatory framework is not unlike what occurs in medical epidemiological work (e.g. identifying the various predictors of heart disease, such as cholesterol levels and exercise).

The present study utilized the Opportunity–Propensity (O–P) Framework (Byrnes, 2003, Byrnes and Miller, 2007, Byrnes and Wasik, 2009, Jones and Byrnes, 2006, Sackes et al., 2011) to study the impact of pre-kindergarten opportunity to learn mathematics and propensity for learning on pre-kindergarten mathematics achievement. To our knowledge, the O–P Framework is the only available theoretical model that effectively combines all of the factors assessed in this and a number of other recent studies. It was specifically designed to explain why some children (e.g., White students) perform significantly better than others (e.g., Black students) on achievement measures. The opportunity component of the O–P Framework refers to variables such as teacher-initiated opportunities to count out loud, use geometric or counting manipulatives and play mathematics games, that offer a context to learn academic content. The propensity component refers to variables such as previous knowledge and achievement motivation that increase the likelihood of children benefiting from opportunities to learn (Byrnes and Miller, 2007, Byrnes and Wasik, 2009). The Opportunity–Propensity Framework suggests that children are more likely to realize their potential for learning a particular subject matter such as mathematics if they are provided opportunities to learn that content at school and in other contexts and have the motivation and capability to benefit from the opportunities provided to them (Byrnes & Wasik, 2009). For low-income children, having school-based or teacher-initiated exposure to mathematics at the pre-kindergarten level is especially relevant and important, given that they are particularly at risk for entering kindergarten and first grade with significantly lower mathematics skills and knowledge than their peers (see Duncan et al., 1998, Jordan et al., 2009). In this framework, variables such as parent expectation and maternal education are considered as indicators of the antecedent factor. The antecedent factor explains why some pre-kindergarten children are more likely than others to benefit from the opportunities provided to them and develop stronger propensities for learning (Byrnes & Wasik, 2009). The antecedent factor can also be used to account for the variation in the outcome variable, allowing for the unique contributions of the propensity factor and opportunity factor on children’s mathematics achievement to be identified (Sackes, Trundle, Bell, & O’Connell, 2011). Thus, research in the O–P Framework confirms prior work demonstrating the important role of factors such as executive function or self-regulation on school readiness. It extends previous studies by embedding such factors into a comprehensive explanatory model that shows how various factors relate to each other and to achievement. It also shows which factors account for the most variance and could be potential targets of intervention.

Researchers have clearly documented that antecedent variables (Byrnes, 2003, Byrnes and Miller, 2007, Byrnes and Wasik, 2009), variables that occur early in a child’s life, predict children’s academic achievement. For early achievement among young children from low-income families, the relationship between age, birth weight, early cognitive functioning, parent expectations, and mothers’ years of education have been examined.

Age has a well-established relationship to children’s mathematical competence (Hindman et al., 2010, Jordan et al., 2006, Ransdell and Hecht, 2003) but has not been assessed in prior studies using the O–P Framework. Age is appropriately construed as an antecedent factor because it can explain why some children were exposed to more opportunities than others (e.g., older children usually have been exposed to more opportunities than younger children) and why some children might be more prone to take advantage of these opportunities (e.g., an increase in cognitive skills due to brain maturation). Generally, very young children begin with basic mathematics competence that over time develops into more complex mathematical skills. This is especially true during the preschool ages, when children’s mathematical competence develops from recognizing small groups of objects to counting the number of objects in correct order and, eventually, developing early arithmetic skills (Geary, 2007). Furthermore, age is an educationally relevant variable since there is an 11–12 month variation in children’s chronological age during each academic year of schooling (Dowker, 2008).

Low birth weight (born less than 2499 g), which is prevalent among low-income children (Collins & David, 1990), has been associated with greater likelihood of increased cognitive delay at 2 years of age (Hillemeier et al., 2010), increased frequency of impaired language functioning (Sajaniemi et al., 2001), lower scores on standardized measures of academic achievement (Bowen et al., 2002), and greater likelihood to fall behind academically (Bowen et al., 2002). Although low birth weight has not been included in prior studies using the O–P Framework, it is a plausible candidate to be an antecedent variable that could explain the emergence of opportunities and propensities.

Higher parental expectations for children have been associated with greater likelihood of attending college (Hossler & Stage, 1992), selection of more core academic courses (Catsambis, 2001), better school attendance (Kurdek & Sinclair, 1988), and stronger academic performance (Fehrmann et al., 1987, Rutchick et al., 2009). Even with socio-economic status controlled, parental expectations have been found to explain significant variance in opportunities, propensities and achievement (Byrnes and Miller, 2007, Byrnes and Wasik, 2009). Parental expectations have been found to influence child expectations (Rutchick et al., 2009) and motivation (Wood, Kurtz-Costes, & Copping, 2011), both of which are associated with better academic performance. Although expectations have been examined in prior studies using the O–P Framework, this variable has not been examined in children prior to kindergarten.

Maternal education has been found to have direct and indirect, positive relationships with achievement (Davis-Kean, 2005, Eccles, 2005, Klebanov et al., 1994, Lee, 1993, Shaff et al., 2008), and explains much of the variance in cognitive outcomes for low-income pre-kindergarten children (Perry & Fantuzzo, 2010). Maternal education has been associated with greater exposure to literacy and numeracy skills at home (Reese, Gallimore, & Goldberg, 1999), more educational opportunities in local communities (Furstenberg, Cook, Eccles, Elder, & Sameroff, 1999), and higher expectations for their children’s education (Davis-Kean, 2005), all three of which are predictive of stronger academic outcomes. Additionally, maternal education has been found to indirectly affect a child’s academic achievement to the extent that it influences family income and residence (Coleman, 1987, Furstenberg et al., 1999).

Early opportunities to learn mathematics are defined in this study as the typical school-based or teacher-initiated opportunities for children to learn mathematics in pre-kindergarten programs. These include teacher-initiated activities for learning basic mathematics skills such as counting out loud and doing shapes and pattern activities, and teacher-initiated, integrated learning activities such as using music or creative movement to learn mathematics concepts.

Even though the year preceding kindergarten has been found to be extremely important in mathematics development (Clements and Sarama, 2009, National Association for the Education of Young Children, 2002), results from the few observational studies of pre-kindergarten teachers and programs suggest that in general, very little mathematics is normally presented during the pre-kindergarten years (Clements and Sarama, 2007, Clements and Sarama, 2008, Early et al., 2005, Graham et al., 1997, Lamy et al., 2004, Tudge and Doucet, 2004). This finding was revealed by evaluation studies of pre-kindergarten curriculums which found that most of these programs were built on literacy goals with minimal time devoted to mathematics (Farran, Lipsey, Watson, & Hurley, 2007). In response, a number of researchers conducted experimental studies to examine the effects of structured early mathematics curriculum on early math knowledge and skill of pre-kindergarten children from low-income families (Chard et al., 2008, Clarke et al., 2011, Clements and Sarama, 2007, Clements and Sarama, 2008, Clements et al., 2011, Starkey et al., 2004).

These structured early mathematics curriculum ranged from the 120, 30-min lessons on numbers and operations, geometry, measurement, and vocabulary of the Early Learning in Mathematics program (Chard et al., 2008, Clarke et al., 2011), to the supplemental Building Blocks for Math program which embedded math learning in children’s daily activities ranging from online math activities to circle and story time (Clements and Sarama, 2007, Clements and Sarama, 2008, Clements et al., 2011), to the Pre-K Mathematics program which included 27, 20-min small-group activities on enumeration, arithmetic reasoning, spatial sense, pattern sense among other concepts (Starkey et al., 2004).

Not surprisingly, these studies have found that students who had the structured early math curricula scored significantly higher than students from comparison or control groups. The average effect size across these studies as measured by Cohen’s d was 0.85 with a range of 0.22–2.16. Differences in the effect size seemed to relate to difference in the curriculum being tested, the condition of the comparison or control group, and the type of assessments being used.

However, most pre-kindergarten curriculum on mathematics are not structured and consisted mostly of a “collection of activities” (Greenes et al., 2004, p. 160). These activities can be roughly grouped into teacher-initiated activities relating to counting, shape identification, the identification and completion of repeating patterns or integrated activities, where mathematics were integrated in everyday activities such as music and creative movement and cooking (Greenes et al., 2004). While many researchers have criticized that the typical mathematics programs at the pre-kindergarten levels are limited (Clements and Sarama, 2007, Greenes et al., 2004), emerging evidence suggests that children from low-income families have differential exposure to even the typical mathematics learning opportunities during the early years (Wang, 2010), and that low-income kindergarten children’s exposure to analytic and reasoning mathematics activities is significantly related to their mathematics test scores (Georges, 2009, Wang, 2010). For instance, Georges (2009) found that while instruction accounted for only 4% of the variance in mathematics test scores attributable to classrooms for all kindergarten students, the portion of variance attributable to classrooms was larger for students in high-poverty classrooms than in low-poverty classrooms. She further found that for high-poverty classrooms, activities that built on students’ analytic and reasoning abilities and worksheet related activities were significantly related to students’ test scores in all subtests (Georges, 2009). However, Wang (2010) found that there were statistically significant variations in the frequency in which children in poverty engaged in analytic and reasoning activities, suggesting that this group of children have differential exposure to typical, early mathematics learning opportunities.

Even though national organizations have affirmed the importance of accessible mathematics education for 3- to 6-year-old children since 2002 (NAEYC, 2002), very little is known about the prevalence and nature of typical mathematics exposure during the year preceding kindergarten among children in poverty and the relationship between pre-kindergarten mathematics exposure and mathematics achievement. The authors were not able to identify studies using nationally representative data that have examined whether children in poverty have differential exposure to typical teacher-initiated mathematics activities during pre-kindergarten and how exposure relates to children’s mathematics achievement. The present study aimed to fill this void.

Many researchers have found that a child’s propensity for learning, which includes prior cognitive skills, motivation, and self-regulation, is an important contributor to academic success (Byrnes and Wasik, 2009, DiPerna et al., 2005). The strongest predictor of later achievement in a domain (e.g., math) is prior achievement in that domain, followed by self-regulation, and motivation (Byrnes, 2011, Byrnes and Miller, 2007, Byrnes and Wasik, 2009, McClelland et al., 2007). General ability (e.g., IQ) also predicts but tends to account for less than 10% of the variance when prior knowledge, self-regulation, and motivation are controlled (Sternberg, Grigorenko, & Bundy, 2001). A study that used structural equation modeling to test the relationship between specific academic enablers (motivation, interpersonal skills, engagement, and study skills) and mathematics achievement for K through grade six students from 21 schools found prior achievement to have moderate to large direct and indirect effects on mathematics achievement (DiPerna et al., 2005).

Research has shown that children’s motivation (including their goals, interests, and self-efficacy) and self-regulation skills and academic achievement are significantly related. Consistent evidence suggests that greater levels of attention, task persistence, and active participation have strong associations with standardized test scores and teacher-rated achievement that is independent of initial cognitive ability and prior basic skills (Alexander et al., 1993, DiPerna et al., 2005, Duncan et al., 2007, Hindman et al., 2010, Matthews et al., 2010). For instance, DiPerna et al. (2005) found that for students in kindergarten through second grade, motivation and engagement were stronger predictors of mathematics achievement than for students in third to sixth grades. Although DiPerna et al. (2005) measured motivation and engagement through teacher-ratings, recent studies suggest using both parental- and teacher-ratings for improved validity of these measures (see Mathieson & Banerjee, 2010).

Although prior studies have examined the individual roles of specific predictors of mathematics skills, or examined the collective influence of sets of predictors using the O–P and other frameworks, it is important to extend this work in the following respects. First, most prior studies have examined the predictive power of various factors for mathematics skills in children older than five. By the time children reach kindergarten, however, significant ethnic and racial differences in cognitive skills already exist (Lee & Burkam, 2002). To know how to mitigate these differences, it is useful to reveal predictive relationships in children younger than five. Second, prior large-scale studies have included both high-income and low-income children from multiple ethnic and racial groups (e.g., Byrnes & Wasik, 2009). We wanted to examine individual differences in mathematics skills within a low-income population, specifically to see what home- and school-based differences correlated with these individual differences. Such a refinement provides greater insight into ways to promote skill development in these children. In addition, since the O–P model has never been applied to such a young (or exclusively low-income) sample, doing so provides a strong test of its applicability. Failures of fit often generate useful new findings and revisions to theoretical frameworks. Finally, all prior tests of the O–P Framework have included variables from specific theoretical or heuristic categories (e.g., “opportunity factors” or “propensity factors”), but never examined whether these sampled factors or indicators of particular categories actually coalesce into coherent latent factors. We wanted to know whether these latent factors could be discovered and be shown to be predictive of early mathematics skills in the manner specified by the O–P Framework.

The current study had two goals and six hypotheses. The first goal was to extend beyond prior accounts by showing that the variables that were assumed to be in the same category in prior studies (e.g., propensity factors) coalesced into a single latent factor. Three hypotheses were derived from this first goal: (1) that the latent antecedent factor would be significantly indicated by the variables birth weight, early cognition, age, parent expectation, and mother’s years of education; (2) that the latent opportunity factor would be significantly indicated by the variables teacher-initiated activities for learning basic mathematics skills and teacher-initiated, game-like mathematics learning activities. Teacher-initiated activities for learning basic mathematics skills and teacher-initiated, integrated learning activities would be significantly indicated by items from previous research; and (3) that the latent propensity factor would be significantly indicated by the variables preexisting cognition, parent rating of child’s self-regulation, and teacher rating of child’s self-regulation. Pre-kindergarten parent and teacher ratings would also be significantly indicated by items from previous research.

A second goal of the current study was to determine whether the latent antecedent, opportunity, and propensity factors were significantly associated with early mathematics skills in low-income children in the same manner as in prior studies grounded in the Opportunity–Propensity (O–P) Framework. The O–P Framework suggests that low-income pre-kindergarten children are more likely to acquire early mathematics competencies if (a) they are provided school-based opportunities to learn mathematics during pre-kindergarten (because they may not have these opportunities at home), (b) they have the motivation and capability to benefit from the opportunities provided to them, and, if early on, (c) antecedent factors lead to exposure to more opportunities that promote stronger propensities for learning (Byrnes & Wasik, 2009). Three hypotheses emerged from this second goal: (4) that the antecedent factor, as a latent variable, would have direct, indirect (through the opportunity and propensity factors), and combined effects on low-income children’s early mathematics skills, (5) that the propensity factor, as a latent variable, would have a direct effect on low- income children’s early mathematics skills, and (6) that the opportunity factor, as a latent variable, would have a direct effect on low-income children’s early mathematics skills. However, these predictions would only hold true if the indicators of all latent factors measured similar sorts of indices used in prior studies of older children. As we discuss below, the architects of ECLS-B did not include pretest scores of early mathematics skills, only post-test scores. As a result, we expected our findings to largely conform to those of prior studies, but the young age of children, low-income status of children, and the absence of the prior knowledge score might lead to somewhat different results than found in prior studies.

Section snippets

Participants

This study was based on secondary analysis of data from the restricted use Early Childhood Longitudinal Study-Birth (ECLS-B) database, the only large scale database that measures frequency of being exposed to mathematics content at pre-kindergarten and kindergarten for a nationally representative sample of 14,000 children born in 2001 (U.S. Department of Education (USDOE), National Center for Education Statistics (NCES), 2009). ECLS-B collected data about child development, school readiness,

Confirmatory factor analysis of the measurement model

In order to validate the multi-dimensional latent constructs of antecedent, opportunity, and propensity factors, first-order confirmatory factor analysis (CFA) was conducted for the measurement model of antecedent, and second-order CFA was conducted for the measurement models of opportunity and propensity. The evaluation of the factor loadings showed that the observed indicators had high factor loadings on their common factors, indicating that they adequately reflected their underlying latent

Discussion

The present study had two goals, (1) to extend beyond prior accounts of the O–P Framework by showing that the variables that were assumed to be in the same category (e.g., propensity factor) would form a single latent factor and (2) to determine whether the latent antecedent, opportunity, and propensity factors were significantly associated with early mathematics skills in low-income children in the same manner found in prior studies grounded in the Opportunity–Propensity (O–P) Framework.

Acknowledgments

This research was supported by a grant from the American Educational Research Association which receives funds for its “AERA Grants Program” from the National Science Foundation under NSF Grant #DRL-0941014. Opinions reflect those of the authors and do not necessarily reflect those of the granting agencies. Special thanks are extended to the AERA Grants Program Governing Board for their support of this study, Henry May for his expert suggestions on multiple imputation, and to Joshua Power and

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