The Role of Mediators in the Development of Longitudinal Mathematics Achievement Associations

Abstract

Despite research demonstrating a strong association between early and later mathematics achievement, few studies have investigated mediators of this association. Using longitudinal data (n=1362), we tested the extent to which mathematics self-concepts, school placement, executive functioning, and proficiency in fractions and division account for the association between mathematics achievement in first grade and at age 15. As hypothesized, a strong longitudinal association between first grade and adolescent mathematics achievement was present (β= .36) even after controlling for a host of background characteristics, including cognitive skills and reading ability. The mediators accounted for 39% of this association, with mathematics self-concept, gifted and talented placement, and knowledge of fractions and division, serving as significant mediators.

Duncan and colleagues (2007) analyzed 6 longitudinal studies in order to assess the predictive power of a number of school-entry measures of achievement and socio-emotional skills. They found that after controlling for a variety of prior child and family background characteristics, mathematics knowledge at school-entry appeared to be the strongest predictor of later mathematics achievement. This finding was robust across the 6 datasets. However, important questions regarding the underlying mechanisms that produced these predictive relations remained. These questions motivated the current paper’s mediational approach to modeling the association between early and later mathematics achievement. By pursuing a comprehensive investigation of a set of distinct mediators, we may better understand the specific pathways that contribute to stability over time of individual differences in mathematics knowledge.

Models of Mathematics Achievement

No current theoretical model incorporates the wide array of factors likely to contribute to long-run mathematics achievement. Consequently, most empirical investigations of mathematics development focus on one or two specific factors, such as motivation (e.g. Marsh et al., 1999), executive function (e.g. Blair & Razza, 2007), or math-specific skills (e.g. Siegler et al., 2012). Although such examinations provide information about specific correlates of mathematics achievement, more encompassing analyses are needed if we are to understand how a theoretically-diverse set of math-related factors act in concert. After all, as children develop mathematical skills, a set of cognitive, socio-emotional, motivational and environmental characteristics all operate simultaneously. Thus, a comprehensive analytic approach needs to incorporate a broad range of potentially important variables.

Our selection of possible mediators was guided by a broad bio-ecological perspective (Bronfenbrenner & Morris, 2006) that led us to model mathematics achievement as a function of both personal and environmental characteristics. Further, in order to design our empirical model, we set forth with a criteria that guided our selection of each mediator. First, candidate mediators needed to have a straightforward theoretical link to mathematics achievement. Second, this theoretical link needed to be supported by convincing empirical evidence. Third, the potential mediators must have been measured quantitatively in a manner that allowed for investigation in a longitudinal analysis. Fortunately, we were able to use a data set that provided timely measures of all but one of our desired set of mediators.

These criteria led us to simultaneously consider four distinct domains, each of which has been shown individually to influence mathematics achievement, but which have not been considered together. Based on the bio-ecological framework, we first examine characteristics and skills of the child and then move outward to the environmental context. We began with the child’s mathematical knowledge. Because mathematics is an unusually hierarchical area in which later knowledge incorporates and builds on earlier knowledge, we hypothesized that certain mathematics skills thought to act as gate-keepers to higher-level mathematics achievement would be important mediators between first-grade and age-15 mathematical skills. The second mediator consists of the domain-general cognitive skills covered by the executive functioning umbrella, as these intellectual processes have been found to be predictive of later mathematics achievement, above and beyond earlier mathematical knowledge (Fuhs et al., 2014).

The third potential mediator that we examined involved motivation. Our main index of motivation was self-concept regarding mathematics ability (on the logic that higher self-concepts regarding math ability both reflect and inspire greater motivation to exercise this ability). Math self-concept has been shown to be related to the development of mathematics achievement (e.g., Bong & Skaalvik, 2003). In particular, children who perceive themselves as good at mathematics typically fare better on subsequent exams.

Finally, moving from inside to outside the child, the fourth potential mediator that we examined was school and classroom environments. In particular, school placements into gifted and talented programs and special education have been linked to subsequent mathematics achievement (Bhatt, 2009; Morgan, Frisco, Farkas & Hibel, 2010), and operate by taking children out of their regular classrooms to provide specialized instruction in mathematics.

We hypothesized that each of these four factors could substantially mediate the association between early and later mathematics achievement. However, we recognize that these four potential mediators are not the only factors that influence the growth of mathematical knowledge. Quality of classroom instruction and teacher characteristics were among the variables we hoped to pursue, but empirical limitations did not allow us to investigate them. Although the present study is not fully-comprehensive, it does include a diverse set of mediators that encompass math-specific competencies, domain-general cognitive skills, motivation, and school-related structures. We know of no other study that has examined these four factors in one analysis, even though this comprehensive approach more closely resembles the complex nature of how these variables influence development.

Our empirical work extends previous longitudinal analyses (e.g., Claessens & Engel, 2013; Duncan et al., 2007; Watts, Duncan, Siegler, & Davis-Kean, 2014) by modeling age-15 mathematics achievement as a function of first-grade academic and cognitive skills, using data drawn from the National Institute of Child Health and Human Development (NICHD) Study of Early Childcare and Youth Development (SECCYD). We control for a host of important variables, all measured at or before first grade, including child temperament, socio-emotional skills, cognitive functioning, and family and child background characteristics. We then assess the extent to which the associations between early and later mathematics achievement can be accounted for by our mediational pathways. In the following sections, we briefly review literature on each mediator, and discuss implications of the study for theory and practice.

Background

Fractions and division knowledge

Because of the hierarchical structure of mathematical skill development, prior knowledge of specific mathematical competencies is crucial for the development of later mathematics achievement. Although a variety of mathematical skills have been implicated in the development of long-run mathematics achievement [e.g. math reasoning (Nunes et al., 2012); number system knowledge (Geary et al., 2013); numeracy skills (Jordan et al., 2009)], knowledge of fractions and division appear to be particularly crucial. Siegler and colleagues hypothesized that understanding fractions plays a central role in developing high school mathematics capacity (Siegler et al., 2012; Siegler, Thompson, & Schneider, 2011). If first-grade mathematics achievement is strongly associated with the development of fraction and division skills, and if these skills are a gateway to algebraic proficiency, then fractions and division knowledge should mediate the relation between first grade mathematics achievement and adolescent mathematical ability.

When students encounter fractions, they are forced to reorganize their understanding of the properties of numbers, from one that applies only to whole numbers to a more general understanding. Unlike whole numbers, fractions do not consistently increase when multiplied, do not consistently decrease when divided, do not have unique successors, and do not have unique symbolic representations. Gaining a broader understanding of what all real numbers have in common (e.g., that they can represent magnitudes) is crucial for success in high school mathematics (Martin, Struchens, & Elliott, 2007). Solving even simple algebraic equations (e.g. 2/3X = 8) is a formidable task if students do not understand the basic properties of fractions (e.g., that X will be larger than 8).

Siegler and colleagues’ (2012) examination of two longitudinal datasets revealed that fractions and division knowledge in fifth grade were uniquely predictive of high school algebra proficiency, even after accounting for domain general cognitive functioning, reading ability, and whole number skills. Although both division skills and fractions skills were found to be predictive, they are closely related concepts, as a fraction is a representation of a division problem (e.g. 3/4 can be thought of as 3 divided by 4). Further, fifth graders’ whole number addition, subtraction, and multiplication were less predictive of high school algebraic ability. Similar shorter-term relations between fraction knowledge and later mathematics learning have been reported elsewhere (Booth & Newton, 2012), and Bailey and colleagues (2014a) found that first-grade proficiency with whole numbers predicted eighth grade proficiency with fractions.

Executive function

Another mediator that could account for the longitudinal relation between early and later mathematics achievement is executive functioning (EF) -- higher-order cognitive and regulatory processes that involve working memory, inhibitory control, attention regulation, set shifting, and complex planning (e.g., Clark, Pritchard, & Woodward, 2010; Fuhs, Nesbitt, Farran, & Dong, 2014). If early mathematical proficiency leads students to practice EF skills, then we may detect an association between early mathematics achievement and growth in EF. Further, if these EF skills play a role in developing later mathematical skills, then EF should account for some portion of the association between early and later math achievement.

EF skills develop during early childhood and grow over the course of childhood and adolescence (Gathercole, Pickering, Ambridge, & Wearing, 2004). It is not difficult to imagine how EF could play a key mediational role in the long-term development of mathematical skills. For example, consider the cognitive processes required to solve a fractions addition problem that calls for derivation of the least common denominator. The student would need high inhibitory control to resist the temptation to treat the fractions as whole numbers and add numerators and denominators separately. The student would also need working memory and attention regulation skills to hold potential solutions in mind, while considering possible common denominators. Set shifting would be needed to alternate focus between the two denominators, and cognitive flexibility would be important to finding the lowest multiple common of the denominators. Finally, as students become more experienced solving such problems, they may develop more advanced strategies that require a certain amount of forethought and planning (e.g. prime factorization).

Indeed, research investigating concurrent associations between EF and mathematical skills have found fairly strong correlations (e.g. Blair & Razza, 2007; St. Clair-Thompson & Gathercole, 2006), but far less research has investigated the predictive relation between EF and subsequent measures of mathematics achievement. Recent studies suggest that such an association may exist. Clark, Pritchard, and Woodward (2010) found that a host of EF skills measured during preschool, including set shifting and inhibitory control, positively predicted mathematics achievement measured two years later. This result held after controlling for general cognitive ability and reading achievement.

Studies have also found longitudinal associations between early measures of working memory and math achievement measured in childhood (Mazzocco & Kover, 2007; Welsh et al., 2010) and adolescence (Watts et al., 2014). Similar findings have been reported when EF skills have been examined as one, latent construct. Most recently, Fuhs and colleagues’ (2014) used a single-factor design to measure a broad array of EF skills during preschool, including inhibitory control, working memory, and attention flexibility, finding that these competencies predicted mathematical skill growth during kindergarten.

For EF to qualify as a mediator of the association between early and later mathematics achievement, early mathematics ability must also lead to growth in subsequent EF. This might occur if those who excel academically from an early age seek new situations that further stimulate their cognitive growth (Fuhs et al., 2014). Consistent with this hypothesis, Welsh et al. (2010) found that academic skills at preschool entry were correlated with end-of-preschool working memory and attention control. However, Fuhs and colleagues (2014) found that although mathematical skills measured at the beginning of preschool predicted end-of-preschool EF, this relation did not hold when EF was measured at the end of kindergarten. These conflicting findings indicate a need for longer-term longitudinal studies of the relation between early mathematical skills and later EF.

Self-concept

Self-concept of ability (SCA), which describes an individual’s knowledge and perceptions regarding their academic capabilities (Bong & Skaalvik, 2003), has been hypothesized to influence students’ motivation and affect toward engaging in particular subjects (Marsh, Byrne, & Yeung, 1999). If a student with high mathematics achievement favorably compares herself with lower-achieving peers, she may begin to view herself as especially competent in mathematics. This perception could also arise through positive feedback from parents and teachers. In turn, this enhanced self-concept could lead her to seek situations in which her self-perception would be affirmed, such as through taking more advanced mathematics classes (Bong & Skaalvik, 2003). Thus, self-perceptions that reflect responses to early-grade achievement patterns in mathematics could account for the relation between early and later mathematics achievement if students internalize these perceptions which, in turn, alter students’ approach to mathematics in a way that either enhances or diminishes their acquisition of new mathematical skills (Marsh & Craven, 2006).

Findings from empirical research investigating the link between SCA and mathematics achievement have produced mixed results. It has been hypothesized that self-perceptions arise in domain-specific contexts (e.g., mathematics), as students evaluate their performance in specific subjects against their peers (Bong & Skaalvik, 2003; Marsh & Craven, 2006). Over time, self-perceptions across domains may even act in opposition to one another. Indeed, Marsh, Byrne, and Shavelson (1988) found that verbal and mathematical self-concepts of high school students were uncorrelated, and although mathematics achievement positively predicted subsequent mathematics SCA, it negatively predicted verbal SCA. Moreover, although the domain-specific link between mathematics achievement at time 1 and SCA at time 2 has been described in both high school (Marsh et al., 1999; Marsh & Yeung, 1997) and elementary school students (Skaalvik & Valas, 1999), it is unclear whether SCA continues to influence subsequent mathematics achievement. Multiple studies have failed to detect such an association in mathematics (Helmke & van Aken, 1995; Skaalvik & Valas, 1999), though other longitudinal investigations have found SCA to be a significant pathway to math achievement (Marsh & Yeung, 1997). Some have argued that the failures to detect a predictive link could be due to methodological shortcomings (Marsh et al., 1999), indicating a need for more rigorous empirical work investigating the longitudinal associations between self-concept and mathematics achievement.

School placement

Another possible mediator of relations between first grade and adolescent achievement is placement into special education or gifted and talented programs. This pathway would be supported if early achievement gains alter the chance of placement into gifted and talented programs or special education services, and if these school structures in turn boost or hinder subsequent mathematics achievement gains. Such relations could arise through several different mechanisms.

First, placement into special school programs could expose students to either higher or lower quality instruction. Some research has suggested that students placed into gifted and talented programs received higher-quality instruction and better curricula (Bui, Craig, & Imberman, 2011), though the effect of such placement on achievement was small. Similarly, some investigators have argued that students placed into special education programs may receive more empirically-supported and individualized instruction (Fuchs & Fuchs, 1998), but it is unknown if these techniques are consistently implemented across schools (Cook & Schirmer, 2003). Further, special education programs have been criticized for subjecting students to lower expectations (Horn & Tynan, 2001), and recently, there has been considerable pressure to move students with learning disabilities into “mainstream education” (Cook & Schirmer, 2003). School placements could also alter achievement trajectories through peer effects. To the extent that academically stronger peers improve the achievement of classmates (e.g. Imberman, Kugler, & Sacerdote, 2009), time spent in gifted and talented programs should support learning, and time spent in special education should impair it.

Some longitudinal evidence supports our hypothesis that placement into special academic programs mediates the relation between early and later mathematics achievement. Early-grade mathematics achievement predicts selection into both gifted and talented programs (McClain & Pfeiffer, 2012) and special education services (Hibel, Farkas, & Morgan, 2010). However, the effect of these programs on subsequent mathematics achievement has not been established.

Special education programs typically aim to provide students specialized services to help mitigate the impact of their disability (Morgan, Frisco, Farkas & Hibel, 2010). Some studies indicate that students with disabilities benefit from such services (Blackorby & Wagner, 1996; Hanushek, Kain, & Rivkin, 2002), but there is mounting evidence to the contrary. Morgan and colleagues’ (2010) analysis of nationally representative data from the Early Childhood Longitudinal Study-Kindergarten Cohort (ECLS-K) found that placement into special education services had a negative association with subsequent mathematics achievement in some, but not all, model specifications. Similarly, Lane, Wehby, Little, and Cooley (2005) found that children assigned to special education classes often had lower scores in both mathematics and reading at the end of the school year than they did before receiving special education services.

Studies investigating the effects of placement into gifted and talented programs have also produced mixed results. Vaughn, Feldhusen, and Asher’s (1991) meta-analysis of correlational studies examining the effect of “pull-out” programs, which periodically remove academically gifted students from their regular classes to receive enrichment, reported that such programs had a positive effect size of .65 sd (SE= .19) on student achievement. However, little research has taken selection bias into account, and strong experimental evidence remains virtually nonexistent.

Quasi-experimental studies have yielded conflicting results. Bui and colleagues (2011) employed a regression-discontinuity approach to examine the effect of gifted and talented placement on middle school students; they found no impact on mathematics achievement. In contrast, Bhatt (2009) used instrumental variables to approximate the causal impact of gifted and talented placement on math achievement among middle and high school students, and found a large effect of program placement on mathematics achievement. Thus, substantial questions remain regarding the impact of gifted and talented program placement on student achievement.

Current Study

The current study investigates the extent to which mathematics-SCA, EF, school placements, and fraction and division knowledge mediate the association between first grade and age-15 mathematics achievement. It adds to previous research in several ways. First, we draw on multi-site national data that include well-validated measures of achievement, allowing us to study achievement trajectories in unusual detail from first grade through the beginning of high school. Second, we employ an extensive set of control variables, including measures of socio-emotional functioning and problem behaviors, cognitive functioning, and individual and familial background characteristics. Inclusion of these control variables should help to reduce bias in the estimation of the unique contributions that each proposed mediator makes to the longitudinal mathematics achievement trajectories we investigate.

Finally, we simultaneously test multiple hypotheses regarding underlying mechanisms that could account for the robust association reported between early and later math skills. Instead of taking the more common piecemeal approach that considers each mediator individually, our analysis estimates the relative contribution of each of the mediators controlling for the others – a strategy that should reduce potential bias in the estimated role of each mediator. We expect that these mediators will account for a substantial portion of the association between first-grade and age-15 mathematics achievement.

Method

Data

Data for our study came from the NICHD SECCYD. Babies born in designated 24-hour periods in 1991 in 10 locations around the United States were eligible to participate if their mothers were over 18 years old, spoke English as a first language, did not have any serious health conditions, did not plan to relinquish parental rights, did not plan to move in the next three years, and resided within an hour of a study site. The study sample was diverse both economically and ethnically, although it was not nationally representative. The recruitment process suffered from nearly 50% attrition, which was concentrated among low-SES mothers and children. More detail on recruitment procedures, data collection, and study procedures is provided by the NICHD Early Child Care Research Network (NICHD ECCRN, 2002).

Reflecting the middle-class nature of the sample, the average family income-to-needs ratio measured repeatedly between birth and 1 month of age was 3.5. Further, 76% of the sample identified as Non-Hispanic White, 13% as African American, and study mothers reported an average of 14 years of education. More details regarding participant background can be found in the Supplementary Information file.

The current study used a subsample of children from the SECCYD (n = 1362). Among the 1364 children in the full sample, 2 had been in both special education and gifted and talented programs between second and fourth grades and were excluded from this study. To account for missing data, multiple imputation was used. All variables were included in the imputation models, and we also included a set of auxiliary variables to help ensure unbiased estimation. A total of 25 data sets were imputed using the ICE procedure in Stata 13.1. Stata MI commands were used for model estimation and statistical tests. Regression estimates were combined using Rubin’s rules of combination (Rubin, 1987).

Measures

Academic achievement in first grade and at age 15

Children were administered the Woodcock-Johnson (WJ-R) Psycho-Educational Battery-Revised (Woodcock, McGrew & Mather, 2001) as a measure of academic achievement at multiple times throughout the study. The current study focused on first-grade and age-15 scores on the Applied Problems subtest of the WJ-R. This commonly used measure of mathematics ability assessed knowledge of a wide range of mathematical topics and took about 15 minutes to complete. Students progressed through the exam until they incorrectly answered 6 consecutive questions. In first grade, most questions concerned principles of counting and simple addition and subtraction; at age 15, questions concerned algebraic concepts and procedures, among other topics.

We also included a measure of first-grade reading achievement in the analysis, as a covariate. Our measure of first-grade reading achievement involved questions taken from 3 components of the WJ-R: Word Attack, Letter-Word Identification, and Picture Vocabulary. Word Attack measured ability to read unfamiliar words using phonic and structural analysis; Letter-Word Identification asked students to identify and read aloud isolated letters and words; Picture Vocabulary measured recognition and naming of familiar objects.

For all subtests taken from the WJ-R, we used W scores, which were centered at a mean of 500. This value represents the average score for a fifth-grade student. We then standardized the W scores based on our sample’s mean and standard deviation. Testing procedures, reliability and validity of the WJ-R subtests are widely available.

Mediators

Our mediators were drawn from child assessments taken between second and sixth grade. Although we would have preferred for all mediators to have been measured in the same waves, the pattern of data collection in the study dictated that we draw measures from several different waves. Key mediators of the associations between first-grade and age-15 math achievement include fractions and division knowledge in fifth grade, EF in third and fourth grade, academic SCA in sixth grade, and school placements between second and fourth grade.

Knowledge of fractions and division

Our measure of division and fractions knowledge was taken from the WJ-R Calculation subtest, which was administered at fifth grade. The Calculation subtest assessed arithmetic and computational skills. We used the proportion of correct responses to items 16, 22, 25, 26, 27, 28, 29, 30, 31, 38 and 42 of the test to derive our measure of division and fractions ability. We excluded questions that involved using division or fractions to solve algebra problems, as we did not want our measure of division and fractions knowledge to overlap conceptually with our age-15 measure of mathematics achievement.

Executive functioning

EF was assessed across three separate measures in third and fourth grade. The first component of EF, working memory, was measured using the WJ-R Memory for Sentences task. The measure was administered in the lab during third grade; students were asked to listen to a tape player and repeat words, phrases and sentences. As with the first-grade WJ-R subtests, we used the W score for the Memory for Sentences measure.

Third graders also were presented the Tower of Hanoi (TOH) task as a broad assessment of their EF skills. Research has indicated that the TOH task assesses planning and problem solving, working memory functioning, and inhibition (see Bull, Espy, & Senn, 2004). Children were directed to plan a sequence of moves to transport a set of disks stacked on one of three pegs, ordered from smallest to largest, to another peg, without ever placing a larger disk on a smaller one. The TOH included six increasingly difficult problems. Students were given an efficiency score based on how many trials they needed to complete each task (Borys, Spits, & Dorans, 1982). The current analysis used the total number of moves across the six trials. The TOH is a well validated and commonly used measure of EF skills in developmental psychology.

To measure sustained attention and inhibitory control, 4^th graders were presented the Continuous Performance Task (CPT; Rosvold et al., 1956). The task involved viewing pictures of common objects on a computer screen and pressing a button when each time a target stimulus appeared. This task was repeated across 10 trials. We used the proportion of correct responses to the target stimulus as a measure of attention regulation, a common use of the CPT (e.g. Duncan et al., 2007).

Self-concept

Our SCA measures were taken from the “How I Do in School Scale,” administered in a lab during sixth grade. The SCA items were adapted from the Self and Task Perception Questionnaire (Jacobs et al., 2002); on them, students were asked to indicate their beliefs regarding their abilities in mathematics (5 items). Students responded to questions such as “How good at math are you?” using a 7-point Likert scale ranging from 1= “not at all” to 7= “very.” Students were also administered similar SCA items designed to measure English self-perceptions (5 items), which we used in order to test the domain-specificity of math SCA. The composite scores for math and English were calculated by taking the mean across the 5 items for each scale. The math and English self-concept scales have reported alphas of .82.

School placement

Selection into special education or gifted and talented programs was measured through a teacher report survey at grades 2, 3 and 4. Teachers were asked to report how many hours per week the child was currently receiving special education or gifted and talented services. Both the “gifted and talented” measure and the “special education” measure were coded 1 if the student received any such services in any of the three grades and 0 otherwise.

Additional covariates

To reduce bias in the estimated effects of first grade academic achievement on subsequent EF and age-15 achievement, we controlled for first grade EF, as measured by the WJ-R Memory Sentences test, the TOH, and the CPT. To account for bias in our estimates due to correlations between achievement and behavioral problems, we also controlled for first-grade measures of inattentiveness, aggressive behavior, and internalizing behavior, as assessed by the Teacher Report Form, an instrument based on the Child Behavior Checklist. We also included an assessment of child social skills, measured by teacher report on the Social Skills Rating System (Gresham & Elliot, 1990).

Controls taken prior to first grade included scores on the Surgency, Negative Affectivity, and Effortful Control subscales of the Child Behavior Questionnaire. These measures of temperament reflected mothers’ reports at the 54-month child interview. We controlled for early cognitive functioning using the Bracken Basic Concept Scale School Readiness score taken at 36 months, and the Bayley Mental Development Index taken at 24 months.

Child and family demographic characteristics

To account for child and family characteristics that could be correlated with achievement and our proposed mediators, we included a host of child and family background covariates. Child demographic controls included dummy variables for the sites at which the family was recruited to participate in the study; birth weight in grams; gender (1=boy); race (Non-Hispanic African American, Hispanic or other, with Non-Hispanic White as the reference group); and age in years at the time of first-grade testing. Family background characteristics included the mean number of children in the household between the child’s 24-month and first-grade assessments; the natural logarithm of family income averaged between the one month and first grade assessments; the education of the mother in years at the time of the one month assessment; and mother’s age at the birth of the target child.

Results

Descriptive statistics for key variables are shown in Table 1 (for descriptive statistics for all analysis variables, including controls, see Table A1 in the online Supporting Information file). Recall that our measures of student achievement were WJ-R W-scores centered at a value of 500 (the average score for a typically-achieving fifth grader). Participants’ growth between first grade and age 15 in both mathematics and reading is shown in Table 1. Not surprisingly, students scored well below 500 in first grade, and well above this score at age 15.

Table 1.

Descriptive Statistics for Primary Analysis Variables

	Mean	S.D.	Min	Max
Age 15 Achievement
Mathematics	524	16	444	591
First Grade Achievement
Mathematics	470	15	408	516
Reading	470	15	417	512
Mediators:
Math SCA (6th grade)	5.76	1.01	1	7
School Placements (2nd to 4th grade)
Special Education	.10	.30	0	1
Gifted and Talented	.15	.35	0	1
Executive Functions (3rd-4th grade)
Memory for Sentences	494	14	409	539
Tower of Hanoi	17	8	0	35
Continuous Performance Task	.95	.07	.36	1
Division and Fraction Knowledge (5th grade)	.44	.22	0	1
First Grade Executive Function Controls
Memory for Sentences	481	15	368	529
Tower of Hanoi	14	7	0	34
Continuous Performance Task	.95	.07	.41	1

Note. Descriptive statistics generated from 25 MI datasets (n=1362 each). Achievement measures, mathematics and reading, and Memory for Sentences (working memory) were drawn from Woodcock Johnson (WJ-R) W-scores. Math SCA (Self Concept of Ability) is scaled from 1-7, with a score of 7 indicating more confidence in one's ability. School placements were measured by the proportion of years between 2nd and 4th grade the student was placed in a special education or gifted and talented program. The Tower of Hanoi is considered to measure planning and a broad array of EF skills. The planning score indicates the total number of moves a child needed to complete the task across six trials. Continuous Performance Task is measured by calculating the proportion of correct responses to target stimuli, and it is understood to be a measure of attention regulation. Division and Fraction knowledge was calculated as a proportion of items correctly answered on the WJ-R Calculation subtest.

Table 1 also presents descriptive statistics for our proposed mediators. Participants scored rather high on the SCA scale for mathematics, as the average score on this scale (ranging from 1 to 7, with 7 indicating very high mathematics self-concept of ability) was 5.76 (SD= 1.01). Between second and fourth grade, 10% of our sample had been designated for special education services and 15% had been recommended for a gifted and talented program. Indeed, our data differ from current estimates of the percentage of students placed in such programs (13% for special education services; 7% for gifted and talented programs; Snyder & Dillow, 2013). Thus, our results should not be generalized to the national level, as our data are not nationally representative. Among our executive function measures, the average proportion of correct responses to target stimuli on the CPT task was high (M= .95), and participants in our sample who took the WJ-R working memory measure during third grade scored near the grade 5 average (M= 494; SD= 14). Finally, on average, participants correctly answered 44% of the fraction and division questions presented on the WJ-R Calculation subtest.

Zero-order correlations among the measures of academic achievement and the mediators taken between first grade and age 15, inclusive, are presented in Table 2 (correlations between key variables and all analysis variables can be found in the Supplementary Information). As might be expected, correlations among the various Woodcock-Johnson achievement subscales were the highest, ranging from .49 to .74. Among our mediators, fifth-grade fraction and division knowledge had the highest correlation with age-15 mathematics achievement (r (1362)= .54, p < .05), followed by proficiency on the Memory for Sentences task in third grade (r (1362)= .48, p < .05). Interestingly, mathematics SCA and English SCA were positively correlated (r (1362)= .28, p < .05); high self-perception in one domain did not predict a lower self-perception in the other. Finally, many mediators had moderate to high correlations with one another and with age-15 achievement. This demonstrates the bias-reducing utility of accounting for multiple intermediary pathways when engaging in mediational analyses.

Table 2.

Correlations Between Mathematics Achievement and Mediators

		(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)
Math (age 15)	(1)	1
Math (1st grade)	(2)	.59	1
Reading (1st grade)	(3)	.50	.60	1
Division and Fraction Skills (5th grade)	(4)	.54	.50	.47	1
Math Self Concept Ability (6th grade)	(5)	.34	.26	.11	.29	1
Special Education (2nd to 4th grades)	(6)	−.27	−.30	−.36	−.27	−.10	1
Gifted and Talented (2nd to 4th grades)	(7)	.26	.27	.29	.21	.06	−.14	1
Memory for Sentences (3rd grade)	(8)	.48	.50	.50	.32	.15	−.28	.22	1
Tower of Hanoi (3rd grade)	(9)	.29	.34	.23	.29	.18	−.19	.14	.25	1
Continuous Performance Task (4th grade)	(10)	.26	.25	.24	.22	.09	−.17	.10	.24	.22	1
Memory for Sentences (1st grade)	(11)	.45	.50	.50	.28	.12	−.23	.24	.74	.22	.18	1
Tower of Hanoi (1st grade)	(12)	.21	.25	.20	.18	.07	−.16	.12	.19	.38	.13	.18	1
Continuous Performance Task (1st grade)	(13)	.25	.21	.24	.20	.10	−.25	.10	.22	.18	.31	.14	.13	1

Note: All correlations presented in this table were obtained from 25 multiply imputed data sets and were statistically significant at .05 level.

Tables 3 and 4 summarize results from OLS and logistic regression analyses in which mediational hypotheses were tested in two steps. First, we tested the relation between each mediator and first grade mathematics achievement, controlling for other cognitive abilities and background characteristics. Next, we related age-15 mathematics achievement to first-grade mathematical skills and controls. We then introduced our set of mediators into the model. We present alternative analytic approaches, including results estimated via path analysis, in the Supplementary Information (Tables A2 & A3). We also describe attempts to estimate our data via SEM, and the analytical problems that led us to prefer the estimates presented here. In brief, we found no evidence that our EF measures should be combined as a single-latent factor, which would lead an SEM analysis to directly replicate the OLS results. Further, the SEM approach introduced problems with missing data, so we chose OLS as our preferred estimation strategy.

Table 3.

OLS/Multinomial Logistic Regression Estimates of the Association Between First Grade Mathematics Achievement and Analysis Mediators

Dependent variables	Coefficients and standard errors for 1st Grade Math
	B	s.e.
	OLS regression coefficients and standard errors
Self Concept
Math Self Concept Ability (SCA)	.29 ***	(.04)
Reading Self Concept Ability (SCA)	−.06	(.05)
Executive functions
Memory for Sentences	.10 *	(.05)
Tower of Hanoi	.19 ***	(.04)
Continuous Performance Task	.12	(.06)
Division and Fraction Knowledge	.31 ***	(.04)
	Multinomial logistic regression coefficients and standard errors
School Placements (MNL)
Special Education	−.54 *	(.22)
Gifted and Talented	.38 *	(.15)

Note. Dependent variables are listed on the left (rows). All coefficients listed were produced from fully controlled models. All continuous variables were standardized to z-scores using the means and standard deviations from each imputed data set (25 MI datasets; n=1362). The likelihood of placement into special education and gifted and talented programs was estimated using multinomial logistic regression, with placement into no program as the comparison group. Standard errors were obtained by robust estimator. Covariates included first grade measures of reading achievement, working memory, planning, sustained attention, externalizing, internalizing, and social skills; 54-month measures of surgency, negative affectivity, and effortful control; 36-month measure of school readiness (Bracken Basic Concept Scale), and 24-month measure of mental development (Bayley Mental Development Index); birth weight, ethnicity, gender, age at first grade test, number of children in the home, average family income between 1-month and first grade, mother's education level, mother's age at birth, and site.

^*p ≤ .05;

** p ≤ .01;

^***p ≤ .001

Table 4.

Regression Adjusted Estimates of the Associations Between First Grade Achievement and Age-15 Mathematics Achievement

	Model 1		Model 2		Model 3		Model 4
	Age 15 Math (baseline)		Age 15 Math (mediators)		Age 15 Math (mediators + div/frac knowledge)		Coefficient Reduction from Model 1- Model 3
	b	s.e.	b	s.e.	b	s.e.	b	s.e.
First Grade Achievement
Mathematics	.36 ***	(.05)	.27 ***	(.05)	.22 ***	(.05)	.13 ***	(.02)
Reading	.10 *	(.04)	.10 *	(.04)	.05	(.04)
Mediators:
Math Self Concept Ability			.20 ***	(.02)	.16 ***	(.02)
School placements (2nd-4th grade)
Special education			−.13	(.09)	−.10	(.09)
Gifted and talented			.20 **	(.07)	.17 *	(.07)
Executive Function (3rd-4th grade)
Memory for Sentences			.08	(.05)	.08	(.05)
Tower of Hanoi			.03	(.03)	.01	(.03)
Continuous Performance Task			.04	(.04)	.03	(.03)
Division and Fractions Knowledge (5th grade)					.21 ***	(.03)
First Grade Executive Function
Memory for Sentences	.10 *	(.04)	.05	(.06)	.07	(.06)
Tower of Hanoi	.03	(.03)	.01	(.03)	.01	(.03)
Continuous Performance Task	.08 *	(.04)	.04	(.03)	.04	(.03)
Covariates	Inc.		Inc,		Inc.

Note. Continuous variables were standardized to z-scores using the means and standard deviations from each imputed data set (25 MI datasets; n=1362). Standard errors were obtained by the robust estimator. All coefficients were generated from fully controlled models. Covariates included first grade measures of externalizing, internalizing, and social skills; 54-month measures of surgency, negative affectivity, and effortful control; 36-month measure of school readiness (Bracken Basic Concept Scale), and 24-month measure of mental development (Bayley Mental Development Index); birth weight, ethnicity, gender, age at first grade test, number of children in the home, average family income between 1-month and first grade, mother's education level, mother's age at birth, and site. Model 4 was obtained by using Seemingly Unrelated Estimation (suest).

^*p ≤ .05;

^**p ≤ .01;

^***p ≤ .001.

Predicting the Mediators

Table 3 displays regression results from a series of OLS and logistic regression models in which each mediator was used as the dependent variable and first-grade mathematics achievement was the key independent variable. The results shown are from fully-controlled models that included reading achievement, other first-grade characteristics, and measures of family background (see table note for full list of covariates). We used OLS regression to relate third-grade EF, fifth-grade fractions and division knowledge, and sixth-grade SCA to first-grade mathematics achievement and covariates. A corresponding multinomial logistic model regressed special education and gifted and talented placements on the same predictors.

First grade mathematics achievement was a significant predictor of each of the mediators (Table 3). First-grade mathematics achievement strongly predicted later math-SCA (β= .29, SE= .04, p < .001), and first-grade reading was not a significant predictor of math-SCA. Thus, we did not find evidence of an opposing relation between achievement in reading and math-SCA. As a further robustness check to examine the domain-specificity of SCA, we also related sixth-grade English SCA to first grade mathematics achievement and covariates. In results not shown in Table 3, mathematics achievement was not a significant predictor of later English SCA, but first-grade reading achievement was (β= .11, SE= .05, p < .01). These results suggest that children develop domain-specific SCAs, but in our data, high achievement in one school subject did not affect SCA in other areas.

First graders’ math knowledge strongly predicted EF skills, including performance on the Memory for Sentences task (β= .10, SE= .05, p < .05) and on the TOH (β= .19, SE= .04, p < .001). It negatively predicted special education placement (β= −.54, SE= .22, p < .05), and positively predicted gifted and talented placement (β= .38, SE= .15, p < .05), and fifth-grade fraction and division knowledge (β= .31, SE= .04, p < .001).

Predicting Age-15 Achievement

Regression results from a series of models estimating the association between first-grade mathematics and age-15 achievement are shown in Table 4. All models include the extensive set of controls listed at the bottom of the table. Coefficients on these covariates can be found in Table A4 of the Supplementary Information.

Results in Model 1 of Table 4 show that, after adjusting for our extensive set of baseline controls, a one standard deviation increase in first-grade mathematics was associated with a .36 sd (SE= .05, p < .001) increase in age-15 math achievement. First-grade measures of reading (β= .10, SE= .04, p < .05), Memory for Sentences (β= .10, SE= .04, p < .05), and CPT (β= .08, SE= .04, p < .05) were significant, but weaker, predictors of age-15 math achievement. These coefficients are remarkably similar to the averages reported in the six-study synthesis of school-readiness skills in Duncan et al. (2007), which examined the association between a similar set of predictors and late-elementary school achievement.

Model 2 displays results of age-15 mathematics achievement models that includes three of our four mediators: EF, SCA, and school placements. A comparison of coefficients on the early achievement measures between Models 1 and 2 showed that these mediators accounted for about 25% of the early-to-later mathematics associations and none of the early reading to later math associations. The .09 reduction in the math coefficient was statistically significant (p <.01). Two of these mediators had significant associations with math outcomes – math-SCA (β= .20, SE= .02, p < .001) and placement into gifted and talented programs (β= .20, SE= .07, p < .01).

To further test our hypothesis that SCA contributes to achievement through domain-specific channels, we also tested a mediational model (not shown in Table 4) in which age-15 mathematics achievement was related to first grade characteristics, all four mediators, and English-SCA. English SCA failed to produce a statistically significant coefficient, and including it did not diminish the coefficient produced by math-SCA (β= .20, SE= .02, p < .001).

Model 3 of Table 4 adds grade-5 fraction and division knowledge to the other three mediators and shows that it also significantly predicts later mathematics achievement. Moreover, when all of our mediators are considered in one model, 39% of the early-to-later mathematics effect is accounted for. This amounts to a .13 reduction in the first-grade mathematics achievement coefficient (p < .001; see Model 4).

Contrary to expectations, none of the three components of third or fourth grade EF was predictive of later math achievement. Note that our set of control variables includes first-grade EF. Substantial correlations (r’s = .25 −50) were present between third- and fourth-grade EF and age-15 mathematics score, but the corresponding correlations between first-grade EF and age-15 mathematics were almost as high (Table 2). When both were included in the regressions, the associations between age-15 mathematics and third- and fourth-grade EF were no stronger than the associations between age-15 mathematics and first-grade EF.

We also estimated models using an aggregated measure of EF, which averaged the three measures of EF into one variable. These models did not produce a substantially different mediational effect; third- and fourth-grade EF was still an insignificant predictor of age-15 mathematics. Models with the aggregated measure of EF are presented in Table A5 of the Supplementary Information.

Discussion

The current study tested for mediators of the strong relation between early and later mathematics achievement. We found that the relation was mediated by fractions and division knowledge, math self-concept, and placement into gifted and talented programs. Together, these mediators accounted for over one-third of the association between mathematics achievement in first grade and at age 15.

Our study was conducted within a bio-ecological framework (Bronfenbrenner & Morris, 2006), as we hypothesized that a combination of environmental and personal characteristics would shape long-run mathematics achievement trajectories. Indeed, we found evidence that both environmental (placement into gifted and talented programs) and personal (mathematics-SCA and fraction and division knowledge) factors influenced long-run achievement in mathematics.

Although cognitive skill-building theories have received much recent attention in regards to the development of academic achievement (e.g. Cunha & Heckman, 2008; Siegler et al., 2013), our results suggest that factors beyond mathematical skill attainment play a role in the development of mathematics achievement. Even when controlling for fraction and division knowledge in fifth grade, which appears to be crucial for later mathematics, mathematics-SCA and participation in gifted and talented programs were still strong predictors of later achievement. Thus, although earlier mathematical skills are critical for obtaining later ones, the influence of other factors, including motivation and school context, should not be overlooked.

Our findings regarding SCA supported our hypothesis that early mathematics achievement would lead to the development of positive SCA in mathematics, which would, in turn, support students’ later math achievement. It has been theorized that SCA may bolster subsequent achievement in mathematics by altering student motivation and mathematics-related affect, thereby increasing the likelihood of engaging in mathematics and spending time studying (e.g. Bong & Skaalvik, 2003). Although our results appear to support this hypothesis, we cannot rule out other avenues through which SCA may contribute to subsequent achievement (e.g. parental or teacher influence).

Our findings also give some indication of how self-perceptions in mathematics arise. The finding that early mathematics achievement predicted mathematics SCA in sixth grade, after controlling for other relevant variables, suggests that students assess their own mathematics abilities with some accuracy. We also found evidence for the domain-specificity of SCA, as early mathematics achievement predicted subsequent math-specific self-concepts, but not subsequent English self-perceptions. However, we found a positive correlation between mathematics SCA and English SCA in grade 6. Some have argued that SCA in math and English should have a null, or even negative, correlation (e.g. Marsh et al., 1988), as children may view themselves as “good” in one subject, but less equipped in others. Yet, mathematics and reading achievement are highly correlated, and if students only saw themselves as proficient in one subject, they would have an inaccurate perception of their abilities in the other subject.

Our findings suggest that this was not the case. First, we found that previous achievement in each domain best predicted later SCA. Second, previous achievement in mathematics, controlling for reading, had no relation to later reading SCA, and the same was true of early reading and later math-SCA. Thus, although we found evidence of domain specificity, we did not find an opposing relation between SCA in mathematics and English.

We also found that placement into gifted and talented programs positively predicted later achievement, even after controlling for a host of background variables and concurrent mediators. This finding suggests the value of such programs at a time that funding reductions threaten their continued existence (Fleming, 2013). Importantly, experimental research is needed to estimate the causal connections between participation in gifted and talented programs and student achievement, especially because selection into such programs could bias the associations reported by studies such as this one. Nonetheless, reducing or eliminating funding without evidence from experimental studies seems premature in light of the positive effects found here and in quasi-experimental work (e.g. Bhatt, 2009; but see Bui et al., 2011). Relatedly, future work should attempt to uncover the mechanisms through which such programs support achievement development, as the “gifted and talented” label undoubtedly describes a heterogeneous set of programs. Improving our understanding of which types of gifted and talented programs are most successful can further our ability to tailor services for these students.

Unsurprisingly, we found that early mathematics achievement negatively predicted placement into special education. Indeed, schools have increasingly relied upon measures of early academic skills for diagnosing students at risk for underachievement, as response to intervention (RTI) and multi-tiered systems of instruction have become prevalent across the country (see Fuchs, Fuchs, & Stecker, 2010). Our results provide some support for this practice, as first-grade mathematics ability strongly predicted high school mathematics achievement. However, our study did not find any relation between placement into special education services and later achievement. This could be seen as evidence supporting efforts to “mainstream” children out of special education programs, but null-effects on student achievement in a correlational model could also be a sign of the programs’ successes. Depending on each student’s learning disadvantages, special education programs may be effective if they merely prevent students from falling further behind. Thus, estimates from correlational studies could contain downward bias, as students placed into special education are more likely to experience negative growth in achievement compared with other students. If this were the case, and special education programs did nothing to curb this trend, then we would expect a negative association between placement into services and later achievement. As with gifted and talented programs, this issue further illuminates the strong need for experimental research in order to draw clear conclusions regarding the effectiveness of these services.

Our findings regarding the lack of mediation for the EF measures converge with recent experimental research. Taken together, this work indicates that attempting to change specific aspects of EF with the hope of affecting mathematics achievement may not be a useful strategy. Experimental evaluations of Tools of the Mind, a curriculum meant to boost young children’s EF, found that the curriculum failed to change EF skills or mathematics achievement (Clements, et al., under review; Wilson & Farran, 2012). Clements and colleagues did find some indication that experimentally-induced boosts in early mathematics achievement may cause later improvements in EF. Although the possibility of boosting EF through early mathematics intervention may be theoretically interesting, most educational interventions hope to change academic skills. Our results imply that improved EF skills do not readily transfer to improved mathematics ability.

The present results also suggest that efforts to improve math achievement through EF interventions in later elementary school would probably have little effect. Our results likely represent upper-bound estimates of the causal effect of each of our respective EF measures on later achievement, as any bias due to omitted variables is likely positive; plausible confounding variables (e.g. parental support for cognitive development, interest in mathematics) are probably positively correlated with the EF measures and adolescent mathematics achievement. However, measurement error could bias the effect of EF downward, and if EF describes a single latent construct, there is little reason to expect that our three measures perfectly captured it. Nonetheless, we employed well-validated measures to assess various aspects of EF (e.g. Bull et al., 2004; Duncan et al., 2007; Watts et al., 2014). Although the Tower of Hanoi has been criticized for its poor test-retest reliability (estimates range from .50 to .75; see Beck, Schaefer, Pang, & Carlson, 2011), there was no indication that the other two, more reliable, measures of working memory (Memory for Sentences) and sustained attention (CPT) positively affected later mathematics achievement.

On the other hand, our findings illustrate the importance of domain-specific knowledge in the development of long-term mathematics achievement, as knowledge of fractions and division substantially mediated long-run mathematics achievement associations. This finding converges with results from prior studies (e.g. Beier & Ackerman, 2005) that have demonstrated the relative importance of content knowledge over domain-general cognitive competencies. Thus, finding crucial areas of content knowledge that strongly relate to subsequent mathematics achievement may provide a fruitful means through which educational practitioners can intervene to improve long-term achievement trajectories. Indeed, practitioners and researchers have designed interventions and instructional practices that improve content knowledge of fractions in late elementary and early middle school and that transfer to overall mathematics achievement (for a review see Siegler, Fazio, Bailey, & Zhou, 2013).

Although our set of mediators had some success in accounting for the association between first-grade and age-15 mathematics achievement, the majority of this association remained unexplained in our models. We certainly did not consider every possible mediator, as we were limited by the measures included in the data. In particular, classroom instructional practices and classroom climate were not measured in our study. Although certain instructional practices, such as emphasizing advanced content (Claessens, Engel, & Curran, 2014; Engel, Claessens, & Finch, 2013), have been linked to elementary school mathematics gains, any mediational analyses covering a time-span of multiple years would need to consider instruction in each year. Further complicating matters, many students have multiple teachers in any given year. Unfortunately, analyses of such factors were not possible with our data.

It could also be argued that the current study was limited by the narrow time window during which each mediator was measured. With the exception of school placements, our mediators were only assessed at one point between first grade and age 15, and the mediators were not assessed at the same time. Future research should examine mediators measured at multiple time points in order to understand the longitudinal pathways that may be occurring. At the same time, previous research has indicated that individual differences in both SCA (Davis-Kean et al., 2008) and EF (Mazzocco & Kover, 2007) are quite stable throughout elementary school. Thus, although we cannot rule out the possibility, we doubt that our results would differ substantially if our measurements of these constructs had been obtained in multiple grades.

Additionally, our measures of school placements were taken over the course of three years, yet we only measured whether a student was ever placed in a special education or gifted and talented program during this time span. Future research should investigate possible dosage effects of school placements on mathematics achievement, as spending more time in one of these programs could have a compounding effect on mathematical development.

Our inability to fully account for the early- to later-mathematics association may also indicate that some of this association is not causal. Indeed, recent work by Bailey, Watts, Littlefield, and Geary (2014b) suggests that even highly controlled longitudinal examinations of mathematics achievement fail to account for all of the underlying personal and environmental characteristics that contribute to achievement trajectories. Consequently, such analyses may overestimate the causal effect of early achievement on later achievement. Moreover, recent experimental work has raised questions regarding the causal association between early and later measures of achievement. Clements and colleagues (2013) found that a preschool mathematics intervention produced a large impact on end-of-preschool mathematics achievement, but this effect faded by over 60% by the end of first grade. Thus, although highly controlled, correlational studies (e.g. Duncan et al., 2007; Watts et al., 2014) have found strong associations between early and later measures of achievement, it remains unclear whether intervention-produced gains in early knowledge produce similar effects.

A final implication of these findings is that experimental studies with longitudinal follow-ups are greatly needed. We found that individual differences in mathematics achievement between first grade and high school are quite stable, and we identified several processes, acting at both the personal and environmental levels, that contribute to stability and change in mathematics achievement. Although our models shed some light on how these long-run trajectories develop, only experimental research can determine whether these trajectories can be meaningfully altered by interventions. Our results may indicate possible areas in which future experimental work could be conducted. However, our study illustrates that a number of processes (e.g. self-concept, school placements, specific math skills) appear to simultaneously influence mathematics achievement development. Thus, interventions should be conducted with realistic expectations, as they are likely to be unsuccessful if they fail to take into account the diverse processes involved in children’s mathematical development and in their cognitive development more generally.