

How Important Is Where You Start? Early Mathematics Knowledge and Later School Successby Amy Claessens & Mimi Engel  2013 Background: Children’s early skills are essential for their later success in school. Recent evidence highlights the importance of early mathematics, relative to reading and socioemotional skills, for elementary school achievement. Key advocacy groups for both early childhood and mathematics education have issued position statements on the importance of early mathematics, arguing that mathematics education for 3 to 6 year olds is essential to promoting future mathematics achievement. Focus of Study: Despite the fact that advocates and researchers are focusing on early math skills, we are still learning about the mathematics knowledge and skills young children typically have and how these early skills affect later academic achievement and school success. This study aims to address these gaps in the extant research by investigating how early math skills predict later school success. We explore how early math skills relate to achievement, from kindergarten through eighth grade, across reading, math, and science test score outcomes, as well as grade retention and teacherreported academic achievement. We also explore whether there is variation in the relationship between early math skills and later outcomes for children who enter school with limited math skills. Research Design: We conduct secondary analysis with data from the Early Childhood Longitudinal Study–Kindergarten Cohort, a longitudinal, nationally representative sample of children who were in kindergarten in 19981999 and were followed through eighth grade. Results: We find that early math skills predict reading, math, and science achievement as well as grade retention from kindergarten through eighth grade. Results show that kindergarten math skills in pattern recognition, measurement, and advanced number are most predictive of eighthgrade outcomes overall and for subgroups including students who enter school with low math skills. The importance of these math skills for subsequent achievement increases or is maintained over time. Conclusions: The results reported here have implications for education policy regarding mathematics instruction in the earliest years of schooling. The fact that early mathematics knowledge and skills are the most important predictors not only for later math achievement but also for achievement in other content areas and grade retention supports a greater emphasis on mathematics than is currently the case in many kindergarten classrooms. It also suggest the possibility that focusing more on advanced number, pattern recognition, and measurement might develop skills that will benefit students in the later years of schooling. Childrens early skills and knowledge are essential for their later success in school. Recently, key advocacy groups for both early childhood and mathematics educationthe National Association for the Education of Young Children (NAEYC) and the National Council of Teachers of Mathematics (NCTM)have issued position statements on the importance of early mathematics (NAEYC, 2002; NCTM, 2007). Further, the National Mathematics Advisory Panel (NMAP), formed in 2006, focused on mathematics learning in Grades PreK8 (NMAP, 2008), emphasizing the advantage that children have if they receive a highquality mathematics education in the earliest years of school. The National Research Council (NRC) also issued a set of recommendations in 2009 for early childhood mathematics teaching and learning. All four groups issued reports arguing that mathematics education for 3 to 6yearolds is essential to promoting future mathematics achievement and that very young children are ready to learn a broad array of mathematics content. Researchers in both psychology and education have investigated the importance of early childhood mathematics education (Baroody, 2003; Clements & Sarama, 2004, 2009, 2011; Ginsburg & Amit, 2008). Further, two recent studies highlight the predictive power of general schoolentry mathematics knowledge, relative to other schoolentry skills, for elementary school achievement (Authors, 2009; Authors, 2007). Despite the fact that advocates and researchers are focusing on early math skills, we are still learning about the mathematics knowledge and skills young children typically have and how these early skills affect later academic achievement and school success. This study aims to address these gaps in the extant research through secondary data analysis that investigates how early math skills predict later school success using a nationally representative sample of kindergartners. We explore how early math skills relate to achievement from kindergarten through eighth grade, across reading, math, and science test score outcomes, as well as grade retention and teacherreported academic achievement. We also explore whether there is variation in the relationship between early math skills and later outcomes for children who enter school with limited math skills. EARLY SKILLS AND LATER SCHOOL SUCCESS A growing body of research highlights the importance of childrens early skills for later school and life success. Childrens early skills are linked to subsequent success because they provide the foundation for more advanced skills (Entwisle, Alexander, & Olson, 2005; Cunha, Heckman, Lochner, & Masterov, 2006. A childs predispositions, knowledge, and skills contribute to his own learning and to the environment in which he operates; in turn, the child receives feedback from others in the environment (Meisels, 1998). This complex interaction between the child and his environment affects his developmental trajectory (Bronfenbrenner & Ceci, 1994; Bronfenbrenner & Morris, 1998). For example, a child who enters kindergarten with high levels of mathematics skills may build upon this knowledge with additional instruction, receive reinforcement from the teacher, or be placed in a higherability group. Thus, childrens schoolentry skills and abilities have the potential to shape their development both within and across domains throughout their schooling. MATHEMATICS AT SCHOOL ENTRY Children enter school with varied levels of early math skills. Some children have very high levels of early math achievement, while others enter with more limited skills (Baroody, 1987; Bodovski & Farkas, 2007; Brannon & Van de Walle, 2001; Clements, 2004; Clements & Sarama, 2011; Crosnoe et al., 2010; Huttenlocher, Jordan, & Levine, 1994; Jordan & Levine, 2009; Levine, Jordan, & Huttenlocher, 1992; Mix, Huttenlocher, & Levine, 2002; Morgan, Farkas, & Wu , 2009; Wynn, 1990). General math achievement measured around kindergarten entry has been found to be highly predictive of subsequent mathematics achievement, measured around third grade (Aubrey, Dahl, & Godfrey, 2006; Authors, 2007, 2009; Jordan, Kaplan, Ramineni, & Locuniak, 2009; Stevenson & Newman, 1986). However, little is known about how predictive early math skills are of middle school success or which sets of early math skills are predictive of later math outcomes. Two recent studies link schoolentry math skills to elementary school achievement in both math and reading (Authors, 2007, 2009), finding that schoolentry math skills predict later math and reading achievement, measured by both test scores and teacher reports. Both studies found that early math skills, measured by tests of general mathematics abilities, were more important for later math and reading achievement than were schoolentry reading skills. However, these studies focused only on reading and math outcomes and did not follow children beyond fifth grade. SPECIFIC EARLY MATHEMATICS SKILLS Research focused specifically on early math skills shows continuity in math achievement over time. Number sense and counting measured in preschool or kindergarten predict later elementary school math achievement test scores (Aunola, Leskinen, Lerkkanen, & Nurmi, 2004; Bodovski & Farkas, 2007; Jordan et al., 2009; Jordan, Kaplan, Locuniak, , & Ramineni, 2007). Children who have difficulty counting tend to have later difficulties in math (Clements & Sarama, 2011; Geary, Hoard, & Hamson,1999). These foundational math skills are important for later levels and gains in mathematics in elementary school (Aunola et al., 2004; Jordan et al., 2007, 2009; Jordan, Kaplan, Olah, & Locuniak, 2006). Foundational concepts, such as knowledge of numbers, allow for deeper understanding of more complex mathematical problems and flexible problemsolving techniques (Baroody, 2003; Clements & Sarama, 2011; Ferrari & Sternberg, 1998; Fuchs et al., 2010; Hiebert & Wearne, 1996; NRC, 2009). Taken together, these findings suggest that focusing on number sense and counting could be an important avenue for improving later achievement. Over the course of early and middle childhood, childrens conventional mathematics skills develop, and these skills are likely essential for the development of later, increasingly more complex mathematical skills. Conventional mathematical skills include the foundational skills of number such as counting, number facts, and baseten structure and more advanced skills such as calculations and measurement (Mix et al., 2002). Mathematical concepts and mathematical conventions are interrelated, but the direction of the relationship between them is the subject of much debate (Clements & Sarama, 2004; Gelman, 1972; Mix et al., 2002; NRC, 2009). For example, children acquire the words of counting almost from the time they begin to speak but before they understand onetoone correspondence, a conventional mathematics skill. At the same time, conventional units are key for a childs ability to represent quantity, and learning these math conventions might expand childrens understanding of measurement. DIFFERENCES ACROSS GROUPS A large body of research documents race and income gaps in schoolentry academic skills, including math skills (e.g. Bodovski & Farkas, 2007; Fryer & Levitt, 2004; Magnuson & Duncan, 2006; Magnuson, Ruhm, & Waldfogel, 2007; Phillips, Crouse, & Ralph, 1998). Lowincome and minority children are at particular risk for underachievement in mathematics (Clements & Sarama, 2011; Jordan et al., 2007, 2009; Jordan & Levine, 2009; Siegler, 2009; Siegler & Ramani, 2008; Starkey, Klein, & Wakeley, 2004). Children from disadvantaged backgrounds lag behind their more advantaged peers on several measures of early mathematics skills (Griffin & Case, 1997; Jordan, Huttenlocher, & Levine, 1992; Jordan & Levine, 2009; Saxe et al., 1987; Starkey & Klein, 1992) and have more varied mathematics skills than do middleincome children (Wright, 1991). Evidence suggests that all children learn basic counting skills, although at different rates, but that there are wide gaps by income in more advanced early mathematics skills, such as sequencing and comparison of shapes (Clements & Sarama, 2004; Sarama & Clements, 2008). A related question is whether the predictive power of early math skills for later outcomes varies across different groups of children. We examine whether math skills are more important predictors of later outcomes for students who begin kindergarten with low math skills. THE PRESENT STUDY Using a largescale, nationally representative longitudinal study of kindergarteners, we examine the relationship between measures of schoolentry mathematics skill sets and subsequent success in school. We focus on how early math skills relate to reading, math, and science achievement as well as grade retention through elementary and middle school; which aspects of early mathematics knowledge are most important for later success; and how this varies across children at risk of school failure. We expect that early math achievement will be highly predictive of middle school reading and math achievement, as found in prior research that examined earlier outcomes (Authors, 2007, 2009). We also expect that early math skills will strongly predict other measures of school success, including science achievement and grade retention. We expect that particular aspects of early math knowledge will be critical to later math achievement, as found in prior studies (Aunola et al., 2004; Bodovski & Farkas, 2007; Jordan et al., 2006, 2007, 2009), as well as to other measures of school success. We contribute to this body of research by using a nationally representative sample of children with measures of school success through eighth grade, by focusing on a range of both early mathematics proficiency areas and later school outcomes, and by examining whether this relationship differs for children with low math skills at kindergarten entry. DATA The data used in this study come from the Early Childhood Longitudinal StudyKindergarten (ECLSK) cohort. The ECLSK is a nationally representative sample of children followed from the fall of kindergarten through eighth grade. Designed to focus on early school experiences, the study sampled children who were in kindergarten in the 199899 school year. For the current study, we use longitudinal measures from the fall and spring of kindergarten and spring of first, third, fifth, and eighth grades. Data were collected from multiple sources, including direct cognitive assessments of children in math, reading, and, beginning in third grade, science; interviews with parents; and teacher surveys. The sample used in the present study includes the 7,655 children for whom there is complete data on both fallofkindergarten and springofeighthgrade achievement tests and valid longitudinal weights. To retain as large a sample as possible, we used multiple imputation for cases with complete information on all dependent and key independent variables. For missing data on covariates such as household income, parental expectations, and parental education on the baseline survey, we followed von Hippels (2007) recommendation and multiply imputed these variables. Thus, missing values on covariates, but not on the dependent variables (achievement test scores, teacher reports, grade retention) or independent variables of interest (achievement test scores), were imputed. Multiple imputation was conducted using the ICE command in Stata 11.0 (Lunt, 2009; Royston, 2004; StataCorp, 2009). Following conventional guidelines (McCartney, Burchinal, & Bub, 2006), five imputed datasets were generated. MEASURES ACHIEVEMENT TESTS Children were given achievement tests in a range of subjects from kindergarten through eighth grade. They were tested in general knowledge in the fall and spring of kindergarten and the spring of first grade. Children were tested in language and literacy (reading) and math in the fall and spring of kindergarten and in reading and math in the spring of first, third, fifth, and eighth grades. They were tested in science during third, fifth, and eighth grades. All of the ECLSK assessments were designed using Item Response Theory (IRT) to allow for the examination of growth over time (Tourangeau et al., 2009). The ECLSK created and provides proficiency probability scores for groups of test items in both reading and math achievement. Ten proficiency levels in reading and nine in math were created to span achievement from kindergarten through eighth grade. We use the measures of proficiency probability in mathematics at the fall of kindergarten to examine mastery rates both overall and within population subgroups (Tourangeau et al., 2009). Math Test The ECLSK math assessment measured childrens conceptual and procedural knowledge and problem solving skills, and the IRTbased scores range in reliability from .91 to .95 (Tourangeau et al., 2009). The ECLSK created and makes available five proficiency areas in math, measured at kindergarten entry. These include: identifying some onedigit numerals, recognizing geometric shapes, and onetoone counting up to 10 objects (Proficiency Level 1); reading all onedigit numerals, counting beyond 10, recognizing a sequence of patterns, and using nonstandard units of length to compare objects (Proficiency Level 2); reading twodigit numerals, recognizing the next number in a sequence, identifying the ordinal position of an object, and solving a simple word problem (Proficiency Level 3); solving simple addition and subtraction problems (Proficiency Level 4); and solving multiplication and division problems and recognizing more complex number patterns (Proficiency Level 5). These Proficiency Levels are measured using proficiency probability scores, variables included in the ECLSK dataset, which range from 0 to 1. In kindergarten, scores are available for Proficiency Levels 15. However, because less than five percent of the analytic sample is proficient, according to the ECLSK measure, on Proficiency Level 4, and virtually no students are proficient at Proficiency Level 5 at kindergarten entry (see Table 1), we include only levels 13 in the analyses that follow. The reliability for the fall kindergarten proficiency probability scores is .91 (Tourangeau et al., 2009). Language and Literacy Test (Reading) The language and literacy test measured childrens basic skills, vocabulary, and comprehension (Tourangeau et al., 2001), and the reliability of the IRTbased scores ranges from .87.96 (Tourangeau et al., 2009). The assessments from kindergarten to eighth grade cover a total of 10 different Proficiency Levels: identifying letters, identifying beginning sounds and ending sounds, reading words by sight and in context, literal inference, extrapolation, evaluation, evaluating nonfiction, and evaluating complex syntax. Science Test Students were given a direct assessment of science in third, fifth, and eighth grades. The science test placed equal emphasis on life science, earth and space science, and physical science. Children were asked to demonstrate understanding of the physical and natural world, draw inferences, and comprehend relationships. In addition, they had to interpret scientific data, formulate hypotheses, and identify the best plan to investigate a given question. Reliabilities range from .84.87. The science test does not include proficiency levels. General Knowledge Test The test, administered in fall and spring of kindergarten and spring of first grade, assessed knowledge of science and social studies and evaluated childrens conception and understanding of the social, physical, and natural world and their ability to draw inferences and comprehend implications. It also measured childrens skills in establishing relationships between and among objects, events, or people and in making inferences and comprehending the implications of verbal and pictorial concepts. The reliability of the general knowledge test ranges from .85.88. We use childrens fallofkindergarten scores on the general knowledge test as a control for cognitive ability in our models (Authors, 2007, 2009). OTHER SCHOOL OUTCOMES In addition to achievement test scores, this study also explores measures of school success, which include retention in grade and teacherrated achievement. For retention in grade, the ECLSK provides a measure of the childs current grade level for each data collection wave. We created measures of retention for different points in time across elementary and middle school. Surveyed teachers reported on sample childrens achievement using the Academic Rating Scale (ARS). The ARS asks teachers to rate the childs skills, knowledge, and behaviors within two areas of academic learning: (1) language and literacy and (2) mathematical thinking in kindergarten through fifth grade. ARS questions use examples to help the teacher think of the range of situations in which a specific child may demonstrate similar skills and behaviors. Each questionnaire assesses skills, knowledge, and behaviors that are appropriate for each grade level in school. The reported reliabilities for the language and literacy scales are .94.95; they are .91.92 for mathematical thinking. The ARS is not used in eighth grade, but different measures of teacherrated school performance were collected. Eighthgradeteacher ratings of a students written expression skills, oral expression skills, math skills, and science skills with testretest reliabilities of .93.96 are used. CHILD CHARACTERISTICS Child characteristics that are likely correlated with initial and later measures of mathematics skills and achievement are included in this analysis. We use measures of the sample childs sex and race/ethnicity as well as information about child health, birth weight, birth date, and whether or not the child was premature. Parents also reported on their childs preschool and childcare experiences. This information was collected in the fall of kindergarten. HOME AND FAMILY BACKGROUND This study also includes a wide range of family and home characteristics that might influence a childs initial skill levels and subsequent outcomes. These measures include maternal and paternal demographic characteristics such as age, education, work status, and income. In addition, we account for measures of home language, immigrant status, and poverty as well as a measure of family structure, which captures parental marital status and number of siblings. The ECLSK also surveyed parents about the number of books in the home and the frequency with which they engage in reading and other activities with the child. A complete list of all of the control variables we include in our analyses is included in Table 1 and Appendix A. ANALYTIC PLAN To answer our primary research question, we relate math proficiency probability scores measured at kindergarten entry to childrens achievement test scores and other measures of school success measured in eighth grade (and at other points throughout elementary school). Schoolentry skills are measured in the fall of kindergarten (FK), while outcomes are measured in the spring of eighth grade (8th). The estimating equation is as follows: (1) SO_{i8th} = a_{1} + ß_{1}Math1_{iFK} + ß_{2}Math2_{iFK} + ß_{3}Math3_{iFK} +ß_{4}Reading_{iFK} + ß_{5 }+ GenKnow_{iFK} + ß_{6} Fam_{i} + ß_{7} Child_{i} + e_{i} where SO_{i8th} is the school outcome measure (e.g. math, grade retention) of child i in the spring of eighth grade; Math1_{IFK}, Math2_{IFK}, and Math3_{IFK} are the math proficiency probability scores 13 for child i at kindergarten entry; and Reading_{iFK} and GenKnow_{iFK} are measures of child is reading and general knowledge, assessed by achievement tests in the fall of kindergarten. We include the last two measures to control for initial reading skills and cognitive ability. Fam_{i} and Child_{i} are sets of family background and child characteristics included to control for individual differences that might influence math achievement before and after school entry; a_{1} is a constant and e_{i} is a stochastic error term. Our interest is in estimating ß_{1}, ß_{2}, _{}and ß_{3,} which, if correctly modeled, can be interpreted as the relationship between particular math skill sets at kindergarten entry and subsequent school outcomes. A key challenge in this approach is ensuring that we have accounted, as much as possible, for the possibility of omitted variable bias, which occurs if family or child characteristics are correlated both with childrens early math skills and their later achievement and are omitted from our model. Our strategy for reducing bias in ß_{1}ß_{3} is to estimate a model of the form of equation (1) that includes as many prior measures of relevant child and family characteristics as possible. The ECLSK data include multiple students per classroom at the fall of kindergarten. We account for this nonindependence or clustering in all of our models. Further, all results are weighted using the appropriate longitudinal weights. RESULTS Table 1 shows descriptive statistics for the background characteristics and kindergarten achievement test scores for both the main analytic sample (N=7,154) and the subgroup of children who were lowachieving in mathematics at the fall of kindergarten (n=2,329). This subgroup is defined as children who have not mastered Proficiency Level 2 at the fall of kindergarten. As shown in Table 1, column 1, children in the main analysis sample have an average score of .95 (out of 1) on Proficiency Level 1identifying some onedigit numerals, recognizing shapes, and onetoone counting 10 objectsat kindergarten entry. Even among the subsample of students who are lower performing in mathematics, the average score in fall of kindergarten on Proficiency Level 1 is .86 out of 1. Students in the analysis sample had an average score of .62 on Proficiency Level 2 and .26 on Proficiency Level 3. Further, as can be seen in the table, students in the analytic sample who were lower achieving in mathematics (column 2) had lower scores on Proficiency Level 2, with an average score of .23. For analyses with this subgroup, we use only Math Proficiency Levels 1 and 2, as there is essentially no variation for these students on Proficiency Level 3, with an average score of only .01 and a standard deviation of .02.
The analytic sample is a little more than twothirds white (69%), and 73% of the children lived in twoparent married families in the fall of kindergarten. Comparing the main analytic sample and the subgroup of children with low math achievement in the fall of kindergarten (columns 1 and 2), we see that the lowerachieving children are more likely to be living with a single parent, on average. Children with low math skills are more likely to be Black or Hispanic (nearly 40%), compared with 24% of the full analysis sample. Table 2 presents descriptive statistics for the outcomes of interest for the primary analytic sample and for children with low mathematics skills. Achievement test scores increase over time for students in both groups, as we would expect. Unlike achievement test scores, teacherrated achievement does not increase monotonically over time. Rates of grade retention increase over time. While only three percent of the sample had been retained by first grade, 13% had been retained by eighth grade. The sample with low math achievement at the fall of kindergarten consistently scores lower than does the full analytic sample on all outcomes of interest. These students are also about twice as likely to have experienced grade retention in any given year, with 26% having been retained by eighth grade.
Table 3 presents results for the full analytic sample using fallofkindergarten math proficiency probability scores to predict eighthgrade achievement test scores in reading, mathematics, and science. Results indicate that fallofkindergarten math skills are important for eighthgrade test scores. The first column for each dependent variable (columns (1), (4), and (7)) shows the bivariate relationship between each of the three math proficiency scores and that outcome. As shown in Table 3, all three fallofkindergarten math proficiency scores are highly correlated with subsequent achievement test scores, with coefficients ranging from .46.65.
Table 3 also provides estimates of two additional specifications of the relationship between early mathematics skills and eighthgrade reading, math, and science test scores. All variables have been standardized by fullsample standard deviations so that coefficients can be compared across models and interpreted in standard deviation increments. Columns (2) and (5) present results from models including math proficiency probability scores and the language and literacy (reading) test score without additional controls. As shown in these models, math proficiency probability scores remain predictive of later reading (column 2) and math (column 5) test scores when included in the same model, although the magnitude of the coefficients drops substantially, highlighting the correlation among these test scores. Interestingly, these results indicate that kindergarten entry math scores on Proficiency Levels 1 and 2 are more predictive of math and reading achievement in eighth grade than are the reading test score at kindergarten entry. Columns (3), (6), and (8) present models that add child and family background control variables and the General Knowledge test score (see Appendix A for a complete list of control variables included in these models). Results indicate that kindergarten entry math skills are more predictive than are early reading skills for all three achievement test score outcomes. Coefficient estimates on math proficiency scores range in size from .0220 for reading test outcomes, .08.29 for math test scores, and .05.17 for science achievement, all in standard deviation units. Results also suggest that kindergarten entry scores on Proficiency Levels 1 and 2 are particularly predictive of eighthgrade achievement in reading, math, and science. Postestimation tests of the differences between coefficients show that for eighthgrade math test score outcomes, fallofkindergarten Proficiency Level 2 is more predictive than are Proficiency Level 3 and fallofkindergarten reading achievement. Further, for both reading and science outcomes, Math Proficiency Levels 1 and 2 are almost always more predictive than is Proficiency Level 3 or the full reading test score. Table 4 shows results, again for the full analytic sample, using fallofkindergarten math proficiency probability scores to predict teacherreported reading and math achievement as well as whether the student had repeated a grade by eighth grade. Models are specified identically to those in Table 3 for the three additional outcome measures. The pattern of results for teacherreported achievement is similar to test score results shown in Table 3. The full models, reported in columns (3) and (6), indicate that kindergarten entry scores on Proficiency Level 2, conditional on other variables, are highly predictive of both teacherreported reading and teacherreported math achievement in eighth grade, with an effect that is onefifth of a standard deviation in magnitude in math. Coefficients for Math Proficiency Levels 1 and 3 are smaller, and fall kindergarten reading scores predict teacherreported reading and math as well.
Columns (7) and (8) present the results from models examining whether or not the child had been retained in grade by the eighth grade year. Coefficients and standard errors are from logistic regression models, with odds ratios provided to the right of coefficients. Column (7) indicates that all three math proficiency probability scores predict whether or not a child experienced grade retention by the end of eighth grade, in an unconditional model. However, once background characteristics are added and the math proficiency measures are included in a single model (column (8)), only scores on Math Proficiency Level 1 predict grade retention. As would be expected, we find a negative relationship, with lower test scores indicating a higher likelihood of having been retained. Tables 5 and 6 show results for the same set of analyses shown in Tables 3 and 4 for the subgroup of children who had not mastered Math Proficiency Level 2 at kindergarten entry. One important difference between these analyses and those described above is that Math Proficiency Level 3 is omitted from these models. Children who had not mastered Proficiency Level 2 essentially scored zero on Level 3 (See Table 1). As a result of the lack of variation on Proficiency Level 3 for these students, we omit it from the analyses. Results for Math Proficiency Levels 1 and 2, however, are similar to those presented for the full sample, with coefficients on the Math Proficiency Levels ranging in size from .0932 in the fullcontrol models (columns (3), (6), and (8)) for test score outcomes reported in Table 5. Table 6 shows results for teacher reports and grade retention. Kindergarten entry scores on Math Proficiency Levels 1 and 2 are, again, highly predictive of eighthgrade teacher reports of reading. Although the magnitude of the coefficients for math is similar, the reduced sample size limits our ability to detect statistically significant effects. Results for grade retention show a similar pattern for this subgroup of children with lower mathematics test scores at kindergarten entry. In the fully controlled model (column (8)), Math Proficiency Level 1 significantly predicts retention by eighth grade, as does reading achievement.
In addition to examining eighthgrade school outcomes, we explore how particular early math skills predict shorterrun outcomes and whether their importance for school success changes over time. Tables 7 and 8 present fully controlled models (e.g. Table 3, column (3)) using fall of kindergarten measures of achievement to predict test scores, teacher reports of achievement, and grade retention at spring of kindergarten, first, third, and fifth grades for the full analytic sample. Table 7 presents results for reading, math, and, when available, science test score outcomes at various points throughout elementary and middle school. The first panel of Table 7 shows that all three math proficiency probability scores are consistently predictive of subsequent reading outcomes, excepting Proficiency Level 2 at the spring of kindergarten (column (1)) and Proficiency Level 3 at the spring of eighth grade (column (5)). Beginning in third grade, Math Proficiency Levels 1 and 2 are more predictive of subsequent reading test scores than is Proficiency Level 3, with coefficients ranging from .11.20. For math test score outcomes, the overall pattern of results is similar. Math Proficiency Level 2 grows in magnitude across elementary school going from .14 for the spring of kindergarten math outcomes (column (6)) to .29 by eighth grade (column (10)). For science test score outcomes, all three math proficiency measures are equally important through fifth grade; however, in eighth grade the coefficient on Proficiency Level 2 is three times the size of that on Proficiency Level 3.
Interestingly, the pattern of results for schoolentry language and literacy (reading) achievement shows that its ability to predict both reading and math achievement declines over the course of elementary school. While fallofkindergarten language and literacy achievement is highly predictive of spring reading test scores, with a coefficient of .74, by eighth grade it drops to .07. We see a similar, albeit less dramatic, pattern in math, with coefficients ranging from .15 in spring of kindergarten to a low of .02 in spring of third grade. Table 8 presents results for teacherrated achievement and grade retention, measured across elementary school using fallofkindergarten math proficiencies. The overall pattern of results in Table 8 is similar to results shown in Table 7. For reading outcomes (columns (1)(5)), Math Proficiency Levels 1 and 2 are more predictive than is Proficiency Level 3. In almost all years, for both math and reading, the coefficient for Proficiency Level 2 is larger in magnitude than the coefficients for both Math Proficiency Level 1 and fallofkindergarten reading test scores. Finally, Math Proficiency Level 1 is the most consistent predictor of ever being retained in grade among the math and reading skills measured at kindergarten entry. We replicated the analyses presented in Tables 7 and 8 for the sample of children with low math achievement at kindergarten entry (results not shown). The pattern of results for children with low math achievement is similar to the pattern shown for the full sample.
SUBGROUPS In addition to examining the relationship between early math skills and later academic achievement and grade retention for the full sample and for students with low math achievement, we examined this relationship for a number of different subgroups. It is well documented that particular groups of children are more at risk of later school failure and that these same groups, on average, have lower schoolentry skills. We did not, however, have any reason to anticipate subgroup differences in the importance of these school entry skills for later schoolrelated outcomes. Despite the fact that we had no a priori expectation of differences, we thought it important to document results for various subgroups. Specifically, we conducted subgroup analyses for students in the top and bottom quartiles of the SES distribution; boys and girls; White, African American, and Hispanic students; students with limited English proficiency; students from singleparent homes; and students whose mothers reported not having completed high school. We found no systematic differences in the relationship between early math skills and later outcomes across these groups of children (results not shown). The pattern of results described above for both the full analytic sample and for students who enter kindergarten with low math skills held for all subgroups. These findings indicate that early mathematics skills, in particular those measured by Proficiency Levels 1 and 2, are consistently important for childrens school success, as measured by achievement test scores, teacher reports, and grade retention. DISCUSSION Using a nationally representative sample of kindergartners, this study explored which early math skills were most important for later school success. We examined reading, math, and science achievement test score outcomes at five points across elementary school and into middle school. We focused first on the eighthgrade, schoolrelated outcomes, the latest available data point, and also examined shorterrun outcomes to understand the relationship between early math skills and achievement over time. We also examined teacherrated achievement outcomes and retention in grade at each available data collection wave. Across all the models, except for reading achievement in the spring of kindergarten and first grade, childrens early math skills were more important predictors of later achievement than were their early language and literacy (reading) skills. Further, the pattern of results indicates that math Proficiency Level 2, which measures a childs ability to read all onedigit numerals, count beyond 10, recognize a sequence of patterns, and use nonstandard units of length to compare objects, is the most consistent and important predictor of later achievement test scores in both reading and math across elementary school. It is more important for fifth and eighthgrade math achievement than is Proficiency Level 1, which measures a childs ability to identify some onedigit numerals, recognize geometric shapes, and count onetoone up to 10 objects. Both Proficiency Levels 1 and 2 are equally important for eventual reading and science achievement test scores. For teacherrated achievement, Math Proficiency Level 2 is the most important predictor by eighth grade. Early mathematics achievement, particularly Proficiency Level 1, is the most consistent predictor of grade retention. Consistent with other studies of early math skills and later achievement (Aunola et al., 2004; Bodovski & Farkas, 2007; Jordan et al., 2007, 2009), we find fallofkindergarten math skills to be important for subsequent elementary school math achievement test scores. The prior studies found evidence that number sense and competence were important for later math achievement. In the current study, we find that number recognition, counting, shapes, and patterns are important predictors of success at various points in elementary school. Unlike the prior studies, this study also looked at outcomes beyond subsequent math test scores, showing that early math skills are important for a broad range of measures of school success, including reading, science, and grade retention. We find an intriguing, suggestive pattern of results when we look across the primary grades. Math Proficiency Level 2 gained in importance throughout elementary school, becoming the most important predictor of eighthgrade math achievement. Both Proficiency Levels 1 and 2 were important predictors of reading and science achievement, with Math Proficiency Level 2 gaining or maintaining its importance over time. The magnitude of the coefficient on Proficiency Level 3, which was very large in the spring of kindergarten and first grade for math, declined over time. Despite the hierarchical structure of the proficiency probability scores, at the fall of kindergarten, Math Proficiency Levels 1 through 3 all had reasonably large standard deviations (see Table 1), suggesting that estimates were not constrained due to a lack of variation. In addition, all three proficiency levels predicted later outcomes, with some changing in magnitude over time. Further, while fallofkindergarten reading achievement was important for the shorterrun measures of achievement test scores, its importance waned over the course of elementary school. The math proficiency levels and reading test scores were similarly predictive of teacherreported outcomes. The fact that we were able to replicate results using nontestscore measures of academic success provides additional evidence of the importance of early math proficiency. WHY MIGHT DIFFERENT SETS OF MATH SKILLS BE IMPORTANT AT PARTICULAR POINTS IN TIME? Both theory and research in mathematics indicate that over the course of early and middle childhood, childrens conventional mathematics skills develop, and these skills are likely essential for learning later, increasingly more complex mathematical skills. We find suggestive evidence that particular sets of early math skills are important predictors of later academic outcomes, while foundational skills in number knowledge predict success across all time points. The present study shows that a large majority of children already have basic knowledge in counting and shapes (Proficiency Level 1) at school entry but that childrens skills in pattern recognition, measurement, and more advanced number knowledge (Proficiency Level 2) vary substantially when they begin formal schooling. Here, we find that childrens basic skills predict both short and longerterm school success but that more complex skill sets are predictive of longerrun outcomes, particularly in mathematics. This might be due to the changing nature of the content taught over the course of elementary school and into middle school. For example, while the primary years of school might be focused on foundational reading and mathematics skills, by eighth grade the focus in reading and mathematics may have shifted to an emphasis on abstraction and extrapolation. These more advanced skills require children to use more abstract thinking, and, therefore, Math Proficiency Level 2which taps skills in patterns, measurement, and more advanced numbermay be measuring the antecedents of these more complex skills.
DO MATH SKILLS MATTER MORE FOR VARIOUS SUBGROUPS OF STUDENTS? We find a similar pattern of results for children who enter kindergarten with lower mathematics achievement as well as for a wide range of subgroups by gender, race/ethnicity, English language learners, income, and maternal education. The similar pattern of results across these groups indicates that these foundational and more advanced mathematics skills are important predictors of academic outcomes for all students. Our results provide a different angle from which to view the concerning and welldocumented achievement gaps that exist between children of different racial/ethnic backgrounds and different socioeconomic status. The fact that these early skills are equally important for all children reinforces the need to continue to work to reduce the achievement gap and to ensure that all children 1) begin school with the skills necessary to succeed and 2) are supported adequately in their efforts to continue to learn.
LIMITATIONS The results presented here are not without limitations. First, the data we use are nonexperimental, and, thus, all of the results are correlational and subject to concern about selection and omitted variable bias. It is also possible that measurement error in the probability proficiency scores may be biasing our results, although the direction of bias would likely be downward. The math proficiency probability scores available in the data do not allow us to further deconstruct early math skills. For example, ideally, we would like to examine pattern recognition, advanced number skills, and measurement separately, but these are combined into a single Proficiency Level in the ECLSK. Replicating these analyses with data that contain more finegrained measures of childrens early math achievement that allow for the isolation of particular math skills as well as longerrun outcomes will be crucial for both confirming and expanding upon the current study. CONCLUSION The results reported here have implications for education policy regarding mathematics instruction in the earliest years of schooling. The fact that early mathematics knowledge and skills are the most important predictors not only for later math achievement but also for achievement in other content areas and grade retention indicates that math should be a primary area of academic focus during the kindergarten year. These findings, which confirm and expand upon earlier work (Authors, 2007, 2009) indicate that particular early math skills grow in importance as children progress through school. Thus, our findings support a greater emphasis on mathematics than is currently the case in many kindergarten classrooms. These results also suggest the possibility that focusing more on advanced number, pattern recognition and measurement during kindergarten might develop skills that will benefit students in the later years of schooling across multiple subject areas. The fact that we find that math skills such as patterns, measurement, and more advanced number become more important over time for measures of school achievement highlights the need to better understand how these early skills develop and are promoted both at home and in school. For example, recent research finds that playing with puzzles in early childhood is a strong predictor of childrens spatial skills at around age 5 (Levine, Ratliff, Huttenlocher, & Cannon, 2012). In addition, several recent early childhood mathematics interventions have shown that childrens skills in measurement, space, and mathematical thinking can, in fact, be improved (Clements & Sarama, 2011). If these skills are both malleable and, as the present study suggests, particularly important, promoting these skills could result in lasting benefits for school achievement across subject areas. 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