foundations of mathematics achievement

Foundations of Mathematics Achievement: Instructional Practices and Diverse KindergartenStudentsAuthor(s): Martha Cecilia Bottia, Stephanie Moller, Roslyn Arlin Mickelson, and ElizabethStearnsSource: The Elementary School Journal, Vol. 115, No. 1 (September 2014), pp. 124-150Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/10.1086/676950 .

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FOUNDATIONS OF MATHEMATICS

ACHIEVEMENT

Instructional Practices and Diverse KindergartenStudents

Martha Cecilia BottiaStephanie MollerRoslyn Arlin MickelsonElizabeth Stearnsuniversity of northcarolina at charlotte

abstractAnalyzing Early Childhood Longitudinal Survey—Kindergarten (ECLS-K) data, we examine how expo-sure to instructional practices influences math testscores at the end of kindergarten for children fromdifferent racial/ethnic and socioeconomic back-grounds, and for children with different levels of mathskills at kindergarten entry. We also analyze the rela-tionship between socioeconomic background andmath academic readiness within racial/ethnic catego-ries. Our results demonstrate that race/ethnicity andlevels of math academic readiness moderate the rela-tionship between instructional practices and mathachievement. While we find that interactive group ac-tivities enhance students’ mathematics achievementin kindergarten and that drills enhance math aca-demic achievement of students with high math aca-demic preparedness in kindergarten, we also find thatuse of manipulatives as well as music and movementhave significant negative effects on mathematicsachievement of Black students. Given the importanceof kindergarten for launching children onto success-ful academic trajectories, the findings have implica-tions for addressing racial/ethnic and socioeconomicstatus gaps in mathematics achievement.

the elementary school journal volume 115 , number 1© 2014 by The University of Chicago. All rights reserved. 0013-5984/2014/11501-0006 $10.00



MA T H E M A T I C S education has become a top national priority in ef-forts to advance the nation’s technical and scientific literacy (NationalResearch Council, 2009). The mathematics education children receivein the early elementary grades sets them on pathways for academic suc-

cess or struggle for the remainder of their formal education. Mathematics perfor-mance in the early grades influences individuals’ achievement trajectories and, con-sequently, their eventual status attainment. Teachers’ instructional practices areessential components of this early mathematics education.

Drawing upon the theoretical framework that holds that national mathematicsstandards represent the formal mathematics curriculum, instructional practices re-flect the implemented math curriculum, and student achievement manifests theattained mathematics curriculum (Suter, 2000; Travers & Westbury, 1989), we ex-amine how aspects of the implemented mathematics curriculum affect the achievedcurriculum among a nationally representative sample of kindergarten students. Pre-vious research has shown that individual characteristics such as race/ethnicity, so-cioeconomic status (SES), and math skills at school entry (math academic readiness)help explain the link between curriculum and students’ math achievement(Bodovski & Farkas, 2007a, 2007b; Lubienski, 2002, 2006; Palardy & Rumberger,2008). Consistent with these relationships, it is likely that diversity in socioeconomicbackground and academic readiness within race and SES groups plays an importantrole in the potential impact of instructional practices on mathematics achievement.Yet, scholars have not thoroughly assessed whether these practices differentially im-pact students’ achievement depending on their race/ethnicity, socioeconomic status,and math academic readiness.

Our article examines whether teachers’ instructional practices differentially affectthe mathematics achievement of kindergarten students whose backgrounds differ interms of their race/ethnicity, socioeconomic status (SES), and math academic read-iness. We focus on “how” mathematics is taught—that is, instructional practices—because we recognize the potential for instructional practices to help diminishachievement gaps within schools (Wenglinsky, 2004). In addition, instructionalpractices are elements of the curriculum that teachers are best positioned to influ-ence (Lubienski, 2006).

We concentrate on the kindergarten curriculum because a strong mathematicsfoundation at the onset of formal schooling is essential for a student’s long-termsuccess. Indeed, within mathematics there is a specific progression of concepts thatmust be mastered before the next concepts can be presented by the teacher andlearned by the student. The earliest years of a child’s education are the most appro-priate years to start building a solid mathematics foundation (Clements & Sarama,2007; Waterford Institute, 2008). Identifying differences in the impact of “how”mathematics material is taught across classrooms can offer valuable clues regardinghow to design policies that reduce educational inequalities and improve the overallachievement of students with various racial/ethnic, SES, and math academic readinessbackgrounds. Unlike most other studies of early mathematics performance, we exploredifferences in socioeconomic and math academic readiness within racial/ethnic catego-ries. This is an important line of inquiry given the increasing diversity of the U.S. studentpopulation and the relatively high rates of growth among the subpopulations that tend toperform poorly in mathematics (Mickelson, Bottia, & Lambert, 2013). We pursue this

math instruction and diversity � 125



research through multilevel modeling techniques using data from the Early ChildhoodLongitudinal Study—Kindergarten Cohort (ECLS-K).

Theoretical Background

Instructional Practices and Mathematics Achievement

Previous studies by the International Association for the Evaluation of EducationalAchievement determined that any country’s national curriculum can be defined by top-ics that are intended by the school system, implemented in the classroom, and attained bythe students (Suter, 2000; Travers & Westbury, 1989). The intended, implemented, andattained curricula are developed simultaneously within an education system and to-gether play a crucial role in the development of students’ education. Each curriculumshapes the next, and the success of one establishes the potential for the others. The in-tended or official curriculum is the desired curriculum based on national or state stan-dards and the opinions of educators and experts in any given discipline. This curriculumdetermines the concepts to be learned and their sequence.

Importantly, the formal curriculum can be modified by different aspects of teachingpractices and consequently results in the implemented curriculum, which often is tovarying degrees distinct from, but related to, the intended one. The implemented curric-ulum is the one actually presented to the students, and the one that more directly reflectsthe information to which students are exposed. Lastly, the attained (or achieved) curric-ulum is the portion of the intended and implemented curricula that the students learn.This is the curriculum that is reflected in students’ test scores (Juenemann, 2004). Thus,in one sense, achievement gaps reflect differences in the attained curriculum. This re-search focuses on the implemented curriculum, operationalized as mathematics instruc-tional practices teachers perform in kindergarten classes, as it influences the attainedcurriculum, operationalized as students’ test scores.

Previous research has investigated how instruction (including curriculum char-acteristics, context, and teachers characteristics) affects student learning (Alexander,2000; Bargagliotti, Guarino, & Mason, 2009; Kessenich, 2006; Palardy & Rumberger,2008; Xue & Meisels, 2004). Researchers identified important relationships betweeninstructional practices and children’s academic achievement. In general, the Na-tional Mathematics Advisory Panel (NMAP) (2008) says that an effective instruc-tional approach with some students is an explicit and systematic approach withteacher modeling. There is no single ideal approach to teaching mathematics; thestudents, the mathematical goals, the teacher’s background and strengths, and theinstructional context all matter.

Specific Instructional Practices

Kindergartners are exposed to various instructional approaches in order to gain nec-essary math knowledge.1 First, the use of manipulatives is common in kindergarten class-rooms. Manipulatives are defined as “physical objects that are used as teaching tools toengage students in the hands-on learning of mathematics” (Teacher Vision, 2013) thatallow children to use concrete objects to observe, model, and internalize abstract con-cepts, therefore providing a common language with which to communicate these modelsto other students and the teacher (Ruzic & O’Connell, 2001). Manipulatives are believedto bridge the gap between the world in which children live and the abstract world of

126 � the elementary school journal september 2014



mathematics (Dienes, 1960). Manipulatives engage students and increase their enjoy-ment and interest in mathematics, all of which have positive effects on students’ achieve-ment (Sutton & Krueger, 2002). In fact, a study of elementary school teachers found that85% of them rated use of manipulatives as a highly effective instructional tool—ratedhigher than textbooks and handouts.

Drilling practice worksheets, workbooks, and measuring exercises are also com-mon in kindergarten classrooms and can be applied universally to a variety of math-ematics problems. These practices are linked to formal procedures of algebra andcalculus, thereby giving children hands-on experience with formal procedures nec-essary in advanced math (Scott-Clayton, 2012). “Drills” have been found to positivelypredict math achievement (Milesi & Gamoran, 2006).

In addition to drills, students are often taught through interactive group practicesin kindergarten. By interacting in groups, children give and receive help— both ofwhich are positively related to mathematics achievement (Webb, 2008). Group/in-teractive activities have also been positively associated with kindergarten mathemat-ics gains (Bodovski & Farkas, 2007a). The benefits of group interaction for students’math achievement might occur through different mechanisms: (1) by directly affect-ing cognitive processes, (2) by mediating variables that could enhance an emotionalor intellectual climate to be conducive to learning, and (3) by the sheer act of verbal-izing information. Additionally, the presence of group feedback and resource shar-ing in interactive group activities helps group members reshape their ideas and learnnovel information that they are unlikely to discover on their own (Slavin, 1977).

More recently, teachers in kindergarten classrooms have started using music andmovement to teach math. Existing research suggests that there are many benefits ofusing music, and many means of incorporating it into mathematics instruction(Yoho, 2011). Music keeps students alert, ready to learn, and actively engaged. Musicprovides children with strategies to increase their memory and improve math skills,and it strengthens the spatial reasoning essential to math skills (Jensen, 2005). Pre-vious literature also suggests that music and movement combined with rhythm,melody, lyrics, and motion affect many of the areas children love and involve more oftheir senses; the more senses involved in learning, the greater the understanding(Palmer, 2001). In fact, Southgate and Roscigno (2009) found that there are clearbenefits of music involvement (measured as weekly in-school music class participa-tion) in school for the math achievement of small children.

Research utilizing ECLS-K data has focused specifically on the importance ofinstructional practices for math gains in first grade (Palardy & Rumberger, 2008).This study found that teachers’ instructional practices, specifically, frequency of useof math worksheets and frequency of work on problems with calendars, had a sig-nificant positive relationship with math achievement gains. Yet, in aggregate, thecorpus of research on instructional practices and mathematics achievement does notprovide much insight into how instructional practices affect the math achievementof students from diverse racial, ethnic, SES, and academic readiness backgrounds.

The Moderating Role of Student Attributes on the Relationship betweenInstructional Practices and Mathematics Achievement

Student attributes, including race/ethnicity, socioeconomic status, and math ac-ademic readiness, moderate the relationship between instructional practices and




math achievement. Theories of cultural and/or linguistic mismatch offer an expla-nation for differences in curricular effects by race/ethnicity. Since teaching andlearning are cultural activities, one might expect that students with different ethnicand cultural backgrounds respond differently to the same curriculum (Farber &Klein, 1999). Research shows that there are cultural traits that have direct implica-tions for teaching and learning. For example, different ethnic groups (a) prioritizecommunal living and cooperative problem solving, and these preferences affect ed-ucational motivation, aspiration, and task performance; and (b) have norms forappropriate ways for children to interact with the adults they encounter in instruc-tional settings. In addition, different cultures may place different values on mathe-matics education and have different ideas of parental roles in children’s learning(Fuligni & Fuligni, 2007; Kaplan, 1991). A linguistic mismatch between home andschool may lead to a lack of parental involvement (Espinosa, 2005) and weakstudent-teacher and student-student relationships (García & Levin, 2001; Ramirez,2003), both of which are important factors for children’s academic achievement.More interactive group practices could be undermined if racial, ethnic, or SES-basedlanguage/cultural differences interfere with the interaction.

Differences by race. In general, research indicates that learning styles character-ized by factors with social and affective emphasis, expressive creativity, and nonver-bal communication might be more successful with African American students, whotend to be more flexible and fluid rather than structured in their perception of ideasbecause their culture emphasizes interaction with the environment (Malloy & Jones,1998). Stiff (1999) and Gilbert and Gay (1985) found that many African Americanstudents prefer learning in more relational, holistic ways, including solving contex-tualized problems and participating in classroom discourse. Wenglinsky (2004), us-ing National Assessment of Educational Progress (NAEP) data from a sample offourth graders, found that an emphasis on topics of measurement and estimation“was the most beneficial practice” for Black students, while an emphasis on dataanalysis appeared to be beneficial for Latino/a students. Utilizing the same NAEPdata for fourth graders, Lubienski (2006) found that the factor related to collabora-tive problem solving more often had a positive correlation with the achievement ofBlack and Latino/a students than White students. Scholars have also found thatinteractive group activities are better suited for ethnic groups whose cultural envi-ronments value the welfare of the group over the individual and where individualsare taught to pool their resources to solve problems (Gay, 2002). In fact, the positivebenefits of communities of learners and cooperative efforts on student achievementpreviously have been validated for Latino/a (Escalanté & Dirmann, 1990), AfricanAmerican (Fullilove & Treisman, 1990), Chinese American (Fullilove & Treisman,1990), and Native Hawaiian (Tharp & Gallimore, 1988) students.

Research has also shown that motion and movement, music, frequent vari-ability in tasks and formats, novelty, and dramatic elements in instructionalpractices improve the academic performance of African Americans (Allen &Boykin, 1992; Allen & Butler, 1996; Boykin, 1982; Guttentag & Ross, 1972; Hanley,1998). However, a more recent study by Southgate and Roscigno (2009) foundthat there are clear benefits of music involvement in school on math achievementof small children, with White students receiving more benefits than AfricanAmerican, Latino/a, and Asian students from music involvement during earlychildhood and high school years.




Differences by socioeconomic status. Turning to socioeconomic status and ac-ademic readiness, social class or SES differences reflect the unequal resources parentspossess that affect the capacities children will have to take advantage of what is taughtin schools and to comply with the requests of teachers. Among the important socialclass differences are variable levels of cultural capital (Lareau, 2000a, 2000b) andfamily wealth (Conley, 1999, 2001). Socioeconomic status has multiple ways of af-fecting the relationship between instructional practices and mathematics achieve-ment. Family investment theory suggests that higher-SES parents invest more inchildren’s learning before entering kindergarten and during the kindergarten year.Consequently, children of higher SES have higher levels of academic readiness thatincrease their chances of obtaining benefits from instructional practices. On theother hand, stress models argue that children from lower-SES backgrounds haveparents who are less effective, and they are more prone to health problems thatdirectly and indirectly affect kindergarten students’ levels of academic readiness andthe context in which students learn during their first year. As a consequence, childrenfrom lower-SES backgrounds have fewer resources with which to take advantage ofinstructional practices. Bodovski and Farkas (2007b) found that academic achieve-ment is influenced by the academic and social abilities that different students bring toschools at entry that are correlated with race and SES. Disadvantaged children startkindergarten with significantly lower skills than their more privileged counterpartsand are therefore unevenly equipped to initiate their learning processes.

Hickey, Moore, and Pellegrino (2001) analyzed how instructional practices affectstudents from different socioeconomic groups and found that reform-oriented in-struction (which includes interactive group practices and music and movement)improved low- and high-SES students’ problem-solving skills, but the same instruc-tion increased the SES-related gap in students’ performance on the concepts andestimation portion of the Iowa Test of Basic Skills. However, other research foundthat reform-minded practices are particularly beneficial for lower-SES and minoritystudents (Boaler, 2002; Stiff, 1999).

Differences by academic readiness. Math academic readiness is related to SESbackground. Academic readiness refers to a number of language, mathematics, smallmotor, and personal/interpersonal skills among young children entering kindergar-ten. Students’ varying levels of academic readiness condition how much children arelikely to understand and benefit from the curriculum to which they are exposed inclassrooms. As such, academic readiness becomes a key predictor of long-termachievement trajectories (Bodovski & Farkas, 2007a). Indeed, previous research hasshown that math academic readiness is an important predictor of subsequent schoolachievement in math (Duncan et al., 2007).

Although the importance of academic readiness on mathematics achievement hasbeen recognized extensively in the past, research that specifically focuses on thepotential moderating role of academic readiness on the relationship between in-structional practices and math achievement is scant. Bodovski and Farkas (2007b)found that the level of mathematics knowledge at the beginning of students’ schoolcareers is associated with students’ subsequent gains. Students who began with themost limited knowledge had the smallest gains.

Most prior studies have examined race, SES, and academic readiness indepen-dently. Only a few studies have examined how they interactively moderate the rela-tionship between instructional practices and students’ mathematics achievement.




Bodovski and Farkas (2007a) used ECLS-K data to study children in kindergartenand in first grade and found that certain instructional practices can produce modestreductions in achievement gaps between African American and White students inkindergarten. However, they found no significant effects of instruction on theachievement gaps between White and Latino/a or lower social class students.Wenglinsky (2004) analyzed NAEP data for eighth graders and found that instruc-tional practices can reduce the African American and Latino within-school achieve-ment gap. Similarly, Lubienski (2006) analyzed NAEP data from students in fourthand eighth grade and found, in contrast to Wenglinsky (2004), that the relationshipbetween various instructional practices and achievement was roughly similar forWhite, Black, and Latino/a students.

We build on the previously discussed research by examining the intersection ofrace/ethnicity, SES, and academic readiness as possible moderating factors betweeninstructional practices and mathematics learning. We do so by analyzing the rela-tionship between instructional practices and mathematics achievement of a nation-ally representative sample of kindergarten students who are either White, AfricanAmerican, Latino/a, or Asian American, from either low-, middle-, or high-SESfamilies, and who have low, middle, and high levels of math academic readiness. Wepredict that (Hypothesis 1) instructional practices significantly affect the mathemat-ics achievement of kindergartners; (Hypothesis 2) race/ethnicity moderates the re-lationship between instructional practices and mathematics achievement; (Hypoth-esis 2a) practices that involve more social and affective emphasis (such as interactivegroup activities) are more beneficial for African American students than for Whitestudents; (Hypothesis 2b) interactive group activities that emphasize the welfare ofthe group are more beneficial for Latino/a and Asian students than for White stu-dents; (Hypothesis 2c) music and movement practices that express creativity andnonverbal communication are more beneficial for African American students thanfor White students; (Hypothesis 3) socioeconomic status and math academic readi-ness moderate the relationship between instructional practices and mathematicsachievement; (Hypothesis 3a) drills, which require more previous knowledge, ben-efit better prepared students; and (Hypothesis 3b) use of manipulatives, a fairlysimple instructional practice that requires little previous knowledge, is more bene-ficial for less academically ready students than for students with higher academicreadiness.

Because there is little research looking at the interactions between race, socioeco-nomic status, and academic readiness, we investigated Hypotheses 4 and 5 in anexploratory manner. (Hypothesis 4) The relationship between instructional prac-tices and mathematics achievement should vary among the combined SES and racial/ethnic categories. (Hypothesis 5) The relationship between instructional practicesand mathematics achievement should vary among the combined academic readinessand racial/ethnic categories.

Method

Data Source

To test the hypotheses above, we analyze data from the U.S. Department of Edu-cation’s Early Childhood Longitudinal Study (ECLS-K) because it focuses on chil-




dren’s early school experiences. This data set began in 1998 with a nationally repre-sentative sample of 22,670 kindergartners and provides descriptive information onfamily, school, community, and individual factors associated with the performanceof students at schools (ECLS-K website).

Given our research interests, we limit the sample to White, Black, Latino/a, andAsian students.2 Doing so narrows our sample to 15,840 students (61% White, 15%Black, 18% Latino/a, and 6% Asian). We also limit our sample to students who arenot repeating kindergarten because their experiences and needs are different fromfirst-time kindergarten students. This further limits our sample to 15,020 students(61% White, 15% Black, 18% Latino/a, 6% Asian).

Any missing data are imputed through multiple imputation because this ap-proach is far superior to listwise deletion of missing data (Allison, 2002; Schafer1997).3 In order to ensure high efficiency, we determined a priori to impute onlyvariables that are missing less than 20% of cases within waves. Most of our variableshave less than 10% missing data and are thus imputed. The imputation is more than93% efficient for all imputed variables in all waves.

After listwise deletion of cases whose missing data could not be imputed, our finalsample includes 13,670 White, Black, Latino/a, and Asian students who attendedkindergarten in 1998. A comparison of this final sample to the initial sample indicatedthat the final sample is not dramatically different from the initial sample in terms ofrace, SES, and achievement (13% Black and 16% Latino/a, 65% White, 6% Asian),socioeconomic status (30% of the final sample are lower SES, compared to 32% priorto sample selection), and math scores (the average kindergarten scores were 36.6 inthe initial sample, and 37.1 in the final sample).

Outcome Variable

The main dependent variable for this study is students’ mathematics achievementin the spring of kindergarten. Math achievement is measured through item responsetheory (IRT) scale scores, which assess the probability of a correct response by esti-mating the number of correct answers expected if the student had answered allquestions for the math test in multiple waves (Tourangeau, Nord, Le, Pollack, &Atkins-Burnett, 2009).4 We analyze spring IRT scores because these scores permitevaluation of achievement trajectories over time with age-appropriate tests. In thisway, these measures can be compared over time.

Predictor Variables

The key independent variables of interest are the frequency of use of certainmathematics instructional practices for specific curricular content areas at the kin-dergarten level. These data come from the ECLS-K spring teacher questionnairewhere teachers are asked to respond to the following questions: “How often is each ofthe following MATH skills taught in your class?” and “How often do children in thisclass do each of the following MATH activities?” Teachers choose from the optionsnever, once a month, two or three times a month, once or twice a week, three or fourtimes a week, or daily (see App. Table A1). There are 17 process variables and 29content variables in ECLS-K data. We focus on the 17 process variables that reflectinstructional practices teachers use in their classrooms and better reflect the imple-




mented curriculum. Many of these instructional practices have been analyzed inearlier research, but here we add to the body of knowledge by intersecting racial/ethnic, socioeconomic, and academic readiness categories (e.g., Bodovski & Farkas,2007b; Palardy and Rumberger, 2008).

Moderator variables. For our analysis we utilize a categorical race variable (Whiteis the omitted reference category). ECLS-K provides a continuous measure of socio-economic status that utilizes family income, parental education, and occupation asinputs. For ease of interpretation of the analyses, we created an ordinal measure ofSES (low, middle, and high SES) based on the distribution of the continuous SESmeasure (high SES is the omitted category). We also created an ordinal measure ofmath academic readiness by dividing the sample of students into terciles based on themath IRT scores students have at the beginning of the kindergarten year (low aca-demic readiness is the omitted category). Since there is substantial variability withinterciles, we also acknowledged the presence of this variability by controlling forprevious math score, where math scores are centered within terciles. To further testour hypotheses regarding the intersection between race, SES, and math academicreadiness categories, we created categorical variables for race socioeconomic (White,high SES is the omitted reference category), race by academic readiness (White, highacademic readiness is the omitted reference category), SES by academic readiness(high SES, high academic readiness is the omitted category), and race by socioeco-nomic by math academic readiness categories (White, high SES, high academic read-iness is the omitted category).5

Control variables. In all models, we control for variables at the individual, class-room, and school levels that could be correlated with math scores and our primaryindependent variables. Our individual-level controls include gender, age, and mea-sures of cultural capital, including English as a second language and socioeconomicstatus of the child in kindergarten. We also control for reading scores to account forthe academic preparation that students bring when they enter school. Classroom-level controls include whether the classroom is a full-day class or not (coded 1 for fullday), time spent in math, teacher’s race (Black or Latino/a, with White as the omittedcategory), teacher enjoys teaching (1 � yes), and teacher’s highest education (1 �high school to 7 � doctorate). Lastly, the school-level controls are school size(logged), percent of students in the school who are Black, percent of students in theschool who are Latino/a, region (south is omitted category), rural/suburban (urbanis omitted category), school is private or not, and whether or not the school was amagnet school or a charter school. Control variables are explained in detail in Table1.

Analytic Strategy

We conducted our analyses in three stages. First we ran 322 models with each ofthe 46 curriculum variables for each race and SES to clarify which practices are moreclosely related to students’ mathematics achievement by race/ethnic and SES back-ground. In this stage, we maintain the original ordinal nature of the data (responseoptions ranged from 1 to 6) to permit detailed nonlinear results. We do not reviewthose results in detail given the sheer complexity of discussing 322 models. We in-clude both teaching practices and content because practices are partially determinedby content. These detailed analyses of each curriculum variable separately pro-




vided two important insights: (1) 17 of these curricular practices and contentvariables are not associated with the mathematics achievement of any racial/ethnic or socioeconomic group (we drop these variables from the second stage ofour analysis), and (2) in most models, the significant curriculum variables havethe greatest effect on achievement when students are exposed to the curricularprocess or content at least once a week. Therefore, prior to moving on to thesecond stage of our analyses, we dichotomize the 29 significant curriculum vari-ables revealed in the first stage, coding them 1 if they are used once per week ormore and 0 for less than once per week.

In the analyses’ second stage, we combined the significant process and contentvariables from stage 1 into a smaller set of variables based on results from a factoranalysis. This is necessary because instructional practices and content do not happenindependently and they must be considered jointly. A factor was extracted using a

Table 1. Control Variables by Level of Analysis

Variable Description

School level:Private school 1 � private, 0 � publicPercent African American Percentage of African American students in schoolPercent Latino/a in school Percentage of Latino/a students in schoolSchool size Category of school size, 0–149, 150–299, 300–499, 500–749, and 750 and

aboveRural 1 � rural, 0 � not ruralRegion of the country Dummy variables for Midwest, West, and Northeast; South is the

reference categoryMagnet 1 � magnet school, 0 � not magnet schoolCharter 1 � charter school, 0 � not charter school

Classroom level:Full-day class 1 � full-day class, 0 � part-time classTime spent in math Time spent in mathematics instructionAfrican American teacher 1 � teacher was African American, 0 � teacher was not African AmericanLatino/a teacher 1 � teacher was Latino/a, 0 � teacher was not Latino/aEnjoys teaching Continuous variable from 1 to 5 that tells whether teacher strongly

disagrees, disagrees, neither agrees or disagrees, agrees, or stronglyagrees with the statement: “I really enjoy my present teaching job.”

Teacher’s education Category of highest educational level teacher achieved: 1 � high school,2 � associate’s degree, 3 � bachelor’s degree, 4 � more than 1 year ofcoursework beyond bachelor’s, 5 � master’s, 6 � education specialist/professional diploma, 7 � doctorate

Student level:Race White (non-Latino/a), African American (non-Latino/a), Latino/a, and

Asian AmericanSocioeconomic status Composite of five variables: father’s education and occupation, mother’s

education and occupation, and household income. SES is categorizedas low SES (the lower two quintiles), middle SES (the third quintile),and high SES (the upper two quintiles)

Math academic readiness Categorized as low readiness, middle readiness, and high readiness; basedon the previous math item response theory (IRT) score

Previous mathematics score Fall IRT scores for kindergartners centered by math academic readinessterciles

Previous reading score Fall reading IRT scores for kindergartnersAge The number of months of life at entry to kindergartenMale 1 � male, 0 � femaleNot English at home 1 � child does not speak English at home, 0 � child speaks English at

home




maximum-likelihood exploratory factor analysis with promax rotation on a tetra-choric correlation matrix. We then validated these results through a confirmatoryfactor analysis with robust weighted least-squares and a polychoric correlation ma-trix.6 The results indicated that there were eight factors. We identified four of thesefactors as instructional practices factors (only process variables loaded on these fac-tors) and labeled them Interactive Group Activities, Manipulatives, Drills, and Mu-sic/Movement. Interactive Group Activities include solving math in small groups orwith a partner, solving real-life math, explaining/solving math problems, and peertutoring. Manipulatives included the practices of using geometric and counting ma-nipulatives and using math-related games. Drills contain doing math worksheets andusing math textbooks. Finally, Music/Movement includes using movement and us-ing music to learn math. Table 2 provides a full description of the factors and theirloadings.

In the final stage of the analyses, we run multilevel regressions to test the effects ofthe extracted instructional practices factors on mathematics achievement acrossrace/ethnicity, SES, and levels of academic readiness. We present interactions be-tween the factors and racial/ethnic categories, SES categories, and math academicreadiness categories. These interactions permit us to identify whether race/ethnicity,SES, and math academic readiness have a significant moderating role in the relation-ship between instructional practices and math achievement. Each regression in-cludes all four instructional practices factors, controlling for other variables pre-sented in Table 1 and discussed below.

Disaggregating our sample into race-by-SES cohorts reveals approximately 950low-SES Black, 300 high-SES Black, 600 middle-SES Black, 1,170 low-SES Latino, 380high-SES Latino, 660 mid-SES Latino, 240 low-SES Asian, 370 high-SES Asian, 210middle-SES Asian, 1,840 low-SES White, 3,810 high-SES White, and 3,140 middle-SESWhite students. These students vary in their levels of academic readiness. Table 3presents data from achievement tests given to children in the fall and spring ofkindergarten as part of the ECLS-K by racial, socioeconomic, and racial-socioeconomic groups. We see that White low-SES and Black mid-SES studentsenter kindergarten with similar achievement (both groups averaged 24 points on thefall kindergarten mathematics achievement test). White high-SES students begin

Table 2. Factors with Variables and Loadings

Interactive GroupActivities Manipulatives Drills Music/Movement

Frequency geometric manipulatives �8 70 a �8 6Frequency counting manipulatives 6 85 a 1 �6Frequency math-related games 16 61 a �12 13Frequency music to learn math 3 1 7 78 a

Frequency movement to learn math �3 7 �2 93 a

Frequency explain/solve math problems 56 a �5 10 3Frequency do math worksheets 5 �2 79 a �2Frequency use math textbooks 9 �19 60 a 5Frequency solve math with partner 66 a 19 12 �4Frequency solve real life math 82 a �9 2 �1Frequency peer tutoring 49 a 14 8 1

Note.—Results of exploratory factor using a tetrachoric correlation matrix and promax rotation.aIndicates highest loading for each item.




school with a 7-point advantage on the math achievement test, compared to theirlow-SES counterparts. Similarly, there is an 8-point differential in initial achieve-ment between low and high SES among Asian and Latino/a students, yet there is onlya 5-point differential between low- and high-SES Black students. Table 4 provides thesample sizes when levels of math academic readiness were crossed with SES and race.

Because students are nested within classrooms, and classrooms are nested withinschools, we utilize three-level hierarchical linear models with random slopes (HLM).HLM models are appropriate because they adjust errors to account for the lack ofindependence among students and classrooms (Raudenbush & Bryk, 2002). We runregressions with the BYCOMW0 longitudinal weight—appropriate when examiningassessment data from the fall and spring of kindergarten—to ensure generalizabilityof the results. The equations for the models with categories of race, SES, and mathreadiness are shown below.

Level 1 Model (Child):

Math ijk � �0jk � �1jkx � �� ijkchild control variables � � ijk.

The dependent variable is mathematics achievement in the spring of kindergarten.The x reflects categories of students’ race/ethnicity, SES, and math academic readi-ness. Other child-level control variables include initial math scores centered aroundacademic readiness terciles; and gender, age in months, and English not spoken athome centered around their grand means.

Level 2 Model (Classroom):

�0jk � �00k � �01kCURR_FACTORS � �n0nkclassroom variables � �0jk,

�1jk � �10k � �11kCURR_FACTORS � �1jk.

Table 3. Average IRT Mathematics Scores (Means) by Racial/Ethnic Category andSocioeconomic Status (Low, Middle, High)

White Black Latino/a Asian

Low Middle High Low Middle High Low Middle High Low Middle High

N 1,840 3,140 3,810 950 600 300 1,170 660 380 240 210 370Spring

kindergarten 34.0 37.8 43.1 28.9 32.8 36.1 28.7 34.2 38.6 34.6 35.4 44.5Fall kindergarten 23.9 27.1 31.5 20.7 23.8 26.4 19.8 23.7 27.5 24.1 25.1 32.3

Table 4. Sample Sizes by Levels of Math Academic Readiness (Low, Middle, High) Crossed withRacial/Ethnic Category and Socioeconomic Status (SES)

Low Middle High

Black 870 670 310Latino 1,150 660 400Asian 210 280 330White 1,990 3,020 3,780Low SES 2,190 1,360 650Middle SES 670 1,550 2,640High SES 1,350 1,730 1,530




In the second-level equation, �0jk is the random intercept, and �0jk is the second-level error term associated with variation across classrooms (see Guo & Hongxin,2000). �01k is the extent to which the curriculum, across classrooms, predicts themathematics achievement of students; �0nk represents the extent that other grandmean centered classroom-level variables (including teacher education, teacher race,teacher satisfaction, time devoted to math in the classroom, full-day kindergartenclassroom) predict, across classrooms, the average mathematics achievement of stu-dents. Our model is specified in such a way that allows us to test whether there aresignificant differences in the impact of the curriculum factors on a student’s math-ematics achievement by race, SES, and math readiness of the student. We run cross-level interactions between the categories of race, SES, and math readiness and cur-riculum factors. �1jk reflects the effects of the categories of race, SES, and mathreadiness on mathematics achievement, which at level 2 is also a function of anintercept and the curriculum factors7. We also include a random slope, �1jk.

Level 3 Model (School):

�00k � �000 � �k00nschool variables � 00k.

In the third-level equation, �000 is the random intercept and 00k is the third-levelerror term associated with variation across schools. k00n represents the extent thatschool-level variables (private vs. public, region of the country, urbanicity, percent-age Black in school, percentage Latino/a in school) predict, across schools, the aver-age mathematics achievement of students. Therefore, our model predicts how race,math academic readiness, factors of instructional practices, and other control vari-ables are related to the mathematics achievement of the students in the spring ofkindergarten, considering the nesting of students into classrooms, and the nesting ofclassrooms into schools. In addition, our model also predicts how the associationbetween factors of instructional practices and mathematics achievement of studentsvaries by race, SES, levels of math academic readiness, race by SES, race by mathacademic readiness, and SES by math academic readiness by including interactionterms between these categorical variables and the factors of instructional practices.

Results

Before testing the hypotheses, we assess how frequently children of different racial/ethnic, socioeconomic, and academic readiness categories are exposed to instruc-tional practices at least once a week (see Table 5). We see that manipulatives arecommonly used in kindergarten classrooms, as most students are regularly exposedto this practice. It is important to note, however, that a smaller proportion of high-SES students and high-math-readiness students are regularly exposed to this prac-tice, compared to low-SES and low-math-readiness students. Music and movementare less common teaching practices, although approximately one-third of studentsare regularly exposed to these practices. Again, a larger proportion of students withlimited math academic readiness and low SES are exposed to music to learn math,compared to students with high academic readiness and high SES, respectively. Theproportion of students exposed to drills and interactive group activities also variesacross categories of students. The variation in exposure to instructional practices by




Tab

le5.

Pro

port

ion

ofSt

ude

nts

Exp

osed

toIn

stru

ctio

nal

Pra

ctic

esby

Rac

ial/

Eth

nic

Cat

egor

y,A

cade

mic

Rea

din

ess,

and

Soci

oeco

nom

icSt

atu

s(S

ES)

Mat

hA

cade

mic

Rea

din

ess

Bla

ckW

hit

eLa

tin

o/a

Asi

anLo

wM

iddl

eH

igh

Low

SES

Mid

dle

SES

Hig

hSE

S

N1,

850

8,79

02,

210

820

4,21

04,

640

4,82

04,

200

4,61

04,

860

Inst

ruct

ion

alpr

acti

ce:

Man

ipu

lati

ves:

Geo

met

ric

man

ipu

lati

ves

.83

.81

.79

.73

.79

.77

.73

.80

.75

.74

Cou

nti

ng

man

ipu

lati

ves

.93

.93

.89

.92

.93

.92

.91

.94

.92

.91

Mat

h-r

elat

edga

mes

.88

.81

.87

.83

.85

.84

.83

.86

.83

.83

Mu

sic/

mov

emen

t:M

usi

cto

lear

nm

ath

.35

.35

.32

.27

.33

.30

.27

.35

.29

.27

Mov

emen

tto

lear

nm

ath

.29

.31

.30

.23

.27

.26

.24

.27

.25

.25

Dri

lls:

Do

mat

hw

orks

hee

ts.7

6.7

1.6

8.6

7.7

2.6

9.6

7.7

3.7

0.6

5U

sem

ath

text

book

s.3

1.2

6.2

4.2

4.2

4.2

5.2

6.2

5.2

5.2

5D

om

ath

onch

alkb

oard

.47

.42

.34

.33

.39

.36

.34

.40

.36

.33

Inte

ract

ive

grou

pac

tivi

ties

:So

lve

mat

hw

/par

tner

.62

.58

.54

.49

.56

.52

.50

.56

.50

.51

Solv

ere

al-l

ife

mat

h.6

8.6

2.6

4.5

9.6

3.6

1.6

0.6

3.5

9.6

1P

eer

tuto

rin

g.5

1.4

6.4

4.3

8.4

5.4

1.3

9.4

7.4

0.3

8E

xpla

in/s

olve

mat

h.6

7.6

3.5

7.5

9.6

0.6

0.6

3.6

1.6

1.6

1



group indicates the feasibility of testing for significant differences in the role thatinstructional practices may play in mathematics performance.

Next, we analyze the effects of instructional practices on mathematics achieve-ment. The results presented in Table 6 examine the effects of instructional practiceson mathematics achievement (without interactions). Table 6 illustrates partial sup-port for hypothesis 1 as students’ mathematics achievement is higher in the spring ofkindergarten when they study in classrooms where teachers frequently use drills andinteractive group activities. However, music and movement instructional practicesdo not have an overall effect on students’ math achievement. Therefore, only someinstructional practices enhance mathematics achievement for all students.

Yet, we posit that instructional practices are differentially effective at enhancingmathematics achievement across groups of students (see Hypotheses 2 and 3). To testthese hypotheses, we ran hierarchical models and examine F-tests to assess the sig-nificance of interactions between instructional practices and categories of students.Table 7 includes these F-tests from four models that examine the interactions be-tween instructional practices and race categories (Model 1), SES categories (Model2), and academic readiness categories (Model 3). Once we determined that the over-all interactions were significant in Table 7, we further assessed the direction of effectsby examining slopes (in Table 8). Therefore, Table 8 only includes slopes for modelsthat had significant interactions in Table 7.

The results presented in Model 1 in Tables 7 and 8 do not offer support for thesecond set of hypotheses. In contrast, interactive group activities do not significantlyinteract with racial categories to predict mathematics achievement (see Table 7,Model 1). Additionally, while the interactions between racial/ethnic categories and

Table 6. Regression Coefficients from HLM Analysis of MathAchievement in Kindergarten

Variable Coefficient (SE)

Intercept 35.89(.43) ***Racial/ethnic category: a

Latino/a �.70(.23) ***Asian .39 (.41)Black �1.76 (.24) ***

Math academic readiness: b

Low academic readiness �1.57 (.26) ***Middle academic readiness �.42 (.19) **

Instructional practice:Interactive group activities .14 (.04) ***Manipulatives �.04(.03)Drills .20(.03) ***Music/movement practices �.17 (.14)

Random effects:Teacher intercept .36School intercept 2.03 ***Race intercept 2.16 ***SES intercept 2.12 ***

Note.—Controls for all variables described in Table 1.aWhite is excluded category.

bHigh academic readiness is excluded category.

**p � .01.

***p � .001.




music/movement practices are significant (as seen in Table 7, Model 1), the effect isopposite than predicted in Hypothesis 2c (as seen in Table 8, Model 4). Instructionalstrategies that include the use of music and movement are not beneficial for AfricanAmerican students; it fact, they are detrimental to their mathematics achievement.

Hypothesis 3 focuses on SES and academic readiness. This hypothesis is tested inModels 2 and 3 in Table 7. We find partial support. The interaction between instruc-tional practices and SES is not significant (see Model 2); the interaction betweenmanipulatives and academic readiness is also not significant (failing to support Hy-pothesis 3b). However, the interaction between drills and academic readiness is sig-nificant (see Model 3). This latter effect is further clarified in Table 8, Model 5. Wefind that drills benefit students with low academic preparedness (as the slope fordrills is significant), and the degree of benefit is comparable for students with me-dium readiness (as the interaction between drills and medium readiness is nonsig-nificant). Furthermore, in support of Hypothesis 3a, students with high academicreadiness benefit the most from drills (as the interaction is significant and positive),suggesting that drills require more previous knowledge.

The results presented in Table 9 assess the final three hypotheses by presentingresults from 24 subsamples. Separating the sample in this way is necessary becausethe hypotheses require three- and four-way interactions. Presenting the models sep-arately for different subgroups overcomes issues with model instability that arisewith four-way interaction terms. Furthermore, we presented these hypotheses asexploratory because the literature does not generate a priori expectations.

The results presented in Table 9 explore the effects of instructional practices forgroups based on socioeconomic status and math academic readiness within racial/ethnic categories. These results illustrate partial support for Hypotheses 4 and 5.Drills are beneficial for most categories of White students, but among White stu-dents, interactive group activities are only beneficial for mid- and high-SES students,

Table 7. F-Tests for Interactions between Instructional Practice Factors and Racial/EthnicCategory, Socioeconomic Status, and Math Academic Readiness Categories from HLM Analysisof Math Achievement in Kindergarten

F-Value

Model 1: Interaction between race/ethnicity and each instructional practice factor:Drills .22Interactive group activities .69Music/movement 2.78 *Manipulatives 2.09 �

Model 2: Interaction between SES and each instructional practice factor:Drills .36Interactive group activities .16Music/movement .27Manipulatives .19

Model 3: Interaction between math academic readiness and each instructional practice factor:Drills 4.40 *Interactive group activities 1.23Music/movement .34Manipulatives .18

Note.—Controls for all variables described in Table 1.�

p � .10.

*p � .05.




and for students with high academic readiness at the beginning of kindergarten.Despite the clear benefits of interactive group activities for some White students, it isparticularly interesting to realize that they are exposed to these practices with lessfrequency than are the other racial/ethnic groups (see Table 5).

Table 8. Regression Coefficients and Standard Errors from Models in Table 7 with SignificantResults; from HLM Analysis of Math Achievement in Kindergarten

Model 4:Race Interactions

Model 5:Math Academic

Readiness Interactions

Intercept 35.20 (.50) 35.14 (.59) ***Racial/ethnic category: a

Latino/a �1.12 (.89) �.68 (.21) ***Asian �.98 (1.91) .52 (.39)Black �.21 (1.00) �1.78 (.23) ***

Math academic readiness: b

Middle academic readiness – .66(.81)High academic readiness – .66(.79)

Instructional practice:Interactive group activities .12 (.04) *** .13 (.05) ***Manipulatives �.02 (.04) �.03 (.05)Drills .19 (.04) *** .11 (.04) ***Music/movement practices �.05 (.16) �.30 (.05)

Race/ethnicity � instructional practice interactions: –Latino/a � interactive group activities .01 (.06) –Latino/a � manipulatives �.01 (.08) –Latino/a � drills .07 (.08) –Latino/a � music/movement practices �.05 (.29) –Asian � interactive group activities .08 (.12) –Asian � manipulatives .14 (.15) –Asian � drills �.10 (.16) –Asian � music/movement practices .60 (.58) –Black � interactive group activities .08 (.09) –Black � manipulatives �.18 (.08) ** –Black � drills �.01 (.07) –Black � music/movement practices �.87 (.33) ** –

Math academic readiness � instructional practice interactions:Medium readiness � interactive group activities – .05 (.07)Medium readiness � manipulatives – .00(.06)Medium readiness � drills – .01 (.05)Medium readiness � music/movement practices – .11 (.28)High readiness � interactive group activities – �.04(.07)High readiness � manipulatives – �.02 (.06)High readiness � drills – .15 (.05) ***High readiness � music/movement practices – �.09(.27)

Random effects:Teacher intercept .31 1.20 ***School intercept 2.13 *** 2.22 ***Race intercept 2.20 ***SES intercept 2.15 ***Academic readiness intercept 3.78 ***

Note.—Controls for all variables described in Table 1. Standard errors in parentheses.aWhite is excluded category.

bLow academic readiness is excluded category.

**p � .01.

***p � .001.




Tab

le9.

HLM

Coe

ffici

ents

Pre

dict

ing

Mat

hA

chie

vem

ent

inK

inde

rgar

ten

from

24Su

bsam

ples

ofSt

ude

nts

byR

ace/

Eth

nic

ity,

Soci

oeco

nom

icSt

atu

s,an

dM

ath

Aca

dem

icR

eadi

nes

s

Wh

ite

Bla

ckLa

tin

o/a

Asi

an

Inst

ruct

ion

alP

ract

ice

Low SE

SM

iddl

eSE

SH

igh

SES

Low

SES

Mid

dle

SES

Hig

hSE

SLo

wSE

SM

iddl

eSE

SH

igh

SES

Low

SES

Mid

dle

SES

Hig

hSE

S

Dri

lls.2

5***

.18

***

.20

***

.18

*.2

7**

�.0

8.1

5**

.12

.10

.40

*.4

9�

.31�

(.06

)(.

05)

(.05

)(.

09)

(.09

)(.

15)

(.07

)(.

10)

(.13

)(.

21)

(.24

)(.

16)

Inte

ract

ive

grou

pac

tivi

ties

.08

.12

**.1

4**

.12

.32

**.3

8**

.17

*.2

7**

.02

�.0

6�

.03

.10

(.08

)(.

06)

(.07

)(.

10)

(.12

)(.

19)

(.09

)(.

12)

(.17

)(.

26)

(.22

)(.

19)

Mu

sic/

mov

emen

t�

.03

�.0

1�

.03

�.7

5��

1.10

*�

1.72

***

�.1

2.0

1�

.10

1.10

.27

.34

(.30

)(.

24)

(.26

)(.

43)

(.47

)(.

66)

(.32

)(.

48)

(.60

)(.

91)

(.85

)(.

78)

Man

ipu

lati

ves

.05

�.0

1�

.02

�.2

2**

�.1

5�

.32

**�

.02

�.0

7�

.07

�.0

4�

.14

.09

(.08

)(.

06)

(.06

)(.

10)

(.11

)(.

15)

(.09

)(.

12)

(.15

)(.

27)

(.22

)(.

18)

Aca

dem

icR

eadi

nes

s,W

hit

eA

cade

mic

Rea

din

ess,

Bla

ckA

cade

mic

Rea

din

ess,

Lati

no/

aA

cade

mic

Rea

din

ess,

Asi

an

Low

Mid

dle

Hig

hLo

wM

iddl

eH

igh

Low

Mid

dle

Hig

hLo

wM

iddl

eH

igh

Dri

lls.1

1�.3

0**

*.1

7**

.09

.33*

*.1

2.0

8.1

6.3

6**

.08

.39

.25

(.05

)(.

05)

(.06

)(.

07)

(.11

)(.

16)

(.06

)(.

10)

(.14

)(.

17)

(.20

)(.

18)

Inte

ract

ive

grou

pac

tivi

ties

.11�

.04

.15*

*.1

1.1

8.5

3**

.23*

*.0

3.1

6�

.06

.01

.07

(.06

)(.

06)

(.07

)(.

08)

(.13

)(.

23)

(.08

)(.

12)

(.20

)(.

25)

(.22

)(.

21)

Mu

sic/

mov

emen

t�

.03

�.0

3.1

4�

.74

**�

.88

��

1.01

�.0

9�

.43

.55

1.27

�.2

7.8

8(.

25)

(.24

)(.

29)

(.34

)(.

50)

(.80

)(.

29)

(.45

)(.

67)

(.89

)(.

99)

(.80

)M

anip

ula

tive

s.0

6�

.04

.02

�.2

1**

�.0

7�

.34

**�

.02

�.0

8�

.06

.06

�.2

0.3

3(.

06)

(.06

)(.

07)

(.08

)(.

13)

(.19

)(.

08)

(.11

)(.

18)

(.20

)(.

19)

(.21

)

Not

e.—

Con

trol

sfo

ral

lvar

iabl

esde

scri

bed

inT

able

1.St

anda

rder

rors

inpa

ren

thes

es.

�p

�.1

0.

*p�

.05.

**p

�.0

1.

***p

�.0

01.



The results in Table 9 also illustrate that music and movement, and manipulatives,are not effective teaching strategies for White students. In fact, these teaching toolsare not positively associated with achievement for any racial/ethnic group: further-more, both strategies harm the achievement of some Black students. This is partic-ularly problematic given that at least 38% of Black students attend kindergartenclasses that regularly implement “music/movement” practices. Furthermore, over90% of low- and high-SES and low- and high-math-readiness Black students workwith any component of the manipulatives factor at least once a week (see Table 5).The regular exposure of students to manipulatives reflects the widespread perceptionthat manipulatives enhance student engagement resulting in higher achievement. Itis important to note that the nonsignificant effects for Asian students are partiallydriven by sample size. Despite this, low-SES Asian students have higher mathematicsachievement in kindergarten when they study in classrooms where teachers employdrilling.

Lastly, to further investigate whether the moderating effect of race or SES on therelationship between instructional practices and math achievement is independentof the mathematics skill set students bring to the classrooms, and given that mathe-matics has a logical scope and sequence, we placed the students in three math aca-demic readiness categories and tested for differential responses based on race and/orSES (see Table 10). We find that the differential effect of music/movement and ma-nipulatives on the math achievement of students by race holds when the analysis isconducted by math academic readiness categories. Among students with low andhigh levels of math academic readiness, there is a significantly different effect ofmusic/movement and manipulatives by race. Specifically, the math achievement ofBlack students in the categories of low math academic preparedness and high mathacademic preparedness decreases the more these students are exposed to music/movement and manipulatives.

To summarize, we find evidence that in many instances, curriculum delivery isdifferentially associated with students’ mathematics achievement depending upontheir race/ethnicity, socioeconomic status, and math academic readiness status.These findings support our general hypotheses and are consistent with Klein’s (1999)proposition that children with different ethnic and cultural backgrounds are likely torespond differently to the same curriculum. They are also consistent with Bodovskiand Farkas’s (2007b) study that found that academic achievement is influenced bythe race- and SES-correlated academic and social abilities that different studentsbring to schools at entry.

Discussion

Our study recognizes the importance of instructional practices for the mathematicsachievement of kindergartners. Specifically, we find that the instructional practicesof interactive group activities, drills, manipulatives, and music and math have sig-nificant associations with the math achievement of kindergarten students. Our anal-ysis by racial-socioeconomic and racial-math academic preparedness categories al-lowed us to uncover the moderating role of race/ethnicity and levels of mathacademic readiness on the relationship between implemented curriculum and kin-dergartners’ math achievement. Consistent with Webb (2008) and Bodovski andFarkas (2007a), we find that children with more exposure to interactive group activ-




ities have higher mathematics achievement. This finding indicates that interacting ingroups and giving and receiving help is positively associated with the mathematicsachievement of kindergartners. Also consistent with previous literature (Milesi &Gamoran, 2006), we find that more exposure to drills is associated with higher mathachievement of kindergartners. Our findings also show that this instructional tech-nique is particularly effective for children with high levels of math skills at kinder-garten entry (particularly Whites and Latino/as). The results concerning the use ofmanipulatives contrast with findings that appear in prior research. We do not findevidence that manipulatives increase math achievement of students. Rather, we findinsignificant effects for most categories of students, and a very troubling negativeassociation of exposure to manipulatives for Black students’ math achievement.Lastly, findings that Black students’ math achievement is negatively associated withhigher exposure to the math/movement instructional practices also challenges priorresearch on the topic.

It is important to remember that curricular practices may be not implemented inthe same ways in different communities, and this could explain why we find thepuzzling results regarding the effects of manipulatives and music/movement on

Table 10. Coefficients for Race and Curriculum Factors from HLM Analysis of MathAchievement in Kindergarten by Levels of Math Academic Preparedness

Math Academic Readiness

LowEstimate (SE)

MiddleEstimate (SE)

HighEstimate (SE)

Intercept 35.27 (.73) *** 35.82 (.74) *** 35.93 (.73) ***Instructional practice:

Drills .13 (.05) ** .28 (.05) *** .18 (.06) ***Interactive group activities .12 (.06) * .04 (.07) .15 (.07) *Music/movement �.03 (.24) �.10 (.25) .10 (.28)Manipulatives .05 (.06) �.06 (.07) .01 (.07)

Race/ethnicity:Black 1.78 (1.22) �3.00(1.62) � �.60(2.55)Latino/a �.49 (1.13) �.49 (1.48) �2.10 (2.40)Asian 1.88 (3.56) �.09(3.05) �3.18 (2.92)White .00 .00 .00

Instructional practice � race/ethnicity interactions:Drills � Black �.21 (.14) .07 (.11) �.13 (.16)Drills � Latino/a �.18 (.14) �.05 (.10) .16 (.14)Drills � Asian �.55 (.36) .12 (.19) .06 (.22)Interactive group activities � Black �.25 (.18) .10 (.16) .46 (.23) **Interactive group activities � Latino/a �.11 (.30) .02 (.13) .06 (.22)Interactive group activities � Asian �.66 (.47) .07 (.25) �.11 (.26)Music/movement � Black �1.62 (.18) � �.70 (.54) �1.47 (.82) *Music/movement � Latino/a �.61 (.85) �.24 (.47) .34 (.76)Music/movement � Asian �1.07(2.77) �.31 (1.00) 1.02 (.98)Manipulatives � Black �.44 (.02) ** �.01 (.15) �.37 (.20) *Manipulatives � Latino/a �.03 (.10) .01 (.13) �.06 (.20)Manipulatives � Asian �.02 (.30) �.09 (.23) .38 (.26)

Note.—Controls for all variables described in Table 1. Standard errors in parentheses.�

p � .10.

*p � .05.

**p � .01.

***p � .001.




Black students’ mathematics achievement. In fact, the two practices that have a dif-ferential effect on math achievement of kindergartners are the particular practicesthat are subject to more variation from individual teachers’ subjective style. In con-trast, instructional practices like drills are relatively more impervious to teachingstyles and are more standardized among communities. Perhaps the negative effect ofexposure to manipulatives on Black students’ math achievement might be becausethe way that manipulatives may be utilized in classrooms differs across teachersworking with different populations.

Our results using large-scale nationally representative survey data (e.g., ECLS-K)are generally consistent with results from previous experiments and qualitative stud-ies conducted at a smaller scale. Our research also moves beyond most studies byexamining effects across racial/ethnic, socioeconomic, and math academic readinessgroups. This is an important advancement in the research on the topic given the greatdiversity in student preparation and achievement, as well as the evidence offered byour study that examining the relationship between instructional practices and math-ematics achievement for the entire sample of students masks significant variationsfor specific racial categories of students.

Notably, our analyses show that the implementation of instructional practices is acrucial part of the explanation of differences in levels of mathematics achievement,but we know that fidelity of the teachers’ implementation of the instructional mate-rials or instructional strategy is difficult to assess (NMAP, 2008). Therefore, thisimplementation requires more attention from policy makers because the early yearsare essential in the educational process. Children’s achievement trajectories appearto be established very early. Early successes promote later achievement and earlydifficulties become entrenched (Wilson, 2011).

Like all studies, this one has several limitations. The structure of the data was suchthat by factor analyzing the instructional practices we lost much of the importantinformation we had at the beginning. In addition, the use of dichotomous variables(once or more per week; yes or no) for the instructional practices is problematic froma perspective that recognizes that mathematics has a learning progression and be-cause there exists the possibility of significant variance within the same classification.Nevertheless, we simplified the analysis to ease interpretation of results. We had themost power for the analyses with White students due to their larger sample size.Because we obtained similar coefficients for other groups for certain instructionalpractices it is possible that other results could have been significant for other racialgroups if larger samples were available. The study’s focus on the youngest studentsmeans we do not know whether the patterns of findings hold for older youth fromdiverse race, ethnic, SES, and math academic readiness backgrounds. In future re-search we will analyze achievement data and curricular practices for older students todetermine whether and how the relationships between implemented curriculum andmath achievement change as students move through the grade structure.

Conclusions and Policy Implication

The effectiveness of instructional practices does not depend solely on the nature ofthe actual practice. Effectiveness involves a great deal of what each teacher does toimplement a practice in each separate community of students. The focus of thisarticle on how mathematics is taught sought to investigate whether certain instruc-




tional practices diminish or contribute to race- and SES-related gaps within schools(Wenglinsky, 2004). Our goal required us to investigate whether similar instruc-tional practices, independent from children’s levels of math academic preparedness,had a differential effect on the math achievement of children from different race/ethnicity and SES backgrounds.

Using the theoretical framework that holds national mathematics standards rep-resent the formal mathematics curriculum, instructional practices reflect the imple-mented math curriculum, and student achievement manifests the attained mathe-matics curriculum, we examined in great detail how aspects of the implementedmathematics curriculum affect the achieved curriculum among a nationally repre-sentative sample of kindergarten students. We demonstrate that children from di-verse race, SES, and math academic readiness backgrounds learn best from differentaspects of the implemented curriculum.

Our research contributes to the literature on how race, SES, and readiness differ-ences among children moderate the relationship of mathematics curricular practicesto learning among young children. First, it empirically highlights the importance ofinstructional practices needed to close achievement gaps. Next, findings reinforcethe need for early childhood education that readies all children for instruction in theformal curriculum once they arrive in kindergarten. Third, the study highlights notonly racial and socioeconomic differences, but also the socioeconomic differenceswithin races. This is a critical set of findings that demonstrate the origins of thesubstantial differences in the mathematics achievement of older Black high-SES andBlack low-SES students (Moller, Stearns, Blau, & Land, 2006). Fourth, our studyexplores academic readiness differences within racial and socioeconomic groups anddemonstrates how specific instructional practices affect students’ mathematicsachievement among those with various levels of academic readiness. Finally, we showthat students who enter kindergarten with different levels of preparation do notnecessarily benefit equally from all components of the curriculum. Our findings offerpolicy makers and educators a clearer understanding of the importance of instruc-tional practices and how they differentially affect students from different back-grounds.

An often-heard mantra among policy makers and politicians maintains that intoday’s increasingly technological world, it is imperative that schools prepare thenext generation of Americans to excel in mathematics and science. In order to in-crease the achievement of children in American schools, the federal government haslaunched campaigns that focus on increasing the quality of the curriculum for allstudents. This study shows that the quality of the curriculum is only part of theanswer. There are significant differences in the way instructional practices foster orundermine the mathematics achievement of kindergarten students depending ontheir racial, ethnic, socioeconomic, and math academic readiness backgrounds.Considering that by 2030, more than 50% of the U.S. student population will belongto racial/ethnic minority groups, and the relative size of the middle class is shrinkingas a proportion of the population, it is increasingly necessary to instruct mathematicsin ways that maximize all students’ achievement. Moreover, it is especially importantto find ways to boost children’s mathematics achievement in the early grades, giventhat elementary school sets the path for later academic development.

A thorough examination of the implemented mathematics curriculum with a lenstoward diversity is required if we wish to ensure that all students are able to enter the




race to the top. As Kersaint, Thompson, and Petkova (2009) suggest, “teaching inways which are culturally responsive is an environment that enables all students tolearn.” Mathematics instruction should be both consistent with curricular standardsand tailored to benefit the diverse population of children that attend schools. Math-ematics curricula, as currently implemented, seem to leave portions of the studentpopulation behind.

Appendix

Notes

The research reported here was supported by the Institute of Education Sciences, U.S. Departmentof Education, through grant R305A100822 to the University of North Carolina at Charlotte. Theopinions expressed are those of the authors and do not represent the views of the Institute or theU.S. Department of Education.

1. There is a great deal of overlap among the mathematics topics deemed appropriate for kin-dergartners by both the National Council of Teachers of Mathematics (NCTM) and the CommonCore Standards in Mathematics (CCSM) for kindergarten students. The National Council ofTeachers of Mathematics (2006) defined as the most important math topics for lasting learning

Table A1. Question STEM for Instructional Practices in ECLS-K Survey Instrument

1 �Never

2 �Once

a Month

3 �2 or 3Times

a Month

4 �Once orTwice

a Week

5 �3 or 4Times

a Week6 �

Daily

1. Count out loud2. Work with geometric manipulatives3. Work with counting manipulatives to

learn basic operations4. Play math-related games5. Use a calculator for math6. Use music to understand math concepts7. Use creative movement or creative

drama to understand math concepts8. Work with rulers, measuring cups,

spoons, or other measuring instruments9. Explain how a math problem is solved

10. Engage in calendar-related activities11. Do math worksheets12. Do math problems from their textbooks13. Complete math problems on the

chalkboard14. Solve math problems in small groups or

with a partner15. Work on math problems that reflect

real-life situations16. Work in mixed achievement groups on

math activities17. Peer tutoring

Note.—Questions 1, 5, 8, 10, 13, and 16 were not included in our factor analysis because previous analyses had shown that they

had no significant relationship with the math achievement of kindergartners.




during the kindergarten year the following: representing, comparing, and ordering whole numbersand joining and separating sets; describing shapes and space; and ordering objects by measurableattributes. The Common Core Standards in Mathematics (2011) recognized that all kindergartnersshould learn about number names and the count sequence, counting the number of objects,comparing numbers, understanding addition as putting together and “adding to,” understandingsubtraction as taking apart and “taking from,” working with numbers 11–19 to gain foundations forplace value, describe and compare measurable attributes, classifying objects and counting thenumber of objects in categories, identifying and describing shapes, and being able to analyze,compare, create, and compose shapes .

2. We excluded Native Hawaiian, other Pacific Islanders, American Indians, and multiracialstudents due to small sample sizes.

3. Scaled variables are imputed with the Markov Chain Monte Carlo method because we havean arbitrary missing data pattern (Schafer, 1997). Categorical variables are imputed with a logisticregression method.

4. We recognize that math IRT scores do not necessarily represent everything a child knowsabout the subject or how much of the curriculum the child has achieved. Nevertheless, math IRTscores are measures of learning and achievement that are important to parents, teachers, andschool administrators.

5. We include these variables as categorical variables, rather than as interaction terms, becausethese variables will be interacted with factors of instructional practices. This would have necessi-tated four-way interactions (race � SES � academic readiness � instructional practices). Ourapproach requires only two-way interactions, which is much easier for the reader to interpret.

6. Altogether, we identify eight curriculum factors we label as Estimation and Recognition ofMath Concepts, Interactive Group Activities, Adding and Subtracting Single Digits, Manipulatives,Drills, Place Value and Three Digits, Music and Movement, and Adding Two Digit Numbers. Inthis study we focus only on the effect of the factors related to instructional practices because astudent without necessary prerequisite skills would not benefit from instruction that requires thoseskills, and because frequency of instructional practices is something malleable by individual teach-ers at schools rather than the content of the curriculum, which is almost unchangeable withoutmajor policy changes.

7. We included random effect for the categories of race-SES, race-academic readiness, SES-academic readiness, race-SES-academic readiness as necessary in each model.

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