Foundations of Mathematics Achievement

Foundations of Mathematics Achievement: Instructional Practices and Diverse Kindergarten
Students
Author(s): Martha Cecilia Bottia, Stephanie Moller, Roslyn Arlin Mickelson, and Elizabeth
Stearns
Source: The Elementary School Journal, Vol. 115, No. 1 (September 2014), pp. 124-150
Published by: The University of Chicago Press
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FOUNDATIONS OF MATHEMATICS
ACHIEVEMENT
Instructional Practices and Diverse Kindergarten
Students
Martha Cecilia Bottia
Stephanie Moller
Roslyn Arlin Mickelson
Elizabeth Stearns
university of north
carolina at charlotte
abstract
Analyzing Early Childhood Longitudinal Survey—
Kindergarten (ECLS-K) data, we examine how expo-
sure to instructional practices influences math test
scores at the end of kindergarten for children from
different racial/ethnic and socioeconomic back-
grounds, and for children with different levels of math
skills at kindergarten entry. We also analyze the rela-
tionship between socioeconomic background and
math academic readiness within racial/ethnic catego-
ries. Our results demonstrate that race/ethnicity and
levels of math academic readiness moderate the rela-
tionship between instructional practices and math
achievement. While we find that interactive group ac-
tivities enhance students’ mathematics achievement
in kindergarten and that drills enhance math aca-
demic achievement of students with high math aca-
demic preparedness in kindergarten, we also find that
use of manipulatives as well as music and movement
have significant negative effects on mathematics
achievement of Black students. Given the importance
of kindergarten for launching children onto success-
ful academic trajectories, the findings have implica-
tions for addressing racial/ethnic and socioeconomic
status 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

M
A 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 (National
Research Council, 2009). The mathematics education children receive
in 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 are
essential components of this early mathematics education.
Drawing upon the theoretical framework that holds that national mathematics
standards represent the formal mathematics curriculum, instructional practices re-
flect the implemented math curriculum, and student achievement manifests the
attained mathematics curriculum (Suter, 2000; Travers & Westbury, 1989), we ex-
amine how aspects of the implemented mathematics curriculum affect the achieved
curriculum 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 socioeconomic
background and academic readiness within race and SES groups plays an important
role 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 affect
the mathematics achievement of kindergarten students whose backgrounds differ in
terms 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 diminish
achievement gaps within schools (Wenglinsky, 2004). In addition, instructional
practices are elements of the curriculum that teachers are best positioned to influ-
ence (Lubienski, 2006).
We concentrate on the kindergarten curriculum because a strong mathematics
foundation at the onset of formal schooling is essential for a student’s long-term
success. Indeed, within mathematics there is a specific progression of concepts that
must be mastered before the next concepts can be presented by the teacher and
learned 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 regarding
how to design policies that reduce educational inequalities and improve the overall
achievement of students with various racial/ethnic, SES, and math academic readiness
backgrounds. Unlike most other studies of early mathematics performance, we explore
differences 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. student
populationandtherelativelyhighratesofgrowthamongthesubpopulationsthattendto
perform poorly in mathematics (Mickelson, Bottia, & Lambert, 2013). We pursue this
math instruction and diversity ⅐ 125

research through multilevel modeling techniques using data from the Early Childhood
Longitudinal Study—Kindergarten Cohort (ECLS-K).
Theoretical Background
Instructional Practices and Mathematics Achievement
Previous studies by the International Association for the Evaluation of Educational
Achievement determined that any country’s national curriculum can be defined by top-
icsthatareintendedbytheschoolsystem,implementedintheclassroom,andattainedby
the students (Suter, 2000; Travers & Westbury, 1989). The intended, implemented, and
attained curricula are developed simultaneously within an education system and to-
gether play a crucial role in the development of students’ education. Each curriculum
shapes 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 curriculum
determines the concepts to be learned and their sequence.
Importantly, the formal curriculum can be modified by different aspects of teaching
practices and consequently results in the implemented curriculum, which often is to
varying 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 reflects
the 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-
searchfocusesontheimplementedcurriculum,operationalizedasmathematicsinstruc-
tional practices teachers perform in kindergarten classes, as it influences the attained
curriculum, 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 between
instructional 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 with
teacher modeling. There is no single ideal approach to teaching mathematics; the
students, the mathematical goals, the teacher’s background and strengths, and the
instructional context all matter.
Specific Instructional Practices
Kindergartners are exposed to various instructional approaches in order to gain nec-
essarymathknowledge.1
First,theuseofmanipulativesiscommoninkindergartenclass-
rooms. Manipulatives are defined as “physical objects that are used as teaching tools to
engage students in the hands-on learning of mathematics” (Teacher Vision, 2013) that
allow children to use concrete objects to observe, model, and internalize abstract con-
cepts,thereforeprovidingacommonlanguagewithwhichtocommunicatethesemodels
to other students and the teacher (Ruzic & O’Connell, 2001). Manipulatives are believed
to bridge the gap between the world in which children live and the abstract world of
126 ⅐ 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 that
85% of them rated use of manipulatives as a highly effective instructional tool—rated
higher 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 and
calculus, thereby giving children hands-on experience with formal procedures nec-
essary in advanced math (Scott-Clayton, 2012). “Drills” have been found to positively
predict math achievement (Milesi & Gamoran, 2006).
In addition to drills, students are often taught through interactive group practices
in kindergarten. By interacting in groups, children give and receive help—both of
which 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 emotional
or 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 learn
novel information that they are unlikely to discover on their own (Slavin, 1977).
More recently, teachers in kindergarten classrooms have started using music and
movement to teach math. Existing research suggests that there are many benefits of
using music, and many means of incorporating it into mathematics instruction
(Yoho, 2011). Music keeps students alert, ready to learn, and actively engaged. Music
provides 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 of
their senses; the more senses involved in learning, the greater the understanding
(Palmer, 2001). In fact, Southgate and Roscigno (2009) found that there are clear
benefits 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 of
instructional practices for math gains in first grade (Palardy & Rumberger, 2008).
This study found that teachers’ instructional practices, specifically, frequency of use
of math worksheets and frequency of work on problems with calendars, had a sig-
nificant positive relationship with math achievement gains. Yet, in aggregate, the
corpus of research on instructional practices and mathematics achievement does not
provide much insight into how instructional practices affect the math achievement
of students from diverse racial, ethnic, SES, and academic readiness backgrounds.
The Moderating Role of Student Attributes on the Relationship between
Instructional 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 and
learning are cultural activities, one might expect that students with different ethnic
and 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) prioritize
communal living and cooperative problem solving, and these preferences affect ed-
ucational motivation, aspiration, and task performance; and (b) have norms for
appropriate 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 and
school may lead to a lack of parental involvement (Espinosa, 2005) and weak
student-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-based
language/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, who
tend to be more flexible and fluid rather than structured in their perception of ideas
because their culture emphasizes interaction with the environment (Malloy & Jones,
1998). Stiff (1999) and Gilbert and Gay (1985) found that many African American
students 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 of
fourth graders, found that an emphasis on topics of measurement and estimation
“was the most beneficial practice” for Black students, while an emphasis on data
analysis appeared to be beneficial for Latino/a students. Utilizing the same NAEP
data for fourth graders, Lubienski (2006) found that the factor related to collabora-
tive problem solving more often had a positive correlation with the achievement of
Black and Latino/a students than White students. Scholars have also found that
interactive group activities are better suited for ethnic groups whose cultural envi-
ronments value the welfare of the group over the individual and where individuals
are taught to pool their resources to solve problems (Gay, 2002). In fact, the positive
benefits of communities of learners and cooperative efforts on student achievement
previously have been validated for Latino/a (Escalanté & Dirmann, 1990), African
American (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 instructional
practices 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) found
that there are clear benefits of music involvement in school on math achievement
of small children, with White students receiving more benefits than African
American, Latino/a, and Asian students from music involvement during early
childhood 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 parents
possess that affect the capacities children will have to take advantage of what is taught
in schools and to comply with the requests of teachers. Among the important social
class differences are variable levels of cultural capital (Lareau, 2000a, 2000b) and
family 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 in
children’s learning before entering kindergarten and during the kindergarten year.
Consequently, children of higher SES have higher levels of academic readiness that
increase their chances of obtaining benefits from instructional practices. On the
other hand, stress models argue that children from lower-SES backgrounds have
parents who are less effective, and they are more prone to health problems that
directly and indirectly affect kindergarten students’ levels of academic readiness and
the context in which students learn during their first year. As a consequence, children
from lower-SES backgrounds have fewer resources with which to take advantage of
instructional practices. Bodovski and Farkas (2007b) found that academic achieve-
ment is influenced by the academic and social abilities that different students bring to
schools at entry that are correlated with race and SES. Disadvantaged children start
kindergarten with significantly lower skills than their more privileged counterparts
and are therefore unevenly equipped to initiate their learning processes.
Hickey, Moore, and Pellegrino (2001) analyzed how instructional practices affect
students 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 and
estimation portion of the Iowa Test of Basic Skills. However, other research found
that reform-minded practices are particularly beneficial for lower-SES and minority
students (Boaler, 2002; Stiff, 1999).
Differences by academic readiness. Math academic readiness is related to SES
background. Academic readiness refers to a number of language, mathematics, small
motor, and personal/interpersonal skills among young children entering kindergar-
ten. Students’ varying levels of academic readiness condition how much children are
likely to understand and benefit from the curriculum to which they are exposed in
classrooms. As such, academic readiness becomes a key predictor of long-term
achievement trajectories (Bodovski & Farkas, 2007a). Indeed, previous research has
shown that math academic readiness is an important predictor of subsequent school
achievement in math (Duncan et al., 2007).
Although the importance of academic readiness on mathematics achievement has
been recognized extensively in the past, research that specifically focuses on the
potential 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’ school
careers is associated with students’ subsequent gains. Students who began with the
most 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 kindergarten
and in first grade and found that certain instructional practices can produce modest
reductions in achievement gaps between African American and White students in
kindergarten. However, they found no significant effects of instruction on the
achievement 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 fourth
and eighth grade and found, in contrast to Wenglinsky (2004), that the relationship
between various instructional practices and achievement was roughly similar for
White, Black, and Latino/a students.
We build on the previously discussed research by examining the intersection of
race/ethnicity, SES, and academic readiness as possible moderating factors between
instructional 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, African
American, Latino/a, or Asian American, from either low-, middle-, or high-SES
families, and who have low, middle, and high levels of math academic readiness. We
predict 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 interactive
group activities) are more beneficial for African American students than for White
students; (Hypothesis 2b) interactive group activities that emphasize the welfare of
the group are more beneficial for Latino/a and Asian students than for White stu-
dents; (Hypothesis 2c) music and movement practices that express creativity and
nonverbal communication are more beneficial for African American students than
for White students; (Hypothesis 3) socioeconomic status and math academic readi-
ness moderate the relationship between instructional practices and mathematics
achievement; (Hypothesis 3a) drills, which require more previous knowledge, ben-
efit better prepared students; and (Hypothesis 3b) use of manipulatives, a fairly
simple instructional practice that requires little previous knowledge, is more bene-
ficial for less academically ready students than for students with higher academic
readiness.
Because there is little research looking at the interactions between race, socioeco-
nomic status, and academic readiness, we investigated Hypotheses 4 and 5 in an
exploratory 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 practices
and mathematics achievement should vary among the combined academic readiness
and 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 on
family, school, community, and individual factors associated with the performance
of students at schools (ECLS-K website).
Given our research interests, we limit the sample to White, Black, Latino/a, and
Asian 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 are
not repeating kindergarten because their experiences and needs are different from
first-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; Schafer
1997).3
In order to ensure high efficiency, we determined a priori to impute only
variables that are missing less than 20% of cases within waves. Most of our variables
have less than 10% missing data and are thus imputed. The imputation is more than
93% efficient for all imputed variables in all waves.
After listwise deletion of cases whose missing data could not be imputed, our final
sample includes 13,670 White, Black, Latino/a, and Asian students who attended
kindergarten in 1998. A comparison of this final sample to the initial sample indicated
that the final sample is not dramatically different from the initial sample in terms of
race, 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% prior
to sample selection), and math scores (the average kindergarten scores were 36.6 in
the initial sample, and 37.1 in the final sample).
Outcome Variable
The main dependent variable for this study is students’ mathematics achievement
in the spring of kindergarten. Math achievement is measured through item response
theory (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 all
questions for the math test in multiple waves (Tourangeau, Nord, Le, Pollack, &
Atkins-Burnett, 2009).4
We analyze spring IRT scores because these scores permit
evaluation of achievement trajectories over time with age-appropriate tests. In this
way, these measures can be compared over time.
Predictor Variables
The key independent variables of interest are the frequency of use of certain
mathematics instructional practices for specific curricular content areas at the kin-
dergarten level. These data come from the ECLS-K spring teacher questionnaire
where teachers are asked to respond to the following questions: “How often is each of
the following MATH skills taught in your class?” and “How often do children in this
class do each of the following MATH activities?” Teachers choose from the options
never, once a month, two or three times a month, once or twice a week, three or four
times a week, or daily (see App. Table A1). There are 17 process variables and 29
content variables in ECLS-K data. We focus on the 17 process variables that reflect
instructional practices teachers use in their classrooms and better reflect the imple-

mented curriculum. Many of these instructional practices have been analyzed in
earlier 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 (White
is the omitted reference category). ECLS-K provides a continuous measure of socio-
economic status that utilizes family income, parental education, and occupation as
inputs. For ease of interpretation of the analyses, we created an ordinal measure of
SES (low, middle, and high SES) based on the distribution of the continuous SES
measure (high SES is the omitted category). We also created an ordinal measure of
math academic readiness by dividing the sample of students into terciles based on the
math IRT scores students have at the beginning of the kindergarten year (low aca-
demic readiness is the omitted category). Since there is substantial variability within
terciles, we also acknowledged the presence of this variability by controlling for
previous math score, where math scores are centered within terciles. To further test
our hypotheses regarding the intersection between race, SES, and math academic
readiness categories, we created categorical variables for race socioeconomic (White,
high SES is the omitted reference category), race by academic readiness (White, high
academic 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 primary
independent variables. Our individual-level controls include gender, age, and mea-
sures of cultural capital, including English as a second language and socioeconomic
status of the child in kindergarten. We also control for reading scores to account for
the 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 full
day), time spent in math, teacher’s race (Black or Latino/a, with White as the omitted
category), teacher enjoys teaching (1 ϭ yes), and teacher’s highest education (1 ϭ
high school to 7 ϭ doctorate). Lastly, the school-level controls are school size
(logged), percent of students in the school who are Black, percent of students in the
school who are Latino/a, region (south is omitted category), rural/suburban (urban
is omitted category), school is private or not, and whether or not the school was a
magnet school or a charter school. Control variables are explained in detail in Table
1.
Analytic Strategy
We conducted our analyses in three stages. First we ran 322 models with each of
the 46 curriculum variables for each race and SES to clarify which practices are more
closely related to students’ mathematics achievement by race/ethnic and SES back-
ground. In this stage, we maintain the original ordinal nature of the data (response
options ranged from 1 to 6) to permit detailed nonlinear results. We do not review
those results in detail given the sheer complexity of discussing 322 models. We in-
clude both teaching practices and content because practices are partially determined
by content. These detailed analyses of each curriculum variable separately pro-

vided two important insights: (1) 17 of these curricular practices and content
variables are not associated with the mathematics achievement of any racial/
ethnic or socioeconomic group (we drop these variables from the second stage of
our analysis), and (2) in most models, the significant curriculum variables have
the greatest effect on achievement when students are exposed to the curricular
process or content at least once a week. Therefore, prior to moving on to the
second 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 or
more and 0 for less than once per week.
In the analyses’ second stage, we combined the significant process and content
variables from stage 1 into a smaller set of variables based on results from a factor
analysis. This is necessary because instructional practices and content do not happen
independently 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 ϭ private, 0 ϭ public
Percent African American Percentage of African American students in school
Percent Latino/a in school Percentage of Latino/a students in school
School size Category of school size, 0–149, 150–299, 300–499, 500–749, and 750 and
above
Rural 1 ϭ rural, 0 ϭ not rural
Region of the country Dummy variables for Midwest, West, and Northeast; South is the
reference category
Magnet 1 ϭ magnet school, 0 ϭ not magnet school
Charter 1 ϭ charter school, 0 ϭ not charter school
Classroom level:
Full-day class 1 ϭ full-day class, 0 ϭ part-time class
Time spent in math Time spent in mathematics instruction
African American teacher 1 ϭ teacher was African American, 0 ϭ teacher was not African American
Latino/a teacher 1 ϭ teacher was Latino/a, 0 ϭ teacher was not Latino/a
Enjoys teaching Continuous variable from 1 to 5 that tells whether teacher strongly
disagrees, disagrees, neither agrees or disagrees, agrees, or strongly
agrees with the statement: “I really enjoy my present teaching job.”
Teacher’s education Category of highest educational level teacher achieved: 1 ϭ high school,
2 ϭ associate’s degree, 3 ϭ bachelor’s degree, 4 ϭ more than 1 year of
coursework beyond bachelor’s, 5 ϭ master’s, 6 ϭ education specialist/
professional diploma, 7 ϭ doctorate
Student level:
Race White (non-Latino/a), African American (non-Latino/a), Latino/a, and
Asian American
Socioeconomic status Composite of five variables: father’s education and occupation, mother’s
education and occupation, and household income. SES is categorized
as 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; based
on the previous math item response theory (IRT) score
Previous mathematics score Fall IRT scores for kindergartners centered by math academic readiness
terciles
Previous reading score Fall reading IRT scores for kindergartners
Age The number of months of life at entry to kindergarten
Male 1 ϭ male, 0 ϭ female
Not English at home 1 ϭ child does not speak English at home, 0 ϭ 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 confirmatory
factor 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 these
factors 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 or
with a partner, solving real-life math, explaining/solving math problems, and peer
tutoring. Manipulatives included the practices of using geometric and counting ma-
nipulatives and using math-related games. Drills contain doing math worksheets and
using 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 their
loadings.
In the final stage of the analyses, we run multilevel regressions to test the effects of
the extracted instructional practices factors on mathematics achievement across
race/ethnicity, SES, and levels of academic readiness. We present interactions be-
tween the factors and racial/ethnic categories, SES categories, and math academic
readiness 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 950
low-SES Black, 300 high-SES Black, 600 middle-SES Black, 1,170 low-SES Latino, 380
high-SES Latino, 660 mid-SES Latino, 240 low-SES Asian, 370 high-SES Asian, 210
middle-SES Asian, 1,840 low-SES White, 3,810 high-SES White, and 3,140 middle-SES
White students. These students vary in their levels of academic readiness. Table 3
presents data from achievement tests given to children in the fall and spring of
kindergarten as part of the ECLS-K by racial, socioeconomic, and racial-
socioeconomic groups. We see that White low-SES and Black mid-SES students
enter kindergarten with similar achievement (both groups averaged 24 points on the
fall kindergarten mathematics achievement test). White high-SES students begin
Table 2. Factors with Variables and Loadings
Interactive Group
Activities Manipulatives Drills Music/Movement
Frequency geometric manipulatives Ϫ8 70a
Ϫ8 6
Frequency counting manipulatives 6 85a
1 Ϫ6
Frequency math-related games 16 61a
Ϫ12 13
Frequency music to learn math 3 1 7 78a
Frequency movement to learn math Ϫ3 7 Ϫ2 93a
Frequency explain/solve math problems 56a
Ϫ5 10 3
Frequency do math worksheets 5 Ϫ2 79a
Ϫ2
Frequency use math textbooks 9 Ϫ19 60a
5
Frequency solve math with partner 66a
19 12 Ϫ4
Frequency solve real life math 82a
Ϫ9 2 Ϫ1
Frequency peer tutoring 49a
14 8 1
Note.—Results of exploratory factor using a tetrachoric correlation matrix and promax rotation.
a
Indicates highest loading for each item.

school with a 7-point advantage on the math achievement test, compared to their
low-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 only
a 5-point differential between low- and high-SES Black students. Table 4 provides the
sample sizes when levels of math academic readiness were crossed with SES and race.
Because students are nested within classrooms, and classrooms are nested within
schools, we utilize three-level hierarchical linear models with random slopes (HLM).
HLM models are appropriate because they adjust errors to account for the lack of
independence among students and classrooms (Raudenbush & Bryk, 2002). We run
regressions with the BYCOMW0 longitudinal weight—appropriate when examining
assessment data from the fall and spring of kindergarten—to ensure generalizability
of the results. The equations for the models with categories of race, SES, and math
readiness are shown below.
Level 1 Model (Child):
Mathijk ϭ ␤0jk ϩ ␤1jkx ϩ ͚␤ijkchild control variables ϩ ␧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 around
academic readiness terciles; and gender, age in months, and English not spoken at
home centered around their grand means.
Level 2 Model (Classroom):
␤0jk ϭ ␩00k ϩ ␩01kCURR_FACTORS ϩ ͚n0nkclassroom variables ϩ ␣0jk,
␤1jk ϭ ␩10k ϩ ␩11kCURR_FACTORS ϩ ␣1jk.
Table 3. Average IRT Mathematics Scores (Means) by Racial/Ethnic Category and
Socioeconomic 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 1,170 660 380 240 210 370
Spring
kindergarten 34.0 37.8 43.1 28.9 32.8 36.1 28.7 34.2 38.6 34.6 35.4 44.5
Fall 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 with
Racial/Ethnic Category and Socioeconomic Status (SES)
Low Middle High
Black 870 670 310
Latino 1,150 660 400
Asian 210 280 330
White 1,990 3,020 3,780
Low SES 2,190 1,360 650
Middle SES 670 1,550 2,640
High 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 the
mathematics achievement of students; ␩0nk represents the extent that other grand
mean centered classroom-level variables (including teacher education, teacher race,
teacher satisfaction, time devoted to math in the classroom, full-day kindergarten
classroom) 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 are
significant 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 math
readiness on mathematics achievement, which at level 2 is also a function of an
intercept and the curriculum factors7
. We also include a random slope, ␣1jk.
Level 3 Model (School):
␩00k ϭ ␥000 ϩ ͚k00nschool variables ϩ ␸00k.
In the third-level equation, ␥000 is the random intercept and ␾00k is the third-level
error term associated with variation across schools. k00n represents the extent that
school-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 of
kindergarten, considering the nesting of students into classrooms, and the nesting of
classrooms into schools. In addition, our model also predicts how the association
between factors of instructional practices and mathematics achievement of students
varies by race, SES, levels of math academic readiness, race by SES, race by math
academic readiness, and SES by math academic readiness by including interaction
terms 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 are
commonly used in kindergarten classrooms, as most students are regularly exposed
to 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 movement
are less common teaching practices, although approximately one-third of students
are regularly exposed to these practices. Again, a larger proportion of students with
limited math academic readiness and low SES are exposed to music to learn math,
compared to students with high academic readiness and high SES, respectively. The
proportion of students exposed to drills and interactive group activities also varies
across categories of students. The variation in exposure to instructional practices by

Table5.ProportionofStudentsExposedtoInstructionalPracticesbyRacial/EthnicCategory,AcademicReadiness,andSocioeconomicStatus(SES)
MathAcademicReadiness
BlackWhiteLatino/aAsianLowMiddleHighLowSESMiddleSESHighSES
N1,8508,7902,2108204,2104,6404,8204,2004,6104,860
Instructionalpractice:
Manipulatives:
Geometricmanipulatives.83.81.79.73.79.77.73.80.75.74
Countingmanipulatives.93.93.89.92.93.92.91.94.92.91
Math-relatedgames.88.81.87.83.85.84.83.86.83.83
Music/movement:
Musictolearnmath.35.35.32.27.33.30.27.35.29.27
Movementtolearnmath.29.31.30.23.27.26.24.27.25.25
Drills:
Domathworksheets.76.71.68.67.72.69.67.73.70.65
Usemathtextbooks.31.26.24.24.24.25.26.25.25.25
Domathonchalkboard.47.42.34.33.39.36.34.40.36.33
Interactivegroupactivities:
Solvemathw/partner.62.58.54.49.56.52.50.56.50.51
Solvereal-lifemath.68.62.64.59.63.61.60.63.59.61
Peertutoring.51.46.44.38.45.41.39.47.40.38
Explain/solvemath.67.63.57.59.60.60.63.61.61.61

group indicates the feasibility of testing for significant differences in the role that
instructional 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 practices
on mathematics achievement (without interactions). Table 6 illustrates partial sup-
port for hypothesis 1 as students’ mathematics achievement is higher in the spring of
kindergarten when they study in classrooms where teachers frequently use drills and
interactive group activities. However, music and movement instructional practices
do not have an overall effect on students’ math achievement. Therefore, only some
instructional practices enhance mathematics achievement for all students.
Yet, we posit that instructional practices are differentially effective at enhancing
mathematics achievement across groups of students (see Hypotheses 2 and 3). To test
these 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 (Model
2), 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 effects
by examining slopes (in Table 8). Therefore, Table 8 only includes slopes for models
that had significant interactions in Table 7.
The results presented in Model 1 in Tables 7 and 8 do not offer support for the
second set of hypotheses. In contrast, interactive group activities do not significantly
interact 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 Math
Achievement in Kindergarten
Variable Coefficient (SE)
Intercept 35.89(.43)***
Racial/ethnic category:a
Latino/a Ϫ.70(.23)***
Asian .39 (.41)
Black Ϫ1.76(.24)***
Math academic readiness:b
Low academic readiness Ϫ1.57 (.26)***
Middle academic readiness Ϫ.42(.19)**
Instructional practice:
Interactive group activities .14 (.04)***
Manipulatives Ϫ.04(.03)
Drills .20(.03)***
Music/movement practices Ϫ.17 (.14)
Random effects:
Teacher intercept .36
School intercept 2.03***
Race intercept 2.16***
SES intercept 2.12***
Note.—Controls for all variables described in Table 1.
a
White is excluded category.
b
High academic readiness is excluded category.
**p Ͻ .01.
***p Ͻ .001.

music/movement practices are significant (as seen in Table 7, Model 1), the effect is
opposite than predicted in Hypothesis 2c (as seen in Table 8, Model 4). Instructional
strategies that include the use of music and movement are not beneficial for African
American students; it fact, they are detrimental to their mathematics achievement.
Hypothesis 3 focuses on SES and academic readiness. This hypothesis is tested in
Models 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 between
manipulatives 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. We
find that drills benefit students with low academic preparedness (as the slope for
drills 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 academic
readiness 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 presenting
results from 24 subsamples. Separating the sample in this way is necessary because
the hypotheses require three- and four-way interactions. Presenting the models sep-
arately for different subgroups overcomes issues with model instability that arise
with four-way interaction terms. Furthermore, we presented these hypotheses as
exploratory because the literature does not generate a priori expectations.
The results presented in Table 9 explore the effects of instructional practices for
groups 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/Ethnic
Category, Socioeconomic Status, and Math Academic Readiness Categories from HLM Analysis
of Math Achievement in Kindergarten
F-Value
Model 1: Interaction between race/ethnicity and each instructional practice factor:
Drills .22
Interactive group activities .69
Music/movement 2.78*
Manipulatives 2.09ϩ
Model 2: Interaction between SES and each instructional practice factor:
Drills .36
Interactive group activities .16
Music/movement .27
Manipulatives .19
Model 3: Interaction between math academic readiness and each instructional practice factor:
Drills 4.40*
Interactive group activities 1.23
Music/movement .34
Manipulatives .18
Note.—Controls for all variables described in Table 1.
ϩ
p Ͻ .10.
*p Ͻ .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 is
particularly interesting to realize that they are exposed to these practices with less
frequency than are the other racial/ethnic groups (see Table 5).
Table 8. Regression Coefficients and Standard Errors from Models in Table 7 with Significant
Results; 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 Ϫ1.12 (.89) Ϫ.68(.21)***
Asian Ϫ.98(1.91) .52 (.39)
Black Ϫ.21 (1.00) Ϫ1.78 (.23)***
Math academic readiness:b
Middle academic readiness – .66(.81)
High academic readiness – .66(.79)
Interactive group activities .12 (.04)*** .13 (.05)***
Manipulatives Ϫ.02 (.04) Ϫ.03 (.05)
Drills .19 (.04)*** .11 (.04)***
Music/movement practices Ϫ.05 (.16) Ϫ.30 (.05)
Race/ethnicity ϫ instructional practice interactions: –
Latino/a ϫ interactive group activities .01 (.06) –
Latino/a ϫ manipulatives Ϫ.01 (.08) –
Latino/a ϫ drills .07 (.08) –
Latino/a ϫ music/movement practices Ϫ.05 (.29) –
Asian ϫ interactive group activities .08 (.12) –
Asian ϫ manipulatives .14 (.15) –
Asian ϫ drills Ϫ.10 (.16) –
Asian ϫ music/movement practices .60 (.58) –
Black ϫ interactive group activities .08 (.09) –
Black ϫ manipulatives Ϫ.18 (.08)** –
Black ϫ drills Ϫ.01 (.07) –
Black ϫ music/movement practices Ϫ.87 (.33)** –
Math academic readiness ϫ instructional practice interactions:
Medium readiness ϫ interactive group activities – .05 (.07)
Medium readiness ϫ manipulatives – .00(.06)
Medium readiness ϫ drills – .01 (.05)
Medium readiness ϫ music/movement practices – .11 (.28)
High readiness ϫ interactive group activities – Ϫ.04(.07)
High readiness ϫ manipulatives – Ϫ.02(.06)
High readiness ϫ drills – .15 (.05)***
High readiness ϫ music/movement practices – Ϫ.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.
a
White is excluded category.
b
Low academic readiness is excluded category.
**p Ͻ .01.
***p Ͻ .001.

Table9.HLMCoefﬁcientsPredictingMathAchievementinKindergartenfrom24SubsamplesofStudentsbyRace/Ethnicity,SocioeconomicStatus,andMathAcademic
Readiness
WhiteBlackLatino/aAsian
InstructionalPractice
Low
SES
Middle
SES
High
SES
Low
SES
Middle
SES
High
SES
Low
SES
Middle
SES
High
SES
Low
SES
Middle
SES
High
SES
Drills.25***.18***.20***.18*.27**Ϫ.08.15**.12.10.40*.49ϩ
.31ϩ
(.06)(.05)(.05)(.09)(.09)(.15)(.07)(.10)(.13)(.21)(.24)(.16)
Interactivegroupactivities.08.12**.14**.12.32**.38**.17*.27**.02Ϫ.06Ϫ.03.10
(.08)(.06)(.07)(.10)(.12)(.19)(.09)(.12)(.17)(.26)(.22)(.19)
Music/movementϪ.03Ϫ.01Ϫ.03Ϫ.75ϩ
Ϫ1.10*Ϫ1.72***Ϫ.12.01Ϫ.101.10.27.34
(.30)(.24)(.26)(.43)(.47)(.66)(.32)(.48)(.60)(.91)(.85)(.78)
Manipulatives.05Ϫ.01Ϫ.02Ϫ.22**Ϫ.15Ϫ.32**Ϫ.02Ϫ.07Ϫ.07Ϫ.04Ϫ.14.09
(.08)(.06)(.06)(.10)(.11)(.15)(.09)(.12)(.15)(.27)(.22)(.18)
AcademicReadiness,WhiteAcademicReadiness,BlackAcademicReadiness,Latino/aAcademicReadiness,Asian
LowMiddleHighLowMiddleHighLowMiddleHighLowMiddleHigh
Drills.11ϩ
.30***.17**.09.33**.12.08.16.36**.08.39.25
(.05)(.05)(.06)(.07)(.11)(.16)(.06)(.10)(.14)(.17)(.20)(.18)
Interactivegroupactivities.11ϩ
.04.15**.11.18.53**.23**.03.16Ϫ.06.01.07
(.06)(.06)(.07)(.08)(.13)(.23)(.08)(.12)(.20)(.25)(.22)(.21)
Music/movementϪ.03Ϫ.03.14Ϫ.74**Ϫ.88ϩ
Ϫ1.01Ϫ.09Ϫ.43.551.27Ϫ.27.88
(.25)(.24)(.29)(.34)(.50)(.80)(.29)(.45)(.67)(.89)(.99)(.80)
Manipulatives.06Ϫ.04.02Ϫ.21**Ϫ.07Ϫ.34**Ϫ.02Ϫ.08Ϫ.06.06Ϫ.20.33
(.06)(.06)(.07)(.08)(.13)(.19)(.08)(.11)(.18)(.20)(.19)(.21)
Note.—ControlsforallvariablesdescribedinTable1.Standarderrorsinparentheses.
ϩ
pϽ.10.
*pϽ.05.
**pϽ.01.
***pϽ.001.

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 tools
are 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 kindergarten
classes that regularly implement “music/movement” practices. Furthermore, over
90% of low- and high-SES and low- and high-math-readiness Black students work
with any component of the manipulatives factor at least once a week (see Table 5).
The regular exposure of students to manipulatives reflects the widespread perception
that manipulatives enhance student engagement resulting in higher achievement. It
is important to note that the nonsignificant effects for Asian students are partially
driven by sample size. Despite this, low-SES Asian students have higher mathematics
achievement in kindergarten when they study in classrooms where teachers employ
drilling.
Lastly, to further investigate whether the moderating effect of race or SES on the
relationship between instructional practices and math achievement is independent
of 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/or
SES (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 is
conducted by math academic readiness categories. Among students with low and
high levels of math academic readiness, there is a significantly different effect of
music/movement and manipulatives by race. Specifically, the math achievement of
Black students in the categories of low math academic preparedness and high math
academic preparedness decreases the more these students are exposed to music/
movement and manipulatives.
To summarize, we find evidence that in many instances, curriculum delivery is
differentially associated with students’ mathematics achievement depending upon
their 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 to
respond differently to the same curriculum. They are also consistent with Bodovski
and Farkas’s (2007b) study that found that academic achievement is influenced by
the race- and SES-correlated academic and social abilities that different students
bring to schools at entry.
Discussion
Our study recognizes the importance of instructional practices for the mathematics
achievement of kindergartners. Specifically, we find that the instructional practices
of 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 math
academic readiness on the relationship between implemented curriculum and kin-
dergartners’ math achievement. Consistent with Webb (2008) and Bodovski and
Farkas (2007a), we find that children with more exposure to interactive group activ-

ities have higher mathematics achievement. This finding indicates that interacting in
groups and giving and receiving help is positively associated with the mathematics
achievement of kindergartners. Also consistent with previous literature (Milesi &
Gamoran, 2006), we find that more exposure to drills is associated with higher math
achievement 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 of
manipulatives contrast with findings that appear in prior research. We do not find
evidence that manipulatives increase math achievement of students. Rather, we find
insignificant effects for most categories of students, and a very troubling negative
association of exposure to manipulatives for Black students’ math achievement.
Lastly, findings that Black students’ math achievement is negatively associated with
higher exposure to the math/movement instructional practices also challenges prior
research on the topic.
It is important to remember that curricular practices may be not implemented in
the same ways in different communities, and this could explain why we find the
puzzling results regarding the effects of manipulatives and music/movement on
Table 10. Coefficients for Race and Curriculum Factors from HLM Analysis of Math
Achievement in Kindergarten by Levels of Math Academic Preparedness
Math Academic Readiness
Low
Estimate (SE)
Middle
Estimate (SE)
High
Estimate (SE)
Intercept 35.27 (.73)*** 35.82 (.74)*** 35.93 (.73)***
Drills .13 (.05)** .28 (.05)*** .18 (.06)***
Interactive group activities .12 (.06)* .04 (.07) .15 (.07)*
Music/movement Ϫ.03 (.24) Ϫ.10 (.25) .10 (.28)
Manipulatives .05 (.06) Ϫ.06 (.07) .01 (.07)
Race/ethnicity:
Black 1.78 (1.22) Ϫ3.00(1.62)ϩ
Ϫ.60(2.55)
Latino/a Ϫ.49 (1.13) Ϫ.49(1.48) Ϫ2.10 (2.40)
Asian 1.88 (3.56) Ϫ.09(3.05) Ϫ3.18 (2.92)
White .00 .00 .00
Instructional practice ϫ race/ethnicity interactions:
Drills ϫ Black Ϫ.21 (.14) .07 (.11) Ϫ.13 (.16)
Drills ϫ Latino/a Ϫ.18 (.14) Ϫ.05 (.10) .16 (.14)
Drills ϫ Asian Ϫ.55 (.36) .12 (.19) .06 (.22)
Interactive group activities ϫ Black Ϫ.25 (.18) .10 (.16) .46 (.23)**
Interactive group activities ϫ Latino/a Ϫ.11 (.30) .02 (.13) .06 (.22)
Interactive group activities ϫ Asian Ϫ.66 (.47) .07 (.25) Ϫ.11 (.26)
Music/movement ϫ Black Ϫ1.62 (.18)ϩ
Ϫ.70 (.54) Ϫ1.47 (.82)*
Music/movement ϫ Latino/a Ϫ.61 (.85) Ϫ.24 (.47) .34 (.76)
Music/movement ϫ Asian Ϫ1.07(2.77) Ϫ.31 (1.00) 1.02 (.98)
Manipulatives ϫ Black Ϫ.44 (.02)** Ϫ.01 (.15) Ϫ.37 (.20)*
Manipulatives ϫ Latino/a Ϫ.03 (.10) .01 (.13) Ϫ.06 (.20)
Manipulatives ϫ Asian Ϫ.02 (.30) Ϫ.09 (.23) .38 (.26)
Note.—Controls for all variables described in Table 1. Standard errors in parentheses.
ϩ
p Ͻ .10.
*p Ͻ .05.
**p Ͻ .01.
***p Ͻ .001.

Black students’ mathematics achievement. In fact, the two practices that have a dif-
ferential effect on math achievement of kindergartners are the particular practices
that are subject to more variation from individual teachers’ subjective style. In con-
trast, instructional practices like drills are relatively more impervious to teaching
styles and are more standardized among communities. Perhaps the negative effect of
exposure to manipulatives on Black students’ math achievement might be because
the way that manipulatives may be utilized in classrooms differs across teachers
working 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 by
examining effects across racial/ethnic, socioeconomic, and math academic readiness
groups. This is an important advancement in the research on the topic given the great
diversity in student preparation and achievement, as well as the evidence offered by
our study that examining the relationship between instructional practices and math-
ematics achievement for the entire sample of students masks significant variations
for specific racial categories of students.
Notably, our analyses show that the implementation of instructional practices is a
crucial 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, this
implementation requires more attention from policy makers because the early years
are essential in the educational process. Children’s achievement trajectories appear
to be established very early. Early successes promote later achievement and early
difficulties become entrenched (Wilson, 2011).
Like all studies, this one has several limitations. The structure of the data was such
that by factor analyzing the instructional practices we lost much of the important
information 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 from
a 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 the
most power for the analyses with White students due to their larger sample size.
Because we obtained similar coefficients for other groups for certain instructional
practices it is possible that other results could have been significant for other racial
groups if larger samples were available. The study’s focus on the youngest students
means we do not know whether the patterns of findings hold for older youth from
diverse race, ethnic, SES, and math academic readiness backgrounds. In future re-
search we will analyze achievement data and curricular practices for older students to
determine whether and how the relationships between implemented curriculum and
math 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 of
the actual practice. Effectiveness involves a great deal of what each teacher does to
implement a practice in each separate community of students. The focus of this
article 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 implemented
mathematics 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 different
aspects 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 practices
to learning among young children. First, it empirically highlights the importance of
instructional practices needed to close achievement gaps. Next, findings reinforce
the need for early childhood education that readies all children for instruction in the
formal curriculum once they arrive in kindergarten. Third, the study highlights not
only racial and socioeconomic differences, but also the socioeconomic differences
within races. This is a critical set of findings that demonstrate the origins of the
substantial differences in the mathematics achievement of older Black high-SES and
Black low-SES students (Moller, Stearns, Blau, & Land, 2006). Fourth, our study
explores academic readiness differences within racial and socioeconomic groups and
demonstrates how specific instructional practices affect students’ mathematics
achievement among those with various levels of academic readiness. Finally, we show
that students who enter kindergarten with different levels of preparation do not
necessarily benefit equally from all components of the curriculum. Our findings offer
policy 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 in
today’s increasingly technological world, it is imperative that schools prepare the
next generation of Americans to excel in mathematics and science. In order to in-
crease the achievement of children in American schools, the federal government has
launched campaigns that focus on increasing the quality of the curriculum for all
students. This study shows that the quality of the curriculum is only part of the
answer. There are significant differences in the way instructional practices foster or
undermine the mathematics achievement of kindergarten students depending on
their racial, ethnic, socioeconomic, and math academic readiness backgrounds.
Considering that by 2030, more than 50% of the U.S. student population will belong
to racial/ethnic minority groups, and the relative size of the middle class is shrinking
as a proportion of the population, it is increasingly necessary to instruct mathematics
in ways that maximize all students’ achievement. Moreover, it is especially important
to find ways to boost children’s mathematics achievement in the early grades, given
that elementary school sets the path for later academic development.
A thorough examination of the implemented mathematics curriculum with a lens
toward 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 in
ways which are culturally responsive is an environment that enables all students to
learn.” Mathematics instruction should be both consistent with curricular standards
and tailored to benefit the diverse population of children that attend schools. Math-
ematics curricula, as currently implemented, seem to leave portions of the student
population behind.
Appendix
Notes
The research reported here was supported by the Institute of Education Sciences, U.S. Department
of Education, through grant R305A100822 to the University of North Carolina at Charlotte. The
opinions expressed are those of the authors and do not represent the views of the Institute or the
U.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 Common
Core Standards in Mathematics (CCSM) for kindergarten students. The National Council of
Teachers 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 ϭ
Never
2 ϭ
Once
a Month
3 ϭ
2 or 3
Times
a Month
4 ϭ
Once or
Twice
a Week
5 ϭ
3 or 4
Times
a Week
6 ϭ
Daily
1. Count out loud
2. Work with geometric manipulatives
3. Work with counting manipulatives to
learn basic operations
4. Play math-related games
5. Use a calculator for math
6. Use music to understand math concepts
7. Use creative movement or creative
drama to understand math concepts
8. Work with rulers, measuring cups,
spoons, or other measuring instruments
9. Explain how a math problem is solved
10. Engage in calendar-related activities
11. Do math worksheets
12. Do math problems from their textbooks
13. Complete math problems on the
chalkboard
14. Solve math problems in small groups or
with a partner
15. Work on math problems that reflect
real-life situations
16. Work in mixed achievement groups on
math activities
17. 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 numbers
and joining and separating sets; describing shapes and space; and ordering objects by measurable
attributes. The Common Core Standards in Mathematics (2011) recognized that all kindergartners
should learn about number names and the count sequence, counting the number of objects,
comparing numbers, understanding addition as putting together and “adding to,” understanding
subtraction as taking apart and “taking from,” working with numbers 11–19 to gain foundations for
place value, describe and compare measurable attributes, classifying objects and counting the
number 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 multiracial
students due to small sample sizes.
3. Scaled variables are imputed with the Markov Chain Monte Carlo method because we have
an arbitrary missing data pattern (Schafer, 1997). Categorical variables are imputed with a logistic
regression method.
4. We recognize that math IRT scores do not necessarily represent everything a child knows
about the subject or how much of the curriculum the child has achieved. Nevertheless, math IRT
scores are measures of learning and achievement that are important to parents, teachers, and
school administrators.
5. We include these variables as categorical variables, rather than as interaction terms, because
these variables will be interacted with factors of instructional practices. This would have necessi-
tated four-way interactions (race ϫ SES ϫ academic readiness ϫ instructional practices). Our
approach 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 of
Math Concepts, Interactive Group Activities, Adding and Subtracting Single Digits, Manipulatives,
Drills, Place Value and Three Digits, Music and Movement, and Adding Two Digit Numbers. In
this study we focus only on the effect of the factors related to instructional practices because a
student without necessary prerequisite skills would not benefit from instruction that requires those
skills, 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 without
major 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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