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Foundations of Mathematics Achievement: Instructional Practices and Diverse Kindergarten Students

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Title: Foundations of Mathematics Achievement: Instructional Practices and Diverse Kindergarten Students
Language: English
Authors: Bottia, Martha Cecilia, Moller, Stephanie, Mickelson, Roslyn Arlin, Stearns, Elizabeth
Source: Elementary School Journal. Sep 2014 115(1):124-150.
Availability: University of Chicago Press. Journals Division, P.O. Box 37005, Chicago, IL 60637. Tel: 877-705-1878; Tel: 773-753-3347; Fax: 877-705-1879; Fax: 773-753-0811; e-mail: subscriptions@press.uchicago.edu; Web site: http://www.press.uchicago.edu
Peer Reviewed: Y
Page Count: 27
Publication Date: 2014
Sponsoring Agency: Institute of Education Sciences (ED)
Contract Number: R305A100822
Document Type: Journal Articles
Reports - Research
Education Level: Kindergarten
Primary Education
Early Childhood Education
Descriptors: Teaching Methods, Mathematics Instruction, Kindergarten, Student Diversity, Correlation, Socioeconomic Status, Mathematics Achievement, School Readiness, Racial Differences, Ethnic Groups, African American Students, Hispanic American Students, White Students, Asian American Students, Hypothesis Testing, Longitudinal Studies, Item Response Theory, Scores, Predictor Variables, Statistical Analysis
Assessment and Survey Identifiers: Early Childhood Longitudinal Survey
DOI: 10.1086/676950
ISSN: 0013-5984
Abstract: Analyzing Early Childhood Longitudinal Survey--Kindergarten (ECLS-K) data, we examine how exposure to instructional practices influences math test scores at the end of kindergarten for children from different racial/ethnic and socioeconomic backgrounds, and for children with different levels of math skills at kindergarten entry. We also analyze the relationship between socioeconomic background and math academic readiness within racial/ethnic categories. Our results demonstrate that race/ethnicity and levels of math academic readiness moderate the relationship between instructional practices and math achievement. While we find that interactive group activities enhance students' mathematics achievement in kindergarten and that drills enhance math academic achievement of students with high math academic 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 successful academic trajectories, the findings have implications for addressing racial/ethnic and socioeconomic status gaps in mathematics achievement.
Abstractor: As Provided
Number of References: 67
IES Funded: Yes
Entry Date: 2014
Accession Number: EJ1035691
Database: ERIC
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  Value: <anid>AN0097451024;esj01sep.14;2018Oct23.11:16;v2.2.500</anid> <title id="AN0097451024-1">FOUNDATIONS OF MATHEMATICS ACHIEVEMENT. </title> <p>Analyzing Early Childhood Longitudinal Survey—Kindergarten (ECLS-K) data, we examine how exposure to instructional practices influences math test scores at the end of kindergarten for children from different racial/ethnic and socioeconomic backgrounds, and for children with different levels of math skills at kindergarten entry. We also analyze the relationship between socioeconomic background and math academic readiness within racial/ethnic categories. Our results demonstrate that race/ethnicity and levels of math academic readiness moderate the relationship between instructional practices and math achievement. While we find that interactive group activities enhance students' mathematics achievement in kindergarten and that drills enhance math academic achievement of students with high math academic 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 successful academic trajectories, the findings have implications for addressing racial/ethnic and socioeconomic status gaps in mathematics achievement.</p> <p>Mathematics education has become a top national priority in efforts to advance the nation's technical and scientific literacy (National Research Council, [<reflink idref="bib44" id="ref1">44</reflink>]). The mathematics education children receive in the early elementary grades sets them on pathways for academic success or struggle for the remainder of their formal education. Mathematics performance in the early grades influences individuals' achievement trajectories and, consequently, their eventual status attainment. Teachers' instructional practices are essential components of this early mathematics education.</p> <p>Drawing upon the theoretical framework that holds that national mathematics standards represent the formal mathematics curriculum, instructional practices reflect the implemented math curriculum, and student achievement manifests the attained mathematics curriculum (Suter, [<reflink idref="bib55" id="ref2">55</reflink>]; Travers & Westbury, [<reflink idref="bib60" id="ref3">60</reflink>]), we examine how aspects of the implemented mathematics curriculum affect the achieved curriculum among a nationally representative sample of kindergarten students. Previous research has shown that individual characteristics such as race/ethnicity, socioeconomic status (SES), and math skills at school entry (math academic readiness) help explain the link between curriculum and students' math achievement (Bodovski & Farkas, [<reflink idref="bib7" id="ref4">7</reflink>], [<reflink idref="bib8" id="ref5">8</reflink>]; Lubienski, [<reflink idref="bib36" id="ref6">36</reflink>], [<reflink idref="bib37" id="ref7">37</reflink>]; Palardy & Rumberger, [<reflink idref="bib45" id="ref8">45</reflink>]). 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 impact students' achievement depending on their race/ethnicity, socioeconomic status, and math academic readiness.</p> <p>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 readiness. 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, [<reflink idref="bib64" id="ref9">64</reflink>]). In addition, instructional practices are elements of the curriculum that teachers are best positioned to influence (Lubienski, [<reflink idref="bib37" id="ref10">37</reflink>]).</p> <p>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 appropriate years to start building a solid mathematics foundation (Clements & Sarama, [<reflink idref="bib10" id="ref11">10</reflink>]; Waterford Institute, [<reflink idref="bib61" id="ref12">61</reflink>]). 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 categories. This is an important line of inquiry given the increasing diversity of the U.S. student population and the relatively high rates of growth among the subpopulations that tend to perform poorly in mathematics (Mickelson, Bottia, & Lambert, [<reflink idref="bib39" id="ref13">39</reflink>]). We pursue this research through multilevel modeling techniques using data from the Early Childhood Longitudinal Study—Kindergarten Cohort (ECLS-K).</p> <hd id="AN0097451024-2">Theoretical Background</hd> <p></p> <hd id="AN0097451024-3">Instructional Practices and Mathematics Achievement</hd> <p>Previous studies by the International Association for the Evaluation of Educational Achievement determined that any country's national curriculum can be defined by topics that are intended by the school system, implemented in the classroom, and attained by the students (Suter, [<reflink idref="bib55" id="ref14">55</reflink>]; Travers & Westbury, [<reflink idref="bib60" id="ref15">60</reflink>]). The intended, implemented, and attained curricula are developed simultaneously within an education system and together 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 intended or official curriculum is the desired curriculum based on national or state standards and the opinions of educators and experts in any given discipline. This curriculum determines the concepts to be learned and their sequence.</p> <p>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 curriculum 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) curriculum 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, [<reflink idref="bib29" id="ref16">29</reflink>]). Thus, in one sense, achievement gaps reflect differences in the attained curriculum. This research focuses on the implemented curriculum, operationalized as mathematics instructional practices teachers perform in kindergarten classes, as it influences the attained curriculum, operationalized as students' test scores.</p> <p>Previous research has investigated how instruction (including curriculum characteristics, context, and teachers characteristics) affects student learning (Alexander, [<reflink idref="bib1" id="ref17">1</reflink>]; Bargagliotti, Guarino, & Mason, [<reflink idref="bib5" id="ref18">5</reflink>]; Kessenich, [<reflink idref="bib32" id="ref19">32</reflink>]; Palardy & Rumberger, [<reflink idref="bib45" id="ref20">45</reflink>]; Xue & Meisels, [<reflink idref="bib66" id="ref21">66</reflink>]). Researchers identified important relationships between instructional practices and children's academic achievement. In general, the National Mathematics Advisory Panel (NMAP) ([<reflink idref="bib43" id="ref22">43</reflink>]) says that an effective instructional 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.</p> <hd id="AN0097451024-4">Specific Instructional Practices</hd> <p>Kindergartners are exposed to various instructional approaches in order to gain necessary math knowledge.[<reflink idref="bib2" id="ref23">2</reflink>] First, the use of manipulatives is common in kindergarten classrooms. Manipulatives are defined as "physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics" (Teacher Vision, [<reflink idref="bib57" id="ref24">57</reflink>]) that allow children to use concrete objects to observe, model, and internalize abstract concepts, therefore providing a common language with which to communicate these models to other students and the teacher (Ruzic & O'Connell, [<reflink idref="bib49" id="ref25">49</reflink>]). Manipulatives are believed to bridge the gap between the world in which children live and the abstract world of mathematics (Dienes, [<reflink idref="bib14" id="ref26">14</reflink>]). Manipulatives engage students and increase their enjoyment and interest in mathematics, all of which have positive effects on students' achievement (Sutton & Krueger, [<reflink idref="bib56" id="ref27">56</reflink>]). 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.</p> <p>Drilling practice worksheets, workbooks, and measuring exercises are also common in kindergarten classrooms and can be applied universally to a variety of mathematics problems. These practices are linked to formal procedures of algebra and calculus, thereby giving children hands-on experience with formal procedures necessary in advanced math (Scott-Clayton, [<reflink idref="bib51" id="ref28">51</reflink>]). "Drills" have been found to positively predict math achievement (Milesi & Gamoran, [<reflink idref="bib40" id="ref29">40</reflink>]).</p> <p>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, [<reflink idref="bib63" id="ref30">63</reflink>]). Group/interactive activities have also been positively associated with kindergarten mathematics gains (Bodovski & Farkas, [<reflink idref="bib7" id="ref31">7</reflink>]). The benefits of group interaction for students' math achievement might occur through different mechanisms: (<reflink idref="bib1" id="ref32">1</reflink>) by directly affecting cognitive processes, (<reflink idref="bib2" id="ref33">2</reflink>) by mediating variables that could enhance an emotional or intellectual climate to be conducive to learning, and (<reflink idref="bib3" id="ref34">3</reflink>) by the sheer act of verbalizing information. Additionally, the presence of group feedback and resource sharing in interactive group activities helps group members reshape their ideas and learn novel information that they are unlikely to discover on their own (Slavin, [<reflink idref="bib52" id="ref35">52</reflink>]).</p> <p>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, [<reflink idref="bib67" id="ref36">67</reflink>]). 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, [<reflink idref="bib28" id="ref37">28</reflink>]). Previous 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, [<reflink idref="bib46" id="ref38">46</reflink>]). In fact, Southgate and Roscigno ([<reflink idref="bib53" id="ref39">53</reflink>]) found that there are clear benefits of music involvement (measured as weekly in-school music class participation) in school for the math achievement of small children.</p> <p>Research utilizing ECLS-K data has focused specifically on the importance of instructional practices for math gains in first grade (Palardy & Rumberger, [<reflink idref="bib45" id="ref40">45</reflink>]). This study found that teachers' instructional practices, specifically, frequency of use of math worksheets and frequency of work on problems with calendars, had a significant 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.</p> <hd id="AN0097451024-5">The Moderating Role of Student Attributes on the Relationship between Instructional Practices...</hd> <p>Student attributes, including race/ethnicity, socioeconomic status, and math academic readiness, moderate the relationship between instructional practices and math achievement. Theories of cultural and/or linguistic mismatch offer an explanation 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, [<reflink idref="bib18" id="ref41">18</reflink>]). Research shows that there are cultural traits that have direct implications for teaching and learning. For example, different ethnic groups (<emph>a</emph>) prioritize communal living and cooperative problem solving, and these preferences affect educational motivation, aspiration, and task performance; and (<emph>b</emph>) have norms for appropriate ways for children to interact with the adults they encounter in instructional settings. In addition, different cultures may place different values on mathematics education and have different ideas of parental roles in children's learning (Fuligni & Fuligni, [<reflink idref="bib19" id="ref42">19</reflink>]; Kaplan, [<reflink idref="bib30" id="ref43">30</reflink>]). A linguistic mismatch between home and school may lead to a lack of parental involvement (Espinosa, [<reflink idref="bib17" id="ref44">17</reflink>]) and weak student-teacher and student-student relationships (García & Levin, [<reflink idref="bib21" id="ref45">21</reflink>]; Ramirez, [<reflink idref="bib47" id="ref46">47</reflink>]), 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.</p> <hd id="AN0097451024-6">Differences by race.</hd> <p>In general, research indicates that learning styles characterized by factors with social and affective emphasis, expressive creativity, and nonverbal 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, [<reflink idref="bib38" id="ref47">38</reflink>]). Stiff ([<reflink idref="bib54" id="ref48">54</reflink>]) and Gilbert and Gay ([<reflink idref="bib23" id="ref49">23</reflink>]) found that many African American students prefer learning in more relational, holistic ways, including solving contextualized problems and participating in classroom discourse. Wenglinsky ([<reflink idref="bib64" id="ref50">64</reflink>]), using 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 ([<reflink idref="bib37" id="ref51">37</reflink>]) found that the factor related to collaborative 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 environments value the welfare of the group over the individual and where individuals are taught to pool their resources to solve problems (Gay, [<reflink idref="bib22" id="ref52">22</reflink>]). In fact, the positive benefits of communities of learners and cooperative efforts on student achievement previously have been validated for Latino/a (Escalanté & Dirmann, [<reflink idref="bib16" id="ref53">16</reflink>]), African American (Fullilove & Treisman, [<reflink idref="bib20" id="ref54">20</reflink>]), Chinese American (Fullilove & Treisman, [<reflink idref="bib20" id="ref55">20</reflink>]), and Native Hawaiian (Tharp & Gallimore, [<reflink idref="bib58" id="ref56">58</reflink>]) students.</p> <p>Research has also shown that motion and movement, music, frequent variability in tasks and formats, novelty, and dramatic elements in instructional practices improve the academic performance of African Americans (Allen & Boykin, [<reflink idref="bib2" id="ref57">2</reflink>]; Allen & Butler, [<reflink idref="bib3" id="ref58">3</reflink>]; Boykin, [<reflink idref="bib9" id="ref59">9</reflink>]; Guttentag & Ross, [<reflink idref="bib25" id="ref60">25</reflink>]; Hanley, [<reflink idref="bib26" id="ref61">26</reflink>]). However, a more recent study by Southgate and Roscigno ([<reflink idref="bib53" id="ref62">53</reflink>]) 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.</p> <hd id="AN0097451024-7">Differences by socioeconomic status.</hd> <p>Turning to socioeconomic status and academic 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, [<reflink idref="bib34" id="ref63">34</reflink>], [<reflink idref="bib35" id="ref64">35</reflink>]) and family wealth (Conley, [<reflink idref="bib12" id="ref65">12</reflink>], [<reflink idref="bib13" id="ref66">13</reflink>]). Socioeconomic status has multiple ways of affecting the relationship between instructional practices and mathematics achievement. 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 ([<reflink idref="bib8" id="ref67">8</reflink>]) found that academic achievement 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.</p> <p>Hickey, Moore, and Pellegrino ([<reflink idref="bib27" id="ref68">27</reflink>]) analyzed how instructional practices affect students from different socioeconomic groups and found that reform-oriented instruction (which includes interactive group practices and music and movement) improved low- and high-SES students' problem-solving skills, but the same instruction 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, [<reflink idref="bib6" id="ref69">6</reflink>]; Stiff, [<reflink idref="bib54" id="ref70">54</reflink>]).</p> <hd id="AN0097451024-8">Differences by academic readiness.</hd> <p>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 kindergarten. 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, [<reflink idref="bib7" id="ref71">7</reflink>]). Indeed, previous research has shown that math academic readiness is an important predictor of subsequent school achievement in math (Duncan et al., [<reflink idref="bib15" id="ref72">15</reflink>]).</p> <p>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 instructional practices and math achievement is scant. Bodovski and Farkas ([<reflink idref="bib8" id="ref73">8</reflink>]) 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.</p> <p>Most prior studies have examined race, SES, and academic readiness independently. Only a few studies have examined how they interactively moderate the relationship between instructional practices and students' mathematics achievement. Bodovski and Farkas ([<reflink idref="bib7" id="ref74">7</reflink>]) 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 ([<reflink idref="bib64" id="ref75">64</reflink>]) analyzed NAEP data for eighth graders and found that instructional practices can reduce the African American and Latino within-school achievement gap. Similarly, Lubienski ([<reflink idref="bib37" id="ref76">37</reflink>]) analyzed NAEP data from students in fourth and eighth grade and found, in contrast to Wenglinsky ([<reflink idref="bib64" id="ref77">64</reflink>]), that the relationship between various instructional practices and achievement was roughly similar for White, Black, and Latino/a students.</p> <p>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 relationship between instructional practices and mathematics achievement of a nationally 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 mathematics achievement of kindergartners; (Hypothesis 2) race/ethnicity moderates the relationship between instructional practices and mathematics achievement; (Hypothesis 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 students; (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 readiness moderate the relationship between instructional practices and mathematics achievement; (Hypothesis 3a) drills, which require more previous knowledge, benefit better prepared students; and (Hypothesis 3b) use of manipulatives, a fairly simple instructional practice that requires little previous knowledge, is more beneficial for less academically ready students than for students with higher academic readiness.</p> <p>Because there is little research looking at the interactions between race, socioeconomic status, and academic readiness, we investigated Hypotheses 4 and 5 in an exploratory manner. (Hypothesis 4) The relationship between instructional practices 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.</p> <hd id="AN0097451024-9">Method</hd> <p></p> <hd id="AN0097451024-10">Data Source</hd> <p>To test the hypotheses above, we analyze data from the U.S. Department of Education's Early Childhood Longitudinal Study (ECLS-K) because it focuses on children's early school experiences. This data set began in 1998 with a nationally representative 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).</p> <p>Given our research interests, we limit the sample to White, Black, Latino/a, and Asian students.[<reflink idref="bib3" id="ref78">3</reflink>] 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).</p> <p>Any missing data are imputed through multiple imputation because this approach is far superior to listwise deletion of missing data (Allison, [<reflink idref="bib4" id="ref79">4</reflink>]; Schafer [<reflink idref="bib50" id="ref80">50</reflink>]).[<reflink idref="bib4" id="ref81">4</reflink>] 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.</p> <p>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).</p> <hd id="AN0097451024-11">Outcome Variable</hd> <p>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 estimating 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, [<reflink idref="bib59" id="ref82">59</reflink>]).[<reflink idref="bib5" id="ref83">5</reflink>] 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.</p> <hd id="AN0097451024-12">Predictor Variables</hd> <p>The key independent variables of interest are the frequency of use of certain mathematics instructional practices for specific curricular content areas at the kindergarten 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 implemented 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, [<reflink idref="bib8" id="ref84">8</reflink>]; Palardy and Rumberger, [<reflink idref="bib45" id="ref85">45</reflink>]).</p> <hd id="AN0097451024-13">Moderator variables.</hd> <p>For our analysis we utilize a categorical race variable (White is the omitted reference category). ECLS-K provides a continuous measure of socioeconomic 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 academic 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 socioeconomic by math academic readiness categories (White, high SES, high academic readiness is the omitted category).[<reflink idref="bib6" id="ref86">6</reflink>]</p> <hd id="AN0097451024-14">Control variables.</hd> <p>In all models, we control for variables at the individual, classroom, and school levels that could be correlated with math scores and our primary independent variables. Our individual-level controls include gender, age, and measures 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 = 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 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.</p> <p>Graph</p> <p>Table 1.  Control Variables by Level of Analysis</p> <p> <ephtml> <table><tr><td>Variable</td><td valign="top">Description</td></tr><tr><td valign="top">School level:</td><td /></tr><tr><td valign="top"> Private school</td><td>1 = private, 0 = public</td></tr><tr><td valign="top"> Percent African American</td><td>Percentage of African American students in school</td></tr><tr><td valign="top"> Percent Latino/a in school</td><td>Percentage of Latino/a students in school</td></tr><tr><td valign="top"> School size</td><td>Category of school size, 0–149, 150–299, 300–499, 500–749, and 750 and above</td></tr><tr><td valign="top"> Rural</td><td>1 = rural, 0 = not rural</td></tr><tr><td valign="top"> Region of the country</td><td>Dummy variables for Midwest, West, and Northeast; South is the reference category</td></tr><tr><td valign="top"> Magnet</td><td>1 = magnet school, 0 = not magnet school</td></tr><tr><td valign="top"> Charter</td><td>1 = charter school, 0 = not charter school</td></tr><tr><td valign="top">Classroom level:</td><td /></tr><tr><td valign="top"> Full-day class</td><td>1 = full-day class, 0 = part-time class</td></tr><tr><td valign="top"> Time spent in math</td><td>Time spent in mathematics instruction</td></tr><tr><td valign="top"> African American teacher</td><td>1 = teacher was African American, 0 = teacher was not African American</td></tr><tr><td valign="top"> Latino/a teacher</td><td>1 = teacher was Latino/a, 0 = teacher was not Latino/a</td></tr><tr><td valign="top"> Enjoys teaching</td><td>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."</td></tr><tr><td valign="top"> Teacher's education</td><td>Category of highest educational level teacher achieved: 1 = high school, 2 = associate's degree, 3 = bachelor's degree, 4 = more than 1 year of coursework beyond bachelor's, 5 = master's, 6 = education specialist/professional diploma, 7 = doctorate</td></tr><tr><td valign="top">Student level:</td><td /></tr><tr><td valign="top"> Race</td><td>White (non-Latino/a), African American (non-Latino/a), Latino/a, and Asian American</td></tr><tr><td valign="top"> Socioeconomic status</td><td>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)</td></tr><tr><td valign="top"> Math academic readiness</td><td>Categorized as low readiness, middle readiness, and high readiness; based on the previous math item response theory (IRT) score</td></tr><tr><td valign="top"> Previous mathematics score</td><td>Fall IRT scores for kindergartners centered by math academic readiness terciles</td></tr><tr><td valign="top"> Previous reading score</td><td>Fall reading IRT scores for kindergartners</td></tr><tr><td valign="top"> Age</td><td>The number of months of life at entry to kindergarten</td></tr><tr><td valign="top"> Male</td><td>1 = male, 0 = female</td></tr><tr><td valign="top"> Not English at home</td><td>1 = child does not speak English at home, 0 = child speaks English at home</td></tr></table> </ephtml> </p> <hd id="AN0097451024-15">Analytic Strategy</hd> <p>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 background. 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 include both teaching practices and content because practices are partially determined by content. These detailed analyses of each curriculum variable separately provided two important insights: (<reflink idref="bib1" id="ref87">1</reflink>) 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 (<reflink idref="bib2" id="ref88">2</reflink>) 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 variables 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.</p> <p>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 maximum-likelihood exploratory factor analysis with promax rotation on a tetrachoric correlation matrix. We then validated these results through a confirmatory factor analysis with robust weighted least-squares and a polychoric correlation matrix.[<reflink idref="bib7" id="ref89">7</reflink>] The results indicated that there were eight factors. We identified four of these factors as instructional practices factors (only process variables loaded on these factors) and labeled them Interactive Group Activities, Manipulatives, Drills, and Music/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 manipulatives and using math-related games. Drills contain doing math worksheets and using math textbooks. Finally, Music/Movement includes using movement and using music to learn math. Table 2 provides a full description of the factors and their loadings.</p> <p>Graph</p> <p>Table 2.  Factors with Variables and Loadings</p> <p> <ephtml> <table><tr><td /><td>Interactive Group Activities</td><td>Manipulatives</td><td>Drills</td><td>Music/Movement</td></tr><tr><td valign="top">Frequency geometric manipulatives</td><td valign="top">−8</td><td valign="top">70a</td><td valign="top">−8</td><td valign="top">6</td></tr><tr><td valign="top">Frequency counting manipulatives</td><td valign="top">6</td><td valign="top">85a</td><td valign="top">1</td><td valign="top">−6</td></tr><tr><td valign="top">Frequency math-related games</td><td valign="top">16</td><td valign="top">61a</td><td valign="top">−12</td><td valign="top">13</td></tr><tr><td valign="top">Frequency music to learn math</td><td valign="top">3</td><td valign="top">1</td><td valign="top">7</td><td valign="top">78a</td></tr><tr><td valign="top">Frequency movement to learn math</td><td valign="top">−3</td><td valign="top">7</td><td valign="top">−2</td><td valign="top">93a</td></tr><tr><td valign="top">Frequency explain/solve math problems</td><td char="." valign="top">56a</td><td valign="top">−5</td><td valign="top">10</td><td valign="top">3</td></tr><tr><td valign="top">Frequency do math worksheets</td><td valign="top">5</td><td valign="top">−2</td><td valign="top">79a</td><td valign="top">−2</td></tr><tr><td valign="top">Frequency use math textbooks</td><td valign="top">9</td><td valign="top">−19</td><td valign="top">60a</td><td valign="top">5</td></tr><tr><td valign="top">Frequency solve math with partner</td><td valign="top">66a</td><td valign="top">19</td><td valign="top">12</td><td valign="top">−4</td></tr><tr><td valign="top">Frequency solve real life math</td><td valign="top">82a</td><td valign="top">−9</td><td valign="top">2</td><td valign="top">−1</td></tr><tr><td valign="top">Frequency peer tutoring</td><td valign="top">49a</td><td valign="top">14</td><td valign="top">8</td><td valign="top">1</td></tr></table> </ephtml> </p> <ulist> <item>21 Note Results of exploratory factor using a tetrachoric correlation matrix and promax rotation.</item> <item>2 a Indicates highest loading for each item.</item> </ulist> <p>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 between 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 relationship between instructional practices and math achievement. Each regression includes all four instructional practices factors, controlling for other variables presented in Table 1 and discussed below.</p> <p>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 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 achievement 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.</p> <p>Graph</p> <p>Table 3.  Average IRT Mathematics Scores (Means) by Racial/Ethnic Category and Socioeconomic Status (Low, Middle, High)</p> <p> <ephtml> <table><tr><td /><td>White</td><td>Black</td><td>Latino/a</td><td>Asian</td></tr><tr><td /><td>Low</td><td>Middle</td><td>High</td><td>Low</td><td>Middle</td><td>High</td><td>Low</td><td>Middle</td><td>High</td><td>Low</td><td>Middle</td><td>High</td></tr><tr><td valign="top">N</td><td valign="top">1,840</td><td valign="top">3,140</td><td valign="top">3,810</td><td valign="top">950</td><td valign="top">600</td><td valign="top">300</td><td valign="top">1,170</td><td valign="top">660</td><td valign="top">380</td><td valign="top">240</td><td valign="top">210</td><td valign="top">370</td></tr><tr><td valign="top">Spring kindergarten</td><td valign="bottom">34.0</td><td valign="bottom">37.8</td><td valign="bottom">43.1</td><td valign="bottom">28.9</td><td valign="bottom">32.8</td><td valign="bottom">36.1</td><td valign="bottom">28.7</td><td valign="bottom">34.2</td><td valign="bottom">38.6</td><td valign="bottom">34.6</td><td valign="bottom">35.4</td><td valign="bottom">44.5</td></tr><tr><td valign="top">Fall kindergarten</td><td valign="top">23.9</td><td valign="top">27.1</td><td valign="top">31.5</td><td valign="top">20.7</td><td valign="top">23.8</td><td valign="top">26.4</td><td valign="top">19.8</td><td valign="top">23.7</td><td valign="top">27.5</td><td valign="top">24.1</td><td valign="top">25.1</td><td valign="top">32.3</td></tr></table> </ephtml> </p> <p>Graph</p> <p>Table 4.  Sample Sizes by Levels of Math Academic Readiness (Low, Middle, High) Crossed with Racial/Ethnic Category and Socioeconomic Status (SES)</p> <p> <ephtml> <table><tr><td /><td>Low</td><td>Middle</td><td>High</td></tr><tr><td valign="top">Black</td><td valign="top">870</td><td valign="top">670</td><td valign="top">310</td></tr><tr><td valign="top">Latino</td><td valign="top">1,150</td><td valign="top">660</td><td valign="top">400</td></tr><tr><td valign="top">Asian</td><td valign="top">210</td><td valign="top">280</td><td valign="top">330</td></tr><tr><td valign="top">White</td><td valign="top">1,990</td><td valign="top">3,020</td><td valign="top">3,780</td></tr><tr><td valign="top">Low SES</td><td valign="top">2,190</td><td valign="top">1,360</td><td valign="top">650</td></tr><tr><td valign="top">Middle SES</td><td valign="top">670</td><td valign="top">1,550</td><td valign="top">2,640</td></tr><tr><td valign="top">High SES</td><td valign="top">1,350</td><td valign="top">1,730</td><td valign="top">1,530</td></tr></table> </ephtml> </p> <p>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, [<reflink idref="bib48" id="ref90">48</reflink>]). 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.</p> <p>Level 1 Model (Child):</p> <p></p> <p>The dependent variable is mathematics achievement in the spring of kindergarten. The <emph>x</emph> reflects categories of students' race/ethnicity, SES, and math academic readiness. 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.</p> <p>Level 2 Model (Classroom):</p> <p></p> <p>In the second-level equation, is the random intercept, and is the second-level error term associated with variation across classrooms (see Guo & Hongxin, [<reflink idref="bib24" id="ref91">24</reflink>]). is the extent to which the curriculum, across classrooms, predicts the mathematics achievement of students; 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 students. 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 mathematics 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 curriculum factors. 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 factors[<reflink idref="bib8" id="ref92">8</reflink>]. We also include a random slope, .</p> <p>Level 3 Model (School):</p> <p></p> <p>In the third-level equation, is the random intercept and is the third-level error term associated with variation across schools. represents the extent that school-level variables (private vs. public, region of the country, urbanicity, percentage Black in school, percentage Latino/a in school) predict, across schools, the average mathematics achievement of students. Therefore, our model predicts how race, math academic readiness, factors of instructional practices, and other control variables 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.</p> <hd id="AN0097451024-16">Results</hd> <p>Before testing the hypotheses, we assess how frequently children of different racial/ethnic, socioeconomic, and academic readiness categories are exposed to instructional 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 practice, 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 group indicates the feasibility of testing for significant differences in the role that instructional practices may play in mathematics performance.</p> <p>Graph</p> <p>Table 5.  Proportion of Students Exposed to Instructional Practices by Racial/Ethnic Category, Academic Readiness, and Socioeconomic Status (SES)</p> <p> <ephtml> <table><tr><td /><td /><td /><td /><td /><td>Math Academic Readiness</td><td /><td /><td /></tr><tr><td /><td>Black</td><td>White</td><td>Latino/a</td><td>Asian</td><td>Low</td><td>Middle</td><td>High</td><td>Low SES</td><td>Middle SES</td><td>High SES</td></tr><tr><td valign="top">N</td><td valign="bottom">1,850</td><td valign="bottom">8,790</td><td valign="bottom">2,210</td><td valign="bottom">820</td><td valign="bottom">4,210</td><td valign="bottom">4,640</td><td valign="bottom">4,820</td><td valign="bottom">4,200</td><td valign="bottom">4,610</td><td valign="bottom">4,860</td></tr><tr><td valign="bottom">Instructional practice:</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom"> Manipulatives:</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">  Geometric manipulatives</td><td valign="bottom">.83</td><td valign="bottom">.81</td><td valign="bottom">.79</td><td valign="bottom">.73</td><td valign="bottom">.79</td><td valign="bottom">.77</td><td valign="bottom">.73</td><td valign="bottom">.80</td><td valign="bottom">.75</td><td valign="bottom">.74</td></tr><tr><td valign="bottom">  Counting manipulatives</td><td valign="bottom">.93</td><td valign="bottom">.93</td><td valign="bottom">.89</td><td valign="bottom">.92</td><td valign="bottom">.93</td><td valign="bottom">.92</td><td valign="bottom">.91</td><td valign="bottom">.94</td><td valign="bottom">.92</td><td valign="bottom">.91</td></tr><tr><td valign="bottom">  Math-related games</td><td valign="bottom">.88</td><td valign="bottom">.81</td><td valign="bottom">.87</td><td valign="bottom">.83</td><td valign="bottom">.85</td><td valign="bottom">.84</td><td valign="bottom">.83</td><td valign="bottom">.86</td><td valign="bottom">.83</td><td valign="bottom">.83</td></tr><tr><td valign="bottom"> Music/movement:</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">  Music to learn math</td><td valign="bottom">.35</td><td valign="bottom">.35</td><td valign="bottom">.32</td><td valign="bottom">.27</td><td valign="bottom">.33</td><td valign="bottom">.30</td><td valign="bottom">.27</td><td valign="bottom">.35</td><td valign="bottom">.29</td><td valign="bottom">.27</td></tr><tr><td valign="bottom">  Movement to learn math</td><td valign="bottom">.29</td><td valign="bottom">.31</td><td valign="bottom">.30</td><td valign="bottom">.23</td><td valign="bottom">.27</td><td valign="bottom">.26</td><td valign="bottom">.24</td><td valign="bottom">.27</td><td valign="bottom">.25</td><td valign="bottom">.25</td></tr><tr><td valign="bottom"> Drills:</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">  Do math worksheets</td><td valign="bottom">.76</td><td valign="bottom">.71</td><td valign="bottom">.68</td><td valign="bottom">.67</td><td valign="bottom">.72</td><td valign="bottom">.69</td><td valign="bottom">.67</td><td valign="bottom">.73</td><td valign="bottom">.70</td><td valign="bottom">.65</td></tr><tr><td valign="bottom">  Use math textbooks</td><td valign="bottom">.31</td><td valign="bottom">.26</td><td valign="bottom">.24</td><td valign="bottom">.24</td><td valign="bottom">.24</td><td valign="bottom">.25</td><td valign="bottom">.26</td><td valign="bottom">.25</td><td valign="bottom">.25</td><td valign="bottom">.25</td></tr><tr><td valign="bottom">  Do math on chalkboard</td><td valign="bottom">.47</td><td valign="bottom">.42</td><td valign="bottom">.34</td><td valign="bottom">.33</td><td valign="bottom">.39</td><td valign="bottom">.36</td><td valign="bottom">.34</td><td valign="bottom">.40</td><td valign="bottom">.36</td><td valign="bottom">.33</td></tr><tr><td valign="bottom"> Interactive group activities:</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">  Solve math w/partner</td><td valign="bottom">.62</td><td valign="bottom">.58</td><td valign="bottom">.54</td><td valign="bottom">.49</td><td valign="bottom">.56</td><td valign="bottom">.52</td><td valign="bottom">.50</td><td valign="bottom">.56</td><td valign="bottom">.50</td><td valign="bottom">.51</td></tr><tr><td valign="bottom">  Solve real-life math</td><td valign="bottom">.68</td><td valign="bottom">.62</td><td valign="bottom">.64</td><td valign="bottom">.59</td><td valign="bottom">.63</td><td valign="bottom">.61</td><td valign="bottom">.60</td><td valign="bottom">.63</td><td valign="bottom">.59</td><td valign="bottom">.61</td></tr><tr><td valign="bottom">  Peer tutoring</td><td valign="bottom">.51</td><td valign="bottom">.46</td><td valign="bottom">.44</td><td valign="bottom">.38</td><td valign="bottom">.45</td><td valign="bottom">.41</td><td valign="bottom">.39</td><td valign="bottom">.47</td><td valign="bottom">.40</td><td valign="bottom">.38</td></tr><tr><td valign="bottom">  Explain/solve math</td><td valign="bottom">.67</td><td valign="bottom">.63</td><td valign="bottom">.57</td><td valign="bottom">.59</td><td valign="bottom">.60</td><td valign="bottom">.60</td><td valign="bottom">.63</td><td valign="bottom">.61</td><td valign="bottom">.61</td><td valign="bottom">.61</td></tr></table> </ephtml> </p> <p>Next, we analyze the effects of instructional practices on mathematics achievement. The results presented in Table 6 examine the effects of instructional practices on mathematics achievement (without interactions). Table 6 illustrates partial support 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.</p> <p>Graph</p> <p>Table 6.  Regression Coefficients from HLM Analysis of Math Achievement in Kindergarten</p> <p> <ephtml> <table><tr><td>Variable</td><td>Coefficient (SE)</td></tr><tr><td valign="bottom">Intercept</td><td char="." valign="bottom">35.89 (.43)***</td></tr><tr><td valign="bottom">Racial/ethnic category:a</td><td char="." valign="bottom" /></tr><tr><td valign="bottom"> Latino/a</td><td char="." valign="bottom">−.70 (.23)***</td></tr><tr><td valign="bottom"> Asian</td><td char="." valign="bottom">.39 (.41)</td></tr><tr><td valign="bottom"> Black</td><td char="." valign="bottom">−1.76 (.24)***</td></tr><tr><td valign="bottom">Math academic readiness:b</td><td char="." valign="bottom" /></tr><tr><td valign="bottom"> Low academic readiness</td><td char="." valign="bottom">−1.57 (.26)***</td></tr><tr><td valign="bottom"> Middle academic readiness</td><td char="." valign="bottom">−.42 (.19)**</td></tr><tr><td valign="bottom">Instructional practice:</td><td char="." valign="bottom" /></tr><tr><td valign="bottom"> Interactive group activities</td><td char="." valign="bottom">.14 (.04)***</td></tr><tr><td valign="bottom"> Manipulatives</td><td char="." valign="bottom">−.04 (.03)</td></tr><tr><td valign="bottom"> Drills</td><td char="." valign="bottom">.20 (.03)***</td></tr><tr><td valign="bottom"> Music/movement practices</td><td char="." valign="bottom">−.17 (.14)</td></tr><tr><td valign="bottom">Random effects:</td><td char="." valign="bottom" /></tr><tr><td valign="bottom"> Teacher intercept</td><td char="." valign="bottom">.36</td></tr><tr><td valign="bottom"> School intercept</td><td char="." valign="bottom">2.03***</td></tr><tr><td valign="bottom"> Race intercept</td><td char="." valign="bottom">2.16***</td></tr><tr><td valign="bottom"> SES intercept</td><td char="." valign="bottom">2.12***</td></tr></table> </ephtml> </p> <ulist> <item>61 Note Controls for all variables described in Table 1.</item> <item>6 a White is excluded category.</item> <item>6 b High academic readiness is excluded category.</item> <item>6 ** <emph>p</emph> <.01.</item> <item>6 *** <emph>p</emph> <.001.</item> </ulist> <p>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 <emph>F</emph>-tests to assess the significance of interactions between instructional practices and categories of students. Table 7 includes these <emph>F</emph>-tests from four models that examine the interactions between instructional practices and race categories (Model 1), SES categories (Model 2), and academic readiness categories (Model 3). Once we determined that the overall 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.</p> <p>Graph</p> <p>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</p> <p> <ephtml> <table><tr><td /><td>F-Value</td></tr><tr><td valign="bottom">Model 1: Interaction between race/ethnicity and each instructional practice factor:</td><td char="." valign="bottom" /></tr><tr><td valign="bottom"> Drills</td><td char="." valign="bottom">.22</td></tr><tr><td valign="bottom"> Interactive group activities</td><td char="." valign="bottom">.69</td></tr><tr><td valign="bottom"> Music/movement</td><td char="." valign="bottom">2.78*</td></tr><tr><td valign="bottom"> Manipulatives</td><td char="." valign="bottom">2.09+</td></tr><tr><td valign="bottom">Model 2: Interaction between SES and each instructional practice factor:</td><td char="." valign="bottom" /></tr><tr><td valign="bottom"> Drills</td><td char="." valign="bottom">.36</td></tr><tr><td valign="bottom"> Interactive group activities</td><td char="." valign="bottom">.16</td></tr><tr><td valign="bottom"> Music/movement</td><td char="." valign="bottom">.27</td></tr><tr><td valign="bottom"> Manipulatives</td><td char="." valign="bottom">.19</td></tr><tr><td valign="bottom">Model 3: Interaction between math academic readiness and each instructional practice factor:</td><td char="." valign="bottom" /></tr><tr><td valign="bottom"> Drills</td><td char="." valign="bottom">4.40*</td></tr><tr><td valign="bottom"> Interactive group activities</td><td char="." valign="bottom">1.23</td></tr><tr><td valign="bottom"> Music/movement</td><td char="." valign="bottom">.34</td></tr><tr><td valign="bottom"> Manipulatives</td><td char="." valign="bottom">.18</td></tr></table> </ephtml> </p> <ulist> <item>71 Note Controls for all variables described in Table 1.</item> <item>7 + <emph>p</emph> <.10.</item> <item>7 * <emph>p</emph> <.05.</item> </ulist> <p>Graph</p> <p>Table 8.  Regression Coefficients and Standard Errors from Models in Table 7 with Significant Results; from HLM Analysis of Math Achievement in Kindergarten</p> <p> <ephtml> <table><tr><td /><td>Model 4:Race Interactions</td><td>Model 5:Math Academic Readiness Interactions</td></tr><tr><td valign="bottom">Intercept</td><td valign="bottom">35.20 (.50)</td><td valign="bottom">35.14 (.59)***</td></tr><tr><td valign="bottom">Racial/ethnic category:a</td><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom"> Latino/a</td><td valign="bottom">−1.12 (.89)</td><td valign="bottom">−.68 (.21)***</td></tr><tr><td valign="bottom"> Asian</td><td valign="bottom">−.98 (1.91)</td><td valign="bottom">.52 (.39)</td></tr><tr><td valign="bottom"> Black</td><td valign="bottom">−.21 (1.00)</td><td valign="bottom">−1.78 (.23)***</td></tr><tr><td valign="bottom">Math academic readiness:b</td><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom"> Middle academic readiness</td><td valign="bottom">–</td><td valign="bottom">.66 (.81)</td></tr><tr><td valign="bottom"> High academic readiness</td><td valign="bottom">–</td><td valign="bottom">.66 (.79)</td></tr><tr><td valign="bottom">Instructional practice:</td><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom"> Interactive group activities</td><td valign="bottom">.12 (.04)***</td><td valign="bottom">.13 (.05)***</td></tr><tr><td valign="bottom"> Manipulatives</td><td valign="bottom">−.02 (.04)</td><td valign="bottom">−.03 (.05)</td></tr><tr><td valign="bottom"> Drills</td><td valign="bottom">.19 (.04)***</td><td valign="bottom">.11 (.04)***</td></tr><tr><td valign="bottom"> Music/movement practices</td><td valign="bottom">−.05 (.16)</td><td valign="bottom">−.30 (.05)</td></tr><tr><td valign="bottom">Race/ethnicity × instructional practice interactions:</td><td valign="bottom" /><td valign="bottom">–</td></tr><tr><td valign="bottom"> Latino/a × interactive group activities</td><td valign="bottom">.01 (.06)</td><td valign="bottom">–</td></tr><tr><td valign="bottom"> Latino/a × manipulatives</td><td valign="bottom">−.01 (.08)</td><td valign="bottom">–</td></tr><tr><td valign="bottom"> Latino/a × drills</td><td valign="bottom">.07 (.08)</td><td valign="bottom">–</td></tr><tr><td valign="bottom"> Latino/a × music/movement practices</td><td valign="bottom">−.05 (.29)</td><td valign="bottom">–</td></tr><tr><td valign="bottom"> Asian × interactive group activities</td><td valign="bottom">.08 (.12)</td><td valign="bottom">–</td></tr><tr><td valign="bottom"> Asian × manipulatives</td><td valign="bottom">.14 (.15)</td><td valign="bottom">–</td></tr><tr><td valign="bottom"> Asian × drills</td><td valign="bottom">−.10 (.16)</td><td valign="bottom">–</td></tr><tr><td valign="bottom"> Asian × music/movement practices</td><td valign="bottom">.60 (.58)</td><td valign="bottom">–</td></tr><tr><td valign="bottom"> Black × interactive group activities</td><td valign="bottom">.08 (.09)</td><td valign="bottom">–</td></tr><tr><td valign="bottom"> Black × manipulatives</td><td valign="bottom">−.18 (.08)**</td><td valign="bottom">–</td></tr><tr><td valign="bottom"> Black × drills</td><td valign="bottom">−.01 (.07)</td><td valign="bottom">–</td></tr><tr><td valign="bottom"> Black × music/movement practices</td><td valign="bottom">−.87 (.33)**</td><td valign="bottom">–</td></tr><tr><td valign="bottom">Math academic readiness × instructional practice interactions:</td><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom"> Medium readiness × interactive group activities</td><td valign="bottom">–</td><td valign="bottom">.05 (.07)</td></tr><tr><td valign="bottom"> Medium readiness × manipulatives</td><td valign="bottom">–</td><td valign="bottom">.00 (.06)</td></tr><tr><td valign="bottom"> Medium readiness × drills</td><td valign="bottom">–</td><td valign="bottom">.01 (.05)</td></tr><tr><td valign="bottom"> Medium readiness × music/movement practices</td><td valign="bottom">–</td><td valign="bottom">.11 (.28)</td></tr><tr><td valign="bottom"> High readiness × interactive group activities</td><td valign="bottom">–</td><td valign="bottom">−.04 (.07)</td></tr><tr><td valign="bottom"> High readiness × manipulatives</td><td valign="bottom">–</td><td valign="bottom">−.02 (.06)</td></tr><tr><td valign="bottom"> High readiness × drills</td><td valign="bottom">–</td><td valign="bottom">.15 (.05)***</td></tr><tr><td valign="bottom"> High readiness × music/movement practices</td><td valign="bottom">–</td><td valign="bottom">−.09 (.27)</td></tr><tr><td valign="bottom">Random effects:</td><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom"> Teacher intercept</td><td valign="bottom">.31</td><td valign="bottom">1.20***</td></tr><tr><td valign="bottom"> School intercept</td><td valign="bottom">2.13***</td><td valign="bottom">2.22***</td></tr><tr><td valign="bottom"> Race intercept</td><td valign="bottom">2.20***</td><td valign="bottom" /></tr><tr><td valign="bottom"> SES intercept</td><td valign="bottom">2.15***</td><td valign="bottom" /></tr><tr><td valign="bottom"> Academic readiness intercept</td><td valign="bottom" /><td valign="bottom">3.78***</td></tr></table> </ephtml> </p> <ulist> <item>81 Note Controls for all variables described in Table 1. Standard errors in parentheses.</item> <item>8 a White is excluded category.</item> <item>8 b Low academic readiness is excluded category.</item> <item>8 ** <emph>p</emph> <.01.</item> <item>8 *** <emph>p</emph> <.001.</item> </ulist> <p>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 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.</p> <p>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 instructional practices and SES is not significant (see Model 2); the interaction between manipulatives and academic readiness is also not significant (failing to support Hypothesis 3b). However, the interaction between drills and academic readiness is significant (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 medium readiness (as the interaction between drills and medium readiness is nonsignificant). 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.</p> <p>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 separately 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.</p> <p>Graph</p> <p>Table 9.  HLM Coefficients Predicting Math Achievement in Kindergarten from 24 Subsamples of Students by Race/Ethnicity, Socioeconomic Status, and Math Academic Readiness</p> <p> <ephtml> <table><tr><td /><td>White</td><td>Black</td><td>Latino/a</td><td>Asian</td></tr><tr><td>Instructional Practice</td><td>LowSES</td><td>MiddleSES</td><td>HighSES</td><td>LowSES</td><td>MiddleSES</td><td>HighSES</td><td>LowSES</td><td>MiddleSES</td><td>HighSES</td><td>LowSES</td><td>MiddleSES</td><td>HighSES</td></tr><tr><td valign="bottom">Drills</td><td char="." valign="bottom">.25***</td><td char="." valign="bottom">.18***</td><td char="." valign="bottom">.20***</td><td char="." valign="bottom">.18*</td><td char="." valign="bottom">.27**</td><td char="." valign="bottom">−.08</td><td char="." valign="bottom">.15**</td><td char="." valign="bottom">.12</td><td char="." valign="bottom">.10</td><td char="." valign="bottom">.40*</td><td char="." valign="bottom">.49+</td><td char="." valign="bottom">.31+</td></tr><tr><td valign="bottom" /><td char="." valign="bottom">(.06)</td><td char="." valign="bottom">(.05)</td><td char="." valign="bottom">(.05)</td><td char="." valign="bottom">(.09)</td><td char="." valign="bottom">(.09)</td><td char="." valign="bottom">(.15)</td><td char="." valign="bottom">(.07)</td><td char="." valign="bottom">(.10)</td><td char="." valign="bottom">(.13)</td><td char="." valign="bottom">(.21)</td><td char="." valign="bottom">(.24)</td><td char="." valign="bottom">(.16)</td></tr><tr><td valign="bottom">Interactive group activities</td><td char="." valign="bottom">.08</td><td char="." valign="bottom">.12**</td><td char="." valign="bottom">.14**</td><td char="." valign="bottom">.12</td><td char="." valign="bottom">.32**</td><td char="." valign="bottom">.38**</td><td char="." valign="bottom">.17*</td><td char="." valign="bottom">.27**</td><td char="." valign="bottom">.02</td><td char="." valign="bottom">−.06</td><td char="." valign="bottom">−.03</td><td char="." valign="bottom">.10</td></tr><tr><td valign="bottom" /><td char="." valign="bottom">(.08)</td><td char="." valign="bottom">(.06)</td><td char="." valign="bottom">(.07)</td><td char="." valign="bottom">(.10)</td><td char="." valign="bottom">(.12)</td><td char="." valign="bottom">(.19)</td><td char="." valign="bottom">(.09)</td><td char="." valign="bottom">(.12)</td><td char="." valign="bottom">(.17)</td><td char="." valign="bottom">(.26)</td><td char="." valign="bottom">(.22)</td><td char="." valign="bottom">(.19)</td></tr><tr><td valign="bottom">Music/movement</td><td char="." valign="bottom">−.03</td><td char="." valign="bottom">−.01</td><td char="." valign="bottom">−.03</td><td char="." valign="bottom">−.75+</td><td char="." valign="bottom">−1.10*</td><td char="." valign="bottom">−1.72***</td><td char="." valign="bottom">−.12</td><td char="." valign="bottom">.01</td><td char="." valign="bottom">−.10</td><td char="." valign="bottom">1.10</td><td char="." valign="bottom">.27</td><td char="." valign="bottom">.34</td></tr><tr><td valign="bottom" /><td char="." valign="bottom">(.30)</td><td char="." valign="bottom">(.24)</td><td char="." valign="bottom">(.26)</td><td char="." valign="bottom">(.43)</td><td char="." valign="bottom">(.47)</td><td char="." valign="bottom">(.66)</td><td char="." valign="bottom">(.32)</td><td char="." valign="bottom">(.48)</td><td char="." valign="bottom">(.60)</td><td char="." valign="bottom">(.91)</td><td char="." valign="bottom">(.85)</td><td char="." valign="bottom">(.78)</td></tr><tr><td valign="bottom">Manipulatives</td><td char="." valign="bottom">.05</td><td char="." valign="bottom">−.01</td><td char="." valign="bottom">−.02</td><td char="." valign="bottom">−.22**</td><td char="." valign="bottom">−.15</td><td char="." valign="bottom">−.32**</td><td char="." valign="bottom">−.02</td><td char="." valign="bottom">−.07</td><td char="." valign="bottom">−.07</td><td char="." valign="bottom">−.04</td><td char="." valign="bottom">−.14</td><td char="." valign="bottom">.09</td></tr><tr><td valign="bottom" /><td char="." valign="bottom">(.08)</td><td char="." valign="bottom">(.06)</td><td char="." valign="bottom">(.06)</td><td char="." valign="bottom">(.10)</td><td char="." valign="bottom">(.11)</td><td char="." valign="bottom">(.15)</td><td char="." valign="bottom">(.09)</td><td char="." valign="bottom">(.12)</td><td char="." valign="bottom">(.15)</td><td char="." valign="bottom">(.27)</td><td char="." valign="bottom">(.22)</td><td char="." valign="bottom">(.18)</td></tr></table> </ephtml> </p> <p>Table 9.  HLM Coefficients Predicting Math Achievement in Kindergarten from 24 Subsamples of Students by Race/Ethnicity, Socioeconomic Status, and Math Academic Readiness</p> <p> <ephtml> <table><tr><td valign="bottom" /><td valign="bottom">Academic Readiness, White</td><td valign="bottom">Academic Readiness, Black</td><td valign="bottom">Academic Readiness, Latino/a</td><td valign="bottom">Academic Readiness, Asian</td></tr><tr><td valign="bottom" /><td valign="bottom">Low</td><td valign="bottom">Middle</td><td valign="bottom">High</td><td valign="bottom">Low</td><td valign="bottom">Middle</td><td valign="bottom">High</td><td valign="bottom">Low</td><td valign="bottom">Middle</td><td valign="bottom">High</td><td valign="bottom">Low</td><td valign="bottom">Middle</td><td valign="bottom">High</td></tr><tr><td valign="bottom">Drills</td><td char="." valign="bottom">.11+</td><td char="." valign="bottom">.30***</td><td char="." valign="bottom">.17**</td><td char="." valign="bottom">.09</td><td char="." valign="bottom">.33**</td><td char="." valign="bottom">.12</td><td char="." valign="bottom">.08</td><td char="." valign="bottom">.16</td><td char="." valign="bottom">.36**</td><td char="." valign="bottom">.08</td><td char="." valign="bottom">.39</td><td char="." valign="bottom">.25</td></tr><tr><td valign="bottom" /><td char="." valign="bottom">(.05)</td><td char="." valign="bottom">(.05)</td><td char="." valign="bottom">(.06)</td><td char="." valign="bottom">(.07)</td><td char="." valign="bottom">(.11)</td><td char="." valign="bottom">(.16)</td><td char="." valign="bottom">(.06)</td><td char="." valign="bottom">(.10)</td><td char="." valign="bottom">(.14)</td><td char="." valign="bottom">(.17)</td><td char="." valign="bottom">(.20)</td><td char="." valign="bottom">(.18)</td></tr><tr><td valign="bottom">Interactive group activities</td><td char="." valign="bottom">.11+</td><td char="." valign="bottom">.04</td><td char="." valign="bottom">.15**</td><td char="." valign="bottom">.11</td><td char="." valign="bottom">.18</td><td char="." valign="bottom">.53**</td><td char="." valign="bottom">.23**</td><td char="." valign="bottom">.03</td><td char="." valign="bottom">.16</td><td char="." valign="bottom">−.06</td><td char="." valign="bottom">.01</td><td char="." valign="bottom">.07</td></tr><tr><td valign="bottom" /><td char="." valign="bottom">(.06)</td><td char="." valign="bottom">(.06)</td><td char="." valign="bottom">(.07)</td><td char="." valign="bottom">(.08)</td><td char="." valign="bottom">(.13)</td><td char="." valign="bottom">(.23)</td><td char="." valign="bottom">(.08)</td><td char="." valign="bottom">(.12)</td><td char="." valign="bottom">(.20)</td><td char="." valign="bottom">(.25)</td><td char="." valign="bottom">(.22)</td><td char="." valign="bottom">(.21)</td></tr><tr><td valign="bottom">Music/movement</td><td char="." valign="bottom">−.03</td><td char="." valign="bottom">−.03</td><td char="." valign="bottom">.14</td><td char="." valign="bottom">−.74**</td><td char="." valign="bottom">−.88+</td><td char="." valign="bottom">−1.01</td><td char="." valign="bottom">−.09</td><td char="." valign="bottom">−.43</td><td char="." valign="bottom">.55</td><td char="." valign="bottom">1.27</td><td char="." valign="bottom">−.27</td><td char="." valign="bottom">.88</td></tr><tr><td valign="bottom" /><td char="." valign="bottom">(.25)</td><td char="." valign="bottom">(.24)</td><td char="." valign="bottom">(.29)</td><td char="." valign="bottom">(.34)</td><td char="." valign="bottom">(.50)</td><td char="." valign="bottom">(.80)</td><td char="." valign="bottom">(.29)</td><td char="." valign="bottom">(.45)</td><td char="." valign="bottom">(.67)</td><td char="." valign="bottom">(.89)</td><td char="." valign="bottom">(.99)</td><td char="." valign="bottom">(.80)</td></tr><tr><td valign="bottom">Manipulatives</td><td char="." valign="bottom">.06</td><td char="." valign="bottom">−.04</td><td char="." valign="bottom">.02</td><td char="." valign="bottom">−.21**</td><td char="." valign="bottom">−.07</td><td char="." valign="bottom">−.34**</td><td char="." valign="bottom">−.02</td><td char="." valign="bottom">−.08</td><td char="." valign="bottom">−.06</td><td char="." valign="bottom">.06</td><td char="." valign="bottom">−.20</td><td char="." valign="bottom">.33</td></tr><tr><td valign="bottom" /><td char="." valign="bottom">(.06)</td><td char="." valign="bottom">(.06)</td><td char="." valign="bottom">(.07)</td><td char="." valign="bottom">(.08)</td><td char="." valign="bottom">(.13)</td><td char="." valign="bottom">(.19)</td><td char="." valign="bottom">(.08)</td><td char="." valign="bottom">(.11)</td><td char="." valign="bottom">(.18)</td><td char="." valign="bottom">(.20)</td><td char="." valign="bottom">(.19)</td><td char="." valign="bottom">(.21)</td></tr></table> </ephtml> </p> <ulist> <item>91 Note Controls for all variables described in Table 1. Standard errors in parentheses.</item> <item>9 + <emph>p</emph> <.10.</item> <item>9 * <emph>p</emph> <.05.</item> <item>9 ** <emph>p</emph> <.01.</item> <item>9 *** <emph>p</emph> <.001.</item> </ulist> <p>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 students, interactive group activities are only beneficial for mid- and high-SES students, 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).</p> <p>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: furthermore, both strategies harm the achievement of some Black students. This is particularly 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.</p> <p>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 mathematics has a logical scope and sequence, we placed the students in three math academic 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 manipulatives 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.</p> <p>Graph</p> <p>Table 10.  Coefficients for Race and Curriculum Factors from HLM Analysis of Math Achievement in Kindergarten by Levels of Math Academic Preparedness</p> <p> <ephtml> <table><tr><td /><td>Math Academic Readiness</td></tr><tr><td /><td>LowEstimate (SE)</td><td>MiddleEstimate (SE)</td><td>HighEstimate (SE)</td></tr><tr><td valign="bottom">Intercept</td><td char="." valign="bottom">35.27 (.73)***</td><td char="." valign="bottom">35.82 (.74)***</td><td char="." valign="bottom">35.93 (.73)***</td></tr><tr><td valign="bottom">Instructional practice:</td><td char="." valign="bottom" /><td char="." valign="bottom" /><td char="." valign="bottom" /></tr><tr><td valign="bottom"> Drills</td><td char="." valign="bottom">.13 (.05)**</td><td char="." valign="bottom">.28 (.05)***</td><td char="." valign="bottom">.18 (.06)***</td></tr><tr><td valign="bottom"> Interactive group activities</td><td char="." valign="bottom">.12 (.06)*</td><td char="." valign="bottom">.04 (.07)</td><td char="." valign="bottom">.15 (.07)*</td></tr><tr><td valign="bottom"> Music/movement</td><td char="." valign="bottom">−.03 (.24)</td><td char="." valign="bottom">−.10 (.25)</td><td char="." valign="bottom">.10 (.28)</td></tr><tr><td valign="bottom"> Manipulatives</td><td char="." valign="bottom">.05 (.06)</td><td char="." valign="bottom">−.06 (.07)</td><td char="." valign="bottom">.01 (.07)</td></tr><tr><td valign="bottom">Race/ethnicity:</td><td char="." valign="bottom" /><td char="." valign="bottom" /><td char="." valign="bottom" /></tr><tr><td valign="bottom"> Black</td><td char="." valign="bottom">1.78 (1.22)</td><td char="." valign="bottom">−3.00 (1.62)+</td><td char="." valign="bottom">−.60 (2.55)</td></tr><tr><td valign="bottom"> Latino/a</td><td char="." valign="bottom">−.49 (1.13)</td><td char="." valign="bottom">−.49 (1.48)</td><td char="." valign="bottom">−2.10 (2.40)</td></tr><tr><td valign="bottom"> Asian</td><td char="." valign="bottom">1.88 (3.56)</td><td char="." valign="bottom">−.09 (3.05)</td><td char="." valign="bottom">−3.18 (2.92)</td></tr><tr><td valign="bottom"> White</td><td char="." valign="bottom">.00</td><td char="." valign="bottom">.00</td><td char="." valign="bottom">.00</td></tr><tr><td valign="bottom">Instructional practice × race/ethnicity interactions:</td><td char="." valign="bottom" /><td char="." valign="bottom" /><td char="." valign="bottom" /></tr><tr><td valign="bottom"> Drills × Black</td><td char="." valign="bottom">−.21 (.14)</td><td char="." valign="bottom">.07 (.11)</td><td char="." valign="bottom">−.13 (.16)</td></tr><tr><td valign="bottom"> Drills × Latino/a</td><td char="." valign="bottom">−.18 (.14)</td><td char="." valign="bottom">−.05 (.10)</td><td char="." valign="bottom">.16 (.14)</td></tr><tr><td valign="bottom"> Drills × Asian</td><td char="." valign="bottom">−.55 (.36)</td><td char="." valign="bottom">.12 (.19)</td><td char="." valign="bottom">.06 (.22)</td></tr><tr><td valign="bottom"> Interactive group activities × Black</td><td char="." valign="bottom">−.25 (.18)</td><td char="." valign="bottom">.10 (.16)</td><td char="." valign="bottom">.46 (.23)**</td></tr><tr><td valign="bottom"> Interactive group activities × Latino/a</td><td char="." valign="bottom">−.11 (.30)</td><td char="." valign="bottom">.02 (.13)</td><td char="." valign="bottom">.06 (.22)</td></tr><tr><td valign="bottom"> Interactive group activities × Asian</td><td char="." valign="bottom">−.66 (.47)</td><td char="." valign="bottom">.07 (.25)</td><td char="." valign="bottom">−.11 (.26)</td></tr><tr><td valign="bottom"> Music/movement × Black</td><td char="." valign="bottom">−1.62 (.18)+</td><td char="." valign="bottom">−.70 (.54)</td><td char="." valign="bottom">−1.47 (.82)*</td></tr><tr><td valign="bottom"> Music/movement × Latino/a</td><td char="." valign="bottom">−.61 (.85)</td><td char="." valign="bottom">−.24 (.47)</td><td char="." valign="bottom">.34 (.76)</td></tr><tr><td valign="bottom"> Music/movement × Asian</td><td char="." valign="bottom">−1.07 (2.77)</td><td char="." valign="bottom">−.31 (1.00)</td><td char="." valign="bottom">1.02 (.98)</td></tr><tr><td valign="bottom"> Manipulatives × Black</td><td char="." valign="bottom">−.44 (.02)**</td><td char="." valign="bottom">−.01 (.15)</td><td char="." valign="bottom">−.37 (.20)*</td></tr><tr><td valign="bottom"> Manipulatives × Latino/a</td><td char="." valign="bottom">−.03 (.10)</td><td char="." valign="bottom">.01 (.13)</td><td char="." valign="bottom">−.06 (.20)</td></tr><tr><td valign="bottom"> Manipulatives × Asian</td><td char="." valign="bottom">−.02 (.30)</td><td char="." valign="bottom">−.09 (.23)</td><td char="." valign="bottom">.38 (.26)</td></tr></table> </ephtml> </p> <ulist> <item>101 Note Controls for all variables described in Table 1. Standard errors in parentheses.</item> <item>10 + <emph>p</emph> <.10.</item> <item>10 * <emph>p</emph> <.05.</item> <item>10 ** <emph>p</emph> <.01.</item> <item>10 *** <emph>p</emph> <.001.</item> </ulist> <p>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 ([<reflink idref="bib33" id="ref93">33</reflink>]) 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 ([<reflink idref="bib8" id="ref94">8</reflink>]) 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.</p> <hd id="AN0097451024-17">Discussion</hd> <p>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 significant associations with the math achievement of kindergarten students. Our analysis by racial-socioeconomic and racial-math academic preparedness categories allowed us to uncover the moderating role of race/ethnicity and levels of math academic readiness on the relationship between implemented curriculum and kindergartners' math achievement. Consistent with Webb ([<reflink idref="bib63" id="ref95">63</reflink>]) and Bodovski and Farkas ([<reflink idref="bib7" id="ref96">7</reflink>]), we find that children with more exposure to interactive group activities 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, [<reflink idref="bib40" id="ref97">40</reflink>]), we find that more exposure to drills is associated with higher math achievement of kindergartners. Our findings also show that this instructional technique is particularly effective for children with high levels of math skills at kindergarten 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.</p> <p>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 Black students' mathematics achievement. In fact, the two practices that have a differential effect on math achievement of kindergartners are the particular practices that are subject to more variation from individual teachers' subjective style. In contrast, 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.</p> <p>Our results using large-scale nationally representative survey data (e.g., ECLS-K) are generally consistent with results from previous experiments and qualitative studies 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 mathematics achievement for the entire sample of students masks significant variations for specific racial categories of students.</p> <p>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 materials or instructional strategy is difficult to assess (NMAP, [<reflink idref="bib43" id="ref98">43</reflink>]). 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, [<reflink idref="bib65" id="ref99">65</reflink>]).</p> <p>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 because 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 research 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.</p> <hd id="AN0097451024-18">Conclusions and Policy Implication</hd> <p>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 instructional practices diminish or contribute to race- and SES-related gaps within schools (Wenglinsky, [<reflink idref="bib64" id="ref100">64</reflink>]). Our goal required us to investigate whether similar instructional 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.</p> <p>Using the theoretical framework that holds national mathematics standards represent the formal mathematics curriculum, instructional practices reflect the implemented math curriculum, and student achievement manifests the attained mathematics curriculum, we examined in great detail how aspects of the implemented mathematics curriculum affect the achieved curriculum among a nationally representative sample of kindergarten students. We demonstrate that children from diverse race, SES, and math academic readiness backgrounds learn best from different aspects of the implemented curriculum.</p> <p>Our research contributes to the literature on how race, SES, and readiness differences 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, [<reflink idref="bib41" id="ref101">41</reflink>]). 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 instructional practices and how they differentially affect students from different backgrounds.</p> <p>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 increase 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.</p> <p>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 ([<reflink idref="bib31" id="ref102">31</reflink>]) 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. Mathematics curricula, as currently implemented, seem to leave portions of the student population behind.</p> <hd id="AN0097451024-19">Appendix</hd> <p>Graph</p> <p>Table A1.  Question STEM for Instructional Practices in ECLS-K Survey Instrument</p> <p> <ephtml> <table><tr><td /><td>1 = Never</td><td>2 = Once a Month</td><td>3 = 2 or 3 Times a Month</td><td>4 = Once or Twice a Week</td><td>5 = 3 or 4 Times a Week</td><td>6 = Daily</td></tr><tr><td valign="bottom">1. Count out loud</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">2. Work with geometric manipulatives</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">3. Work with counting manipulatives to learn basic operations</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">4. Play math-related games</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">5. Use a calculator for math</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">6. Use music to understand math concepts</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">7. Use creative movement or creative drama to understand math concepts</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">8. Work with rulers, measuring cups, spoons, or other measuring instruments</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">9. Explain how a math problem is solved</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">10. Engage in calendar-related activities</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">11. Do math worksheets</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">12. Do math problems from their textbooks</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">13. Complete math problems on the chalkboard</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">14. Solve math problems in small groups or with a partner</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">15. Work on math problems that reflect real-life situations</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">16. Work in mixed achievement groups on math activities</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr><tr><td valign="bottom">17. Peer tutoring</td><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /><td valign="bottom" /></tr></table> </ephtml> </p> <p>1 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.</p> <ref id="AN0097451024-20"> <title> Notes </title> <blist> <bibl id="bib1" idref="ref17" type="bt">1</bibl> <bibtext> 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.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref23" type="bt">2</bibl> <bibtext> There is a great deal of overlap among the mathematics topics deemed appropriate for kindergartners 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 ([42]) defined as the most important math topics for lasting learning 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 ([11]) 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.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref34" type="bt">3</bibl> <bibtext> We excluded Native Hawaiian, other Pacific Islanders, American Indians, and multiracial students due to small sample sizes.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref79" type="bt">4</bibl> <bibtext> Scaled variables are imputed with the Markov Chain Monte Carlo method because we have an arbitrary missing data pattern (Schafer, [50]). Categorical variables are imputed with a logistic regression method.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref18" type="bt">5</bibl> <bibtext> 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.</bibtext> </blist> <blist> <bibl id="bib6" idref="ref69" type="bt">6</bibl> <bibtext> 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 necessitated four-way interactions (race × SES × academic readiness × instructional practices). Our approach requires only two-way interactions, which is much easier for the reader to interpret.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref4" type="bt">7</bibl> <bibtext> 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. 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  Data: Foundations of Mathematics Achievement: Instructional Practices and Diverse Kindergarten Students
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  Data: <searchLink fieldCode="AR" term="%22Bottia%2C+Martha+Cecilia%22">Bottia, Martha Cecilia</searchLink><br /><searchLink fieldCode="AR" term="%22Moller%2C+Stephanie%22">Moller, Stephanie</searchLink><br /><searchLink fieldCode="AR" term="%22Mickelson%2C+Roslyn+Arlin%22">Mickelson, Roslyn Arlin</searchLink><br /><searchLink fieldCode="AR" term="%22Stearns%2C+Elizabeth%22">Stearns, Elizabeth</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22Elementary+School+Journal%22"><i>Elementary School Journal</i></searchLink>. Sep 2014 115(1):124-150.
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  Data: University of Chicago Press. Journals Division, P.O. Box 37005, Chicago, IL 60637. Tel: 877-705-1878; Tel: 773-753-3347; Fax: 877-705-1879; Fax: 773-753-0811; e-mail: subscriptions@press.uchicago.edu; Web site: http://www.press.uchicago.edu
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  Data: Y
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  Data: 27
– Name: DatePubCY
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  Data: 2014
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  Data: Institute of Education Sciences (ED)
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  Data: R305A100822
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Kindergarten%22">Kindergarten</searchLink><br /><searchLink fieldCode="EL" term="%22Primary+Education%22">Primary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Early+Childhood+Education%22">Early Childhood Education</searchLink>
– Name: Subject
  Label: Descriptors
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  Data: <searchLink fieldCode="DE" term="%22Teaching+Methods%22">Teaching Methods</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Kindergarten%22">Kindergarten</searchLink><br /><searchLink fieldCode="DE" term="%22Student+Diversity%22">Student Diversity</searchLink><br /><searchLink fieldCode="DE" term="%22Correlation%22">Correlation</searchLink><br /><searchLink fieldCode="DE" term="%22Socioeconomic+Status%22">Socioeconomic Status</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Achievement%22">Mathematics Achievement</searchLink><br /><searchLink fieldCode="DE" term="%22School+Readiness%22">School Readiness</searchLink><br /><searchLink fieldCode="DE" term="%22Racial+Differences%22">Racial Differences</searchLink><br /><searchLink fieldCode="DE" term="%22Ethnic+Groups%22">Ethnic Groups</searchLink><br /><searchLink fieldCode="DE" term="%22African+American+Students%22">African American Students</searchLink><br /><searchLink fieldCode="DE" term="%22Hispanic+American+Students%22">Hispanic American Students</searchLink><br /><searchLink fieldCode="DE" term="%22White+Students%22">White Students</searchLink><br /><searchLink fieldCode="DE" term="%22Asian+American+Students%22">Asian American Students</searchLink><br /><searchLink fieldCode="DE" term="%22Hypothesis+Testing%22">Hypothesis Testing</searchLink><br /><searchLink fieldCode="DE" term="%22Longitudinal+Studies%22">Longitudinal Studies</searchLink><br /><searchLink fieldCode="DE" term="%22Item+Response+Theory%22">Item Response Theory</searchLink><br /><searchLink fieldCode="DE" term="%22Scores%22">Scores</searchLink><br /><searchLink fieldCode="DE" term="%22Predictor+Variables%22">Predictor Variables</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+Analysis%22">Statistical Analysis</searchLink>
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  Label: Assessment and Survey Identifiers
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  Data: <searchLink fieldCode="SU" term="%22Early+Childhood+Longitudinal+Survey%22">Early Childhood Longitudinal Survey</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1086/676950
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0013-5984
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Analyzing Early Childhood Longitudinal Survey--Kindergarten (ECLS-K) data, we examine how exposure to instructional practices influences math test scores at the end of kindergarten for children from different racial/ethnic and socioeconomic backgrounds, and for children with different levels of math skills at kindergarten entry. We also analyze the relationship between socioeconomic background and math academic readiness within racial/ethnic categories. Our results demonstrate that race/ethnicity and levels of math academic readiness moderate the relationship between instructional practices and math achievement. While we find that interactive group activities enhance students' mathematics achievement in kindergarten and that drills enhance math academic achievement of students with high math academic 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 successful academic trajectories, the findings have implications for addressing racial/ethnic and socioeconomic status gaps in mathematics achievement.
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  Data: As Provided
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  Label: Number of References
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  Data: 67
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  Data: 2014
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  Data: EJ1035691
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  BibEntity:
    Identifiers:
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        Value: 10.1086/676950
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 27
        StartPage: 124
    Subjects:
      – SubjectFull: Teaching Methods
        Type: general
      – SubjectFull: Mathematics Instruction
        Type: general
      – SubjectFull: Kindergarten
        Type: general
      – SubjectFull: Student Diversity
        Type: general
      – SubjectFull: Correlation
        Type: general
      – SubjectFull: Socioeconomic Status
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      – SubjectFull: Mathematics Achievement
        Type: general
      – SubjectFull: School Readiness
        Type: general
      – SubjectFull: Racial Differences
        Type: general
      – SubjectFull: Ethnic Groups
        Type: general
      – SubjectFull: African American Students
        Type: general
      – SubjectFull: Hispanic American Students
        Type: general
      – SubjectFull: White Students
        Type: general
      – SubjectFull: Asian American Students
        Type: general
      – SubjectFull: Hypothesis Testing
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      – SubjectFull: Longitudinal Studies
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      – SubjectFull: Item Response Theory
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      – SubjectFull: Scores
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      – SubjectFull: Predictor Variables
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      – SubjectFull: Statistical Analysis
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      – SubjectFull: Early Childhood Longitudinal Survey
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      – TitleFull: Foundations of Mathematics Achievement: Instructional Practices and Diverse Kindergarten Students
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