

Educating the Whole Child: Kindergarten Mathematics Instructional Practices and Students’ Academic and Socioemotional Developmentby Anna Bargagliotti, Michael A. Gottfried & Cassandra M. Guarino  2017 Background: To date, school policies and practices have emphasized earlyschooling mathematics instructional practices only as a way to boost academic achievement. However, because young children spend a large part of their formative years in classroom settings, it is important to understand not only the link between instruction and achievement but also the link between instruction and socioemotional development. Our study addressed this issue. Using a nationally representative dataset of kindergarten students, we inquired into which early mathematics instructional practices might be linked to a range of child outcomes, including both achievement and socioemotional development. We also investigated whether these associations varied across different subpopulations of students. Population: The ECLSK: 2011 dataset for a nationally representative sample of kindergartners in 2010–11 was used. Children were assessed in mathematics at kindergarten entry and at the end of the kindergarten year. Their socioemotional development was also rated by teachers and parents. Research design: We used studentlevel multiple regression analyses with school fixed effects, a rich set of individual, family, and classroom covariates, and school clusteradjusted standard errors to estimate the associations between kindergarten mathematics instructional practices and achievement and multiple socioemotional outcomes. Conclusions: We found that several instructional practices were associated with multiple types of outcomes, often with different results for different types of students. Thus, an exclusive focus on achievement likely obscures the full range of influence that teaching practices have on student success. As school systems increasingly seek to foster socioemotional learning, it is important to establish a research base that considers the links between pedagogy and all facets of childhood development. A number of empirical studies have examined the impact of various mathematics instructional practices on mathematics achievement (e.g., Bodovski & Farkas, 2007; Cohen & Hill, 2000; Guarino, Dieterle, Bargagliotti, & Mason, 2013; Guarino, Hamilton, Lockwood, & Rathbun, 2006; Hamilton et al., 2003; Le et al., 2006; Palardy & Rumberger, 2008). An exclusive focus on academic outcomes, however, yields a limited understanding of the potential for mathematics instruction to influence student attainment. That is, a focus on achievement as the outcome may paint a onesided picture of the effects of mathematics instruction and may exclude important facets of overall childhood development, such as the nurture and acquisition of social and emotional skills. Since mathematics is a subject through which children learn to reason, solve problems, think critically, communicate specific types of information, and work with others, the manner in which it is taught may affect the development of socioemotional skills in addition to academic ability. Little empirical work to date has considered this relationship, however. The narrow focus on the association between mathematics pedagogy and mathematics achievement outcomes may obscure the way in which mathematics teaching practices can educate the whole child. Instructional practices are inarguably a key vehicle by which teachers transmit academic content to students. In early schooling, the classroom and instruction are also the primary ways in which young children learn and develop social skills and adopt appropriate behaviors (Corno & Randi, 1999; Haycock, 1998; Wenglinsky, 2002). Therefore, it is quite possible that certain types of instructional practices, such as group work, peer tutoring, or childcentered practices, may stimulate students to develop these socioemotional skills (Blair & Peters Razza, 2007). Instructional practices used to teach mathematics, in particular, may be wellpositioned to affect socioemotional outcomes. Given that the teaching of mathematics simultaneously sharpens critical thinking and induces interpersonal communication (Byrnes, 2008; Schoenfeld, 1992) and that mathematics reform over the past decade has emphasized studentcentered practices and group work, certain pedagogical approaches to teaching mathematics have the potential to exert a positive influence on the development of childrens socioemotional skills (Henry, 2003). At this point, however, little is known as to whether they do. This study fills a gap in the literature by exploring the relationship between kindergarten mathematics instructional practices and a span of outcomes linked to a childs overall success and wellbeing that cover both socioemotional development and mathematics achievement. Taking advantage of the recent release of an extensive new nationally representative dataset on young childrenthe Early Childhood Longitudinal Study of the Kindergarten Class of 201011 (ECLSK: 2011)we study the effects of mathematics instructional practices in kindergarten (KMIP) on the whole child, using data on teaching practices, student mathematics achievement, and a number of wellestablished and validated socioemotional outcomes. We also examine how these relationships vary by student characteristics associated with known mathematics achievement gaps, including gender and socioeconomic status (SES). Our research questions are as follows: 1. Is there a link between KMIP and students academic and socioemotional outcomes in kindergarten? 2. Does the relationship between KMIP and academic and socioemotional outcomes differ by individual student characteristicsspecifically by gender and SES? Our study addresses several notable gaps in the current literature. Few recent empirical studies address the impact of specific instructional practices on early mathematics achievement; most used data that are more than a decade old, and only a small subset focused specifically on kindergarten, as discussed below. This lack of insight into what instructional factors might boost achievement is concerning, particularly given that mathematics achievement levels in kindergarten have been shown to predict early school success in this subject area (Claessens, Duncan, & Engel, 2009; Claessens & Engel, 2013). Second, virtually no existing research addresses the equally important issue of early childhood socioemotional responses to various mathematics teaching practices, and kindergarten is a key year for the development of socioemotional skills (Olson, Sameroff, Kerr, Lopez, & Wellman, 2005; Posner & Rothbart, 2000). Duncan et al. (2007) showed that early attention skills as well as early mathematics skills are strong predictors of later achievement. Other research suggests that highquality teaching in the early grades has an impact on future educational and earnings outcomes that may stem from the development of noncognitive as well as cognitive skills (Chetty et al., 2011). As it is as critical to foster the development of socioemotional skills in the first year of education as it is to foster the development of educational ability (Shonkoff & Phillips, 2000), our study helps document which schooling factors can not only support academic ability but also bolster social competency during this formative period. This research is particularly timely because it follows a decade in which significant reforms in mathematics teaching have been continuously rolled out and are still in flux as instruction grapples with the task of meeting new standards that emphasize critical thinking and communicationi.e., the Common Core or similar standards being adopted throughout the United States. Efficient policies to encourage effective KMIP must be able to identify desirable instructional practices. Moreover, policy implications cannot be fully assessed without investigating the effects of KMIP on all child outcomes and whether there are differential effects for different subpopulations of students. This study aims to address these needs. BACKGROUND MATHEMATICS INSTRUCTION AND ACHIEVEMENT Prior research has examined links between specific types of instructional practices and elementary mathematics achievement. Grade levels vary across studies, and results are somewhat mixed, some showing that studentcentered instruction has a positive effect on achievement and others showing that more traditional pedagogy has a positive effect. For example, Cohen and Hill (2000), using data from approximately 500 California elementaryschool teachers, found small positive associations between students mathematics achievement and the reported use of working in small groups, doing problems that have several solutions, working on projects that take several days, and writing about and discussing how to solve a problem. Hamilton et al. (2003) studied how the use of manipulatives, openended assessments, and group work affected achievement in a study of approximately 500 elementary and middleschool teachers. They found small but positive associations between the use of studentcentered practices and students mathematics performance on an openresponse test as well as on a multiplechoice test. Le et al. (2006), in a longitudinal study of elementary and middle school students, asked teachers to report the frequency of use of working in groups, using openended assessments, assigning problems that extend over several days, explaining mathematics problems, and assigning openended problems with several solutions. Using two forms of assessmenta standardized test and a test consisting of openended questionsthey found few associations between a teachers emphasis on most practices and student mathematics achievement and, interestingly, differences depending on the type of assessment being used. For example, they found that an emphasis on group work was negatively associated with achievement measured on the standardized test but positively associated with achievement on the openended question test. Most relevant to our study, only a few prior studies have investigated the impact of particular pedagogies on kindergarten achievement using ECLSKa dataset similar to the one used in the present study but collected 11 years earlier. Bodovski and Farkas (2007) and Guarino et al. (2006) found that kindergartners mathematics achievement was positively associated with traditional practices, defined as the use of worksheets, texts, and the chalkboard. Palardy and Rumberger (2008) studied the firstgrade wave of these data and found a positive effect from using math worksheets and a negative effect from the use of geometric manipulatives. Guarino et al. (2013) studied both kindergarten and first grade and found that the effectiveness of specific instructional approaches differed in the two grades, suggesting that certain teaching styles better promoted learning in one developmental stage than the other. For example, they found that teaching modalities such as working with counting manipulatives, using mathematics worksheets, and completing problems on the chalkboard had positive effects on achievement in kindergarten, whereas pedagogical practices relating to problemsolving, explanation, and working on problems from textbooks had positive effects on achievement in first grade. Desimone and Long (2010), also using ECLSK data for kindergarten and first grade, studied the extent to which time spent on mathematics instruction and the type of mathematics instruction predicted student achievement. Type of instruction was categorized into basic procedural, conceptual, and advanced procedural. They found no significant differences in the amount of time spent on mathematics or different types of instruction for students of different racial/ethnic backgrounds or SES levels. However, they found that students were initially distributed differently across classrooms and teachers on the basis of abilityteachers who used more advanced procedural instruction and conceptual approaches to mathematics were more likely to have higherachieving kindergartners. Palardy (2015) looked at achievement across different types of students, using firstgrade ECLSK data, to examine how classroom context, access to qualified teachers, and access to effective teachers were associated with achievement gaps. He found that achievement gaps were significantly associated with inequitable access to effective teachers between Black and White students. Finally, although they were not directly linked to a single instructional practice per se, several studies have found that a greater number sense and counting ability in the earliest years of schooling was linked to higher mathematics outcomes later in elementary school (Aunola, Leskinen, Lerkkanen, & Nurmi, 2004; Bodovski & Farkas, 2007; Jordan et al., 2007; Jordan, Kaplan, Ramineni, & Locuniak, 2009). Other studies examining achievement across different types of students find small gender differences, with some evidence that these differences may be explained by societal and parental stereotypes as well as girls lack of confidence in their abilities. The gender gap in mathematics achievement widens in K12 education and becomes more dramatic as students become older (Dee, 2007; Entwisle, Alexander, & Olson, 1994; Leahey & Guo, 2001). When surveying a sample of 194 thirdgrade students, Stipek and Gralinski (1991) found that girls rated their mathematics abilities lower than boys and were more likely to attribute their failures to their abilities. Using ECLSK, Penner and Paret (2008) showed that although gender differences are typically thought to begin appearing in the middle grades, they in fact emerge in the early childhood grades, in particular for students with highly educated parents. The authors attributed these differences to social stereotypes present in families of middle and high SES status, noting the influence of stereotypes in perpetuating negative views of females success in mathematics. In addition, a number of studies examine elementary teachers perceptions of girls and boys performance in mathematics. Such studies repeatedly show that elementary teachers ask boys more difficult math questions, hold stereotypes related to girls not needing mathematics, and, in general, have low selfconfidence in their own mathematics skills (e.g., Bursal & Paznokas, 2010; Handal & Herrington, 2003; Trujillo & Hadfield, 1999). A few studies have also examined the types of classroom activities that promote success for different genders. For example, Peterson and Fennema (1985) worked with 36 fourthgrade classrooms and noted that girls and boys engaged differently with certain types of mathematics activities. They found that mathematics achievement was negatively related to competitive mathematics activities for girls but positively related for boys, and they showed that cooperative activities for girls were positively related for girls but negatively related for boys. Other studies have noted that working in singlegender small groups or classrooms for mathematics can also help girls take leadership roles and ask more questions (Streitmatter, 1997). LowerSES students tend to have lower measured math skills compared with students at higherSES levels. One reason may be that students of lower SES have been found to have fewer opportunities to learn, often being tracked into lowerlevel groups or courses, and the achievement gap is mitigated in some studies when controlling for courses that students have taken (Tate, 1997). SES issues also have particular implications for future outcomes. Boalers work in classrooms examining equitable teaching found that reformbased pedagogy focusing on openended tasks and problem solving led to higher achievement for students of all SES backgrounds (Boaler, 1998, 2000, 2002). Although the studies cited in this section discussed links among mathematics achievement, pedagogy, and different types of students, they have relied entirely on data collected prior to the many accountability and curricular reforms introduced after 2001. The past decade has witnessed important policy changes. For instance, with the passage of the No Child Left Behind legislation resulting in an increase in emphasis on standardized tests, we might expect to see a continued positive association between traditional instructional practices that can promote success on tests that require memorization and computationsuch as working on worksheets or with textsand achievement. In addition to the increased prominence of standardized testing, a push for earlier acquisition of problemsolving skills through new sets of state standardsmost notably the Common Coresuggests that pedagogies aimed at accomplishing this goal may succeed in promoting achievement even at the kindergarten level. At the same time, curricular reforms emphasizing more collaborative learning environments have been implemented over the past decade (Bargagliotti, 2012, Groth & Bargagliotti, 2012), and we thus might expect links to emerge between socioemotional development and mathematics instruction that incorporates classroom interactions through group work or working on openended problems. As we discuss in the next section, however, the research on these associations is scant. MATHEMATICS INSTRUCTION AND SOCIOEMOTIONAL OUTCOMES Little prior research has directly examined how the pedagogy used to teach mathematics relates to socioemotional outcomes, and yet it seems highly likely that many of these instructional practices would be significant predictors of these outcomes. Theorists in educational psychology have long surmised that teacher instruction based on group interaction, learning activities, collaboration, trial and error, and experimentation would have positive effects on socioemotional development (Henry, 2003; Linares et al., 2005; Zimmerman, 2002). Although it is underresearched, the link between mathematics instruction and socioemotional development has the potential to be significant. It is well documented that mathematics as a subject has particular influence on individuals stress, development, social interactions, and enjoyment of learning (Bandura, 1993; Bong & Skaalvik, 2003; Dunn & Kontos, 1997; Fennema & Sherman, 1976, Hembree, 1990; Malmivuori, 2001; Tobias, 1978). For example, anxiety about performing in mathematics is a phenomenon that has been linked to overall stress levels (Malmivuori, 2006) as well as the development of selfefficacy (Bong & Clark, 1999, Zimmerman, 2000). Therefore, instructional practices that induce mathematics anxiety in the classroom likely result in higher stress levels, which in turn have been linked to greater problem behaviors in young children (Campbell, Pierce, Moore, Marakovitz, & Newby, 1996). On the other hand, there may be mathematical practices that could help to reduce these negative outcomes. For instance, it has been documented that childcentered practices in kindergarten have been linked to greater instances of positive individual behavior and prosocial relations, as well as greater motivation to learn (Dunn & Kontos, 1997; Stipek, Feiler, Daniels, & Milburn, 1995). Therefore, it may be the case that developmentally appropriate practices, such as childcentered activities, smallgroup instruction, and individualized attention, are linked to stronger social skills as well as fewer problem behaviors, because the nature of these activities reduces stress and increases opportunities for socialization (NC Regional Educational Laboratory, n.d.). Such socioemotional responses to KMIP may vary for particular subgroups of children, including grouping by gender and by SES. Girls typically show higher levels of mathematics anxiety and stress (Armstrong, 1985; Wigfield & Meece, 1988), and the stereotype threat that operates on girls in mathematics (Steele, Spencer, & Aronson, 2002) may potentially make the classroom less hospitable. Boys in kindergarten are also more likely to engage in problem behaviors (Campbell et al., 1996), and thus the impact of particularly stressful mathematics activities might lead to even higher frequencies of problem behaviors in boys. For students of lower SES, the increased academization of kindergarten may be particularly stressful. Prior research has suggested that children of lowerSES families start school with less proficiency in mathematics than higherSES children (Coley, 2002). Because lowerSES families often lack the resources at home to support their children (Orr, 2003), KMIP that press forward more directly academically might lead to increased stress and frustration from both children and their families. Research outside the realm of mathematics instruction suggests that certain types of pedagogy can influence the development of social skills. For example, in a twoyear study of a cooperative elementary school, Stevens and Slavin (1995) found that students in the cooperativelearning environment exposed to instructional strategies such as peer tutoring had better social relations than students at traditional elementary schools. Other authors have also noted that cooperative learning in the classroom can promote social relationships outside of the classroom (Cowie, 1994; Putnam, 1993). Again, the research investigating links between classroom instruction and socioemotional outcomes is sparse and uses data collected a decade or more ago. None investigated socioemotional responses to pedagogy during the crucial year in which children enter formal schooling. An investigation of these links that is up to date, in depth, and focused on kindergarten is the aim of our study. METHOD DATA The data employed in this study were from the Early Childhood Longitudinal StudyKindergarten Class of 201011 (ECLSK: 2011). This dataset was developed by the U.S. Department of Education via the National Center for Education Statistics (NCES). The collection process included a largescale survey design and assessment data collection pertaining to more than 18,000 children, along with their families, teachers, classrooms, and schools. Children were in kindergarten in the academic year 20102011, which was the first year of data collection. The ECLSK: 2011 used a threestage, stratified sampling strategy, in which geographic region represented the first sampling unit, public and private school represented the second sampling unit, and students stratified by race/ethnicity represented the third sampling unit. Hence, observations in the dataset were from a diversity of school types, student socioeconomic levels, and student racial and ethnic backgrounds. The current study relied on the fall and spring survey waves from the kindergarten year. The large sample size of the ECLSK: 2011 ensured that we had sufficient power to detect effects; that said, it was nonetheless important to account for missing responses. First, we relied on two sampling weightsone that adjusted for teacher nonresponse and another that adjusted for parent nonresponse to preserve a sample that was nationally representative of the population of children in the United States who attended kindergarten in 20102011. Estimations that used teacher responses as outcomes used the teacher weight, whereas those estimations that used parent responses as outcomes used the parent weight. Second, for any single variable, missingness was typically low (about 5%), except for a few cases with the parent surveys in which missingness was as high as 24%. To counter the loss of information stemming from nonresponse on the part of survey subjects, we used STATAs implementation of chained multiple imputation to produce ten imputed data sets. Postimputation estimation was carried out using STATAs mi estimation command. After imputation using the sample weights, we had a total of N = 14,370 for the achievement and teacher outcome estimations and N = 10,920 observations for the parent outcome estimations.^{1} Appendix Table 1 presents the means and standard deviations of all variables utilized in this study post imputation for both sample sizes. OUTCOMES Mathematics Test Scores Itemresponse theory (IRT) scale mathematics scores were utilized in this study as our measure of achievement. The mathematics assessments were conducted with the sampled children through oneonone tests administered by trained individuals to each of the two kindergarten wavesfall and spring (Tourangeau et al., 2015). Assessors asked the children questions related to images, such as pictures or number problems, presented on a small easel, and most of the text on the easel was read to them. Children could respond verbally or through pointing; no writing was required, although paper and pencils as well as wooden blocks were available if a child wished to use them. Each test was conducted using a twostage design. The first stage consisted of a routing section that was administered to all students, and the second stage consisted of one of three alternative forms of different difficulty levels, the choice of which depended on the childs performance on the first stage. The content of the mathematics assessments was designed to measure skills in conceptual knowledge, procedural knowledge, and problemsolving and consisted of questions on number sense, properties, and operations; measurement; geometry and spatial sense; data analysis, statistics, and probability; and patterns, algebra, and functions. Spanishspeaking children who did not pass the language screener completed the full mathematics assessment administered in Spanish. Socioemotional Scales The luxury of utilizing the ECLSK dataset is that it provides us with a rich set of socioemotional measures, incorporating measures regarding social skills as well as problem behaviors. Our first set of socioemotional outcomes was derived from teachers assessments of students in the sample. Teachers were surveyed in the fall and spring. Spring assessments were the outcomes, and fall served as a prior measureanalogous to achievement models. There were five teacherreported socioemotional scales in the dataset, developed from unique question items from the teachers rating of an individual student: NCES (2002) provided detail on the individual survey questions used to create the five separate scales. Based on Gottfried (2014), these five scales can be delineated into two aggregate categories: social skills and problem behaviors. Four of the socioemotional scales found in the ECLSK: 2011 dataset were constructed by NCES by adapting the validated Social Skills Rating System (SSRS) developed by Gresham and Elliott (1990). NCES modified the original scales and created its own Teacher Social Rating Scale (SRS) within the ECLSK data. Meisels, AtkinsBurnett, and Nicholson (1996) provided detail. An additional scale, called approaches to learning, was developed specifically for ECLSK: 2011. Gottfried (2014) has shown that within the ECLSK, there are three scales pertaining to social skills within the classroom. The social skills scales included selfcontrol, interpersonal skills, and approaches to learning. The selfcontrol scale measured the frequency of the students ability to control his or her temper, respect others property, accept peer ideas, and handle peer pressure. The interpersonalskills scale measured the frequency with which a child has been getting along with people; forming and maintaining friendships; helping other children; showing sensitivity to the feelings of others; and expressing feelings, ideas, and opinions. The approachestolearning scale was based on items relating to a childs frequency of organization, eagerness to learn new things, independent work ability, adaptability to change, persistence in completing tasks, and ability to pay attention. Gottfried (2014) has also shown that two scales pertain to problem behaviors: Problembehavior scales included externalizing and internalizing behaviors. The externalizing scale measured the frequency with which a child argues, fights, gets angry, acts impulsively, and disturbs ongoing activities. The internalizing scale rated the presence of anxiety, loneliness, low selfesteem, and sadness. NCES constructed all five teacherreported socioemotional rating scales in the same way: Each was a continuous scale based on averaging a series of survey items regarding the frequency of a students behavior, ranging from 1 (never) to 4 (very often). A higher value on any of the socialskills scales reflected a favorable outcome, in which students displayed these social behaviors more frequently. On the other hand, a higher score on the externalizing and internalizing scales reflected an unfavorable outcome, as a higher value indicated that these problem behaviors occurred more frequently for a student. According to the ECLSK: 2011 users manual, the reliability for these scales ranged from approximately 0.79 to 0.91 (Tourangeau et al., 2015). Note that the ECLSK restricteduse data manual provides additional details on the psychometric properties of these scales, though individual items making up the scales are not available in the manual (Tourangeau et al., 2015). Teachers ratings of individual children may be subjectively reported relative to the average behavior of the class. For example, a generally disruptive child may be rated favorably in a class with numerous unruly peers but unfavorably in a class with few unruly peers. Therefore, we also used a variant on our teacherrated outcomes in sensitivity tests; we used behavioraloutcome measures constructed based on parents responses to questions pertaining to their childrens socioemotional behaviors. As with the teacher SRS, NCES also based the parent SRS on the original SSRS. The four parent scales are selfcontrol (five items), social interaction (three items), sad/lonely (four items), and impulsive/overactive behaviors (two items).^{2} Higher scores indicate greater frequency of that behavior. According to the users manual of the ECLSK: 2011 data, the alphareliability coefficients ranged from 0.58 to 0.72, although no reliability measure was constructed for impulsive/overactive behaviors because it was created from only two items. Additional parentrated socioemotional outcomes. We also included four parentrated survey items, all rated on a scale of 1 (more than once per week) to 3 (not at all). A parent indicated the frequency with which a child (a) complained about school, (b) was upset to go to school, (c) faked being sick to stay home, and (d) said he/she liked the teacher. KMIP The ECLSK: 2011 spring teacher surveys contained several questions that pertained specifically to instructional practices used to teach mathematics. Teachers were asked about the degree to which they emphasized18 specific instructional practices, listed as items under the main question How often do children in the class do each of the following mathematics activities? (e.g., How often do children in this class explain how problems are solved?). We coded teacher responses on all of these items to reflect days per month spent using the instructional practice.^{3} When examining instructional practices, prior studies using similar datasets have used factoranalytic techniques to combine like practice items into scales in order to deal with the large number of practices (Bodovski & Farkas, 2007; Guarino et al., 2006). A common approach is to group items according to datadriven exploratory factor analysis. However, it is also crucial to consider the substantive meaning of the scales formed by grouping particular sets of items together. Grouped instructional items should highlight an underlying mathematical teaching construct. Given the large number of individual practice items, we combined like items into scales to capture meaningful teaching constructs that embodied similar components. We used both factoranalytic and conceptual approaches to group particular items together; we then constructed our final scales by averaging the selected items grouped within each scale. The process resulted in five scales containing groupings of items. (Three additional scales composed of the original single items were retained because they did not group with any other items either conceptually or empirically.) The multiitem scales were traditional, manipulatives, problem solving, together, creative, and tools. The singleitem scales were number line, counting out loud, and calendar. Appendix 2 further details the methodology used for scale construction. Table 1 presents descriptive statistics relating to the scales and individual practices that make up each scale. The means represent the average number of times per month (out of 20 possible times) that the practice or practice scale is employed in the classroom. For example, the together scale was used, on average, 8.26 out of 20 possible times per month. Table 1 reveals that two of the single practices, counting out loud and working with calendars, were, on average, used almost daily. On the other extreme, the practices listed in the creative scale were seldom used, and those in the tools scale were almost never used. The minimal use of the practices listed in these scales indicated that these practices may not be appropriate for kindergarten instruction. For example, kindergarten mathematics does not necessitate the use of calculators or rulers; these are typically introduced in later grades in which the topics of multidigit addition/subtraction and measurement are introduced. In addition, these practices had little variation (standard deviations for these items are small). For these reasons, we chose to drop these infrequently used scales from consideration in the study. We also dropped the almost universally used single items from our analyses. Table 1. Kindergarten Mathematics Instructional Practices (KMIP)
This study therefore focused on the following five KMIP scales: traditional, manipulatives, problemsolving, and together, as well as the singleitem scale number line. The traditional scale was composed of practices that have students work with traditional instructional materials such as textbooks, worksheets, and the chalkboard. The manipulatives scale tied together practices that have students work with manipulatives and games. The problemsolving scale focused on mathematical applications. The together scale was composed of instructional practices that rely on group activities. A few additional pedagogical practices listed in separate survey questions were also used in the study. Teachers were asked the extent to which they utilized divided achievement groupings for mathematics. This was coded as hours per week the students spent in such groups. In addition, we included a measure of time spent on mathematics in our analyses. Teachers were asked how often they taught mathematics and how much time they spent on the subject on the days they taught it. We combined the responses to both questions to estimate the total hours per week a teacher reported spending on mathematics. It is important to acknowledge that retrospective survey items may not always accurately capture frequency of use. Nonetheless, some studies have validated survey responses with classroom observations (Mayer, 1999; Stipek & Byler, 2004). However, some measurement error could still reside in these item responses. We might expect any source of error to be randomly distributed with mean zero because of the number of other teacher characteristics included in our models (described in the next section), thus conforming to a classical measurementerror scenario. Measurement error in independent variables, if classical, can lead to attenuation bias in the coefficients and may be one explanation for the relatively small effect sizes found in this and other studies exploring similar questions using survey data. We also further discuss the magnitude of effect sizes in our results section. ANALYTIC APPROACH The associations between KMIP and our set of outcomes were estimated first using the following baseline regression model: where Y represents one of our dependent variablesmathematics achievement or one of the socioemotional outcomesfor a child i in classroom c in school s. All outcomes were measured in the spring of kindergarten, and all outcomes and KMIP variables were standardized so that the coefficients could be interpreted as effect sizes. The covariates in this model are as follows: KMIP are the mathematics instructional practices, X is a set of individual and family characteristics, and CT represents classroom and teacher characteristics. The μ_{s} refer to school fixed effects. The error term ε represents all other factors relating to the outcome variable. It should be noted that all models utilized adjusted errors clustered at the school level to account for the maximum amount of interdependence across observations in the form of correlation among individuals in the same school. Student characteristics included in the model were a continuous measure of kindergartenentry age (in months) and indicators for gender, race (as well as an indicator for whether the child was the same race as his/her teacher), receipt of an individualized education plan (IEP)which served as our indicator that a student had a disabilityEnglish not being the primary home language, whether the child was repeating kindergarten, whether the child was chronically absent during the year,^{4} and whether the child had switched teachers or schools between the fall and spring survey waves. In the mathematicsachievement regressions, we also included a measure of the time elapsed between tests. Family demographics included in the model were household size, number of siblings, and an indicator for a singleparent household. For SES, we used the continuous composite measure based on family education, employment, and income created by NCES (Tourangeau et al., 2015). Control variables pertaining to the classroom context included class size, percentage of students in the classroom with disabilities, and percentage of students in the classroom who were not White. These measures were reported by the childs teacher. We also included a measure for whether the student was in full versus partday kindergarten. In the sample, 81% of the students were in fullday kindergarten; thus, they drive the results presented in this article. Teacher characteristics included indicators for race/ethnicity, holding a standard statelevel certification, having a masters degree or higher, credential status, and certification by the National Board for Professional Teaching Standards. Continuous measures were included for age, years of experience, and the number of courses completed in methods of teaching mathematics. We used school fixed effects to strengthen the causal associations between our outcomes and independent variables of interest. A key concern was that KMIP may not be randomly assigned to students across schools. Therefore, we investigated the issue of random assignment by regressing each teaching practice on a full set of student (including fall mathematics and reading scores), family, teacher, and school characteristics. The results indicated that teaching practices were not randomly distributed across all schools in the dataset. But when school fixed effects were included in the regressions, we found drastically fewer differences in the ways practices were distributed across student characteristics, indicating that teaching practices differed depending on what school a child attended.^{5} For example, several significant associations between SES and practices disappeared when school fixed effects were used. Simply controlling for the school characteristics available in the dataset did not sufficiently capture the schoolrelated factors that contributed to nonrandom assignment, suggesting that unobserved schoollevel programs and processes are correlated with KMIP and possibly also with our outcomes. For instance, some schools may have highly involved principals who urge teachers to boost various types of KMIP. These same principals may be undertaking other investments to boost student outcomes as well. This finding is consistent with prior work using ECLSK data, including Gottfried (2014), Guarino et al. (2013), Aizer (2009), and Neidell and Waldfogel (2010). Based on our investigations, as well as on prior literature, it was evident that a school fixedeffects analysis should be executed as the main specification. Included as a covariate in the achievement and SRS regressions were fall (i.e., kindergarten entry) measures of the outcomesthat is, lagged dependent variables. Children were assessed in mathematics at kindergarten entry as well as in the spring. Similarly, teachers and parents were asked to rate the socioemotional status of each child in both fall and spring. Thus, the majority of our models fell in the category of a valueadded approach. Valueadded models control for prior achievement and thus account for a host of omitted variables that similarly affect both prior and current outcomes. These are more effective in isolating the impact of instructional practices than models that do not take prior achievement into account. Several studies, such as Hanushek (1979), Sass, Semykina, and Harris (2014), Todd and Wolpin (2003), and Guarino, Reckase, and Wooldridge (2015), carefully outlined the theoretical basis for these specifications and the assumptions made to render them both tractable for analysis and supportive of causal inference. Guarino et al. (2015), in particular, discussed the importance of including prior achievement as a covariate rather than using gain scores as dependent variables. We therefore followed this approach here. The only outcomes for which no corresponding fall measure existed were the parent responses to questions about their childs complaining about school, feeling upset about school, pretending to be sick to avoid school, and liking the teacher. Thus the models for these four outcomes did not have a valueadded structure. RESULTS Table 2 presents findings from regressing achievement and socioemotional outcomes on the set of KMIP and our wide span of control measures, including school fixed effects. The regressions presented in Table 2 were based on our baseline specification, which did not contain interactions. Our inquiry into whether the associations with instructional practices were differentiated by individual student characteristics is presented in Table 3, which displays the coefficients and standard errors for the interactions of practices with gender and SES from the interacted regression models. (Only interactions that are statistically significant at the α = 0.05 level are shown.) In both Tables 2 and 3, standardized regression coefficients^{6} are presented with clusteradjusted standard errors in parentheses. All other control variables were included in the regression models but are not presented in the tables for the sake of brevity.^{7} In this section, we report findings by type of KMIP rather than by outcome. Our discussion is organized around each individual type of practice and its associations with the various outcomes we consider. We report only on coefficients that are significant at the 0.05 level or lower. Table 2. Estimated Associations between KMIP and Achievement and Socioemotional Outcomes from Multiple Regressions
To begin, differences in frequencies on the traditional scale were unrelated to achievement and the set of teacher and parentrated social rating scales, but they were associated with differences in certain parentrated nonacademic socioemotional outcomes. Parents whose children were exposed to higher levels of traditional KMIP reported that their children had higher frequencies of complaining about school, being upset about having to go to school, and faking sickness to stay home from school. In Table 3, where interactions were added into the model, the interaction of traditional with SES was significant. Parents of higher SES levels reported higher levels of social interaction when their children were exposed to higher levels of traditional KMIP. Table 3. Estimated Interactions between KMIP and Student Characteristics from Multiple Regressions of Achievement and Socioemotional Outcomes The problemsolving scale was positively related to mathematics achievement in the noninteracted model in Table 2. It was unrelated to any socioemotional outcomes in these models. In the interaction models (Table 3), however, we found that teachers reported higher levels of externalizing for girls when greater emphasis was placed on problem solving. Parents reported higher levels of positive approaches to learning for girls compared with boys when teachers placed a greater emphasis on problem solving. An increased frequency of utilizing manipulatives was associated with lower mathematics achievement outcomes (Table 2). In the interaction model, results indicated that teachers who used manipulatives more often rated girls as having higher levels of interpersonal skill than boys. The together scale had no association with achievement, teacher or parent socioemotional outcomes, or other nonacademic outcomes in the noninteracted model. In the interaction model, however, we observed that girls experiencing a greater frequency of together KMIP were less likely to display externalizing behaviors from the point of view of their teachers but more likely to be reported as complaining about school by their parents. Working with the number line in the classroom had no direct association with achievement, teacher and parentrated social rating scales, or other parentrated nonacademic outcomes. In the interaction models presented in Table 3, however, we can see that parents of girls exposed to higher frequencies of this practice were more likely to report that their child liked their teacher than parents of boys. The amount of time spent on mathematics in the classroom had no significant associations with outcomes in the noninteracted models, but parents of higher SES levels were more likely to report that their children liked their teachers than those of lower SES levels when their children were exposed to greater amounts of mathematics instruction. A caveat to be applied to these findings is that the effect sizes reported for these pedagogies are fairly small, ranging in absolute value from 0.02 to 0.09. Such effectsize magnitudes, however, are typical for this type of analysis. Effect sizes for pedagogies in achievement regressions in prior studies based on the earlier ECLSK were similar, ranging, for example, from 0.01 to 0.1 in Guarino et al. (2006) and from 0.02 to 0.03 in Palardy and Rumberger (2008). DISCUSSION Our findings suggest several possible strategies to consider for mathematics pedagogy in kindergarten and many important avenues for further exploration. Taken together, the results suggest that pedagogy in mathematics can influence childhood developmentboth learning and social developmentin a number of ways. The mere presence of associations between mathematics pedagogy and childhood development is noteworthy because mathematics is a fundamental subject throughout all grade levels. Thus, responses to instruction in this subject for development, beyond just achievement, are important to consider. In addition to several baseline associations, the interactions revealed several notable links between pedagogies and outcomes, suggesting that different types of practices in the kindergarten grade may be linked to outcomes in different ways for different children. This is not surprising, as kindergarten students are still very young, and their development and maturity levels may vary greatly, thus making some students more receptive to certain types of practices than others. Two types of pedagogy were associated with achievement: problem solving and manipulatives. The problemsolving pedagogy was positively associated with achievement, whereas manipulatives were negatively associated. The problemsolving finding is both noteworthy and encouraging, because problemsolving skills are 21stcentury skills that are heavily emphasized in the current era of Common Core and other state standards. As recent curricular reforms have called for increased attention to critical thinking, this result provides some evidence that, even at the early childhood stage, improvement of student problemsolving skills could lead to a better overall understanding of mathematics. Moreover, we find no negative socioemotional responses to this pedagogy and, in fact, a positive association for girls with respect to their parentreported approaches to learning. The use of manipulatives, on the other hand, appears to be problematic for achievement, yet associated with positive interpersonal ratings of girls by teachers. This finding is somewhat puzzling, but it is possible that, although manipulatives provide students with ways to visualize and connect mathematical concepts, they can be too advanced. Moreover, a wide variety of types of manipulatives are currently in use in kindergarten and elementary classrooms, requiring different types of pedagogical skills. Given that our scale combined two types of manipulativescounting and geometricalong with math games into one scale, we also investigated regressions of outcomes on all 18 singlepractice items as a sensitivity analysis.^{8} These results indicated that geometric manipulatives were driving the negative result for achievement; counting manipulatives and math games had no effect. These findings are more or less in line with prior research conducted with the ECLSK data collected a decade earlier, in which Guarino et al. (2013) found positive associations between counting manipulatives and achievement in kindergarten, and Palardy and Rumberger (2008) found a negative association between geometric manipulatives and achievement in first grade. Given the limitations of using teacherreported practice frequencies, which do not indicate how the manipulatives are used in the classroom, to measure pedagogy, the use of manipulatives is an important area of investigation to explore using classroom observations. Moreover, our results flag this as an area to consider carefully in preservice and inservice professional development. Professionaldevelopment courses for elementaryschool mathematics may not adequately address the proper use of different types of manipulatives at different grade levels and how they may affect, and possibly confuse, younger children. Two pedagogies were linked to socioemotional outcomes in the baseline model. Pedagogy that emphasized traditional didactic techniques was strongly associated with undesirable socioemotional responses to school, such as complaining about school or being upset to have to go to school. The use of divided achievement groups was positively associated with teacher reports of internalizing problems on the part of students. Thus, it appears that mathematics pedagogies of this sort may not always play a helpful role in the development of kindergartners abilities and emotional health; there were no offsetting achievement gains associated with either of these pedagogies. On the other hand, we found a positive interaction between traditional KMIP and SES for sociability, indicating that children of higher SES levels may be more tolerant of the pedagogyenough to overcome a negative main effect. Overall, however, our results provide little encouragement for the use of this pedagogical style to teach mathematics in kindergarten, given a lack of association with achievement and mostly negative socioemotional responses in the main model. Gender differences emerged in socioemotional responses to several pedagogies in addition to problem solving and manipulatives. The together pedagogy presented mixed results for girls. Although our findings suggested that having children work together results in lower frequencies of externalizing problems for girls, working together was also associated with girls complaining more about school than boys. One possible explanation for these phenomena could be that girls, when placed in groups, find it less comfortable to express themselves, resulting in some anxiety about the school context. It is also possible that working together at the kindergarten age level requires a level of behavioral restraint and maturity that girls are more likely to exhibit than boys, with the caveat that such restraint may be accompanied by a decreased enjoyment of school. Interestingly, girls were more likely than boys to be rated as exhibiting externalizing behaviors in the context of an emphasis on problem solving, perhaps engendering an unusual amount of engagement for them. However, parentreported approaches to learning were heightened for girls relative to boys when problem solving was more heavily emphasized. It is possible that both these findings can be viewed in a positive light if girls are responding to problem solving with an increase in expressions of interest. With these data, it is difficult to know how best to interpret these findings, but they signal clear gender sensitivities to this type of pedagogy that warrant further investigation with classroomobservation techniques. Interestingly, children of families with higher socioeconomic status were less likely to be rated highly on the interpersonal skills scale by their teachers when their teachers used more of the together type of pedagogya further indication that this instructional approach may have differential effects on particular subpopulations of students. Two additional pedagogies that appeared to affect girls more than boys were working with the number line and working in dividedachievement groups. The number line represents an opportunity to visualize the progression of numbers and the concept of relative quantity in the abstract, and girls were more apt to like their teachers when the number line was emphasized. Girls were also more likely to receive favorable parent ratings for selfcontrol when dividedachievement grouping was emphasized, suggesting that the peergroup factor could have a positive effect on girls social development. These findings are particularly intriguing given the emphasis on working with the number line in the Common Core and other new state standards. In light of the new curricular push to utilize this type of pedagogy, more indepth, observationbased research will be needed to understand fully the developmental factors driving the gender differences. As mentioned above, students from higherSES backgrounds were more likely to have higher parentreported social interactions, compared with those of lower SES, when exposed to higher amounts of traditional KMIP. In addition, when more time was spent on math, parents of higherSES students were more likely to indicate that their children like their teachers. Although this is speculative, it may be the case that children from higherSES families have more opportunities outside of school to engage in mathematics (at home or in academically oriented preschool centers); therefore, these students may enjoy spending more time on mathematics because they are more prepared for the topic. CONCLUSION As mathematics is increasingly emphasized in the early school years, it is of great value to know that what happens in a mathematics classroom can have implications for overall child development. In this study, we examined the role of mathematical instructional practices on academic and socioemotional outcomes for children in kindergarten. Little work has been conducted in this area; almost nothing has previously been established empirically about the role that KMIP plays on outcomes beyond achievement. To address this critical research void, we investigated whether KMIP was associated with differences in achievement and socioemotional outcomes by employing a nationally representative dataset of U.S. kindergartners from the 20102011 school year. Using largescale national data was advantageous for several reasons. The data contained a rich array of outcome measures, which enabled us to examine the role of KMIP with regard not only to achievement but also to multiple socioemotional outcomes, measured by both teachers and parents, for a single sample of children. Moreover, the dataset contained a wide range of contextual information about children, their families, and their teachers and classrooms; thus, our empirical models were able to control for numerous observable factors. At the same time, because the sampling design included multiple students per school, we were able to account for unobservable schooltoschool variation (hence exploiting classroomtoclassroom variation), as well as the nested structure of the data. Given the generally small effect sizes found in the literature with respect to the impact of instructional practices, as well as the possibility of measurement errorinduced attenuation bias, ECLSK offers the best largescale support currently available on a nationally representative scale to investigate these relationships. This study finds several interesting results that shed light on the links among mathematics teaching practices, achievement, and socioemotional outcomes. Several key takeaway points result from our findings. First and foremost, it is clear that the manner in which mathematics is taught in kindergarten can affect not only how much mathematics children learn but also their socioemotional development and their feelings about school. We found that KMIP is in fact associated with differences across multiple domains of student attainment. Second, certain pedagogical approaches are associated with cognitive and noncognitive development in different ways. Third, children respond differently to certain pedagogical approaches according to their gender and their SES. Gender differences in response to a wide number of instructional practices are particularly notable. One potentially encouraging finding is that an emphasis on problem solvingan approach currently promoted by Common Core and other state standards, as well as concomitant curricular reformsshows a positive association with learning and a possible increase in the engagement of girls. On the other hand, we find that an emphasis on traditional didactic forms of instructionsuch as using worksheets, texts, and the chalkboardshows no positive link to learning but links to a number of negative socioemotional responses to school, with the exception of a positive socioemotional response with regard to social interaction for children of higher SES levels. Considering that the students in our sample were in their first year of schooling, these findings are noteworthy. The instructional experiences that children have at the outset of their education can have implications for their longterm educational and social development (Chetty et al., 2011). Therefore, knowing which pedagogical approaches, particularly for a subject such as mathematics, may improve students potential to grow both academically and developmentally would be especially useful for schools. Our findings help to address the role of classroom instruction in kindergarten in developing the whole child, shedding light on which factors help create supportive environments at the very start of formal education. The information regarding teacher practices in mathematics generated in this study can help school practitioners consider how instruction sets students on a trajectory of wholechild growth, helping to secure both short and longterm school success from the very first year of school. A possible concern is that the relations among our reported results may be an artifact of the large sample size.^{9} Given the generally small effect sizes found in the literature with respect to the impact of instructional practices, as well as the possibility of measurement errorinduced attenuation bias that we discuss in the paper, our view is that large sample data sets are needed to detect evidence of small but significant and potentially important relationships. ECLS offers the best support currently available on a nationally representative scale to investigate these relationships. In addition, this study finds several interesting results that shed light on the links among mathematicsteaching practices, achievement, and socioemotional outcomes that are relevant to policies with respect to gender and SES. As is the case with most research, some of our findings merit further study to discern the mechanisms that produce them. Based on our findings, there are several directions for future research. For instance, because this study focused exclusively on a sample of kindergarten students, one limitation may be that we did not examine growth over time. It would be useful to identify whether KMIP has longterm associations with child outcomesthat is, does KMIP support academic and socioemotional growth over time, or do the effects fade? Second, the majority of outcomes investigated in this study were surveybased teacher or parentreported socioemotional scales or attitudes about school. Hence, there is the possibility of some degree of subjectivity in the ratings of students (DiPerna, Lei, & Reid, 2007; Galindo & Fuller, 2010). These findings also merit further study to discern the mechanisms that produce them. Deeper exploration into the developmental mechanisms that result in socioemotional responses to teaching practicesparticularly by subgroupis needed. Analyses of classroom observations might serve this purpose and delve further into the quality of instruction as well. Third, while this study was the first to link mathematical practice to a range of student outcomes, it was not possible to identify how teachers developed their practices. For instance, no information is provided in the dataset about whether teachers had engaged in any professional development to improve their ability to utilize specific types of practices. Similarly, it was not possible to determine how school leadership or various curricula might encourage certain instructional practices in their schools. Therefore, future work may consider an evaluation of how these practices are developed and refined. That is, this study has established that many KMIP are linked to developing the whole child; a next step would be to determine what factors improve effectiveness of practice. Finally, we wish to emphasize that while our findings carry potential policy implications for the encouragement and discouragement of particular mathematical teaching practices at the kindergarten level, we caution readers against generalizing these findings to grade levels beyond kindergarten. It is possible and entirely plausible, for example, that practices that engender negative socioemotional reactions or achievement outcomes in kindergarten may engender positive responses in first grade, when children are developmentally more advanced. Therefore, an informative direction for future research would be to continue similar investigations at successive grade levels. Importantly, policies that foster particular types of instructional approaches, whether through preservice training or professional development, must be sensitive to the readiness of children to accept them at each developmental stage. Over the past decade, there has been a heightened focus on school accountability. This has frequently materialized as policy dialogue around standardized testing and schoolperformance metrics. Very recently, however, policymakers have considered incorporating socioemotional measures of attainment into schoolaccountability metrics.^{10} Our findings support a continued policy conversation regarding how to best gauge school effectiveness. Namely, having established a research base linking mathematics instructional practices to both achievement and socioemotional outcomes in a manner that includes differential impacts by subgroups, policymakers and practitioners can develop further inquiry as to how to improve the classroom context by focusing efforts on improving multiple facets of childhood development simultaneously. In the interest of the development of the whole child, pedagogy must be sensitive to a range of important responses. In fact, another significant direction for future work includes studying how teaching practices in subjects other than mathematics are linked to student achievement and socioemotional outcomes. Notes 1. To preserve the representativeness of the sample, the parent weight weighted many more observations as 0 than did the teacher weight. This explains the difference in sample size between the estimation of parent outcomes and that of teacher outcomes. 2. As with the teacher SRS, individual items and responses were not available for review. 3. In more detail: The response categories of never, once a month or less, two or three times a month, once or twice a week, three or four times a week, and daily were recoded according to the rubric of Guarino et al. (2013). This assumes that there are four weeks in one month and five instructional days per week, leading to a recoding of the responses as follows: Never = 0 times per month; once a month or less = 1; two or three times per month = 2.5; once or twice a week = 6 (average of once a week [4 times] and twice a week [8 times]); three or four times a week = 14 (average of three times a week [12 times] and four times a week [16 times]); and daily = 20. 4. According to Gottfried (2015), this was best defined as missing more than 10 school days. 5. Results from these regressions are available upon request. 6. Recall that all outcomes and KMIP were standardized so that coefficients could be interpreted as effect sizes. 7. Results from the full models are available upon request. 8. Results not shown but available upon request. 9. To allay this potential concern, we ran a splitsample analysis, in which we performed our analyses on three different random 50% samples. 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Selfefficacy: An essential motive to learn. Contemporary Educational Psychology, 25, 8291. Zimmerman, B. J. (2002). Becoming a selfregulated learner: An overview. Theory Into Practice, 41(2), 6470. APPENDIX 1: DESCRIPTIVE STATISTICS Appendix Table 1.1. Descriptive Statistics for All Variables Included in Regressions
APPENDIX 2: KMIP SCALE CONSTRUCTION <TXT>Prior studies using similar datasets have used factoranalytic techniques to combine like practice items into scales (e.g., Bodovski & Farkas, 2007; Guarino et al., 2006;). A common approach is to group items according to datadriven, exploratory factor analysis. However, it is also crucial to consider the substantive meaning of the scales formed by grouping particular sets of items together. Grouped instructional items should highlight an underlying mathematicalteaching construct. Only if all items within a scale are aligned with similar mathematicalteaching pedagogy will their grouping into a scale provide insight into how particular types of practices can affect student outcomes of interestin our case, mathematics achievement and socioemotional responses. We therefore used both factor analysis and a conceptual approach to grouping items into composite scales. As a first step to help consolidate and understand the practices listed in ECLSK: 2011, we performed an exploratory factor analysis using principalcomponent analysis with orthogonal rotation. We chose to retain five factors by applying the usual criterion of setting the number of factors retained to equal the number of eigenvalues of the correlation matrix that are greater than one (see Appendix Table 2.1 for factor loadings of the 18 practices). All but three KMIP practices loaded to one of the five scales. Because the factor loadings for counting out loud, working with calendars, and working with the number line had low loadings on all of the factors, we chose to keep these three practices as singleitem scales. Appendix Table 2.1. Exploratory Factor Analysis Factor Loadings (Teacher Sample N = 3,131)
Note. Highlighted loading indicates highest loading value for the practice. Source: ECLSK: 2011. Several of the groupings that emerged from the exploratory factor analysis matched well with conceptual groupings of the practice scales. For example, doing mathematics worksheets, working with textbooks, and completing problems at the chalkboard loaded to the same factor. We called this factor traditional pedagogy. Items in the manipulatives, creative, and tools scales also loaded highly with one another. While working in small groups or with partners, working in mixedachievement groups, peer tutoring, explaining how a math problem is solved, and working on math problems that reflect realworld solutions all loaded together into one group, these items conceptually mapped onto two different types of practices. We broke up this grouping into two scalesworking with others (together) and working on problem solving (problem solving). Because problemsolving activities may lend themselves to working with others, teachers may use a lot of group activities while working on explaining mathematics problems and working on realworld problems. However, because the goal of this study is to understand how different types of practices are linked to different student outcomes, we chose to separate this scale into two scales that capture the substantive conceptual grouping of these practices. Our together scale consists of working with partners, working with mixed mathematics groups, and working with peers. Our problem solving scale combines working on explaining mathematics problems and working on realworld problems. In an effort to improve the understanding of how teaching practices were associated with student outcomes, consideration of how to group teachers instructional practices was represented by using both datadriven and conceptdriven groupings. This study paid close attention to what the data told us about the item groupings as well as the underlying mathematical constructs related to the groupings. The correlations (see Appendix Table 2.2) among practices within each of the scales are higher than those among practices across scales. For example, the three traditional practices have correlations ranging from 0.300.39, the correlations between the three manipulatives instruction items range from 0.400.53, and the two creative items have a correlation of 0.6. Appendix Table 2.2. Correlations Among Practices
Note. Source: ECLSK: 2011. The group instruction practices (together) correlations range from 0.360.42, and the problem solving items have a correlation of 0.47. Overall, the correlations suggest that teachers utilizing practices that fall under one of the types of pedagogy captured by the scales are also using other practices of that type of pedagogy. As a final effort to examine whether our practice groupings were appropriate, we performed a confirmatory factor analysis. We ran four specific tests: a chisquare test, the rootmean square residual (RMSR), the comparative fit index (CFI), and the rootmean square error approximation (RMSEA), to evaluate whether the data supported our constructed scales. The chisquare test and the RMSR were not in support of our groupings, and the CFI and the RMSEA were. However, as noted by Gatignon (2014), the chisquare test is particularly sensitive to the size of the sample. With large samples sizes like the one in our study, the chisquare test may in fact reject appropriate models. In addition, since the exploratory factor analysis, correlations, and substantive meeting of the scales were in accordance with our groupings, we are confident that we have created scales that provide accurate groupings for these data for the scope of our study. The scales created in this study were similar to several of those created in prior studies using earlier versions of ECLSK datasets. For example, Bodovski and Farkas (2004), using the kindergarten, firstgrade, and thirdgrade waves of ECLSK: 1998, combined the same three practices to define their traditional scale and combined the same three practices that we include in our together scale in their interactive scale. Our manipulatives scale was also identical to one of their scales. Guarino et al. (2006) also grouped similar traditional practices, group practices, and problemsolving practices together.





