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The relationship between mathematics selfefficacy and mathematics achievement: multilevel analysis with NAEP 2019
Largescale Assessments in Education volume 12, Article number: 16 (2024)
Abstract
Background
This empirical study aims to investigate the association between mathematics selfefficacy and mathematics achievement gaps among students in Grades 4, 8, and 12, utilizing data from the 2019 National Assessment of Educational Progress (NAEP). The study also considers studentlevel (e.g., mathematics selfefficacy, gender, race/ethnicity) and schoollevel (e.g., school location, proportion of underrepresented students) demographics to provide a comprehensive analysis of the factors contributing to mathematics achievement gaps.
Methods
A twolevel crosssectional multilevel modeling approach was employed to analyze the variance in mathematics achievement, partitioning it into within and betweenschool components. This approach allowed for an examination of association between mathematics selfefficacy and achievement gaps while considering various student and school characteristics. The rationale behind this methodology lies in its ability to capture the hierarchical nature of educational data and provide a nuanced understanding of the factors associated with mathematics achievement.
Results
The analysis of the NAEP data revealed substantial variability in mathematics achievement across schools in the United States at all grade levels. Furthermore, mathematics selfefficacy emerged as a robust predictor of students' mathematics achievement, exhibiting significant effect sizes for Grades 4, 8, and 12. Remarkably, when students' mathematics selfefficacy was held constant, the mathematics achievement gaps among different student subgroups by gender, race/ethnicity, ELL, IEP, NSLP status narrowed, highlighting the importance of selfefficacy in addressing these disparities. The study also identified the presence of significant school contextual effects, further emphasizing the role of the educational environment in shaping mathematics achievement.
Conclusions
This study underscores the critical role of mathematics selfefficacy in influencing mathematics achievement gaps among students. By acknowledging the association between selfefficacy beliefs and mathematics achievement, policymakers and educators can develop targeted interventions to enhance students' confidence and motivation in mathematics, ultimately promoting equitable educational outcomes. The findings also emphasize the significance of schoollevel factors, calling for comprehensive approaches that consider both individual and contextual factors in narrowing achievement gaps. The implications of adopting a selfefficacy perspective to address mathematics achievement gaps extend to educational policy, curriculum development, and instructional practices, ultimately fostering more inclusive and effective mathematics education for all students.
Introduction
The disparities in mathematics achievement by student subgroups have been a critical issue in addressing the inequities in mathematics education (Hanushek et al., 2019). In previous achievement gap studies, attention has been given to highlighting persistent achievement gaps by accounting for students’ sociodemographic characteristics and school contextual differences (e.g., Lee & Reeves, 2012; Lubienski & Lubienski, 2006; Young et al., 2017). However, caution must be stressed in overemphasizing the associations between students’ demographic characteristics and mathematics achievement, because it may reinforce the deficit views toward the underrepresented students when years of efforts to enhance the achievement of underrepresented students do not yield desirable educational outcomes (Garcia & Guerra, 2004). Also, overemphasis on the effects of demographic markers may take our attention away from possible omitted variables that are associated with achievement, potentially obscuring the broader landscape of influences, including psychological aspects (Wilms et al., 2021). Therefore, we propose that studies exploring achievement gaps should be reframed from the antideficit and assetoriented approach (Harper, 2010) to identify how some malleable factors, such as selfefficacy, are associated with the observed mathematics achievement gaps.
Mathematics selfefficacy is closely associated with students’ mathematics achievement. Selfefficacy refers to an individual’s beliefs in their capacities to execute courses of action to accomplish and succeed in given tasks (Bandura, 1977). According to social cognitive theory (Bandura, 1986), students’ selfefficacy is associated with individual factors, such as goal setting (Schunk & DiBenedetto, 2021), as well as by environmental factors, such as schools where students belong (Bandura, 2001). Studies (e.g., Sakellariou, 2022) have shown a consistent positive correlation between selfefficacy in mathematics and students’ mathematics achievement.
Nevertheless, at least three gaps in the current studies on the role of mathematics selfefficacy in enhancing mathematics achievement remain. First, methodologically, previous studies (e.g., Soland & Sandilos, 2021) tend to overlook the potential associations between school environments and mathematics selfefficacy or mathematics achievement; as a result, the broader interpretative value of these findings may be constrained. Second, the relationship between mathematics selfefficacy and achievement has been shown to covary when considering some affective variables (e.g., mathematics anxiety, Kalaycıoğlu, 2015; interest, Zhang & Wang, 2020). These variables were often reported to be moderately to highly correlated with selfefficacy (e.g., Hiller et al., 2022). This may result in underappreciation of the extent to which mathematics selfefficacy and other affective variables are associated with mathematics achievement. Finally, while the association between mathematics selfefficacy and achievement may differ by students’ developmental stages (Shell et al., 1995), such potential differences by age or grade levels are rarely discussed. Thus, we designed this study to address these gaps in the literature by providing a better understanding of the association between mathematics selfefficacy and achievement gaps across grades and contexts. More specifically, we used a multilevel modeling (MLM) framework to nest students into schools to explore the association of mathematics selfefficacy with achievement, considering a series of student and schoollevel demographics with data from the National Assessment of Educational Progress (NAEP) in Grades 4, 8, and 12.
With the findings from this study about academic selfefficacy, we aim to shift the attention to psychological variables that are malleable and may correlate with narrower achievement gaps among students from different demographic subgroups. Therefore, in this study, we sought to (1) estimate the degree to which mathematics selfefficacy is associated with student mathematics achievement; (2) examine changes in mathematics achievement gaps among student demographic subgroups when accounting for variations in mathematics selfefficacy; and (3) explore if average mathematics selfefficacy at each school is related to their overall school mathematics achievement.
The association between mathematics selfefficacy and mathematics achievement
Research tends to report a positive association between selfefficacy and mathematics achievement. For example, Kalaycıoğlu, (2015) explored the association of mathematics selfefficacy, socioeconomic status (SES), and mathematics anxiety with secondary school students’ mathematics achievement in PISA 2012 and found mathematics selfefficacy to be a strong correlational predictor across all countries with medium effect sizes. Similarly, in a study using PISA 2012 in Greece, Pitsia et al., (2017) found that mathematics selfefficacy was a significant correlational factor in predicting middleschool students’ mathematics achievement after controlling gender and school mean SES, among a group of noncognitive factors (e.g., mathematics selfconcept, attitudes toward school, intrinsic motivation, and instrumental motivation). Moreover, in a metaanalysis on psychological correlates of academic achievement, Richardson et al., (2012) reported a mediumsized positive correlation between academic selfefficacy and academic achievement. They also found selfefficacy had the strongest correlations with academic achievement among compared to approximately fifty other achievementrelated cognitive and noncognitive variables. Mathematics selfefficacy was associated with a considerable proportion of the variance in mathematics achievement alone, accounting for about 54% in Cheema and Kitsantas, (2014) and 40% in Kitsantas (2011) within U.S. PISA data.
The association of mathematics selfefficacy with mathematics achievement across student subgroups
The literature generally highlights the positive correlation between selfefficacy and academic performance, noting variations in the association among student subgroups. For example, gender has been linked to differences in students’ mathematics selfefficacy, particularly during their high school years. Huang’s metaanalysis (2013) of 187 studies from elementary to high school ages indicated no statistically significant gender differences in mathematics selfefficacy existed among students either at elementary or middle school, but found such difference in high school with males reporting higher selfefficacy in mathematics. Conversely, Schwery, (2015) compared fifth to eighthgrade students and found no significant gender gap in mathematics selfefficacy levels or their mathematics achievement, nor in the strength of the association between mathematics selfefficacy and mathematics achievement. These findings may suggest that the oftenreported gender gap in mathematics achievement in high school or beyond may be associated, to some extent, with the reported difference in mathematics selfefficacy between genders (e.g., Cheema & Galluzzo, 2013).
Likewise, a student’s status as an English language learner (ELL) has been associated with variations in mathematics selfefficacy. Sandilos et al., (2020) investigated elementary ELLs and reported lower levels of selfefficacy and achievement in mathematics, even when student demographics and working memory were accounted for. They also found no significant association between limited English proficiency and either mathematics selfefficacy or mathematics achievement. In contrast, Soland and Sandilos, (2021) found an association between ELL status and the growth trajectory of mathematics achievement through selfefficacy in middle school students. Variations in findings between these studies may be partially linked to their differing sample compositions in terms of race/ethnicity. Sandilos et al., (2020) focused on a sample predominantly composed of White students (43%) and a small proportion of ELLs (16%), whereas Soland & Sadilos, (2021) examined a longitudinal cohort from a large urban middle school district with a significant representation of Hispanic (70%) and ELL students (30%). Therefore, the association involving ELL status reported in these studies might also relate to other sociodemographic factors within the samples.
Students with learning disabilities are often observed to have lower selfefficacy in mathematics. Larsen and Jang, (2022) noted an association between students with IEP in Grade 6 and lower selfefficacy compared to their nonIEP peers. Additionally, they found that when students with IEPs were placed in classes with inquirybased teaching, there was an association with improved mathematics achievement. Similarly, Jungert & Anderson, (2013) reported that the lower mathematics selfefficacy among fifth graders with learning disabilities correlated with a history of challenges in mathematics performance and prolonged experience of difficulty with mathematics learning. Their research suggests that instructional practices aimed at enhancing selfefficacy could be associated with supporting achievement among students with special needs.
The literature consistently reports an association between students’ SES and their selfefficacy in mathematics, which in turn is correlated with their mathematics achievement. Wiederkehr et al., (2015) found a correlation between lower SES and decreased levels of mathematics selfefficacy among elementary and secondary students, with this group also tending to have lower mathematics achievement compared to their higher SES peers. These associations were statistically significant with moderate to strong effect sizes. Similarly, McConney and Perry, (2010) analyzed PISA 2003 data and found that students with a higher level of selfefficacy were associated with better mathematics performance when controlling SES status. Yet, the used measures of students’ SES in these studies are often not consistent, such as using parents’ occupations (e.g., Wiederkehr et al., 2015), or constructed SES index with few variables (e.g., McConney & Perry, 2010). Such variability in measuring SES may relate to the observed strength of its association with selfefficacy or achievement.
Finally, by using a largescale dataset, Kotok, (2017) found that highachieving African Americans selfreported the highest degree of selfefficacy whereas highachieving Asian students had the lowest. After controlling for some student and schoollevel variables (e.g., family background, peer engagement, math efficacy, mathematics identity, school SES, school types, and school locations), inversely, highachieving African American students with higher mathematics selfefficacy were associated with lower mathematics achievement compared to their White and Hispanic counterparts. Although Kotok, (2017) only use the subsample of highachieving students in this study, the result partly supports the results by Cheema and Kitsantas, (2014) that reported a small, but significant negative relationship between mathematics selfefficacy and mathematics achievement among Black students in PISA 2003 U.S. sample. Nevertheless, with NSLS: 2009, Andersen and Ward (2014) did not detect any statistically significant differences in the association between mathematics selfefficacy and achievement among Black, Hispanic, and White highachieving students.
School contexts and their relationship with mathematics achievement
Students develop their academic knowledge through the dynamic interplays between students’ selfefficacy and socialecological factors (Bandura, 2001), and thus, considering school contexts is crucial for a more comprehensive understanding of the association between selfefficacy and mathematics achievement. Nevertheless, findings regarding the significance of school location as related to mathematics achievement vary. Kotok, (2017) did not find a significant association between school location (city, town, rural) and achievement disparities. In contrast, Wu, (2015), using NAEP data from 2000 to 2011 for Native American students at advanced academic levels, reported that regional school location was related to achievement in Grades 4 and 8. With PISA 2000 U.S. data, Williams, (2005) found a marginal ruraltown achievement gap in mathematics but a substantial urbantown achievement gap in favor of town schools. Webster and Fisher, (2010) analyzed the TIMSS Australian sample and found that rural school students were associated with lower mathematics scores compared to urban school students, even when considering differences in socioeconomic status.
While Webster and Fisher, (2010) found a higher availability of mathematics resources in rural schools compared to urban schools, the presence of mathematics resources did not show a substantial association with mathematics achievement. It leaves us to wonder if there are other potential variables might play a role in the relationship between school location and schoolwide achievement. For example, school SES, reflecting the average of all students’ SES in a school (as used by McConney & Perry, 2010), is reported as a significant factor related to a school's overall achievement (e.g., Kotok, 2017; Pitsia et al., 2017), but then the relationships between school SES and mathematics achievement vary by school location as well. For example, Williams, (2005) found that school SES was particularly influential in urban schools, more so than in rural ones, in relation to mathematics achievement. In town schools, a significant correlation was observed between school SES and student mathematics achievement. When school SES was accounted for, the gap in mathematics achievement between rural and town schools diminished, yet the difference between urban and town schools persisted, with town schools having the advantage. McConney and Perry, (2010) implied that regardless of their mathematics selfefficacy levels, students being part of a high SES school group were associated with higher mathematics achievement compared to those in lower SES schools after controlling students’ individual SES status. The strongest association between school SES and mathematics achievement was seen in students with high selfefficacy from less affluent families. Therefore, more thorough investigations of the school’s environmental influences are warranted for a better understanding of student mathematics achievement.
Related to this point, to date, there has been little discussion on how access to gifted education programs at schools is associated with students’ mathematics achievement. Therefore, in this study, we included the variable that indicates access to the gifted and talented programs at school as an additional school contextual variable. We view the percentage of gifted program participation to represent the rate of access to gifted education by following a talent development paradigm that focuses on increasing students’ access to highquality gifted services to develop their talents in certain areas (Gentry et al., 2021). It is commonly accepted to view that schools with a higher percentage of students attending gifted programs tend to have higher average mathematics achievement because only already highachieving students are identified to attend gifted programs. Such a view followed the traditional gifted child paradigm that assumes gifted children are born with high potentials (e.g., IQ, Terman, 1925) and the goal of gifted education is to make the fullest use of these potentials (Dai & Chen, 2013). Researchers (e.g., Gentry et al., 2021) have criticized this paradigm for substantially excluding a broad range of students with gifts and talents in domainspecific areas, which exacerbated the inequity in gifted identification. Gentry et al., (2019) reported that rural and town schools demonstrated considerably less equity in identification for gifted programs than city and suburban schools. They also found that nationally, students in lowSES schools were less likely to be identified as gifted students, specifically; 58% of students were from highSES schools. Thus, after accounting for school location and school SES, examining the association between the rate of gifted program participation at school and mathematics achievement will reflect the importance of access to gifted education services.
Collectively, these studies presented thus far provide evidence that students’ demographics and school contexts are associated with students’ selfefficacy. Given the significance of mathematics selfefficacy in relation to mathematics achievement, it is reasonable to speculate that students’ differences in mathematics selfefficacy may be associated with observed disparities in mathematics achievement among student demographic subgroups. Nonetheless, findings regarding the association of mathematics selfefficacy with achievement for different student subgroups have been variable, potentially due to limitations in sample compositions, research designs, and methodologies. Furthermore, the consistent association of mathematics selfefficacy with achievement across different grade levels has yet to be fully explored, which could be addressed with a systematic investigation with nationally representative student assessment data. As students’ mathematics selfefficacy may vary with their academic or developmental progression (Schunk & Pajares, 2002), exploring the role of mathematics selfefficacy at different academic levels provides additional insight into the relationship between motivation and achievement gaps.
Thus, we examined the association between mathematics selfefficacy and mathematics achievement, taking into account student and schoollevel demographic variables for three grade levels (i.e., Grades 4, 8, and 12). And we analyzed the achievement gaps by subgroups with consideration of variations in mathematics selfefficacy. Sequential twolevel crosssectional multilevel models (MLM) were applied for each grade to address the following research questions with NAEP 2019, a nationally representative largescale dataset. The research questions are:
RQ1. What is the distribution of variability in students’ mathematics achievement within schools and between schools?
RQ2. How are studentlevel characteristics (i.e., gender, ELL status, IEP status, NSLP eligibility, race/ethnicity, and mathematics selfefficacy) associated with studentlevel variations in students’ mathematics achievement?

2.1 What proportion of the variance in mathematics achievement can be associated with mathematics selfefficacy when controlling for studentlevel demographic characteristics?

2.2 How do associations between student subgroups (i.e., by gender, ELL, IEP, NSLP, and race/ethnicity) and mathematics achievement differ when mathematics selfefficacy is considered?
RQ3. What are the associations between schoollevel characteristics and variations in student mathematics achievement?
Method
NAEP Data
This secondary data analysis study used the data from NAEP 2019 mathematics (IES license # 13090032). NAEP is designed to measure the trends in the academic performance of U.S. students in Grades 4, 8, and 12 (National Assessment Governing Board, 2020). To make valid inferences about the mathematics achievement of students in the U.S. population, NAEP employed a twostage stratified sampling design to select students in Grades 4, 8, and 12 who were attending public, private, Bureau of Indian Education (BIE), and Department of Defense Education Activity (DoDEA) schools for assessment (National Center for Education Statistics [NCES], 2022a). The two levels of clustering in NAEP were schools within geographic groups and students within schools. Questionnaires that were administered to students, teachers, and school principals contain questions regarding students’ backgrounds as well as school contexts. NAEP data are crosssectional consisting of student and schoollevel variables that are associated with students’ mathematics achievement at a given point in time. The list of NAEP variable names and descriptions that are used as dependent, independent, and weights variables in this study is presented in Appendix A.
Plausible values for NAEP mathematics achievement
NAEP mathematics achievement was reported as scaled scores.^{Footnote 1} NAEP reported 20 plausible values of students’ mathematics achievement scores (variable name in the dataset: MRPCM120) for estimating population characteristics from the sampled students through marginal maximum likelihood analysis (NCES, 2016). The 20 plausible values of students’ mathematics achievement in NAEP 2019 served as the dependent variable as a set in this study.
Studentlevel predictors
Mathematics selfefficacy
Students’ mathematics selfefficacy was a primary variable of interest. Several items measured students’ selfefficacy in mathematics at each grade, asking students to what extent they could answer subjectspecific questions. For example, there were seven items in Grade 8, one of which asked students whether they could “list all of the different possible outcomes when a coin is flipped three times.” Students answered on a 5point response scale from “I definitely cannot” to “I definitely can.” Twoparameter item response theory model (NCES, 2022b) was applied to estimate scaled scores for students’ selfefficacy, and the variable (SQRPM7)^{Footnote 2} was used for analysis. The higher the score, the higher the students’ mathematics selfefficacy.
Demographic characteristics
We also used students’ background variables, which included gender (DSEX), ELL status (LEP), IEP status (IEP), NSLP eligibility status (SLUNCH), and race/ethnicity (DRACE10). Gender was dummy coded. Female students were coded as 1, and male students were coded as 0. We created a set of dummy coded variables to represent the categories of race/ethnicity with White students as a reference group, that is, Asian students (Asian = 1, White = 0), Hispanic students (Hispanic = 1, White = 0), Black students (Black = 1, White = 0), American Indian/Alaska Native students (American Indian/Alaska Native = 1, White = 0), Native Hawaiian/Pacific Islander (Native Hawaiian/ Pacific Islander = 1, White = 0), Two or more races (Two or more races = 1, White = 0). We collapsed the original NAEP categories of “No, formerly ELL” and “No” into the category of students who were not ELL and dummy coded it as 0 (reference group), ELLs as 1. Variables of students with IEP and 504 plans were also combined and recoded as 1, with students not having either plan as a 0. The categories of “reducedprice eligible” and “free lunch eligible” were collapsed as the group of students not eligible for NSLP (dummy coded as a 1) and students not eligible for any lunch programs as a 0.
Schoollevel predictors
A set of background variables was selected to represent schools’ characteristics. The variables of the proportion of Hispanic students (SSCHHSP), Black students (SSCHBLK), American Indian/Alaska Native students (SSCHIND), and Native Hawaiian/Pacific Islander students (SSCHHPI) are continuous, ranging from 0 to 100 percent. The values of these four variables were added to form the new continuous variable of the proportion of underrepresented students in this study. The proportions of ELLs (C044006), students with IEPs (C044007), students eligible for NSLP (C051651), students receiving targeted Title I services^{Footnote 3} (C051801), and students in gifted and talented programs (C044004) are divided into several categories (for details, see Appendix A). The variable of school location (SULOCAL)^{Footnote 4} was recoded as a set of dummy variables with suburban schools coded as 0 (reference group) that generated three new variables, which were City (city = 1, suburb = 0), Town (town = 1, suburb = 0), and Rural (rural = 1, suburb = 0). Schoollevel mathematics selfefficacy variable was created by taking the average of student mathematics selfefficacy in each school.
Data analysis
Because NAEP student data are nested within the school, we applied a set of twolevel crosssectional multilevel models (Raudenbush & Bryk, 2002) that enables us to examine within and betweencluster relations of selfefficacy to mathematics achievement (Stapleton, 2013). We applied group mean centering and grand mean centering (McCoach, 2010) for independent variables to minimize bias in regression coefficient estimates (Enders & Tofighi, 2007). In this study, group means centering^{Footnote 5} was applied for all studentlevel variables. Grand mean centering^{Footnote 6} was used for all schoollevel variables. We also used the final school weight (SMSRSWT) variable at schoollevel models and student conditional weights (i.e., student original joint weight variable, ORIGWT/ SMSRSWT) at studentlevel models and to obtain unbiased students’ mathematics achievement estimates representative of the target population. To assess the association between selfefficacy and the magnitude of mathematics achievement gaps, we examined the rate of variance reduction between schools captured by models (Raudenbush & Bryk, 2002). We ran all MLM analyses with HLM 8.0 software (Raudenbush & Congdon, 2021). The plausible value feature in HLM 8.0 was used to handle the 20 plausible values of mathematics achievement when running models. Maximum likelihood estimation was used for all models.
MLM Modeling steps
To address the research questions in this study, five models were tested with data from each grade level. The details about model equations for Grades 4, 8 & 12 can be found in Appendix B. First, we fitted an unconditional model (Model 0) to the data without any predictors at student or schoollevel, which helped to determine the extent of variance in student mathematics achievement that is associated with schoollevel grouping. We calculated the intraclass correlation coefficient (ICC) to assess the unit dependence within the group. ICC here represents the proportion of total variance in student mathematics achievement accounted for by betweenschool heterogeneity, which ranges from 0 to 1 (Musca et al., 2011). An ICC value between 0.1 and 0.15 (Scherbaum & Ferreter, 2009) or around 0.2 for schoolbased clustering (Hedges & Hedberg, 2007) will support investigating the cluster effects with multilevel modeling analysis.
Next, we added the student demographic predictors into the Level1 model (Model 1). The effects of all studentlevel demographic predictors were allowed to vary across schools initially but fixed for subsequent analyses when statistically nonsignificant random effects were detected. The students’ level of selfefficacy was then incorporated into Model 1 as Model 2, to observe its association with mathematics achievement alongside demographic variables.
As the next step, we simultaneously entered the schoollevel demographic variables into the random slope models of Model 2 to explain the variation of the effects of the student predictors across schools (Model 3). Additionally, the schoollevel mean selfefficacy was included in Model 3 to observe the association of this contextual variable with the outcomes (Model 4). Finally, in Model 5, we added all the abovementioned schoollevel characteristics into the Level2 slope model for selfefficacy and tested the crosslevel interactions. Note that the above analytical steps with MLM were repeated with data from Grades 4, 8, and 12, and the results were evaluated separately to address research questions. Also, note that the final conditional model was not identical across grades because some effects of studentlevel predictors varied across schools at one grade but not at another. In addition, since no school location information was provided in the NAEP Grade 12 dataset, we did not have the school location variables in Models 3, 4, and 5.
Findings
To minimize the repetition in reporting the findings per grade, we focus on the result of Grade 8 and highlight the key results from Grades 4 and 12. All estimates were computed based on appropriate sample size in each variable per NCES reporting guidelines of statistical results (U.S. Department of Education & Institute of Education Sciences, 2005).
Mathematics achievement variation across schools
Results with the unconditional model (Model 0) in Grade 8 showed that the grand mean of the mathematics achievement scores across schools was 281.41 (SE = 0.57, p < 0.001). However, school averages varied significantly across schools, and on average, school means deviated about 16.45 points from the overall mean (p < 0.001). The calculated ICC value was 0.18, which meant that about 18% of the total variance in the eighthgrade mathematics achievement scores was accounted for by betweenschool heterogeneity. Similar results were found in the 4th and 12th grades. For Grades 4 and 12, the grand mean of mathematics achievement (γ_{00}) is 239.52 (SE = 0.4, p < 0.001) and 148.47 (SE = 0.67, p < 0.001). But school means varied significantly between schools (Grade 4: SD = 14.49, variance = 209.96, p < 0.001, Grade 12: SD = 13.38, variance = 178.98, p < 0.001). More specifically, about 18% and 16% of the variance in NAEP 4th and 12th grade mathematics was attributable to betweenschool differences. While meaningful contextual effects exist across all grade levels, the association of the contexts with student mathematics achievement appears to diminish as the grade level advances.
The effect of mathematics selfefficacy on mathematics achievement
Twolevel conditional MLM models (Models 1 & 2) were applied to examine the association between mathematics selfefficacy and mathematics achievement, alongside student demographic predictors. At Grade 8, no random effect at Level2 was statistically significant, and thus we removed all random effects from models for Level1 coefficients at Level2 and set the effects fixed, indicating the association of these student variables with mathematics achievement was consistent across schools. But in Grades 4 and 12, some random effects of studentlevel demographic predictors were detected, for example, only the intercept of IEP and NSLP demographic variables at Level1 were allowed to vary across schools at Grade 4. Table 1 presents the results of Models 1 and 2 for Grades 4, 8, and 12. The intercept of Model 2 in Grade 8 presented that the estimated mean achievement of a student who was at the level of 0 on all dummycoded demographic variables (i.e., nonELL, nonNSLP While male students without IEP) was 280 points (SE = 0.52). A similar interpretation is applied for the conditional means for Grades 4 and 12.
The unique effect of mathematics selfefficacy after controlling demographics
The results of Model 2 showed that after controlling demographic differences, mathematics selfefficacy was significantly associated with students’ mathematics achievement in Grade 8, as well as in Grades 4 and 12. Similarly, the addition of mathematics selfefficacy accounted for an extra 11.86% of the variance in Grade 4 and 17.66% in Grade 12, suggesting a substantial association between students’ mathematics selfefficacy and their mathematics achievement. Furthermore, the magnitude of this association with mathematics achievement was larger in Grade 8 compared to Grades 4 and 12. An increase of one point in studentlevel mathematics selfefficacy was associated with an increase of 8.63 points (SE = 0.1) in mathematics achievement in Grade 8, 5.84 points (SE = 0.09) in Grade 4, and 7.34 points (SE = 0.15) in Grade 12. The variability in the association between mathematics selfefficacy and mathematics achievement across schools was significant for all grades, implying that this relationship was not uniform and varied by school context.
Mathematics selfefficacy and achievement gaps
Compared to the results with Model 1, the results with Model 2 showed that achievement gaps between all student subgroups (except the gendered gap at Grade 8) shrunk considerably after accounting for mathematics selfefficacy at all grades. However, the gaps were still statistically significant. Further details are described below.
Achievement gaps by gender
Gender achievement gaps in mathematics, although notable, remained relatively small in all grades in Model 1. For example, after accounting for other demographic variables, in Grade 8, the average mathematics achievement of males was 2.68 points (SE = 0.19, p < 0.001) higher than that of females. After accounting for the association with mathematics selfefficacy, the gender gaps narrowed down slightly in Grades 4 and 12 but widened with a slight increase in Grade 8.
Achievement gaps by ELL, IEP, and NSLP
The results in Model 1 also presented a glaring mathematics achievement disparity between ELL and nonELL students in Grades 4, 8, and 12. The disparity in Grade 4 between ELL and nonELL students was about 15.56 points (SE = 0.57) when controlling other demographics. However, the disparity was widened to 28.05 points (SE = 0.47) in Grade 8 and 29.38 points (SE = 1.23) in Grade 12. Even when controlling for their mathematics selfefficacy differences, the achievement gap by ELL status at Grades 4, 8, and 12 remained relatively wide.
The original mathematics achievement gap between IEP and nonIEP students in Grade 4 was 29.65 points (SE = 0.41), which widened significantly with increasing school years, 35.82 points (SE = 0.32) in Grade 8, 32.22 points (SE = 0.86) in Grade 12. After considering students’ mathematics selfefficacy, all gaps narrowed by about 5–9 points, but the disparities between IEP and nonIEP students were still large.
The gap between NSLP and nonNSLP students increased slightly from 9.64 points in Grade 4 (SE = 0.31) to 12.04 points in Grade 8 (SE = 0.22) and decreased to 9.11 points in Grade 12 (SE = 0.57). After controlling mathematics selfefficacy, the gap shrunk slightly.
Achievement gaps by race/ethnicity
The gaps of Black—White and Hispanic—White had been significant and extensive across grades, particularly for the gap of Black—White. Among Black students who had the same degree of selfefficacy in mathematics and were identical on other demographic predictors (i.e., gender, ELL, IEP, NSLP), the gaps between those Black and White students fluctuated across grades, about 9.84 points (SE = 0.43) in Grade 4, 19.15 points (SE = 0.37) in Grade 8, 20.75 points (SE = 0.8) in Grade 12 (see results from Model 1). After the selfefficacy difference between Black and White was controlled, the demographic effects were substantially reduced at all grades and still significant. The gaps between NHPI/AIAN and White were statistically significant regardless of their level of selfefficacy across grade levels. When considering the effects of mathematics selfefficacy, the gap between White—Asian in Grades 4, 8, and 12 remained still wide and significant, but shrunk about 4points in Grades 8 & 12.
School contextual effects on mathematics achievement
Tables 2 and 3 summarized the results of these models for Grades 4, 8, and 12. In Model 3, the inclusion of schoollevel demographics explained 52.17% of the variance in average mathematic scores between schools in Grade 8, 45.9% in Grade 4, and 43.45% in Grade 12, which all suggested that in addition to students’ backgrounds (including mathematics selfefficacy), the demographic composition of schools was also found to be significantly associated with school mathematics achievement. After considering all the studentlevel variables, all schoollevel demographic variables, except the percentage of ELLs and city schools or not in Grade 8, significantly explain the variation in school average mathematics achievement. Taking Grade 8 as an example, as for school location, in comparison with suburban schools with similar school settings (i.e., percentage of ELL, IEP, students receiving Title I service, gifted students, and underrepresented students), school average mathematic achievement was 9.95 points (SE = 1.14) lower for town schools; 8.99 points (SE = 0.94) lower for rural schools. Also, for example, a oneunit increase (e.g., from 26–50% to 51–75%) in the percentage of IEP students in schools would result in a 3.82point (SE = 0.52) decrease in school mean achievement with all other contextual backgrounds equal. A onepoint increase in the percentage of underrepresented students in schools also resulted in 0.33 points (SE = 0.02) decrease in school mean achievement. All differences are statistically significant.
The results with Model 4 indicated that school mean mathematics selfefficacy was significantly associated with the schoollevel mean mathematics achievement after controlling schoollevel demographics across grades. This finding indicates a potential contextual correlation with selfefficacy, as the association of the average school mathematics selfefficacy with achievement was observed to be more pronounced than that of individual selfefficacy variables across the grade levels. For Grade 8, a onepoint change in the average schoollevel mathematics selfefficacy is associated with a 17.4point difference (SE = 0.42) in the average school mathematics achievement; for Grade 4, there is a 14.46point difference (SE = 0.51); and for Grade 12, a 14point difference (SE = 0.76). Including the average school mathematics selfefficacy in Model 4 accounted for an additional 27.56% of the betweenschool variance in Grade 8, 20.42% in Grade 4, and 27.42% in Grade 12, suggesting a substantial correlation. Although we were originally interested in whether schoollevel characteristics, specifically school average selfefficacy, explain the variation in the relationship between mathematics selfefficacy and mathematics achievement across schools, we did not find significant variation in the relationship in Model 2. Accordingly, as reported in Table 3, the results showed trivial changes in variances between schools explained by Model 5 compared to Model 4 across three grades.
Discussion
Mathematics achievement disparities by students’ sociodemographic subgroups have been widely discussed for decades (Hanushek et al., 2019), which instead of contributing to shrinking the achievement gaps but reinforced the negative stereotypes thrown on traditionally underrepresented students (Garcia & Guerra, 2004). In this study, using assetoriented perspectives on reducing achievement gaps by subgroups, we draw attention away to the role of malleable characteristics of students (i.e., mathematics selfefficacy), which is a significant motivational process and predicts subsequent academic outcomes (Schunk & DiBenedetto, 2021). Our aim was to determine how mathematics selfefficacy is associated with mathematics achievement across different developmental stages and to consider its potential role as a factor related to the achievement gap attributable to students’ backgrounds. The findings indicate that mathematics selfefficacy is correlated with achievement, with the association appearing most prominent in Grade 8 compared to Grades 4 and 12. Contextual factors such as the proportion of students eligible for NSLP and from underrepresented groups, as well as school location, are also shown to be related to achievement, resonating with previous research (Pitsia et al., 2017; Wu, 2015). The findings highlight the relevance of learning; or school contexts when understanding students’ achievement. Additionally, the data suggest that mathematics selfefficacy is related to the narrowing of achievement disparities across sociodemographic subgroups. In the subsequent sections, we will delve into these associations in greater detail and discuss the implications of our findings for educational practice and future research, while acknowledging that our crosssectional study design precludes the establishment of causal relationships.
The association between mathematics selfefficacy and achievement
Consistent with previous research (e.g., Bohrnstedt et al., 2020; Keşan & Kaya, 2018), findings from this study highlight the significant correlation between students’ mathematics selfefficacy and mathematics achievement across grades. Notably, results indicate that mathematics selfefficacy had the largest association with achievement in Grade 8. This observation points to the potential importance of supporting middle school students in bolstering their mathematics selfefficacy, which may be related to their current and prospective mathematics performance. We posit that there could be at least two reasons for this observed correlation.
First, as students progress through their academic coursework into middle school, they become more aware of the bombarded feedback cues about their mathematics performance that are from their primary resources (e.g., peers, mathematics teachers, and parents) (Hickman & Sherman, 2019). With the increasing complexity of mathematics in middle school, the positive cues students received from their direct ecological environments that blend with their positive dispositions toward learning mathematics play a determinant role in forming a high level of selfefficacy in mathematics (Hickman & Sherman, 2019). For example, for students who have a history of outstanding mathematics performance and access to mathematics learning facilities/resources, they tend to feel mathematics is becoming manageable in middle school and selfaffirm their mathematics abilities (Usher, 2009). It is posited that the level of mathematics selfefficacy is correlated with students’ selfregulation, persistence, engagement, and effort in tackling challenging mathematics problems, which in turn are associated with differences in mathematics achievement (Bandura, 1977).
Second, when students go to middle school, students’ mathematics experiences start to diverge because of tracking and course placement decisions. Based on Subotnik et al. (2021)’s talent development megamodel, when students enter early adolescence, their mathematics talent begins to flourish with the provision of advanced mathematics learning opportunities and psychosocial skills (e.g., selfefficacy) (Subotnik et al., 2021). Middle school opens the door for able students to embrace multiple spheres of opportunities (e.g., honors classes, math clubs, academic summer camps) from which they build high selfefficacy through discourses and interactions with others. Unfortunately, for students not placed in advanced mathematics classes, they often face the prospects of less rigorous coursework, lack of teacher support, and low expectation, whereas their able peers are taught coherent conceptual understanding and highorder thinking skills (Stiff et al., 2011). Additionally, research has shown a correlation between placement in lower academic tracks and a decrease in selfefficacy, which is associated with changes in cognition and performance (Gray et al., 2002). Furthermore, this situation appears to be correlated with disproportionate effects on underrepresented students (e.g., Black, Hispanic, Native American, students from lowincome families, ELLs, and students with disabilities), with some indicators that placement decisions may be associated with demographic data (Stiff et al., 2011).
Mathematics selfefficacy and achievement gaps
When students have similar levels of selfefficacy in mathematics, we observed that the associations of demographic factors with their achievement were less pronounced; mathematics achievement gaps by subgroups were reduced from small to large extent. This implies that mathematics selfefficacy could be uniformly associated with supporting mathematics achievement, irrespective of subgroup identity. But some student subgroups tend to have low mathematics selfefficacy, which is associated with lower achievement outcomes.
In our study, the correlation between mathematics selfefficacy and the narrowing of gender and racialethnic achievement gaps in mathematics was not significant, whereas a strong positive correlation with selfefficacy was observed in the achievement gaps of ELL, IEP, and NSLP students, particularly in Grade 8, which is in line with the findings of Polat et al. (2016) and Soland and Sandilos (2021). With the statistical control of differences in mathematics selfefficacy, we noted a correlation with the reduction of achievement gaps, leading to the consideration that the gaps often seen between wellrepresented and underrepresented students are associated with differences in selfefficacy levels. According to Bandura’s (1977) sources of selfefficacy, ELLs, students with IEPs, and students eligible for NSLP are traditionally documented to experience persistent underperformance in mathematics (Rodriguez et al., 2022) (mastery experiences), express frustrations after social comparisons to normally functioning peers (Coleman, 2001) (vicarious experiences), receive few encouragements (Solomon et al., 1996) (verbal persuasion), and struggle with physical and psychological illbeing (Campbell & Gilmore, 2014) (emotional and physiological states). These factors might interact with the context of having underproficient mathematics teachers (Abedi et al., 2006), who may not effectively reinforce concepts of mathematics selfefficacy, or being in environments with prevalent negative stereotypes (Steele & Aronson, 1995), or attending schools in highpoverty areas with limited resources (DarlingHammond, 2013), which are all associated with lower selfefficacy appraisals.
In contrast, White and Asian students, nonELLs, students without learning disabilities, and students from families with more resources are often associated with a wealth of reinforcing experiences for selfefficacy at schools and homes, such as skillful mathematics teachers structured routine opportunities for success in mathematics and provide abovegradelevel mathematics instructions (Usher, 2009); these experiences are correlated with higher selfefficacy in mathematics. Consequently, if ELL, IEP, and NSLP students were to exhibit high levels of selfefficacy comparable to their peers, they too might show similar academic achievements, and the observed achievement gaps might be less pronounced. It indicates that exploring how underrepresented groups cultivate an interest in mathematics, form aspirations related to mathematics, construct effective responses to stereotypes, and see themselves as capable mathematicians, be correlated with enhancements in their selfefficacy in mathematics.
Contextual correlations with demographic factors
The findings from this study also indicate a significant association between environmental factors (e.g., proportions of students eligible for NSLP and racially/ethnically underrepresented students, the proportion of gifted students, and school location) and the overall mathematics achievement in school. With the findings indicating that schools with a high proportion of students from lowincome families, as well as Black, Hispanic, and Native American students, tend to have lower mathematics achievement compared to others. This underscores the importance of exploring how certain schools with comparable demographic profiles are associated with higher levels of student success in mathematics despite facing academic, social, and institutional challenges. For example, indepth qualitative research into the student support strategies employed in such schools could yield insights that are correlated with the enhancement of educational outcomes for students from diverse cultural and racial backgrounds.
This underscores the importance of further research into scaling up interventions and practices that are associated with effectiveness in various locales. For example, findings from this study are associated with the idea that increased access to gifted education services correlates with smaller mathematics achievement gaps. It aligns with the finding by Young et al. (2017) with NAEP (2009) data that there is no significant difference in mathematics achievement between Black and White fourthgrade girls when Black students had similar levels of access to participate in gifted and talented programs.
Implications for practices
Previous studies (e.g., Plucker et al., 2013; Yang & Maeda, 2023) have identified the emergence of the mathematics achievement gap as early as fourth grade. This study finds a significant association between mathematics selfefficacy and mathematics achievement beginning from fourth grade. This highlights the correlation between early mathematics selfefficacy and achievement outcomes and suggests the potential benefits of supporting students’ mathematics selfefficacy during early school years in relation to achievement gaps. The change in students’ mathematics selfefficacy is associated with both internal personal and external environmental conditions, which can be variable and controllable (Van der Biji & ShortridgeBaggett, 2001). Likewise, students’ selfefficacy development in mathematics is associated with school contexts and involves interactions between students and educational elements such as effective mathematics teaching, highquality mathematics programs, supportive school environments, and stakeholders (e.g., school leaders, mathematics teachers, and specialists) (e.g., Bobis et al., 2013; Griggs et al., 2013; Johnsen & Sheffield, 2021). Thus, fostering students’ resilient sense of selfefficacy in mathematics is associated with continued support from schools and educators at all levels, from elementary to high schools.
As the correlations observed in our study suggest, there is a potential benefit in focusing efforts toward rehumanizing mathematics education, particularly for students who are traditionally underrepresented in mathematics (Goffney et al., 2018). For example, it may be associated with positive outcomes if public schools frontload highquality mathematics education since early childhood education (Plucker et al., 2017) to build young students’ selfefficacy in mathematics. Teachers create inclusive and culturally responsive mathematics learning environments that involve underrepresented students with high potential in meaningful and rigorous learning activities (Yang & Gentry, 2023), which are correlated with increased mastery experiences and thus may support selfefficacy. In this regard, we suggest revisiting the approach of antideficit achievement theory (Harper, 2010) and concentrating our conversations on closing achievement gaps by understanding how students from diverse cultural, linguistic, and socioeconomic backgrounds, are associated with high levels of selfefficacy and how this relates to navigating the systematic environmental barriers put on them. This would be more meaningful than exhaustive and overt statements of achievement gaps among student subgroups.
Limitations and future research directions
The large sample size and representative nature of the final analytic sample drawn from the NAEP dataset support the validity of the results reported in this study. However, some possible limitations are unavoidable, which may require some caution for implications and may be addressed in future studies.
First, the NAEP dataset used in this study is crosssectional; therefore, we refrain from inferring any longterm influences of mathematics selfefficacy on mathematics achievement over time. Consequently, while our findings reveal various patterns in the association between selfefficacy and achievement, we advise researchers to employ longitudinal largescale assessment data to more accurately discern the trends of demographic characteristics and selfefficacy in relation to mathematics achievement throughout the course of education. Additionally, qualitative research could provide valuable insights into the sources of selfefficacy among underrepresented students in mathematics.
Second, mathematics selfefficacy cannot capture all variations in mathematics achievement disparities within schools. Further research could examine the correlations between the mathematics achievement gaps among student subgroups and additional student motivationrelated variables (e.g., achievement goal orientations, interest, enjoyment in mathematics, persistence, and effort expended). Future studies should also be mindful of the association of collective selfefficacy^{Footnote 7} at the school level. There is a trend that students are engaged in cooperative learning mathematics projects and teacher designed mathematics curricula through collaborative work (Schunk & DiBenedetto, 2021). Therefore, examining how collective selfefficacy is related to students’ mathematics achievement could shed light on promoting students’ selfefficacy and mathematics achievement. In addition, research has indicated that the teachers’ collective selfefficacy^{Footnote 8} in teaching is significantly associated with academic performance and the educational environment of students, even after controlling for previous student accomplishments and crucial demographic variables, such as socioeconomic status (e.g., Bandura, 1993). Schools that achieve success are identified by teachers’ shared beliefs in the capabilities of their staff to assist students in their growth and learning (Klassen et al., 2011). Therefore, investigating the factors contributing to mathematics teachers' collective selfefficacy, especially in schools with scarce resources and a high proportion of underrepresented students, and how this is associated with their students’ mathematics abilities, could contribute valuable insights.
Finally, mathematics performance disparities are associated with more than just the demographic composition of schools. Findings from other NAEP studies have shown that contextual factors, such as teachers’ professional development (Havard et al., 2018), classroom instruction (Lubienski, 2006), and teacher resources (Lee & Reeves, 2012), are also associated with students’ selfefficacy and mathematics achievement. Future researchers may consider including a broader spectrum of schoollevel variables to understand these contextual correlations further. Moreover, we also encourage future researchers to include some equityrelated variables in mathematic learning as part of contextual variables. In this study, we considered the variable representing students’ access to gifted programs, acknowledging that such variables need further careful operationalization and modeling to enhance their interpretative validity.
Conclusion
The purpose of this study was to explore the association between mathematics selfefficacy and mathematics achievement disparities. As an exploratory study, it sheds light on how enhancing mathematics selfefficacy is related to the achievement gaps among student subgroups. We found a significant correlation between mathematics selfefficacy and U.S. students’ mathematics achievement, particularly noting a larger association in Grade 8. It provides empirical evidence for supporting the assetoriented approach to consider students’ mathematics selfefficacy as a potentially influential and malleable factor in relation to achievement gaps.
Availability of data and materials
This study used the restricteduse NAEP data, therefore it is not accessible to public without IES permission.
Notes
The scaled score ranged from 0 to 500 for Grades 4 and 8, 0 to 300 for Grade 12.
NAEP reported the scaled score to represent the degree of selfefficacy.
For those public schools in which students who qualify for free or reduced lunch programs share at least 40 percent of enrollment, such schools are eligible to use federally granted Title I funds to operate schoolwide programs to enhance the achievement of students at risk (U.S. Department of Education, 2018).
The dataset of grade 12 did not have this variable. It did not have any variable to explicitly represent school locale through urbancentric type, either. Therefore, in the data analysis of grade 12, there was no variables to present school locations.
Group means centering refers to subtracting the mean score from the higherlevel group, which is school level in this study, for all students within the same school. Thus, the betweenschool variations are removed from studentlevel predictors. The intercept presents the average score of the group students belong.
Grand mean centering subtracts the overall mean for the variables across all groups from each score. The intercept represents the average score of students’ mathematics achievement for all students in 2019 NAEP.
Bandura (1977) defined collective efficacy as “a group’s shared belief in its conjoint capabilities to organize and execute the courses of action required to produce given levels of attainments” (p. 477).
Teachers' collective efficacy pertains to their conviction in their collaborative capacity to affect the lives of their students (Bandura, 1993).
Abbreviations
 NAEP:

National assessment of educational progress
 ELL:

English language learner
 IEP:

Individualized education plan
 NSLP:

National school lunch program
References
Abedi, J., Courtney, M., Leon, S., Kao, J., Azzam, T. (2006). English language learners and math achievement: A study of opportunity to learn and language accommodation. Technical report 702. National center for research on evaluation, standards, and student testing.
Andersen, L., & Ward, T. J. (2014). Expectancy‐value models for the STEM persistence plans of ninth‐grade, high‐ability students: a comparison between Black, Hispanic, and White students. Science Education, 98(2), 216–242. https://doi.org/10.1002/sce.21092
Bandura, A. (1986). Social foundations of thought and action: A social cognitive theory. PrenticeHall.
Bandura, A. (1977). Selfefficacy: toward a unifying theory of behavioral change. Psychological Review, 84(2), 191–215. https://doi.org/10.1037/0033295X.84.2.191
Bandura, A. (1993). Perceived selfefficacy in cognitive development and functioning. Educational Psychologist, 28, 117–148.
Bandura, A. (2001). Social cognitive theory: An agentic perspective. Annual Review of Psychology, 52(1), 1–26. https://doi.org/10.1146/annurev.psych.52.1.1
Bobis, J., Mulligan, J., & Lowrie, T. (2013). Mathematics for children: Challenging children to think mathematically. London: Pearson.
Bohrnstedt, G. W., Zhang, J., Park, B. J., Ikoma, S., Broer, M., & Ogut, B. (2020). Mathematics identity, selfefficacy, and interest and their relationships to mathematics achievement: A longitudinal analysis. In R. T. Serpe, R. Stryker, & B. Powell (Eds.), Identity and symbolic interaction: Deepening foundations, building bridges (pp. 169–210). Springer.
Campbell, M., & Gilmore, L. (2014). The importance of social support for students with intellectual disability: An intervention to promote mental health and wellbeing. Cypriot Journal of Educational Sciences, 9(1), 21–28.
Cheema, J. R., & Galluzzo, G. (2013). Analyzing the gender gap in math achievement: Evidence from a largescale US sample. Research in Education, 90(1), 98–112.
Cheema, J. R., & Kitsantas, A. (2014). Influences of disciplinary classroom climate on high school student selfefficacy and mathematics achievement: A look at gender and racial–ethnic differences. International Journal of Science and Mathematics Education, 12(5), 1261–1279. https://doi.org/10.1007/s1076301394544
Coleman, M. R. (2001). Surviving or thriving? 21 gifted boys with learning disabilities share their school stories. Gifted Child Today, 24(3), 56–64. https://doi.org/10.4219/gct2001538
Dai, D. Y., & Chen, F. (2013). Three paradigms of gifted education: In search of conceptual clarity in research and practice. Gifted Child Quarterly, 57(3), 151–168. https://doi.org/10.1177/0016986213490020
DarlingHammond, L. (2013). Inequality and school resources. In K. G. Welner & P. L. Carter (Eds.), Closing the opportunity gap: What America must do to give every child an even chance (pp. 77–93). Oxford University Press.
Enders, C. K., & Tofighi, D. (2007). Centering predictor variables in crosssectional multilevel models: A new look at an old issue. Psychological Methods, 12(2), 121–138. https://doi.org/10.1037/1082989X.12.2.121
Garcia, S. B., & Guerra, P. L. (2004). Deconstructing deficit thinking: Working with educators to create more equitable learning environments. Education and Urban Society, 36(2), 150–168.
Gentry, M., Gray, A., Whiting, G. W., Maeda, Y., & Pereira, N. (2019). Access denied/system failure: Gifted education in the United States: Laws, access, equity, and missingness across the country by locale, Title I school status, and race. Purdue University. https://www.education.purdue.edu/geri/newpublications/giftededucationintheunitedstates
Gentry, M., Desmet, O. A., Karami, S., Lee, H., Green, C., Cress, A., Chowkase, A., & Gray, A. (2021). Gifted education’s legacy of high stakes ability testing: Using measures for identification that perpetuate inequity. Roeper Review, 43(4), 242–255. https://doi.org/10.1080/02783193.2021.1967545
Goffney, I., Gutiérrez, R., & Boston, M. (2018). Rehumanizing mathematics for black, indigenous, and Latinx students. Reston: National Council of Teachers of Mathematics.
Gray, J. R., Braver, T. S., & Raichle, M. E. (2002). Integration of emotion and cognition in the lateral prefrontal cortex. Proceedings of the National Academy of Sciences, 99(6), 4115–4120. https://doi.org/10.1073/pnas.06238189
Griggs, M. S., RimmKaufman, S. E., Merritt, E. G., & Patton, C. L. (2013). The responsive classroom approach and fifth grade students’ math and science anxiety and selfefficacy. School Psychology Quarterly, 28(4), 360–373. https://doi.org/10.1037/spq0000026
Hanushek, E. A., Peterson, P. E., Talpey, L. M., & Woessmann, L. (2019). The achievement gap fails to close: Half century of testing shows persistent divide between haves and havenots. Education next, 19(3), 8–17.
Harper, S. R. (2010). An antideficit achievement framework for research on students of color in STEM. New Directions for Institutional Research, 2010(148), 63–74. https://doi.org/10.1002/ir.362
Havard, B., Nguyen, G. N., & Otto, B. (2018). The impact of technology use and teacher professional development on US national assessment of educational progress (NAEP) mathematics achievement. Education and Information Technologies, 23(5), 1897–1918. https://doi.org/10.1007/s1063901896964
Hedges, L. V., & Hedberg, E. C. (2007). Intraclass correlation values for planning group randomized trials in education. Educational Evaluation and Policy Analysis, 29(1), 60–87. https://doi.org/10.3102/0162373707299706
Hickman, C. J., & Sherman, H. J. (2019). Learning mathematics successfully: Raising selfefficacy in students, teachers, and parents. Charlotte: Information Age Publishing.
Hiller, S. E., Kitsantas, A., Cheema, J. E., & Poulou, M. (2022). Mathematics anxiety and selfefficacy as predictors of mathematics literacy. International Journal of Mathematical Education in Science and Technology, 53(8), 2133–2151. https://doi.org/10.1080/0020739X.2020.1868589
Johnsen, S. K., & Sheffield, L. J. (2021). Using the common core state standards for mathematics with gifted and advanced learners. Routledge.
Jungert, T., & Andersson, U. (2013). Selfefficacy beliefs in mathematics, native language literacy and foreign language amongst boys and girls with and without mathematic difficulties. Scandinavian Journal of Educational Research, 57(1), 1–15. https://doi.org/10.1080/00313831.2011.621140
Kalaycioğlu, D. B. (2015). The influence of socioeconomic status, selfefficacy, and anxiety on mathematics achievement in England, Greece, Hong Kong, the Netherlands, Turkey, and the USA. Educational Sciences Theory and Practice, 15(5), 1391–1401.
Keşan, C., & Kaya, D. (2018). Mathematics and science selfefficacy resources as the predictor of academic success. International Online Journal of Educational Sciences, 10(2), 45–58. https://doi.org/10.15345/iojes.2018.02.004
Kitsantas, A., Cheema, J., & Ware, H. W. (2011). Mathematics achievement: The role of homework and selfefficacy beliefs. Journal of Advanced Academics, 22(2), 310–339. https://doi.org/10.1177/1932202X1102200206
Klassen, R. M., Tze, V. M., Betts, S. M., & Gordon, K. A. (2011). Teacher efficacy research 1998–2009: Signs of progress or unfulfilled promise? Educational Psychology Review, 23, 21–43. https://doi.org/10.1007/s1064801091418
Kotok, S. (2017). Unfulfilled potential: Highachieving minority students and the high school achievement gap in math. High School Journal, 100(3), 183–202. https://doi.org/10.1353/hsj.2017.0007
Larsen, N. E., & Jang, E. E. (2022). Instructional practices, students’ selfefficacy and math achievement: A multilevel factor score path analysis. Canadian Journal of Science, Mathematics and Technology Education, 21(4), 803–823. https://doi.org/10.1007/s42330021001813
Lee, J., & Reeves, T. (2012). Revisiting the impact of NCLB highstakes school accountability, capacity, and resources: State NAEP 1990–2009 reading and math achievement gaps and trends. Educational Evaluation and Policy Analysis, 34(2), 209–231. https://doi.org/10.3102/0162373711431604
Lubienski, S. T. (2006). Examining instruction, achievement, and equity with NAEP mathematics data. Education Policy Analysis Archives, 14(14), 1–33. https://doi.org/10.14507/epaa.v14n14.2006
Lubienski, S. T., & Lubienski, C. (2006). School sector and academic achievement: A multilevel analysis of NAEP mathematics data. American Educational Research Journal, 43(4), 651–698. https://doi.org/10.3102/00028312043004651
McCoach, D. B. (2010). Hierarchical linear modeling. In G. R. Hancock, G. R. Hancock, R. O. Mueller, L. M. Stapleton, & R. O. Mueller (Eds.), The reviewer’s guide to quantitative methods in the social sciences (pp. 123–140). Routledge.
McConney, A., & Perry, L. B. (2010). Science and mathematics achievement in Australia: The role of school socioeconomic composition in educational equity and effectiveness. International Journal of Science and Mathematics Education, 8(3), 429–452. https://doi.org/10.1007/s1076301091974
Musca, S. C., Kamiejski, R., Nugier, A., Méot, A., ErRafiy, A., & Brauer, M. (2011). Data with hierarchical structure: Impact of intraclass correlation and sample size on typeI error. Frontiers in Psychology, 2, 1–6. https://doi.org/10.3389/fpsyg.2011.00074
National Center for Education Statistics. (2009). NAEP technical documentation item scaling models. https://nces.ed.gov/nationsreportcard/tdw/analysis/scaling_models.aspx
National Center for Education Statistics. (2016). NAEP technical documentation: Plausible values versus individual scores. https://nces.ed.gov/nationsreportcard/tdw/analysis/est_pv_individual.asp
National Center for Education Statistics. (2022b). NAEP technical documentation: NAEP assessment IRT parameters. https://nces.ed.gov/nationsreportcard/tdw/analysis/scaling_irt.aspx
National Center for Education Statistics. (2022a). NAEP technical documentation NAEP assessment sample design. https://nces.ed.gov/nationsreportcard/tdw/sample_design/
National Assessment Governing Board. (2020). Mathematics framework for the 2019 National Assessment of Educational Progress. Washington, DC: U.S. Department of Education.
Pitsia, V., Biggart, A., & Karakolidis, A. (2017). The role of students’ selfbeliefs, motivation and attitudes in predicting mathematics achievement: A multilevel analysis of the Programme for International Student Assessment data. Learning and Individual Differences, 55, 163–173. https://doi.org/10.1016/j.lindif.2017.03.014
Plucker, J. A., Hardesty, J., & Burroughs, N. (2013). Talent on the sidelines: Excellence gaps and America’s persistent talent underclass. Stamford: University of Connecticut.
Plucker, J. A., Peters, S. J., & Schmalensee, S. (2017). Reducing excellence gaps: A researchbased model. Gifted Child Today, 40(4), 245–250. https://doi.org/10.1177/1076217517723949
Polat, N., ZareckyHodge, A., & Schreiber, J. B. (2016). Academic growth trajectories of ELLs in NAEP data: The case of fourthand eighthgrade ELLs and nonELLs on mathematics and reading tests. The Journal of Educational Research, 109(5), 541–553. https://doi.org/10.1080/00220671.2014.993461
Raudenbush, S.W., & Congdon, R.T. (2021). HLM 8: Hierarchical linear and nonlinear modeling. Scientific Software International.
Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (Vol. 1). Newcastle upon Tyne: Sage.
Richardson, M., Abraham, C., & Bond, R. (2012). Psychological correlates of university students’ academic performance: A systematic review and metaanalysis. Psychological Bulletin, 138(2), 353–387. https://doi.org/10.1037/a0026838
Rodriguez, D., Carrasquillo, A., Garcia, E., & Howitt, D. (2022). Factors that challenge English learners and increase their dropout rates: Recommendations from the field. International Journal of Bilingual Education and Bilingualism, 25(3), 878–894. https://doi.org/10.1080/13670050.2020.1722059
Sakellariou, C. (2022). The reciprocal relationship between mathematics selfefficacy and mathematics performance in US high school students: Instrumental variables estimates and gender differences. Frontiers in Psychology. https://doi.org/10.3389/fpsyg.2022.941253
Sandilos, L. E., Baroody, A. E., RimmKaufman, S. E., & Merritt, E. G. (2020). English learners’ achievement in mathematics and science: Examining the role of selfefficacy. Journal of School Psychology, 79, 1–15. https://doi.org/10.1016/j.jsp.2020.02.002
Scherbaum, C. A., & Ferreter, J. M. (2009). Estimating statistical power and required sample sizes for organizational research using multilevel modeling. Organizational Research Methods, 12(2), 347–367. https://doi.org/10.1177/1094428107308906
Schunk, D. H., & DiBenedetto, M. K. (2021). Selfefficacy and human motivation. In A. J. Elliot (Ed.), Advances in motivation science (pp. 153–179). Amsterdam: Elsevier. https://doi.org/10.1016/bs.adms.2020.10.001
Schunk, D. H., & Pajares, F. (2002). The development of academic selfefficacy. In A. Wigfield & J. S. Eccles (Eds.), Development of achievement motivation (pp. 15–31). Academic Press.
Schwery, D. A. (2015). How do mathematics selfefficacy and gender interact to predict mathematics achievement in fifth through eighth graders? (Publication No. 3723935) [Doctoral dissertation, University of South Dakota]. ProQuest Dissertations and Theses database.
Shell, D. F., Colvin, C., & Bruning, R. H. (1995). Selfefficacy, attribution, and outcome expectancy mechanisms in reading and writing achievement: Gradelevel and achievementlevel differences. Journal of Educational Psychology, 87(3), 386–398. https://doi.org/10.1037/00220663.87.3.386
Soland, J., & Sandilos, L. E. (2021). English language learners, selfefficacy, and the achievement gap: Understanding the relationship between academic and socialemotional growth. Journal of Education for Students Placed at Risk, 26(1), 20–44. https://doi.org/10.1080/10824669.2020.1787171
Solomon, D., Battistich, V., & Hom, A. (1996). Teacher beliefs and practices in schools serving communities that differ in socioeconomic level. The Journal of Experimental Education, 64(4), 327–347.
Stapleton, L. M. (2013). Using multilevel structural equation modeling techniques with complex sample data. In G. R. Hancock & R. O. Mueller (Eds.), Structural equation modeling: A second course (2nd ed., pp. 521–562). Charlotte: Information Age Publishing.
Steele, C. M., & Aronson, J. (1995). Stereotype threat and the intellectual test performance of African Americans. Journal of Personality and Social Psychology, 69(5), 797–811. https://doi.org/10.1037/00223514.69.5.797
Stiff, L. V., Johnson, J. L., Akos, P., Tate, W. F., King, K. D., & Anderson, C. R. (2011). Examining what we know for sure: Tracking in middle grades mathematics. In W. F. Tate, K. D. King, & C. R. Anderson (Eds.), Disrupting tradition: Research and practice pathways in mathematics education (pp. 63–77). National Council of Teachers of Mathematics.
Subotnik, R. F., OlszewskiKubilius, P., & Worrell, F. C. (2021). The talent development megamodel: A domainspecific conceptual framework based on the psychology of high performance. In R. J. Sternberg & D. Ambrose (Eds.), Conceptions of giftedness and talent (pp. 425–442). Cham: Palgrave Macmillan. https://doi.org/10.1007/9783030568696_24
Terman, L. M. (1925). Genetic studies of genius Vol. 1. Mental and physical traits of a thousand gifted children. Redwood City: Stanford University Press.
U.S. Department of Education & Institute of Education Sciences. (2005). IES Style Guide. https://nces.ed.gov/statprog/styleguide/pdf/styleguide.pdf
U.S. Department of Education (2018). Improving basic programs operated by local educational agencies (Title I, part A). https://www2.ed.gov/programs/titleiparta/index.html#:~:text=Schools%20in%20which%20children%20from,of%20the%20lowest%2Dachieving%20students
Usher, E. L. (2009). Sources of middle school students’ selfefficacy in mathematics: A qualitative investigation. American Educational Research Journal, 46(1), 275–314. https://doi.org/10.3102/0002831208324517
Van der Biji, J. J., & ShortridgeBaggett, L. M. (2001). The theory and measurement of the selfefficacy construct. Scholarly Inquiry for Nursing Practice, 15(3), 189–207.
Webster, B. J., & Fisher, D. L. (2010). Accounting for variation in science and mathematics achievement: A multilevel analysis of Australian data Trends in International Mathematics and Science Study (TIMSS). School Effectiveness and School Improvement, 11(3), 339–360. https://doi.org/10.1076/09243453(200009)11:3;1G;FT339
Wiederkehr, V., Darnon, C., Chazal, S., Guimond, S., & Martinot, D. (2015). From social class to selfefficacy: Internalization of low social status pupils’ school performance. Social Psychology of Education, 18(4), 769–784. https://doi.org/10.1007/s1121801593088
Williams, J. H. (2005). Crossnational variations in rural mathematics achievement. Journal of Research in Rural Education, 20(5), 1–18.
Wilms, R., Mäthner, E., Winnen, L., & Lanwehr, R. (2021). Omitted variable bias: A threat to estimating causal relationships. Methods in Psychology, 5, 100075. https://doi.org/10.1016/j.metip.2021.100075
Wu, J. (2015). A comprehensive analysis of the NAEP data from Native American youth concerning excellence gaps (Publication No. 3734112) [Doctoral dissertation, Purdue University]. ProQuest Dissertations and Theses database.
Yang, Y., & Gentry, M. L. (2023). Striving to excel in STEM: Insights from underrepresented, minoritized graduate students with high academic ability. Gifted Child Quarterly, 67(2), 110–136. https://doi.org/10.1177/0016986222111
Yang, Y., & Maeda, Y. (2023). An investigation of the excellence gaps in mathematics education: Evidence from NAEP 2019 [Manuscript in preparation]. Purdue Univeristy.
Young, J. L., Young, J. R., & Ford, D. Y. (2017). Standing in the gaps: Examining the effects of early gifted education on Black girl achievement in STEM. Journal of Advanced Academics, 28(4), 290–312. https://doi.org/10.1177/1932202X17730549
Zhang, D., & Wang, C. (2020). The relationship between mathematics interest and mathematics achievement: Mediating roles of selfefficacy and mathematics anxiety. International Journal of Educational Research, 104, 101648. https://doi.org/10.1016/j.ijer.2020.101648
Acknowledgements
We would like to extend our sincere gratitude to the Institute of Education Sciences (IES) for providing access to the NAEP 2019 dataset. This valuable resource has been instrumental in the advancement of our research. Additionally, we are thankful to the American Institutes for Research (AIR) for their expert guidance in data analysis, which has greatly enhanced our understanding and interpretation of the data. Their support has been crucial in enabling us to conduct a thorough and meaningful analysis.
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MG, who tragically passed away during the course of this study, played a significant role in designing the conceptual framework and research plan. Her invaluable contributions laid the foundation for the study's direction and objectives. YM contributed to the study by designing the data analysis plan and developing the models used for analysis. His expertise in statistical methodologies and modeling techniques ensured the robustness and accuracy of the data analysis process. YY took on the responsibility of analyzing and interpreting all the data collected for this study. With meticulous attention to detail and expertise in data analysis, YY provided insightful interpretations of the findings, contributing to the study's overall conclusions. YM and YY collaborated closely throughout the study, discussing and refining the research methods, analyzing the data, and interpreting the results. They both read and critically reviewed the final manuscript, providing valuable feedback and ensuring its quality. Although MG's untimely passing prevented her from reviewing the final manuscript, her contributions and insights were instrumental in shaping the study. Her dedication and intellectual contributions will forever be remembered and acknowledged. Overall, the study represents a collaborative effort, with MG, YM, and YY each making significant contributions to different aspects of the research process, ultimately leading to the completion of this empirical study.
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Appendices
Appendix 1
Dependent, independent, and weight variables in MLM
Construct  NAEP 2019 data variable name  Description of the variable  Coding/score range 

Mathematics Achievement  MRPCM120  20 mathematics composite plausible values  Gr. 4: 0–500 Gr. 8:0–500 Gr.12: 0–300 
Student level  
Gender  DSEX  Gender  1 = Female, 0 = Male 
Race/ethnicity  DRACE10  Race/ethnicity (studentreported)  1 = Asian, not Hispanic 0 = White; 1 = Hispanic of any race, 0 = White; 1 = Black, not Hispanic, 0 = White; 1 = American Indian/Alaska Native (AIAN), 0 = White; 1 = Native Hawaiian/ Pacific Islander (NHPI), 0 = White; 1 = Two or more races, 0 = White; 
ELL  ELL  Student classified as English language learner  1 = ELL, 0 = Not ELL 
IEP  IEP  Student classified as having a disability  1 = students with IEPs, 0 = students without IEPs 
NSLP  SLUNCH  National School Lunch Program eligibility  1 = Yes (eligible for NSLP program) 0 = No (not eligible for NSLP program) 
Mathematics selfefficacy  SQRPM7  Students’ selfefficacy in mathematics  Gr.4:6.62—12.96 Gr.8:5.63—13.37 Gr.12: 5.54–14.14 
School contextual level  
Proportion_ ELLs  C044006  Percent receiving ESL instruction  0 = None, 1 = 1–5%, 2 = 6–10%, 3 = 11–25%, 4 = 26–50%, 5 = 51–75%, 6 = 76–90%, 7 = Over 90%, 88 = Omitted 
Proportion_ IEPs  C044007  Percent in special education  0 = None, 1 = 1–5%, 2 = 6–10%, 3 = 11–25%, 4 = 26–50%, 5 = 51–75%, 6 = 76–90%, 7 = Over 90%, 88 = Omitted 
Proportion_Title I service  C051801  Percent receiving targeted Title I services  0 = None, 1 = 1–5%, 2 = 6–10%, 3 = 11–25%, 4 = 26–50%, 5 = 51–75%, 6 = 76–90%, 7 = Over 90%, 88 = Omitted 
Proportion_gifted  C044004  Percent in gifted and talented program  0 = None, 1 = 1–5%, 2 = 6–10%, 3 = 11–25%, 4 = 26–50%, 5 = 51–75%, 6 = 76–90%, 7 = Over 90%, 88 = Omitted 
Proportion_Hispanic  SSCHHSP  School percent Hispanic  0–100% 
Proportion_ Black  SSCHBLK  School percent Black  0–100% 
Proportion_ AIAN  SSCHIND  School percent American Indian/Alaska Native  0–100% 
Proportion_ NHPI  SSCHHPI  School percent Native Hawaiian/ Pacific Islander  0–100% 
Proportion_ NSLP  C051651  Percent eligible National School Lunch Program  0 = 0%, 2 = 1–5%,3 = 6–10%,4 = 11–25%,5 = 26 34%,6 = 35–50%,7 = 51–75%,8 = 76–99%, 9 = 100%, 88 = Omitted 
Schoollocation  SULOCAL  Urbancentric type of locale  1 = City, 0 = Suburb; 1 = Town, 0 = Suburb; 1 = Rural, 0 = Suburb 
Mean mathematics selfefficacy*  –  Mean of students’ mathematics selfefficacy  – 
Weights  
ORIGWT  Student original joint weight  1.8751–553.7003  
SMSRSWT  School final weight  1–855.7789 
Appendix 2
Model equations across grades 4, 8 &12
Model 0 equation: unconditional model
Grade 4  Grade 8  Grade 12  

Level1 Model  MRPCM1 = B0 + r  
Level2 Model  B0 = G00 + u0 
Model 1 Equation: add all studentlevel demographic variables
Grade 4  Grade 8  Grade 12  

Level1 Mode  MRPCM1 = B0 + B1* (FEMALE) + B2* (ELL) + B3* (IEP) + B4* (NSLP) + B5* (BLACK) + B6* (HISPANIC) + B7* (ASIAN) + B8* (AIAN) + B9* (NHPI) + B10* (> 1 RACE) + r  
Level2 Model  B0 = G00 + u0  B0 = G00 + u0  B0 = G00 + u0 
B1 = G10  B1 = G10  B1 = G10  
B2 = G20  B2 = G20  B2 = G20  
B3 = G30 + u3  B3 = G30  B3 = G30  
B4 = G40 + u4  B4 = G40  B4 = G40  
B5 = G50  B5 = G50  B5 = G50  
B6 = G60  B6 = G60  B6 = G60  
B7 = G70  B7 = G70  B7 = G70  
B8 = G80  B8 = G80  B8 = G80  
B9 = G90  B9 = G90  B9 = G90  
B10 = G100  B10 = G100  B10 = G100 
Model 2 Equation: add selfefficacy variable
Grade 4  Grade 8  Grade 12  

Level1 Model  MRPCM1 = B0 + B1* (FEMALE) + B2* (ELL) + B3* (IEP) + B4* (NSLP) + B5* (BLACK) + B6* (HISPANIC) + B7* (ASIAN) + B8* (AIAN) + B9* (NHPI) + B10* (> 1 RACE) + B11* (SELFEFFICACY) + r  
Level2 Model  B0 = G00 + u0  B0 = G00 + u0  B0 = G00 + u0 
B1 = G10  B1 = G10  B1 = G10  
B2 = G20  B2 = G20  B2 = G20  
B3 = G30 + u3  B3 = G30  B3 = G30  
B4 = G40 + u4  B4 = G40  B4 = G40  
B5 = G50  B5 = G50  B5 = G50  
B6 = G60  B6 = G60  B6 = G60  
B7 = G70  B7 = G70  B7 = G70  
B8 = G80  B8 = G80  B8 = G80  
B9 = G90  B9 = G90  B9 = G90  
B10 = G100  B10 = G100  B10 = G100  
B11 = G110 + u11  B11 = G110 + u11  B11 = G110 + u11 
Model 3 Equation: add all schoollevel demographic variables to the random slope
Grade 4  Grade 8  Grade 12  

Level1 Model  MRPCM1 = B0 + B1* (FEMALE) + B2* (ELL) + B3* (IEP) + B4* (NSLP) + B5* (BLACK) + B6* (HISPANIC) + B7* (ASIAN) + B8* (AIAN) + B9* (NHPI) + B10* (> 1 RACE) + B11* (SELFEFFICACY) + r  
Level2 Model  B0 = G00 + G01* (%ELL) + G02* (%IEP) + G03* (% TITLE I) + G04* (% GIFTED)  B0 = G00 + G01* (%ELL) + G02* (%IEP) + G03* (% TITLE I) + G04* (% GIFTED)  B0 = G00 + G01* (%ELL) + G02* (%IEP) + G03* (% TITLE I) + G04* (% GIFTED) 
+ G05* (% UNDERREPRESENTED) + G06* (CITY) + G07* (TOWN) + G08* (RURAL) + u0  + G05* (% UNDERREPRESENTED) + G06* (CITY) + G07* (TOWN) + G08* (RURAL) + u0  + G05* (% UNDERREPRESENTED) + u0  
B1 = G10  B1 = G10  B1 = G10  
B2 = G20  B2 = G20  B2 = G20  
B3 = G30 + u3  B3 = G30  B3 = G30  
B4 = G40 + u4  B4 = G40  B4 = G40  
B5 = G50  B5 = G50  B5 = G50  
B6 = G60  B6 = G60  B6 = G60  
B7 = G70  B7 = G70  B7 = G70  
B8 = G80  B8 = G80  B8 = G80  
B9 = G90  B9 = G90  B9 = G90  
B10 = G100  B10 = G100  B10 = G100  
B11 = G110 + u11  B11 = G110 + u11  B11 = G110 + u11 
Model 4 Equation: add schoollevel mean selfefficacy variable to the random slope
Grade 4  Grade 8  Grade 12  

Level1 Model  MRPCM1 = B0 + B1* (FEMALE) + B2* (ELL) + B3* (IEP) + B4* (NSLP) + B5* (BLACK) + B6* (HISPANIC) + B7* (ASIAN) + B8* (AIAN) + B9* (NHPI) + B10* (> 1 RACE) + B11* (SELFEFFICACY) + r  
Level2 Model  B0 = G00 + G01* (%ELL) + G02* (%IEP) + G03* (% TITLE I) + G04* (% GIFTED)  B0 = G00 + G01* (%ELL) + G02* (%IEP) + G03* (% TITLE I) + G04* (% GIFTED)  B0 = G00 + G01* (%ELL) + G02* (%IEP) + G03* (% TITLE I) + G04* (% GIFTED) 
+ G05* (% UNDERREPRESENTED) + G06* (CITY) + G07* (TOWN) + G08* (RURAL)  + G05* (% UNDERREPRESENTED) + G06* (CITY) + G07* (TOWN) + G08* (RURAL)  + G05* (% UNDERREPRESENTED)  
+ G09* (SELFEFFICACY_M) + u0  + G09* (SELFEFFICACY_M) + u0  + G06* (SELFEFFICACY_M) + u0  
B1 = G10  B1 = G10  B1 = G10  
B2 = G20  B2 = G20  B2 = G20  
B3 = G30 + u3  B3 = G30  B3 = G30  
B4 = G40 + u4  B4 = G40  B4 = G40  
B5 = G50  B5 = G50  B5 = G50  
B6 = G60  B6 = G60  B6 = G60  
B7 = G70  B7 = G70  B7 = G70  
B8 = G80  B8 = G80  B8 = G80  
B9 = G90  B9 = G90  B9 = G90  
B10 = G100  B10 = G100  B10 = G100  
B11 = G110 + u11  B11 = G110 + u11  B11 = G110 + u11 
Model 5 Equation: add all schoollevel variables into level2 selfefficacy slope model
Grade 4  Grade 8  Grade 12  

Level1 Model  MRPCM1 = B0 + B1* (FEMALE) + B2* (ELL) + B3* (IEP) + B4* (NSLP) + B5* (BLACK) + B6* (HISPANIC) + B7* (ASIAN) + B8* (AIAN) + B9* (NHPI) + B10* (> 1 RACE) + B11* (SELFEFFICACY) + r  
Level2 Model  B0 = G00 + G01* (%ELL) + G02* (%IEP) + G03* (% TITLE I) + G04* (% GIFTED)  B0 = G00 + G01* (%ELL) + G02* (%IEP) + G03* (% TITLE I) + G04* (% GIFTED)  B0 = G00 + G01* (%ELL) + G02* (%IEP) + G03* (% TITLE I) + G04* (% GIFTED) 
+ G05* (% UNDERREPRESENTED) + G06* (CITY) + G07* (TOWN) + G08* (RURAL)  + G05* (% UNDERREPRESENTED) + G06* (CITY) + G07* (TOWN) + G08* (RURAL)  + G05* (% UNDERREPRESENTED)  
+ G09* (SELFEFFICACY_M) + u0  + G09* (SELFEFFICACY_M) + u0  + G06* (SELFEFFICACY_M) + u0  
B1 = G10  B1 = G10  B1 = G10  
B2 = G20  B2 = G20  B2 = G20  
B3 = G30 + u3  B3 = G30  B3 = G30  
B4 = G40 + u4  B4 = G40  B4 = G40  
B5 = G50  B5 = G50  B5 = G50  
B6 = G60  B6 = G60  B6 = G60  
B7 = G70  B7 = G70  B7 = G70  
B8 = G80  B8 = G80  B8 = G80  
B9 = G90  B9 = G90  B9 = G90  
B10 = G100  B10 = G100  B10 = G100  
B11 = G110 + G111* (%ELL) + G112* (%IEP) + G113* (% TITLE I) + G114* (% GIFTED)  B11 = G110 + G111* (%ELL) + G112* (%IEP) + G113* (% TITLE I) + G114* (% GIFTED)  B11 = G110 + G111* (%ELL) + G112* (%IEP) + G113* (% TITLE I) + G114* (% GIFTED)  
+ G115* (% UNDERREPRESENTED) + G116* (CITY) + G117* (TOWN) + G118* (RURAL)  + G115* (% UNDERREPRESENTED) + G116* (CITY) + G117* (TOWN) + G118* (RURAL)  + G115* (% UNDERREPRESENTED)  
+ G119* (SELFEFFICACY_M) + u11  + G119* (SELFEFFICACY_M) + u11  + G116* (SELFEFFICACY_M) + u11 
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Yang, Y., Maeda, Y. & Gentry, M. The relationship between mathematics selfefficacy and mathematics achievement: multilevel analysis with NAEP 2019. Largescale Assess Educ 12, 16 (2024). https://doi.org/10.1186/s4053602400204z
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DOI: https://doi.org/10.1186/s4053602400204z