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Math achievement and self-efficacy of linguistically and ethnically diverse high school students: their relationships with English reading and native language proficiency

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Math achievement and self-efficacy of linguistically and ethnically diverse high school students: their relationships with English reading and native language proficiency

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Content Running head: NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

Math Achievement and Self-Efficacy of Linguistically and Ethnically Diverse High School
Students: Their Relationships with English Reading and Native Language Proficiency

Elena Son

A Dissertation Presented to the
Faculty of the USC Graduate School
University of Southern California
In Partial Fulfillment of the
Requirements for the Degree of
Doctor of Philosophy in Education

December 2015

NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

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TABLE OF CONTENTS
LIST%OF%TABLES% % % % % % % % % % 5%
%
LIST%OF%FIGURES % % % % % % % % % % 6%
%
ABSTRACT % % % % % % % % % % % 7%
%
CHAPTER%I:%INTRODUCTION % % % % % % % % 8%
% Statement%of%the%Problem% % % % % % % % 8%
% Statement%of%Purpose%% % % % % % % % 14%
%
CHAPTER%II:%LITERATURE%REVIEW % % % % % % % 16%
% Motivation%Research%and%Theory % % % % % % % 16%
Overview%of%Motivation% % % % % % % 16%
Social%Cognitive%Theory%and%SelfRefficacy%% % % % % 17%
Math%SelfRefficacy%and%Related%Factors % % % % % 18%
AchievementRRelated%Outcomes % % % % % 19%
Prior%and%Subsequent%SelfRefficacy%% % % % 20%
Gender% % % % % % % % 20%
Language%Minority%Status,%English%Language%Proficiency,%%
and%L1%(nonREnglish)%Proficiency% % % % % 21%
Mathematics%Research% % % % % % % % % 23%
Math%Learning/Achievement%and%Related%Factors% % % % 23%
Language% % % % % % % % 23%
Reading% % % % % % % % 24%
English%Language%Proficiency% % % % 25%
Native%Language%Proficiency%% % % % 26%
%% Prior%Mathematics%Achievement% % % % % 27%
Socioeconomic%Status% % % % % % 28%
Race/Ethnicity % % % % % % % 28%
Gender% % % % % % % % 29%
% English%Reading/Achievement%and%Second%Language%Learning% %
%%Research % % % % % % % % % % 30%
English%Reading/Achievement%and% Related%Factors % % % 30%
Language%Minority%Status% % % % % % 30%
CrossRLinguistic%Transfer% % % % % % 31%
Native%Language%Proficiency% % % % % 34%
Socioeconomic%Status% % % % % % 37%
Race/Ethnicity % % % % % % % 38%
Moderating%Role%of%First%Language%Proficiency% % % % % % 38%
Mediating%Roles%of%English %Reading%and%Mathematics%Achievement % % 39%
SchoolRLevel%Student%Demographics,%Achievement,%and%Motivation% % 40%
School%Socioeconomic%Status%% % % % % % 41%
School%Proportion%of%Racial/Ethnic%Minority%Students % % % 41%
School%Proportion%of%ELL%students%% % % % % 43%
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

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Gaps%in%Literature% % % % % % % % % 43%
%
CHAPTER%III:%METHODOLOGY % % % % % % % % 46%
Research%Questions%and%Hypotheses % % % % % % 46%
% Education%Longitudinal%Study%(Cohort%2002)% % % % % 50%
% % ELS 2002 Sampling Procedure and Data Collection 50
Strengths%and%Limitations%of%ELS%2002% % % % % 53
% Sample%and%Variables% % % % % % % % 54%
% Aggregated%Variables,%ELS%2002%Sampling%Weights,%and%Imputation% % 61%
% Multilevel%Structural%Equation%Modeling% % % % % % 63%
Specification,%Identification,%Estimation,%and%Evaluation%%
%of%the%Proposed%Multilevel%SEM%model% % % % % 64%
% % Weights,%Mediation,%and%Sensitivity%Analysis% % % % 68%
% % Power%% % % % % % % % % 69%
% % %
CHAPTER%IV:%RESULTS % % % % % % % % % 70%
% Descriptive%Statistics%% % % % % % % % 70%
% Multivariate%Normality%Assumptions% % % % % % 75%
% Outliers,%and%Leverage%and%Influential%Points% % % % % 76%
% Multicollinearity% % % % % % % % % 77%
% Missing%Data% % % % % % % % % % 77%
% Psychometric%Properties%of%Mathematics%SelfRefficacy%Measure%% % 80%
% Pearson%Correlations%and%Intraclass%Correlations%% % % % 81%
Multilevel%Structural%Equation%Modeling%Analysis%% % % % 83%
Findings%pertaining%to%Research%Questions % % % % 83%
% % Additional%Findings% % % % % % % % 94%
%
CHAPTER%V:%DISCUSSION % % % % % % % % % 96%
% Supported%Relationships % % % % % % % % 96%
% % School%SES%and%Mathematics%Achievement%% % % % 96%
% % Reading%and%Mathematics%Achievement % % % % % 96%
Reciprocal%Relationship%between%Mathematics%Achievement% %
%and%SelfRefficacy% % % % % % % % 97%
Mediating%Roles%of%English%Reading%and%Mathematics% %
%Achievement%% % % % % % % % 99%
% Unsupported%Relations % % % % % % % % 100%
School%Proportions%of%Racial/Ethnic%Minority%and%ELL%Students,% %
Math%Achievement,%and%SelfRefficacy% % % % % 100%
% % Relations hips%among%LM%Status,%L1%Proficiency,%and%English%%
Reading % % % % % % % % % 102%
% Comparing%Results%Across%Levels % % % % % % % 107%
% Implications% % % % % % % % % % 109%
Limitations% % % % % % % % % % 111%
% Future%Research % % % % % % % % % 113%
% Contribution%of%Study% % % % % % % % 115%
% Conclusion% % % % % % % % % % 116%
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

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%
REFERENCES% % % % % % % % % % % 117%
APPENDICES%% % % % % % % % % % 143%
Appendix A: Histograms and Plots for Student-Level Outcomes 143
Appendix B: Histograms and Plots for School-Level Outcomes 146
Appendix C: Histograms of Possible Transformations of Student-Level
Variables 149
Appendix D: Histograms of Possible Transformations of School-Level
Variables 152
Appendix E: Bivariate Scatter Plots for Student-Level Variables 153
Appendix F: Residual Plots for Student-Level Variables 155
Appendix G: Bivariate Scatter Plots for School-Level Outcomes 159
Appendix H: Residual Plots for School-Level Variables 162
Appendix I: Missing Data Patterns and Frequencies 167
Appendix J: Percentage of Complete Data (Covariance Coverage) 170
Appendix K: Results of Logistic Multiple Regression Models Predicting
Missingness 171
Appendix L: Unweighted Student-Level Correlations 174
Appendix M: Unweighted School-Level Correlations 175
Appendix N: Mplus Script for Final Weighted Multilevel SEM model (mod1) 176
Appendix O: Mplus Script for Final Unweighted All Cases Multilevel SEM
Model (mod2) 178
Appendix P: Mplus Script for Unweighted Some Cases Multilevel Model
(mod3) 180
Appendix Q: Mplus Script for Final Weighted Model with Mediation Paths
Constrained to Zero (mod4) 182

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LIST OF TABLES
Table 1: Descriptive Statistics 73
Table 2: Math Self-efficacy Items Factor Loadings and Residual Variances
for Grades 10 and 12 81
Table 3: Weighted Student-Level Correlations 82
Table 4: Weighted School-Level Correlations 83
Table 5: Parameter Estimates for Weighted, Unweighted, and Some Cases
Multilevel Models 90

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LIST OF FIGURES
Figure 1: Hypothesized Multilevel Model – School-Level 48
Figure 2: Hypothesized Multilevel Model – Student-Level 49
Figure 3: Final Weighted Multilevel Model – School-Level 85
Figure 4: Final Weighted Multilevel Model – Student-Level 88

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Abstract
The under-preparation in math at the high school and college levels, as well as the low
participation of ethnically and linguistically diverse individuals in STEM fields are concerning
because their preparation for work in these areas is essential for the U.S. to remain competitive in
the innovative knowledge economy. While there is now a substantial body of research on this
group of students, there remain unresolved questions around the role of linguistic factors,
affective variables, and prior achievement. In light of this concern, the purpose of the study was
two-fold. One was to examine the moderating role of first language (L1) proficiency on the
effects of language minority (LM) status in English reading. The second was to investigate the
mediating roles of English reading and math achievement in the relationship between such
interaction and math self-efficacy. The study was a secondary analysis of the Education
Longitudinal Study (ELS 2002, n =16,110). Using a multilevel SEM analysis the study did not
find support for the moderating role of L1 proficiency. However, English reading and math
achievement mediated the relationship between LM status and math self-efficacy. These findings
provide further knowledge for the development of targeted interventions that aim at increasing
the preparation and participation of linguistically and ethnically diverse students in STEM fields.

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Chapter I: Introduction
Statement of the Problem
The U.S. population has become increasingly diverse with regards to the population size
of ethnic and racial groups, socioeconomic status, and home language. Specifically, the
percentage of Hispanic and Asian/Pacific Islander student enrollment in public schools has
increased, whereas the percentage of White student enrollment has decreased from 2001 to 2011
(National Center for Education Statistics (NCES), 2013). For example, the enrollment percentage
for Hispanic students increased from 17.1 to 23.7 and for Asian/Pacific Islander students it
increased from 4.2 to 5.1. In economic terms of poverty, the percentage of public school students
who were eligible for free or reduced-price lunch increased from 28.3 in 2000-01 school year to
49.6 (around 25 million students) in 2011-12 school year (NCES, 2013). Among the 16 million
students under the age of 18 who lived in poverty in 2011, 39% were Black, 34% were Hispanic,
and 13% were Asian (Aud et al., 2013). Similarly, the percentage of students who received
language support in public schools, English language learners
1
(ELLs), has increased from nine
percent of the school-age population (or 4.1 million) in 2002-2003 to 10 percent in 2010-2011
(4.7 million) (Aud et al., 2013). In 2012, about 20% of the U.S. population over the age of five
spoke a language other than English at home (U.S. Census Bureau, 2012). Spanish was the
language most frequently spoken after English. For example, 38 million U.S. individuals spoke
Spanish or Spanish Creole, and from this group almost half (16 million) spoke English “less than
very well”, whereas the other half spoke English “very well” (22 million; U.S. Census Bureau,
2012). The current demographic trends point to a growing population of students of color and
non-native English speakers who are more likely to live in poverty.

1
I define ELLs as students whose first language is other than English, who learn English as a second language, and
who do not yet have full proficiency in English (see Genesee, Lindholm-Leary, Saunders, & Christian, 2005).
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

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The fast growing percentages of students of color, students who live in poverty, and
English language learners bring greater attention to their academic achievement. Although the
reading achievement gaps between Whites and Blacks, and Whites and Hispanics have generally
narrowed since 1990, they still persist (NCES, 2013). For example, in 2012, 17-year-old White
students outperformed Black and Hispanic students in the National Assessment Educational
Progress (NAEP) reading tests by 26 and 21 points, respectively. The achievement gap based on
race/ethnicity persists in the math domain as well. For example, in 2009 the percentages of 12
th

grade students who were at or above the level of proficiency on the NAEP math tests were as
follows: 33% were White, 6% were Black, and 11% were Hispanic (NCES, 2013). Similarly, the
percentage of 12
th
grade students who scored at or above the level of proficiency on the 2009
NAEP math test and were eligible for free or reduced-price lunch was 10%, compared to 32% of
students who were ineligible (NCES, 2013). Many English language learners also underperform
in reading and math compared to native English speakers. For instance, 12
th
grade ELLs scored
around 20 points lower on the 2009 math NAEP tests compared to non-ELLs who had taken the
same level high school mathematics (NCES, 2013). Additionally, 30% of 8
th
grade ELLs scored
at or above basic on the 2013 NAEP reading assessment as compared to 79% of native English
speakers, and 3% of ELLs as compared to 36% native English speakers scored at or above
proficiency on the same test (NCES, 2013). The above achievement gaps indicate that the
academic needs of students of color, students living in poverty, and ELLs in the domains of
reading and mathematics are not currently being met.
Underachievement in K-12 math is particularly problematic because many college
students are underprepared in math and only a small proportion of students have careers in the
science, technology, engineering, and mathematics (STEM) field (National Science Board
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(NSB), 2014). For example, although the majority of first-time, full-time freshmen entering four-
year colleges in 2012 completed algebra II and pre-calculus/trigonometry in high school, less
than a third took probability and statistics courses, and calculus (Pryor et al., 2012). Most
importantly, a third of students enrolled in four-year institutions, and only half of students
enrolled at two-year institutions needed math remediation courses in the 2003-2004 academic
year (NSB, 2014). With regards to degrees in STEM areas, from the total number of first
university degrees in science and engineering provided worldwide, the United States conferred
10% of these degrees, whereas China and the European Union provided 24% and 17% of these
degrees, respectively. In particular, only 5% of total degrees conferred in the U.S. are in
engineering, whereas 31% of total degrees in these fields are given in China. The under-
preparation in math at the high school and college levels, as well as the low number of STEM
degrees and individuals working in these fields in the U.S. are concerning because the
preparation of youth and young adults for work in these areas is essential for the U.S. to remain
competitive in the innovative knowledge economy (NSB, 2014).
The ethnic, racial, and gender diversity in the STEM fields has increased; however, the
proportions of individuals of color and women still remain low (NSB, 2014). Specifically, the
population of individuals with science and engineering careers has become more diverse as
compared to the 1990s, but the percentages of women and individuals from various ethnic/racial
backgrounds remain low (NSB, 2014). For example, in 2010 the percentages of women (30% -
38%), Hispanics, and Blacks (5% - 8%) that work in science and engineering occupations or
hold high degrees in these areas were much lower than the percentages of men and White
individuals (NSB, 2014). The lower participation of women and non-whites in STEM fields has
resulted in the National Research Council (2011) establishing as one of its goals of science
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education to increase the number of students from non-representative demographic groups who
acquire advance degrees and pursue occupations in STEM areas (cf. National Science
Foundation (NSF), 2011). Therefore, examining how to increase the math preparation and
participation of U.S. students, in particular of women and non-whites, in the STEM field is a
pressing priority.
The population size and growth of students of color, students living in poverty, and
ELLs highlight their academic underachievement at the K-12 level. Their academic performance
has important implications for other areas, including college course selection and graduation, as
well as career earnings. For instance, students of color, particularly Hispanics and Blacks have
low college entrance and completion rates (Aud et al., 2013; Ortiz, Valerio, & Lopez, 2012;
Zwick & Sklar, 2005). Moreover, research reporting associations between academic achievement
and school dropout (Rumberger & Lim, 2008); and college enrollment, GPA, and graduation; as
well as career earnings (Hu & Wolniak, 2013; Zarate & Gallinore, 2005; Zwick & Sklar, 2005)
strongly suggest the need to address the academic needs of diverse students, particularly in the
domain of math, and especially in high school when students begin to seriously consider higher
education and career choices.
Several approaches have been taken to address U.S. students’ lower achievement,
particularly in STEM areas including professional developments and school reforms (NRC,
2011). For instance, some researchers (e.g., Lee, Maerten-Rivera, Penfield, LeRoy, & Secada,
2008) have developed professional learning opportunities for teachers aimed at helping
classroom instructors meet the needs and raise the science and literacy achievement scores of
English language learners at the elementary school level. Specific instructional strategies for
improving math learning have been studied as well, such as self-explanation training (Hodds,
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Alcock, & Inglis, 20114), inquiry-based approaches (Laursen, Hassi, Kogan & Weston, 2014),
and literacy instruction within math content learning (Carter & Dean, 2006).
Although achievement outcomes are important in and of themselves, research also
suggests that motivation is significant in learning (Bong, Cho, Ahn, & Kim, 2012; Wolters,
Denton, York, & Francis, 2014). Motivation is reciprocally related to achievement (Williams &
Williams, 2010) and is malleable (Barber et al., 2014; Hulleman, Godes, Hendricks, &
Harackiewicz, 2010; Perry, Stupnisky, Hall, Chipperfield, & Weiner, 2010). Therefore,
examining students’ motivation in addition to their academic achievement seems necessary.
Much is known about students’ motivation, including in the math domain; however, very little
research has specifically examined the motivation of linguistically diverse students (e.g., Barber
& Buehl, 2013; Taboada, Townsend, & Boynton, 2013), particularly those from high school
(Barrett, Barile, Malm, & Weaver, 2012) and in the content area of math (e.g., Guglielmi, 2012;
Lewis et al., 2012; Riconscente, 2014). Because some studies and statistical data indicate that
subgroups of students underperform in reading and math; and because self-efficacy (a
component of motivation) is reciprocally related to achievement including in the math domain,
investigating the relationships among reading, math achievement, and math self-efficacy of
linguistically and ethnically diverse high school students is relevant.
Another important factor that relates to the academic achievement of linguistically
diverse students is their language proficiency and literacy skills in their second language (L2)
and first languages (L1). With regards to L2, math achievement gaps between native English
speakers and ELLs, as well as studies on test accommodations and math achievement of ELLs
underscore the salience of examining the relationship between English language proficiency and
math achievement. More specifically, research shows that ELLs’ English language proficiency
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influences their math learning (e.g., Barbu & Beal, 2010; Barrett et al., 2012; Martiniello, 2008;
Mosqueda, 2010; Mosqueda & Maldonado, 2013). For example, English language proficiency in
reading, listening, speaking, and writing of Latino/a high school Language Minority
2
(LM)
students—who do not use English as their home language and who have different English
proficiency levels (Goodrich, Lonigan, & Farver, 2013)—positively predicted their math
achievement in standardized tests given in English (Mosqueda, 2010; Mosqueda & Maldonado,
2013). Although more research has been done on the role of language in math learning within
ELLs, research with native English speakers indicates that English language ability and skills,
such as reading, are not only important for ELLs but also for native English speakers (e.g.,
Boonen, van der Schoot, van Wesel, de Vries, & Jolles, 2013; Grimm, 2008; Kyttälä & Björn,
2014). For example, vocabulary and listening comprehension are necessary for all students,
including both native English speakers and LM students, to understand mathematical concepts in
data analysis/probability and geometry at the elementary school level (Vukovic & Lesaux, 2013).
Similar to the importance of English language proficiency (L2) for linguistically diverse
students’ learning and achievement, L1 proficiency also appears to influence linguistically
diverse students’ L2 content area learning, including in reading and math (e.g., Goodrich et al.,
2013; Valentino & Reardon, 2015). Specifically, studies suggest that L1 proficiency is positively
related to L2 achievement (Slavin & Cheung, 2005; Umanski & Reardon, 2014), L2 reading and
math achievement (e.g., Genesee, Lindholm-Leary, Saunders, & Christian, 2005; Greene, 1997;
Lindholm-Leary & Block, 2010: Rolstad, Mahoney, & Glass, 2005). For example, fifth grade
Hispanic English proficient students and ELLs in dual language programs performed higher in
the English Language Arts (ELA) and mathematics subtests of the California Standardized Test

2
I distinguish LM students from ELL students because the literature treats them as distinct groups on their
English proficiency (e.g., Genesee et al., 2005; Kieffer, 2008).
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

14
(CST) compared to their counterparts in mainstream classes (Lindholm-Leary & Block, 2010).
However, most of these studies focused on elementary school students and the domain of
reading, whereas there is very limited information on the role of L1 proficiency in L2
achievement in high school students, and in the math domain in particular.
In summary, despite research supporting relationships between language and content
learning, and between academic motivation and learning, there is limited research
simultaneously examining relationships among first language (non-English) proficiency, English
reading, math achievement, and math self-efficacy in linguistically diverse high school students.
More specifically, there is lack of research on the moderating role of first language (non-English)
proficiency on the effects of LM status on English reading achievement; and the mediating roles
of English reading and math achievement between this interaction and math self-efficacy in
linguistically diverse adolescents. Knowledge of the moderating role of L1 language proficiency
could suggest which specific LM student groups need greater academic support, and thus could
guide the development of targeted interventions to increase such students’ participation in STEM
areas, especially in light of the significance of language skills in math learning. In addition,
understanding the mediating roles of English reading and math achievement could provide
mechanisms through which math self-efficacy could be increased, particularly for LM students,
as this may ultimately further their math achievement and participation in the STEM field.
Statement of Purpose
The current study has two purposes:
• to examine the moderating role of first language (non-English) proficiency on the effects
of language minority status in English reading achievement;
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• and to investigate the mediating roles of English reading and math achievement in the
relationship between the above interaction and math self-efficacy while adjusting for
gender, race/ethnicity, and socioeconomic status (SES).
Although this study focuses primarily at student-level data, I also investigate school
factors—e.g., school poverty, and proportions of racial/ethnic minority and ELL students—that
appear to influence school math achievement and math self-efficacy because students are
clustered within schools and research shows the significance of some of these relationships (e.g.,
Kitsantas, Ware, & Cheema, 2010; Niehaus & Adelson, 2014).
In the sections that follow I first provide an overview of motivation, social cognitive
theory, and self-efficacy, in particular. Second, I discuss individual factors that relate to math
self-efficacy, mathematics learning, and English literacy. Third, I describe the moderating and
mediating roles of some constructs, and discuss school-level factors associated with students’
math achievement and motivation. Fourth, I indicate gaps in the literature and describe the
current study, which uses the Education Longitudinal Study (cohort 2002, ELS) dataset to
address the above research questions. Specifically, I discuss the relevant sample and variables,
analyses, and results. I conclude by providing discussions of the results, as well as this study’s
implications, limitations, future research directions, and contributions to the field.

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Chapter II: Literature Review
Motivation Research and Theory
Overview of motivation. Motivation is commonly defined as “the process whereby goal-
directed activity is instigated and sustained” (Schunk, Pintrich, & Meece, 2008, p. 4).
Essentially, motivation addresses questions, such as what moves or energizes individuals, and
towards which tasks (Pintrich, 2003). Contemporary research on motivation is largely based on
the social cognitive perspective (Zusho, Anthony, Hashimoto, & Robertson, 2014). More
specifically, individuals’ cognitive aspects, including their thoughts, goals, and beliefs about
tasks and themselves—as well as personal, social, cultural, and contextual factors—impact their
motivation (Zusho et al., 2014; cf. Wigfield & Cambria, 2010). Indicators of a person’s
motivation are assessed through the individual’s choice, persistence, and effort (Schunk et al.,
2008). Some characteristics of motivation include its domain and context specificity, its
reciprocal relationship with achievement-related outcomes, and its change across developmental
periods (Bong, 2001; Jacobs, Lanza, Osgood, Eccles, & Wigfield, 2002; Linnenbrink & Pintrich;
2003; Marsh, Trautwein, Lüdtke, Kӧller, & Baumert, 2005; Wigfield & Eccles, 2000).
Several social cognitive constructs characterize motivation. For example, according to
Pintrich (2003), adaptive self-efficacy (i.e., confidence in doing a task at a particular level,
Bandura, 1997), attributions (i.e., beliefs about reasons for success and failures; Weiner, 1974),
and control beliefs (i.e., belief that one can accomplish an outcome, Skinner, Chapman, &
Baltes, 1988) influence motivation. High levels of interest (i.e., psychological state and tendency
to engage repeatedly with a specific content over time, Hidi & Renninger, 2006), intrinsic
motivation (i.e., doing a task for its own sake, Deci & Ryan, 2012), value (i.e., ways in which
tasks meet individuals’ needs, Wigfield & Eccles, 1992), and goals also motivate students
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(Pintrich, 2003). Various motivational theories incorporate these constructs into their
frameworks. In particular, the construct of self-efficacy has received substantial attention in the
motivation literature (e.g., Cheema & Kitsantas, 2014; Lee & Stankov, 2013; Liang, 2010), and
it is detailed in relation to Social Cognitive Theory below.
Social cognitive theory and self-efficacy. Social Cognitive Theory (SCT) assumes that
behavior, cognition, and environment interact and influence each other (Bandura, 1986). This
theory also hypothesizes that humans have agency, meaning that they, for example, have the
capacity to symbolize and form action plans, engage in forethought (e.g., visualized goals and
anticipated outcomes), learn vicariously, and regulate and reflect on the self (Bandura, 2006a). In
other words individuals both shape and are shaped by their environments. One significant
construct within SCT is self-efficacy, which serves as the foundation for human agency and
relates to choice, persistence, and effort (Bandura, 1997, 2006a).
Self-efficacy refers to one’s perceived convictions that one can perform a task at
designated levels (Bandura, 1997). Individuals develop their self-efficacy by interpreting
information from sources, such as vicarious experiences (e.g., watching others undertake a task),
verbal persuasion (e.g., feedback on performance of a task), enactive mastery experiences (e.g.,
one’s own success and failure experiences in doing a task), and physiological and affective states
(e.g., anxiety) (Bandura, 1997). Although Bandura (1997) posited that enactive experience
influences one’s self-efficacy the most, recent research has indicated that the influence of the
source also appears to depend on an individual’s race/ethnicity, gender, and academic level
(Usher & Pajares, 2008). For example, men pursuing STEM careers reported that mastery
experiences provided them with confidence information, whereas women noted that vicarious
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18
experiences and social persuasions were the information sources that influenced their self-
efficacy (Zeldin, Britner, & Pajares, 2008).
Self-efficacy is distinct from other self constructs including self-concept (e.g., Lee, 2009;
Parker, Marsh, Ciarrochi, Marshall, & Abduljabbar, 2014), but because some individuals either
confuse them or use them interchangeably, I discuss their distinction here. Self-concept refers to
generalized perceptions about oneself, which develop through interactions with the environment
and are highly susceptible to environmental reinforcements and significant others (Shavelson,
Hubner, & Stanton, 1976). Self-efficacy and self-concept are similar in that both consist of
beliefs about one’s perceived capabilities, incorporate normative comparison, and predict
motivation and performance (Bong & Skaalvik, 2003; Pajares, 2005). Similar to self-efficacy,
self-concept positively relates to academic outcomes (e.g., Green et al., 2012; Marsh et al., 2005;
Parker et al., 2014). One major difference between self-efficacy and self-concept is that self-
concept is not task specific, whereas self-efficacy is (Bandura, 1997). Additional differences are
that academic self-concept focuses on perceived competence and incorporates affective appraisal
of self (Bong & Skaalvik, 2003). Self-concept is also past-oriented and measured at a more
general and global level. In contrast, while academic self-efficacy emphasizes perceived
confidence, it incorporates only cognitive self-appraisal. Self-efficacy is also future-oriented and
measured at a task and situation-specific level (Bong & Skaalvik, 2003). Furthermore, self-
efficacy is a judgment that considers the goals or performance levels of the tasks (Pajares, 2005).
Math self-efficacy and related factors. Various factors influence and/or are influenced
by self-efficacy. In the sections below I discussed how achievement-related outcomes, prior self-
efficacy, gender, and language relate to self-efficacy, and math self-efficacy in particular.
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19
Achievement-related outcomes. Some key characteristics of self-efficacy are its domain
and context specificity (Bandura, 2012) as well as its positive relationship to academic outcomes,
including mathematics engagement and achievement (e.g., Parker et al., 2014; Martin, Anderson,
Bobis, Way, & Vellar, 2011; Riconscente, 2014). For instance, math self-efficacy positively
predicted future intent to continue studying math, but negatively predicted disengagement in
math of middle school students (Martin et al., 2011). In addition, math self-efficacy appears to
have a stronger relationship to math achievement compared to other constructs, such as anxiety
and self-concept (Lee & Stankov, 2013). The strong relationship between math self-efficacy and
achievement has been observed across developments and contexts, and even at the school level
(e.g., Alhija & Amasha, 2012; Liu, 2009; Kitsantas et al., 2010; Panaoura, Gagatsis, Deliyianni,
& Elia, 2010). For example, math self-efficacy of 15-year-olds from U.S., Canada, and Finland
predicted their math achievement in the Program for International Student Assessment (PISA)
assessments (Liang, 2010). Similarly, using the PISA data Kitsantas and colleagues (2010)
reported that school-level math self-efficacy predicted school-level math achievement.
Similar to self-efficacy’s influence on academic achievement, academic performance also
predicts self-efficacy. The studies focusing on the reciprocal relationship between self-efficacy
and academic outcomes suggested that self-efficacy predicted achievement and vice versa across
longitudinal and cross-sectional data (Caprara et al., 2008; Caprara, Vecchione, Alessandri,
Gerbino, & Barbaranelli, 2011; Williams & Williams, 2010). Specific to the area of math,
Williams and Williams (2010) found support for a reciprocal relationship between math
achievement and self-efficacy among students from 24 out of 33 countries that participated in the
2003 PISA. However, it appears unclear whether the strengths of these relationships—from
achievement to self-efficacy, and from self-efficacy to achievement—vary. For instance, Caprara
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20
and colleagues (2011) reported that these relationships were equivalent while examining
academic self-efficacy and school grades of adolescents using longitudinal data. In contrast,
Caprara and colleagues (2008) found that the strengths of these relationships differed,
specifically the relationship from school grades to self-regulatory self-efficacy was stronger than
the reverse relationship.
Prior and subsequent self-efficacy. Research indicates that prior self-efficacy is related
to subsequent self-efficacy (e.g., Caprara et al., 2008). This relationship has been investigated for
academic self-efficacy, self-efficacy for regulated learning, and math self-efficacy, among others
(e.g., Caprara et al., 2008, 2011; Riconscente, 2014). For example, Riconscente (2014) indicated
that prior and subsequent math self-efficacy were related in a sample of 326 Latino high school
students while adjusting for other constructs including interest, teacher caring, and strategy use.
The relationship between prior and subsequent self-efficacy was present even after adjusting for
academic achievement, which was associated with self-efficacy (Caprara et al., 2008, 2011).
Gender. Gender differences in self-efficacy, particularly in math, persist (Huang, 2013;
Kingston & Lyddy, 2013; Louis & Mistele, 2012) despite either slight or non-existent gender
math achievement gaps (Else-Quest, Hyde, & Linn, 2010; Lindberg, Hyde, Petersen, & Linn,
2010). Research shows that boys are more self-efficacious in math as compared to girls (Parker
et al., 2014; Williams & Williams, 2010). For instance, Huang (2013) conducted a meta-analysis
using 187 studies and found small differences (g = .08) in academic self-efficacy favoring males.
Content domain, such as mathematics, was a significant moderator indicating higher math self-
efficacy for males in comparison to females. Moreover, the author found that significant gender
differences in academic self-efficacy began in high school and increased as students became
older (g = .08 to g = .23; Huang, 2013).
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Language minority status, English language proficiency, and L1 (non-English)
proficiency. Research on self-efficacy, including math self-efficacy, of English language learners
shows inconclusive findings (e.g., Barrett et al., 2012; Lewis et al., 2012; Riconscente, 2014).
Some studies suggest that the math self-efficacy of English learners and native English speakers
vary (e.g., Barrett et al., 2012; Lewis et al. 2012), whereas others find no relationship (e.g.,
Riconscente, 2014). For example, fifth and sixth grade Hispanic English language learners
reported lower math self-efficacy compared to Hispanic native English speakers and second
language learners with English proficiency (Lewis et al., 2012). Among language minority
students, those with higher English proficiency have higher levels of self-efficacy. For example,
using the ELS data Barrett and colleagues (2012) found that English language proficiency of
Asian and Hispanic high school LM students positively predicted their academic self-efficacy.
However, in the study by Riconscente (2014) the math self-efficacy of students as measured in
time 3 did not vary by English language proficiency.
In addition to English language proficiency, L1 (non-English) proficiency seems to relate
to self-constructs, such as self-efficacy and self-concept within LM students. Specific to math,
Riconscente (2014) found that Spanish high school speakers fluent in English were more self-
efficacious in math compared to Hispanic native English speakers as measured in time 1.
Because the research on the relationship between L1 and self-efficacy is very limited, and
because self-efficacy shares some similarities with self-concept, some relevant studies on self-
concept and scholastic competence are presented below, with the acknowledgement that these
constructs are different from self-efficacy.
Similar to the research on self-efficacy, studies suggest that ELLs’ self-concept differs
from that of native English speakers, specifically when they receive native language support
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22
(Niehaus & Adelson, 2013; Lopez, 2010). For example, Spanish and Asian language speaking
elementary school ELLs reported higher math self-concept compared to their native English
speakers (Niehaus & Adelson, 2013). Although Spanish-speaking ELLs had lower math
achievement compared to native English speakers, they still indicated higher math self-concept
(Niehaus & Adelson, 2013). Moreover, elementary school Hispanic ELL students receiving
bilingual instruction support had higher scholastic competence in comparison to Hispanic ELLs
who did not receive this support (Lopez, 2010).
Although the math self-efficacy of LM students appears to depend on their English
language proficiency, the relationship between motivation and achievement does not seem to
depend either on LM students’ English language proficiency or the type of native language. In
other words, the relationship between motivation and achievement remains consistent across LM
students’ with varying English language proficiency and type of first language. For instance,
Barrett and colleagues (2012) found that academic motivation predicted math achievement to the
same extent for LM high school students whose native language was either Spanish or those who
spoke an Asian language. Similarly, Lewis and colleagues (2012) reported that the relationship
between math self-efficacy and math achievement of Hispanic middle school students did not
vary by English proficiency. That is, how predictive math self-efficacy is of math achievement
does not depend on students’ English language proficiency.
In sum, based on the above review achievement-related outcomes, prior math self-
efficacy, language minority status, linguistic skills, including reading, and gender all have been
shown to relate to math self-efficacy. More specifically, the above literature on motivation,
particularly math self-efficacy, suggests that 1) LM students with either higher English language
proficiency or first language proficiency seem to have higher levels of self-efficacy; 2) math self-
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23
efficacy and math achievement are related; and 3) mastery experiences are one source of self-
efficacy, indicate the possibility that LM students with greater L1 proficiency and/or English
language proficiency would be more self-efficacious in math through their higher math
achievement. However, this idea has not been tested empirically.
To better understand the connections among LM status, English language proficiency and
reading, math achievement, and math self-efficacy, these relationships are discussed in detail
below. More specifically, I discuss factors that influence math achievement and English
achievement/reading, such as language skills and students’ demographic characteristics. I then
discuss the possible moderating role of L1 proficiency in the relationship between LM status and
English reading achievement. Next, I describe the possible mediating roles of English reading
and math achievement in the relationship between an interaction—LM status and L1 (non-
English) language proficiency—and math self-efficacy.
Mathematics Research
Mathematics learning/achievement and related factors. Among relevant factors that
influence math learning and achievement are language, including English reading and L1
proficiency, prior math achievement, SES, ethnicity, and gender. Each of these factors is
discussed in the sections that follow. Because of the current emphasis of language proficiency
expectations in the Common Core State Standards, including in content areas, such as math, for
all students, but for ELLs in particular, I begin by describing the role of language in math
learning (Council of Chief State School Officers, 2012; see Bunch, 2013).
Language. Content knowledge learning, including in the math domain, depends on
language proficiency (Kieffer, Lesaux, Rivera, & Francis, 2009). That is, language and learning
cannot be taken apart while studying a language specific to a domain (Schleppegrell, 2007).
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Researchers have suggested linguistic challenges in learning math that all students face,
including ELLs (e.g., Schleppegrell, 2007; Vukovic & Lesaux, 2013). That is, in a sense, math
has its own particular linguistic features that must be attended to in order for learning to occur in
this domain. For example, based on the literature, Schleppegrell (2007) synthesized features of
the mathematics register, which refers to meanings that characterize the mathematical language.
Some of these features include multiple meaning creating systems, such as mathematics
symbolic notation (e.g., relationships and patterns) and oral language; and grammatical patterns,
such as technical vocabulary (specific word meaning in math context) and dense noun phrases
(phrases with many nouns to describe attributes). These features appear to present challenges to
all students regardless of their LM status.
Reading. Studies examining the relationship between reading and math have mostly
indicated that reading predicts math achievement. For example, Boonen et al. (2013) examined
whether spatial ability and reading comprehension predict word problem solving directly and
indirectly through production of visual-schematic representations and relational processing,
respectively for 128 6
th
grade students. They found that reading comprehension is significantly
related to word problem solving directly and indirectly through relational processing. Moreover,
the relationship between linguistic and mathematical skills has been investigated across various
areas, such as reading comprehension, general verbal/reading ability, arithmetic, and data
interpretation, among others (e.g., Grimm, 2008; Kyttälä & Björn, 2014; Vukovic & Lesaux,
2013). As expected, this relationship has been studied more extensively within word problem
contexts because they require higher reading comprehension demands than other contexts (e.g.,
Boonen et al., 2013; Fuchs, Fuchs, Compton, Hamlett, & Wang, 2015). Furthermore, the link
between linguistic and math skills has been explored more frequently for English language
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25
learners because of their dual challenge of learning the English language and math content (and
math literacy) through this new language (e.g., Beal, Adams, & Cohen, 2010; Guglielmi, 2012;
Martiniello, 2008, 2009).
Reading also appears to influence math growth from grades 3 to 8 (Grimm, 2008; Shin,
Davison, Long, Chan, & Heistad, 2013) with a longitudinal influence (Watts, Duncan, Siegler, &
Davis-Kean, 2014). For instance, Shin and colleagues (2013) studied the relationship between
reading and math growth using longitudinal data from grades 4 to 7 from a large school district.
They reported that initial reading positively predicts math growth, and initial math negatively
predicts reading growth. In addition, reading and math growths were found to be correlated.
Much of this research has sampled elementary school students (Kyttälä & Björn, 2014; Larwin,
2010) and some have included adolescents (e.g., Goforth, Noltemeyer, Patton, Bush, & Bergen,
2014; Korpershoek, Kuyper, & van der Werf, 2014; Kyttälä & Björn, 2014; Watts et al., 2014).
However, very few studies have investigated these relationships within high school students
from representative samples, and those that have did not adjust for prior math achievement,
which is a good predictor of math achievement and which will be discussed at the end of this
section (e.g., Larwin, 2010).
English language proficiency. Although all students may encounter linguistic challenges
while learning math, it is likely that ELLs face them more frequently because they learn math
content in a language they are not yet proficient in. Research on ELLs’ math learning and
performance, as well as ELL test accommodations have indicated that English language
proficiency of LM students is related to their math learning and achievement (Alt, Arizmendi, &
Beal, 2014; Barbu & Beal, 2010; Beal et al., 2010; Kieffer et al., 2009; Li & Suen, 2012;
Martiniello, 2008; Wolf, Kim, & Kao, 2012). More specifically, ELLs perform better on math
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26
tasks that are worded simply (e.g., Barbu & Beal, 2010). For instance, Barbu and Beal (2010)
conducted an experimental design study in which they manipulated the language and
mathematical complexity of math word problems. Middle school ELL students (n = 41) were
more likely to identify the correct operation, require fewer hints, and commit fewer errors when
the text was easy as compared to a more linguistically complex word problem with the same
mathematics difficulty level. ELLs reported that problems with higher linguistic complexity were
harder to understand and they perceived them as being more difficult than problems written in
easy texts. Furthermore, studies on test accommodation for ELLs suggest that test
accommodations improve ELLs’ academic performance, including in math (Kieffer et al., 2009;
Li & Suen, 2012). For instance, Kieffer and colleagues (2009) found through a meta-analysis of
11 studies that provision of English dictionaries and glossaries increases ELLs’ achievement.
However, other accommodations—including simplified English, bilingual dictionaries or
glossaries, tests in the native language, dual language test booklets, dual language questions for
English passages, and extra time—did not influence ELLs’ achievement.
Native language proficiency. In addition to the influence of English language proficiency
on LM students’ math learning, their native language proficiency seems to plays a role. In
particular, the relationship between native language proficiency and second language
achievement of second language learners has been examined including in the domain of math
(e.g., Lindholm-Leary & Block, 2010). However, results on whether native language relates to
ELLs’ second language learning and achievement, particularly in math, appear to be mixed.
Some studies on bilingual programs (e.g., Rolstad et al., 2005) and ELLs’ test accommodations
(e.g., Alt et al., 2014) suggest that native language proficiency positively impact second language
learning, including math in L2 (Genesee, Lindholm-Leary, Saunders, & Christian, 2006;
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27
Guglielmi, 2008; Lindholm-Leary & Block, 2010; Pennock-Roman & Rivera, 2011; Rolstad et
al., 2005; Slavin & Cheung, 2005; Umanski & Reardon, 2014). For instance with regards to
overall academic achievement, the ELL test accommodation meta-analyses, which were
conducted by Li and Suen (2012) and Pennock-Roman and Rivera (2011), showed that native
language support positively improves ELLs’ general academic achievement. Specifically, test
booklets in the native and second languages have similar and positive, but small effect on ELLs’
academic achievement when compared to other accommodations, such as linguistic
simplification and use of dictionaries and glossaries (Li & Suen, 2012). However, the effect of
native language support (in the form of test booklets in the native language or dual languages,
and bilingual glossaries) depends on the time limit of the exam, as well as on students’ Spanish
and English proficiencies (Pennock-Roman & Rivera, 2011). For example, the effect size for the
native language test version was the highest (Glass d = +1.45) for students with literacy in
Spanish but limited English proficiency (Pennock-Roman & Rivera, 2011). Specific to the math
domain, Guglielmi (2012) found that Spanish proficiency of Hispanic high school English
language learners is related to their math intercept and slope through their English reading
intercept and slope, respectively. However, there is also some evidence indicating that native
language support does not influence ELLs’ achievement in the second language (e.g., Kieffer et
al., 2009). For example, in the meta-analysis conducted by Kieffer and colleagues (2009),
bilingual dictionaries and tests in students’ native language did not have any significant effects
on ELLs’ academic performance. The limited number of studies using native language support as
test accommodations (Pennock-Roman & Rivera, 2011) may explain its lack of effect.
Prior mathematics achievement. A significant predictor of math achievement is prior
math performance. Research indicates that prior math achievement is a strong predictor of future
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28
math performance even when various covariates (e.g., cognitive skills, SES, ethnicity, and
ability) are adjusted for (Hemmings, Grootenboer, & Kay, 2010; Park, 2008; Riegle-Crumb &
Grodsky, 2010; Stevens, Olivarez, Lan & Tallent-Runnels, 2004; Watts et al., 2014). For
example, Watts and colleagues (2014) reported that math skills measured from as early as 54
months and first grade (as well as their change) predicted 15-year-olds’ math achievement while
adjusting for cognitive skills, reading, and several student demographic characteristics.
Socioeconomic status. Another individual characteristic that consistently relates to
achievement including in math is students’ socioeconomic status (Brown-Jeffy, 2009; Burnett &
Farkas, 2009; Caprara et al., 2011; Chiu & Xihua, 2008; Ismail 2009; Levpušček, Zupančič, &
Sočan; 2013; Murayama, Pekrun, Lichtenfeld, & von Hofe, 2013; Parker et al., 2014; Weiss,
Carolan, & Baker-Smith, 2010; Werblow & Duesbery 2009; Zhao et al., 2014). For instance,
using the ELS 2002 dataset, Weiss and colleagues (2010) reported that parent education and
socioeconomic status were positively related to 10
th
grade students’ math achievement on a
standardized test. The relationship between students’ socioeconomic status and their math
achievement has also been consistently reported in diverse contexts, including China, Italy,
Slovenia, and Germany (Caprara et al., 2011; Levpušček et al., 2013; Murayama et al., 2013;
Zhao et al., 2014). For example, Levpušček and colleagues (2012) reported based on a sample of
415 eighth graders from Slovenia that parent education positively predicted math achievement,
which was measured as ninth grade final math grade and scores on a national math test.
Race/ethnicity. Math achievement of Black and Latino students has increased since the
1990s. For example, Brown and Campbell (2009) reported that more Black and Latino students
were taking the Advanced Placement tests in math and that their passing rates were higher in
2003 as compared to years since 1998. However, because the math achievement of White
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29
students has concurrently increased, the achievement gaps between Whites and Blacks, and
Latinos and Blacks remain (Berends & Peñaloza, 2010; Else-Quest, Mineo & Higgins, 2013;
Hemphill & Vanneman, 2011; Riegle-Crumb & Grodsky, 2010: Vanneman, Hamilton, Baldwin
Anderson, & Rahman, 2009). Although reports and studies have indicated that math achievement
gaps persist, it is still unclear whether they have either remained the same or narrowed over time.
For instance, Berends and Peñaloza (2010) used longitudinal national samples, such as the
National Longitudinal Study of High School, High School and Beyond, National Education
Longitudinal Study, and Educational Longitudinal Study to find that the Black-White and
Latino-White math achievement gaps for high school students narrowed from 1972 to 2004. In
contrast, Hemphill and Vanneman (2011), and Vanneman and colleagues (2009) reported that
the math achievement gaps for 8
th
grade students remained unchanged since the 1990s.
Gender. Compared to the racial/ethnic math achievement gaps, the gender math
achievement gap is small (e.g., Else-Quest et al., 2010). Specifically, in two meta-analyses (Else-
Quest et al., 2010; Lindberg et al., 2010), researchers found slight differences in math
achievement based on gender. For example, based on 242 studies and large U.S. adolescents
datasets, such as the NAEP and National Education Longitudinal Study (NELS 88), the authors
reported small average weighted effect sizes ranging from d = .05 to d = .07 (Lindberg et al.,
2010). Through moderator analysis the authors found that problem type (multiple choice, short
answer, open-ended question), selectivity of sample, ethnicity, and grade level are significant
moderators. Particularly, the gender difference in math achievement was higher in high school (d
= + .23). Additionally, gender achievement differences appear only in the math intercept, but not
in the math growth of adolescents (Murayama et al., 2013).
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In summary, language skills in L1 and L2, prior achievement, socioeconomic status,
gender, and race/ethnicity appear to be significant factors that influence math achievement
especially for adolescents. Because English reading and achievement are connected to math
achievement, particularly for LM students, it is relevant to discuss factors that influence them.
English Reading/Achievement and Second Language Learning Research
English Reading/Achievement and Related Factors. In the sections below, I discuss
how LM status, first language proficiency, race/ethnicity, and SES relate to English reading and
achievement. I also include a short discussion on cross-linguistic transfer.
Language minority status. Some language minority students tend to underperform in
reading tests as compared to native English speakers (Han & Bridglall, 2009; Kim & Herman,
2009; Lesaux & Kieffer, 2010; Zarate & Pineda, 2014). However, the reading achievement gap
depends on the time when LM students gain English language proficiency (Halle, Hair,
Wandner, McNamara, & Chien, 2012; Kieffer, 2008), school concentration of ELL students
(Han & Bridglall, 2009), and type of first language (Roberts, Mohammed, & Vaughn, 2010). For
example, if ELL students gain English language proficiency by kindergarten entry, their initial
reading scores are similar to those of native English speakers (Halle et al., 2012). In addition to
the reading average, their reading growth rates differed based on the period LM students
achieved English language proficiency. The reading growth rate is greater when either LM
students attain English proficiency by kindergarten entry or do not become proficient by spring
of first grade, as compared to the reading growth rate of native English speakers (Halle et al.,
2012). In terms of school demographics, LM students underperform in 5
th
grade reading
assessments as compared to native English speakers regardless of whether they attended a high
or low ELL-concentrated school (Han & Bridglall, 2009). However, LM students from schools
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31
without any ELLs had a slightly higher reading achievement when compared to native English
speakers from similar schools. Another factor that influences English proficiency of LM students
is type of first language. For instance, among English proficient students, students whose
primary language was Spanish had lower initial reading scores and slopes when compared to
students whose native language was either English or an Asian language (Roberts et al., 2010).
Students who spoke an Asian language as a native language had higher initial reading scores
compared to native Spanish and English speakers, but similar slopes as native English speakers.
The majority of these studies have focused on young LM students and few studies have
investigated the reading differences between adolescent LM students and native English
speakers, and among ELL students (e.g., Zarate & Pineda, 2014).
In particular for LM students, L1 proficiency is a significant factor that impacts their L2
(English) reading and achievement. Before presenting literature relating L1 proficiency and
English achievement and reading, I provide an overview of second language learning,
specifically cross-linguistic transfer, to build background knowledge on this literature.
Cross-linguistic transfer. The developmental interdependence hypothesis (Cummins,
1979) and common underlying proficiency (Cummins, 1981) are two frequently used
frameworks that provide possible explanations for findings on the transfer of language
knowledge and proficiency from one language to another (Genesee, Geva, Dressler, & Kamil,
2006). The developmental interdependence hypothesis posits that proficiency in L2 depends on
the L1 proficiency level an L2 learner had obtained before intense L2 exposure (Cummins,
1979). Based on this hypothesis, students with higher native language proficiency when they
begin learning the second language are expected to have greater second language proficiency as
compared to students who started learning their L2 with limited native language proficiency. The
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32
common underlying proficiency model states that native- or second-language learning can
develop a proficiency that underlies both languages, if appropriate motivation and exposure to
the languages exist (Cummins, 1981). That is, proficiencies in the native and second languages
are not separate, but rather, they constitute one cross-lingual proficiency. Therefore, concepts
and skills learned in one language can be accessed and used in another language. Once a concept
is learned in one language, it is accessible in the second language as well, although it will have a
different label (cf. Ordóñez, Carlo, Snow, & McLaughlin, 2002). The more extensive the
learning in the native language, the greater the opportunity to amass a store of concepts,
knowledge, and skills would be, which could serve as a foundation in the second language.
These theories highlight the importance of native language proficiency in second language
learning including in content areas, such as reading and math.
Several researchers have asked whether proficiency in one language transfers to another
(cross-linguistic transfer). Various syntheses (e.g., Genesee, Geva et al., 2006; Genesee et al.,
2005), meta-analyses (e.g., Jeon & Yamashita, 2014; Melby-Lervåg & Lervåg, 2011), and single
studies (e.g., Goodrich et al., 2013; Guglielmi, 2008) have examined different aspects of cross-
linguistic transfer. They have covered a variety of topics, including transferable language skills
and types of languages; ideal transfer period; and duration of transfer effects. Much of this work
has been carried out in the domain of reading with elementary school students. Below I briefly
discuss some main patterns related to these topics.
The cross-linguistic literature indicates that not all skills transfer and that, among
transferable skills, specific rather than general skills transfer. L1 language skills that are
explicitly linked to literacy, as well as academic and cognitive language, transfer more readily
(Genesee et al., 2005; Jeon & Yamashita, 2014). Based on the above mentioned reviews and
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33
meta-analyses, phonological awareness (e.g., Genesee et al., 2005; Goodrich et al., 2013)
decoding (e.g., Genesee, Geva et al., 2006; Melby-Lervåg & Lervåg, 2011); vocabulary (e.g.,
Cisco & Padrón, 2012; Genesee et al., 2005), specifically of higher order and cognates; reading
strategies (Cisco & Padrón, 2012; Genesee, Geva, et al., 2006); reading comprehension
(Genesee, Geva, et al., 2006; Jeon & Yamashita, 2014); and meta-cognitive strategies (Genesee
et al., 2005) are all specific examples of transferable language skills.
Most studies investigating cross-linguistic transfer have relied on correlational methods
(Genesee, Geva, et al., 2006), but only a few researchers have studied the causal relationship
between L1 and L2 proficiencies through intervention studies (e.g., Goodrich et al., 2013;
Roberts, 2008). For example, Goodrich and colleagues (2013) reported that Spanish proficiency
moderates an intervention programs’ effects on students’ vocabulary and elision skills (i.e., the
skill of forming new words by deleting parts of it) suggesting cross-linguistic transfer of these
skills. Specifically, initial Spanish receptive vocabulary significantly moderates the relationship
between the intervention (English pull out group) and posttest English receptive vocabulary.
Transfer appears to occur less consistently across modalities—from one language skill to
a different language skill (cf. Genesee, Geva, et al., 2006) and between languages that vary
substantially in their writing systems. For instance, L1 oral proficiencies are not related to word
reading (Genesee, Geva, et al., 2006), and reading comprehension in L2 (Genesee, Geva, et al.,
2006; Melby-Lervåg & Lervåg, 2011). In contrast, phonological and orthographic skills in L1 are
related to L2 word reading (e.g., Carlo 2009; Genesee, Geva, et al., 2006). With regards to type
of languages, transferable language skills, including reading comprehension and orthographic
knowledge, are more correlated across alphabetic languages than between ideographic and
alphabetic languages (e.g., Jeon & Yamashita, 2014; Melby-Lervåg & Lervåg, 2011).
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34
In terms of the period in which transfer is most likely to occur, some scholars (e.g.,
Genesee et al., 2005) suggest that transfer happens readily during the beginning period of L2
acquisition, and the effects are fairly durable. At the initial stage of L2 learning, students
primarily rely on their L1 skills; however, as students acquire more proficiency in their L2, they
tend to rely more on their L2 than L1 skills (Genesee et al., 2005). With regards to the duration
of the transfer effect, a few studies using longitudinal data reported that transfer effects were
present from upwards of 2-10 years (e.g., Reese, Garnier, Gallimore, & Goldenberg, 2000;
Sparks, 2009). For example, Reese et al. (2000) found that early Spanish (L1) literacy measured
in kindergarten predicted grade 7 English reading while adjusting for kindergarten oral English
proficiency for a sample of 121 students. In summary, studies on cross-linguistic transfer
indicate that specific rather than general language-skills transfer happens more frequently in the
beginning period of L2 learning and across the same modalities. Moreover, cross-linguistic
transfer happens more readily across similar languages, and the effects appear to be long lasting.
Native language proficiency. Proficiency in one’s first language can be assessed through
either direct or inferential measures. On the one hand, direct language measures include language
assessments (e.g., Proctor, August, Carlo, & Snow, 2006), as well as teacher-report and student
self-report of language proficiency (e.g., Guglielmi, 2008). Because teacher-report can be
subjective, and student self-report can be influenced by social desirability, language assessments
provide less biased information on students’ language proficiency. On the other hand, inferential
language measures consist of assessments that indirectly assess students’ language proficiency.
One example of an inferential measure is the years of schooling received in the students’ home
country (e.g., Thomas & Collier, 2000). For example, Padilla and Gonzalez (2001) used the
number of years students were schooled in Mexico as a proxy for L1 (Spanish) proficiency to
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35
examine its relationship with L2 measures, such as Grade Point Average. Although inferential
language proficiency measures are informative they are less precise when compared to direct
language measurements because they are less specific and may include other components.
Based on cross-linguistic transfer theories and research, and a desire to maintain one’s
first language (and/or facilitate transition to learning a new language), bilingual programs of
various types have been developed. The majority of research on bilingual education indicates
that ELLs in programs with a native language component outperform their counterparts in
English immersion (English only) programs on several academic outcomes. Some of these
outcomes are standardized tests in English (Greene, 1997), English reading (Greene, 1997;
Rolstad et al., 2005; Slavin & Cheung, 2005; Willig, 1985), math in English (Rolstad et al.,
2005; Willig, 1985), English language skills (Willig, 1985) and writing (Rolstad et al., 2005).
Several meta-analyses (Greene, 1997; Rolstad et al., 2005; Slavin & Cheung, 2005; Willig,
1985) suggest that overall bilingual education is more advantageous for ELLs in comparison to
English-only programs on both L1 and L2 academic outcomes. For example, Slavin and Cheung
(2005) reported that out of the 17 studies examined, 12 studies found positive effects of bilingual
education, whereas five studies found no effects. Although these meta-analyses have limitations
on their definitions of bilingual education and English immersion programs, and length of
students’ participation in these programs, they all consistently report an advantage on the
inclusion of native language instruction.
Recent studies comparing students from various language programs on their
reclassification rates and academic trajectory also showed that programs with native language
components were advantageous. The results appear to differ by ethnicity, however. For example,
initially students in English Immersion (EI) programs compared to two-language programs (e.g.,
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36
dual immersion (DI), transitional bilingual (TB), and maintenance/developmental bilingual
(DB)) had higher log odds of being reclassified as English proficient during the elementary
school years (Umanski & Reardon, 2014). However, over time the log odds of reclassification of
students in two-language programs were higher in either middle or high school. Similar patterns
were found while using the California English Language Development Test and CST in ELA as
criteria for English proficiency. In addition, most of these patterns were consistent while
examining ELLs’ academic trajectory from grades two to seven (Valentino & Reardon, 2015).
ELLs in EI did significantly better in grade 2 ELA as compared to DI students, worse than TB
students, and similarly to DB students. However, the growth rate in ELA scores was significantly
higher for DI and TB students when compared to EI students. In grade six, DI students met the
state ELA average and outperformed students in EI and DB programs. However, these patterns
varied by type of first language. Specifically, although Latino ELLs’ ELA achievement was
greater in the long run when enrolled in a bilingual program, Chinese ELLs’ ELA longitudinal
achievement was similar when they were either in the DI or EI program.
The impact of L1 proficiency appears to differ depending on either the time period in
which bilingual education instruction is received or the developmental period in which this
influence is examined. For example, based on the NELS 1988 dataset, Roscigno, Vélez, and
Ainsworth-Darnell (2001) found differential effects of bilingual education on eight grade
students’ reading and math achievement. In particular, language minority students who received
bilingual instruction between grades 1 and 4 received significantly higher scores in reading and
math compared to their counterparts who did not receive bilingual instruction (or students who
were in English-as-a-Second Language classes). In contrast, receiving bilingual instruction
between grades 5 and 7 did neither predict reading nor math achievement. One limitation of the
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37
study was that the authors did not discuss how bilingual education was defined in the student
survey. Characterizing the bilingual education program students received could have shed more
light into understanding these differential findings. Similarly, findings from the study by Howard
and colleagues (2014) suggested that the relationship between L1 and L2 varies based on
students’ grade levels. In their study Spanish vocabulary predicted English letter word
identification for Spanish-speaking kindergarteners and third graders, and it predicted English
passage comprehension for grade 3 students. However, these relationships were not present in
fifth grade students. Because of their sampling of students from different contexts (English-only
or bilingual instruction), it is unclear whether the relationships between L1 and L2 are different
because of these varying instructional contexts or students’ developmental levels. In contrast to
these studies, using a meta-analysis, Prevoo, Malda, Mesman, and van IJzendoorn (2015) found
that age did not moderate cross-language relationships between oral proficiency and early
literacy/reading. Although the issue of whether cross-linguistic transfer effects vary across
developmental levels has received very limited attention, the above preliminary findings indicate
that these effects could depend on students’ developmental level.
Socioeconomic status. SES is a strong predictor of reading achievement. Studies have
suggested that SES is related to average reading score, rate of change, and acceleration (Halle et
al 2012; Kieffer 2008, 2010; Roberts et al., 2010; Sirin, 2005). For example, Halle and
colleagues (2012) reported that parent education level and family income positively relate to the
initial score and rate of change of reading scores from kindergarten to eighth grade for students
from the Education Childhood Longitudinal Study-Kindergarten Cohort (ECLS-K). Moreover,
some studies have specifically compared and contrasted measures used as proxies for SES (Sirin,
2005; Harwell & LeBeau, 2010), and they have suggested that the validity of measuring access
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38
to economic resources, and the correlation between SES and achievement, varies depending on
the SES measure used. For example, Harwell and LeBeau (2010) indicated that householders’
income, education, and occupation are more accurate measures as compared to eligibility for free
or reduced-priced lunch, number of siblings, and home resources checklist. In addition to
investigating the influence of a student’s SES on his or her own achievement, other researchers
have examined the influence of peers’ SES on students’ reading achievement, and provided
support for this relationship (Chiu & Chow, 2015; van Ewijk, & Sleegers, 2010).
Race/ethnicity. Racial or ethnic differences in reading achievement are evident. For
example, although Blacks and Hispanics in grades 4 and 8 have improved their reading scores in
the NAEP assessments since 1990s, the achievement gap for grade 8 did not decrease since then
(Hemphill & Vanneman, 2011; Vanneman et al., 2009). These findings seem consistent for
average reading scores (e.g., Reardon & Galindo, 2009), even after adjusting for various student,
school, and community covariates (e.g., Drake, 2014). For example, Halle et al., (2012) reported
that compared to Whites, Blacks, and Others, Asian students had significantly higher average
reading achievement. However, they did not find racial/ethnic differences in the reading
achievement rate of change from kindergarten to grade 8 using the ECLS-K.
In sum, factors that are related to English achievement and reading are LM status, first
language proficiency, SES, and race/ethnicity. Based on studies examining the factors
influencing math self-efficacy, math achievement, and English reading and achievement, it is
probable that some of these constructs could act as moderators and mediators in the relationship
between LM status and math self-efficacy.
Moderating Role of First Language Proficiency
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The reviewed studies on LM status, L1 (non-English) proficiency, and English reading
and achievement, and cross-linguistic transfer frameworks suggest that L1 proficiency could
moderate the relationship between LM status and English reading achievement. In other words,
LM students with higher L1 proficiency would be expected to have greater English achievement
than LM students with limited L1 proficiency. In particular, studies indicating the advantage of
native language instruction for ELL students’ L2 achievement (including in English reading)
support this idea (e.g., Umanski & Reardon, 2014; Valentino & Reardon, 2015). However, it has
yet to be tested empirically, particularly with high school students.
The idea of L1 proficiency acting as a moderator is consistent with findings from studies
investigating the moderating role of English language proficiency in the relationship between
LM status and math achievement (Mosqueda, 2010; Mosqueda & Maldonado, 2013), and
research focusing on the role of linguistic skills in math learning (e.g., Fuchs et al. 2015;
Korpershoek et al., 2014). Based on the ELS 2002, Mosqueda (2010) and Mosqueda and
Maldonado (2013) found that LM status of Latino high school students interacts with their
English language proficiency. Subsequently this interaction positively predicts grade 10 and 12
math achievement. These studies, in addition to the cross-linguistic transfer studies reviewed
above, suggest that L1 proficiency likely moderates the relationship between LM status and
English reading.
Mediating Roles of English Reading and Mathematics Achievement
Taken together, the studies covered in this review suggest that LM status, L1 (non-
English) proficiency, English reading, math achievement, and math self-efficacy are related. In
particular, the reviewed literature indicates that English reading and math achievement probably
mediate the relationship between the interaction (LM status and L1 proficiency) and math self-
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40
efficacy. Although this mediation seems evident, few studies have concurrently examined
relationships among all these constructs and specifically investigated this mediation with high
school student samples. One study that comprehensibly included most of these constructs in an
SEM model, except for math self-efficacy and the interaction between first language proficiency
and LM status, was conducted by Guglielmi (2012). Guglielmi (2012) examined the
relationships among L1 proficiency, self-esteem, English and math self-concepts, and academic
achievement in reading, math, and science of limited English language learners from the
NELS:88/2000 dataset. Among the main findings of relevance to the present discussion were that
grade 8 L1 proficiency predicts reading intercept and slope; reading intercept and slope predicts
math intercept and slope, respectively; and self-esteem and math self-concept are not related.
Overall, the study indicated that L1 proficiency is related to math achievement through reading
achievement. Therefore, these results are consistent with the idea that the interaction between
LM status and L1 proficiency is related to math self-efficacy through English reading and math
achievement.
In addition to the individual factors that relate to students’ academic achievement and
motivation, particularly in the math domain, it is necessary to examine school factors that
influence these outcomes because students are nested within schools. Studies suggest that certain
school characteristics, including school poverty, percentage of racial/ethnic minority students,
and ELL students’ proportion, relate to academic performance and motivation (e.g., Benson &
Borman, 2010; Niehaus & Adelson, 2014; Roberts et al., 2010). Failure to account for the nested
structure of the data may result in biased estimates while investigating students’ outcomes
(Kitsantas et al., 2010). These school-level factors are discussed in more detail below.
School-Level Student Demographics, Achievement, and Motivation
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41
School socioeconomic status. With regards to school poverty, most research shows that
the school SES relates to students’ achievement including in math (Kitsantas et al., 2010;
Mosqueda, 2010; Mosqueda & Maldonado, 2013; Werblow & Duesbery, 2009) and reading
(Benner & Crosnoe, 2011; Benson & Borman, 2010; D’Agostino, 2000; Drake, 2014; Perry &
McConney, 2010). For example, based on the ELS 2002 dataset, Werblow and Duesbery (2009)
found that high school SES, which was measured as the percent of students eligible for free or
reduced-price lunch, was negatively related to math achievement growth. In contrast, some
studies report no relationship between school poverty and student math achievement (e.g.,
Brown-Jeffy, 2009; Drake, 2014). For instance, using the same ELS 2002 dataset Drake (2014)
did not find any association between school percentages of students eligible for free or reduced-
price lunch and math achievement. The differences may have occurred because of the different
school-level control variables that were included in the models. With regards to reading
achievement, studies indicated that school SES is related to achievement (e.g., Benner &
Crosnoe, 2011; Benson & Borman, 2010; Drake, 2014; Perry & McConney, 2010). For example,
using the Australian 2003 PISA data, Perry and McConney (2010) reported that school SES
predicts adolescents’ reading performance. Additionally, these authors found that the advantage
of attending a higher SES school is consistently observed for students across varying individual
SES levels.
School proportion of racial/ethnic minority students. Another school factor that
appears to be associated with math achievement is the proportion of racial/ethnic minority
students (Benner & Crosnoe, 2011; Brown-Jeffy, 2009; Han & Bridglall, 2009; Kitsantas et al.,
2010; Newton, 2010). In one such study based on the 2003 PISA U.S. sample of about four
thousand 15-year old students, Kitsantas and colleagues (2010) reported that the percentage of
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42
non-white students negatively predicts math achievement. In addition, Benner and Crosnoe
(2011) suggested that depending on how school racial/ethnic composition is conceptualized, the
direction of this relationship with math achievement changes. More specifically, they reported
that school diversity and proportion of same race/ethnicity peers positively predicts elementary
school students’ math achievement, whereas proportion of racial/ethnic minority students is
negatively related to achievement, while adjusting for several individual, family, teacher, and
school covariates. Although several studies appear to support the relationship between proportion
of racial/ethnic minority student and math achievement, a few studies indicate the contrary. For
instance, in the research by D’Agostino (2000), racial/ethnic minority student percentage did
neither predict math achievement average nor rate of change for students from grades 1 through
6.
In contrast to the research on links between school racial/ethnic minority student
proportion and achievement, much less is known about the relationship between this school-level
factor and motivation (and, in particular math self-efficacy) (e.g., Kitsantas et al., 2010; Niehaus
& Adelson, 2014). For example, Kitsantas and colleagues (2010) reported that the percentage of
non-White students is correlated to the school math self-efficacy of U.S. adolescents who
participated in the 2003 PISA. Similarly, based on the ECLS-K Niehaus and Adelson (2014)
found that the proportion of students of color is positively associated with elementary school
ELLs’ academic self-concept, which included reading and math self-concepts. More research on
how school factors, such as proportion of racial/ethnic minority students, relate specifically to
students’ math self-efficacy longitudinally and while including covariates is needed to
understand how school enrollment of racial/ethnic minority students influences this particular
construct, which is strongly related to achievement at the student-level.
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School proportion of ELL students. A school-level factor related to racial/ethnic
minority student proportion is school ELL composition. Research on the relationship between
school ELL proportion and math performance shows discrepancies in findings. On the one hand,
studies indicate that school ELL percentage is not related to students’ math achievement (Drake,
2014; Mosqueda, 2010; Werblow & Duesbery, 2009). For instance, Mosqueda (2010) did not
find any relationship between percentage of ELLs and Latino/a high school students’ math
achievement on a standardized test using the ELS 2002 dataset. On the other hand, some studies
indicate that school percentage of ELL may indeed influence math performance. For example,
Fry (2008) reported that in California a higher percentage of White and Black third grade
students were classified as proficient or above if they attended schools without the minimum
threshold number of ELL students, as compared to their peers who were in schools with greater
number of ELLs. Moreover, Han and Bridglall (2009) found that math achievement differences
existed while comparing ELLs from schools with various proportions of ELLs to native English
speakers attending schools with no ELLs. However, there were no achievement differences
among native English speakers attending schools with different percentages of ELLs while
adjusting for several school covariates, such as Title I and instructional English-as-a-second
language services. With regards to motivation, school proportion of ELLs positively predicted
academic self-concept of elementary school ELLs sampled from the ECLS-K (Niehaus &
Adelson, 2014). In sum, school poverty, and proportions of racial/ethnic minority and ELL
students seem to influence achievement; and both percentages of school racial/ethnic minority
student and ELLs appear to relate to motivation.
Gaps in the Literature
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This review of the literature indicates that much has been learned with regards to
individual and school factors that influence math self-efficacy, math achievement, and English
reading. However, the review also points to gaps within this literature. For instance, most studies
investigating the relationship between L1 proficiency and L2 achievement including in English
reading have focused on elementary school students. Little is known around whether this
relationship is also prevalent in older students. Similarly, the moderating role of L1 proficiency
on the relationship between LM status and English reading has yet to be investigated with high
school students. Studies investigating these relationships with older students are needed to
determine whether findings with elementary level students are generalizable to other student
populations. This knowledge could contribute in the decisions around the necessity and
significance of providing native language instruction at the high school level, especially in
determining which groups of adolescents need the most support to improve their English reading,
and in developing future interventions aimed at meeting specific subgroups’ academic needs.
Although separate studies suggest relationships among LM status, L1 proficiency,
English reading, math achievement, and math self-efficacy, there have not been any studies
examining these relationships simultaneously. In particular, there is very limited knowledge on
these concurrent relationships within linguistically and ethnically diverse high school students
from a nationally representative sample. Furthermore, the mediation of English reading and math
achievement on the relationship between the interaction (LM and L1 proficiency) and math self-
efficacy has not been explored despite indications of this possible mediation. Another gap exists
in the literature on the reciprocal relationship between math achievement and math self-efficacy
and their relative strengths using longitudinal data. Research exploring these gaps can provide a
comprehensive model to explain how LM status relates to math self-efficacy and to show how
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45
the math achievement and self-efficacy of linguistically and ethnically diverse adolescents could
be improved. This expanded knowledge could be used to develop more targeted interventions
aimed at not only improving student math achievement, but also their motivation.
Much of the research on factors influencing achievement and motivation has been done at
the individual-level. Therefore there is a knowledge gap on whether school factors, such as
racial/ethnic minority and ELL student proportions, relate to motivation, and in particular to
math self-efficacy. Studies that examine these relationships and clarify the inconsistent findings
in the relationship between school ELL percentage and math achievement are needed.
Furthermore, although studies using student-level data appear to indicate a reciprocal
relationship between math achievement and self-efficacy, few studies have investigated this
relationship at the school-level. Knowledge gained from research addressing these school-level
gaps could suggest if and which school-level factors influence school math achievement and
self-efficacy, which in turn could support programs that attempt to desegregate schools by
race/ethnicity, income, and English language learner status. Lastly, this knowledge could be used
to test whether relationships observed at the student-level are also present at the school-level and
therefore support the generalization of student-level findings to higher levels.
In sum, addressing the above gaps would further guide the development of targeted
interventions that could ultimately improve linguistically and ethnically diverse adolescents’
math achievement and motivation. By improving these students’ math self-efficacy and
achievement, their likelihood of choosing and pursuing STEM careers may increase and thus
lead to a greater participation of ethnically, racially, and linguistically diverse individuals in
STEM careers (see Perez, Cromley, & Kaplan, 2014; Wang, 2012; Wang, 2013).

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Chapter III: Methodology
Research Questions and Hypotheses
I addressed the above gaps in the current study by first sampling linguistically and
ethnically diverse high school students from a representative sample and by using longitudinal
achievement and self-efficacy data. Second, I examine both individual- and school-level factors
that relate to English reading, math achievement, and self-efficacy. Third, I concurrently include
LM status, L1 (non-English) proficiency, English reading, math achievement, and math self-
efficacy in one comprehensive model. More specifically, the current study answered the
following research questions:
1. Do schools’ poverty level and percentages of racial/ethnic minority students and ELLs
relate to high school students’ math achievement and math self-efficacy?
2. How are language minority status, first language (non-English) proficiency, English
reading, math achievement, and math self-efficacy related in high school students while
adjusting for race/ethnicity, gender, and SES? More specifically,
a. Does first language (non-English) proficiency relate to English reading?
b. Does first language proficiency moderate the relationship between language minority
status and English reading?
c. If such a moderating relationship exists, does the interaction between language
minority status and first language proficiency predict math self-efficacy through English
reading and math achievement?
For research question one, I expected that school poverty level and the percentages of
racial/ethnic minority and ELL students would be negatively related to math achievement
(Brown-Jeffy, 2009; Drake, 2014; Kitsantas et al., 2010; Mosqueda, 2010; Mosqueda &
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47
Maldonado, 2013; Newton, 2010; Werblow & Duesbery, 2009). I also hypothesized that school
percentage of racial/ethnic minority and ELL students would be positively related to math self-
efficacy based on their relations to self-beliefs (Kitsantas et al., 2010; Niehaus & Adelson, 2014).
With regards to research question 2a, I hypothesized that L1 (non-English) proficiency
would be positively and significantly related to English reading achievement based on cross-
linguistic transfer studies (Rolstad et al., 2005; Slavin & Cheung, 2005; Umanski & Reardon,
2014; Valentino & Reardon, 2015), developmental interdependence hypothesis (Cummins,
1979), and common underlying model (Cummins, 1981). In particular, I expected that students
with higher L1 proficiency would obtain higher English reading achievement scores as compared
to students with lower L1 proficiency.
For research question 2b, I expected L1 proficiency to moderate the relationship between
LM status and English reading achievement. Specifically, I hypothesized that LM students with
higher L1 proficiency would attain higher English reading achievement as compared to LM with
less L1 proficiency. When compared to English native speakers, I expected LM with L1
proficiency to have similar or greater English reading achievement (Lindholm-Leary & Block,
2010; Umanski & Reardon, 2014).
With regards to research question 2c, I hypothesized a mediation path from the
interaction between LM status and L1 (non-English) proficiency to math self-efficacy through
English reading and math achievement. This mediation was expected based on findings in the
literature, in particular that: (1) prior achievement influences self-efficacy including in the math
domain, (2) LM status is related to English achievement, reading in particular, and (3) linguistic
skills influence math achievement (Han & Bridglall, 2009; Kim & Herman, 2009; Schleppegrell,
2007; Stevens, Olivárez, & Hamman, 2006; Stevens et al., 2004; Vukovic & Lesaux, 2013). In
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48
addition, there is some preliminary evidence that LM students have lower math self-efficacy
compared to native English speakers, but this difference lessens or even disappears when LM
students are proficient in English (Riconscente, 2014).
These specific hypotheses are depicted in the multilevel structural model in Figures 1 and
2. Figure 1 shows relationships among variables at the school-level, and Figure 2 displays
relations at the student-level. I statistically adjusted the student-level model for SES, gender, and
race/ethnicity by including them as covariates because the reviewed literature has reported
differences in competence beliefs and achievement based on these covariates (e.g., Berends &

Figure 1. Hypothesized multilevel model displaying only school-level relationships among
school-level student demographics, achievement, and self-efficacy. % MIN = percentage of
racial/ethnic minority students; % ELL = percentage of English language learners in grade 10; %
FRL = percentage of sophomores eligible for free or reduced-price lunch; RA10 = grade 10
English reading achievement; MA10 = grade 10 math achievement; MA12 = grade 12 math
achievement; MSE12 = grade 12 math self-efficacy; and MSE10 = grade 10 math self-efficacy.
The letter “B” in front of a variable name represents a school-level variable.
BMA10 BMA12
BMSE10 BMSE12
BRA10
% FRL
% MIN
% ELL
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49

Figure 2. Hypothesized multilevel model displaying only student-level relationships among
student characteristics, achievement, and self-efficacy including moderation and mediation. SES
= socioeconomic status; RA10 = grade 10 English reading achievement; MA10 = grade 10 math
achievement; MA12 = grade 12 math achievement; MSE12 = grade 12 math self-efficacy;
MSE10 = grade 10 math self-efficacy; LM = language minority status; L1PROF = L1 (non-
English) proficiency; LMXL1PROF = interaction term between LM and L1PROF; G = gender;
HISP = Hispanic; ASI = Asian; BLK = Black; OTH = Other. White students were used as
reference group. Red arrows indicate hypothesized mediation path.

RA10
MA10 MA12
MSE 10 MSE 12
LM
L1PROF
LMXL1PROF
SES
G
HISP
ASI
BLK
OTH
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Peñaloza, 2010; Else-Quest et al., 2013; Hemphill & Vanneman, 2011; Huang, 2013; Kingston
& Lyddy, 2013; Watt, 2008). To address the current study’s research questions, I analyzed
secondary data, specifically the Educational Longitudinal Study (ELS, cohort 2002).
Educational Longitudinal Study (Cohort 2002)
For the current study, I conducted a secondary data analysis using the restricted-use
version of the ELS 2002. The ELS 2002 is a nationally representative and longitudinal study of
students who were in the 10th grade in the U.S. in 2002 (Ingels, Pratt, Rogers, Siegel, & Stutts,
2004). The purpose of the ELS 2002 was to provide longitudinal data that documents students’
transitions from grade 10 to either higher education or the work force. To date the following data
waves have been collected: base-year (Spring 2002, 10
th
grade), first follow-up (Spring 2004,
12
th
grade) high school transcript (fall 2004 and 2005), second follow-up (2006), third follow-up
(2012), and postsecondary transcript (2013) (Ingels et al., 2014). For the current study, I used the
student, parent, and administrator data from the base-year and student data from follow-up year
when students were in 10
th
and 12
th
grades, respectively.
ELS 2002 Sampling Procedure and Data Collection
The ELS 2002 utilized a two-stage sample selection process, in which schools, including
public, Catholic, and other privates schools, with 10
th
grades in the 2001-02 school year were
first randomly selected using a stratified cluster probability proportional to size sampling method
(1,221 schools from a total of 2,700 schools with 10
th
graders), and second, sophomores from
these schools (about 26 sophomores per school from the 752 participating schools) were selected
systematically from a race/ethnicity stratified sample (Ingels et al., 2004). More specifically, the
Common Core of Data and Private School Survey were used as sampling frames for school
sample selection. After stratifying the school sample by U.S. Census divisions and metropolitan
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51
status separately for public and private schools including Catholic schools, the sample was
divided by stratum at random into three pools. Schools were randomly selected from these pools.
With regards to student selection, based on the participating schools’ 10
th
grade enrollment lists,
students were classified by race/ethnicity (Hispanic, Asian, Black, and other race/ethnicity) to
achieve a nationally representative sample in terms of race/ethnicity. Students were sampled on a
flow basis as student enrollment lists were received and fixed student sample rates were used.
Additionally, Asian and Hispanic students, as well as students attending private schools, were
oversampled to reflect the population of sophomores. English language learners who had either
received at least three years of academic instruction mainly in English or determined by school
staff that they were able to participate despite receiving fewer than three years of instruction
were included in the sample. In particular, 44 ELLs in the base-year and 10 ELLs in the follow-
up year were excluded because of language barriers. Students’ with Individualized Education
Plans (IEP) were included as long as their IEP did not prevent them from participating in
assessment programs and schools could provide the appropriate accommodations (Ingels, Pratt,
Rogers, Siegel, & Stutts, 2005).
Of the 17,591 eligible selected students, 15,362 students completed the base-year survey.
About 87 percent of parents/guardians of participating students (13,488 parents) and 743
principals completed the base-year survey (Ingels et al., 2004). In the first follow-up the sample
was refreshed by recruiting seniors who were new at the base-year schools and 1000 students,
who were among non-respondents in the base-year, were subsampled. From the base-year
respondents, 14,062 students (92.35 % of original sample) completed the first follow-up survey.
In the follow-up year ELS 2002 study also surveyed base-year participants, who had either
transferred, dropped out, graduated early, or were homeschooled.
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In the base-year, students, parents, teachers, school administrators, and librarians
completed the surveys (Ingels et al., 2004). Student questionnaires, which lasted about 45
minutes, were administered in a group setting in students’ classrooms. Students self-reported
their math self-efficacy among other information, such as use of non-English language and
demographic information, on questionnaires that were in English. During this period, they also
took reading and math tests. Parent questionnaires were mailed to students’ home addresses and
were completed by the parent/guardian who was most familiar with the student’s school
experience through self-selection. Spanish and English forms as well as hard copies and
electronic versions of the parent questionnaires were available. The questionnaire contained
items about students’ family background, school and family lives, and parents’ thoughts about
the child’s school, and their future expectations about their child. The school administrator
survey covered information about characteristics of the school, student body, and faculty, as well
as their school policies, programs, technology, and climate. Teacher questionnaires asked for
English and math teachers’ evaluations of participating students’ behaviors, academic
achievement, and educational and work plans. The teacher questionnaire also included items on
teachers’ background, such as teaching experience and professional development.
In the follow-up year only students and administrators were surveyed. Various versions
of student surveys were developed and administered to account for students’ characteristics
including dropout, transfer, homeschool, and membership in freshened sample. Some common
items were present across different student questionnaire versions, such as academic coursework,
extracurricular activities, and purposes of computer use.
Cognitive assessments were provided in both base- and follow-up years. However,
English reading was only assessed in grade 10, whereas math achievement was assessed in both
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years. In the follow-up year, dropouts and early graduates along with homeschooled and transfer
students were not tested, although the math scores for homeschooled and transfer students were
imputed. More information on these assessments is provided in later sections.
Strengths and limitations of ELS 2002. The ELS 2002 was selected as an appropriate
dataset for addressing the current study’s research questions. In particular, this dataset contains
information on a large nationally representative sample from U.S. high school students,
including a substantive proportion of LM students (about 17%). Available are demographic
information, such as the sample’s socioeconomic status (SES), gender, number of grade levels
completed outside of the U.S., and number of years non-U.S. born students have been in the U.S.
In addition, the dataset includes target variables, such as English reading and math achievement,
and math self-efficacy. The target constructs were assessed with measures used in other studies
that reported high reliability and validity for these measures (Barrett et al., 2012; Fan, 2011;
Marsh, Hau, Artelt, Baumert, & Peschar, 2006). Moreover, ELS 2002 has continuous data for
achievement, item-level information for math self-efficacy, and longitudinal math achievement
and self-efficacy data which variable characteristics enable the use of sophisticated statistical
methods.
Despite the inclusion of a vast amount of relevant and reliable data, the ELS 2002 dataset
is not perfect and exhibits several shortcomings. First, L1 (non-English) proficiency was not
directly measured using cognitive tests, which would have provided a more valid and reliable
measure of this proficiency instead of the number of grade levels completed outside of the U.S.
Second, the limited number of students selected from each school did not allow for
disaggregation of LM students by either native language (Spanish vs. Asian language) or country
of origin. Missing data is substantive for grade 12 math self-efficacy because participating
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54
students who were no longer in the base schools were not surveyed on this construct. Lastly, the
ELS 2002 excluded English learners with very limited English proficiency. Therefore, the results
of the proposed study would not be generalizable to this specific subgroup, which may need the
most academic and motivation support.
Although the ELS 2002 dataset has the above limitations, it still represents a suitable
dataset because of the previously outlined strengths and its prior use in investigating students’
motivation and achievement, particularly of ELLs (Barrett et al., 2012; Mosqueda, 2010) in the
domain of math (Fan, 2011; Mosqueda & Maldonado, 2013). Consequently, I argue that the ELS
2002 dataset could answer the questions the current study addresses.
The current study was informed by Social Cognitive Theory (SCT; Bandura, 1986),
specifically the construct of self-efficacy, developmental interdependence hypothesis (Cummins,
1979), and the common underlying model (Cummins, 1981). I have selected SCT as one of the
theoretical frameworks because first, the theory suggests that enactive experiences (i.e.. prior
achievement experiences) influence self-efficacy (Bandura, 1997; Usher & Pajares, 2008). That
is, previous academic successes and failures of students influence their self-efficacy. Second, the
high predictability of self-efficacy has been widely supported by research including in the math
domain (e.g., Caprara et al., 2011; Carroll et al., 2009; Lee, 2009). The developmental
interdependence hypothesis and common underlying proficiency model also guided the current
study because they have been frequently used to interpret studies on cross-linguistic transfer
(e.g., Genesee, Geva, et al., 2006; Goodrich et al., 2013).
Sample and Variables
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Initially, I had planned to use all observations from the ELS 2002 dataset (about 16, 200
students and 750 schools)
3
. However, school percentage of ELLs perfectly predicted missingness
in school-level grade 12 math achievement and math self-efficacy while adjusting for several
covariates (i.e., school percentage of students eligible for free or reduced-price lunch (FRL) and
racial/ethnic minority students, school-level grade 10 math achievement and self-efficacy,
reading achievement, and grade 12 math achievement (only used as predictor for missing grade
12 math self-efficacy)). Therefore, I dropped these fewer than 10 schools and with them about 90
students as well. The final sample size was composed of about 16,110 students and 740 schools.
Student demographic information including their LM status, SES, race/ethnicity, gender,
first language (non-English) proficiency, and math self-efficacy data were measured through
items from student and parent questionnaires. Student achievement in English reading and math
were measured through cognitive tests, and school demographic information was collected
through administrator questionnaires. More detailed information about the measures used to
collect this data is provided in the section below.
Race/ethnicity. Dummy race/ethnicity variables were created based on student’s
race/ethnicity composite variables (restricted-use) from the base- and follow-up years
(BYRACE_R and F1RACE_R). The ELS 2002 study imputed these variables if they were
missing in the student questionnaire by using information from sampling roster, parent
questionnaire, or other questionnaire items. If base-year values were missing, I replaced them
with follow-up year information if available
4
. The race/ethnicity dummy variables were White,
Black/African American, Hispanic (race specified and not specified combined), Asian (including

3
NCES requires rounding of sample sizes and degrees of freedom to the ten for analyses that use ELS 2002
restricted-use data. For numbers below 10 including those below 5 were rounded them up to 10.
4
For all variables, if students had base-year missing information but this data was available for the follow-
up year, I replaced the missing base-year values with the follow-year data.
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Hawaiian and Pacific Islander), and Other (including multi-race, American Indian, and Alaska
native). White was the reference group.
Gender. Students’ gender was surveyed with the question, “What is your sex?” The
composite variables for both years (BYSEX and F1SEX) were logically and statistically
imputed. About .07% of this data in total was imputed for both years. To facilitate analysis, I
recoded females as zero and males as one.
Socioeconomic status (BYSES1 and F1SES1). SES was a composite variable based
primarily on base-year parent survey information and student questionnaire if information was
missing. It included father’s/guardian’s education and occupation, mother’s/guardian’s education
and occupation, and family income, all of which were equally weighted and standardized. If
missing, each of these variables was imputed before creating the composite variable. In the base-
year the percentages of imputation for the SES composite variable components were 3.65% for
mother education, 9.25% for father education, 5.19% for mother’s occupation, 13.58% for
father’s occupation, and 26.5% for income. In the follow-up year the percentage of imputation
were .8% for mother’s education, 1.25% for father’s education, 1.15% for mother’s occupation
and 1.59% for father’s occupation.
Language minority (LM) status. This variable was created based on students’ home
language information (composite variables BYSTLANG and F1STLANG). Students were asked,
“Is English your native language (the first language you learned to speak when you were a
child)?” The response options were either yes or no. About 2% of students had their base-year
data imputed, and .4% had their follow-up data imputed. If students responded that their home
language was English, they were considered native English speakers (coded as zero), but if they
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answered that their home language was other than English, they were considered LM students
(coded as one).
First language (non-English) proficiency. Information on the number of grade levels (K
through 10) completed outside of the U.S. (BYP25, BYP26A-K) was used as a proxy for first
language proficiency. In the base-year, parents were asked, “What grade(s) has your tenth grader
completed outside the United States?” The response options were each grade level from
kindergarten to 10
th
grade with respondents marking all applicable options. I used this
information to create a dummy variable for L1 proficiency. If students had completed at least
grades 1 through 5 subsequently outside of the U.S., they were coded as 1 (L1 proficient),
otherwise they were coded as zero. Consequently, the reference group included native English
speakers. I utilized five grade levels as a cut off point based on the finding that a minimum of
four years of primary language schooling was required to positively influence second language
achievement (Thomas & Collier, 2002). Because parents were not surveyed in the follow-up
year, students who did not participate in the base-year did not have this data available.
Language minority X L1 proficiency (interaction term). The construct was created by
multiplying values from LM and L1 proficiency variables. Students who were LM and
completed at least grades 1 through 5 subsequently outside of the U.S. were coded as one,
whereas others were coded as zero.
Math self-efficacy (BYS89A, BYS89B, BYS89L, BYS89R, BYS89U, F1S18A,
F1S18B, F1S18C, F1S18D, F1S18E). The math self-efficacy measure for each grade level
consisted of five identical items. Students were asked to rate how often the following items
applied to them on a four-point Likert scale (1= almost never to 4 = almost always): “I’m
confident that I can do an excellent job on my math tests”; “I’m certain I can understand the most
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difficult material presented in math texts”; “I’m confident I can understand the most complex
material presented by my math teacher”; “I’m confident I can do an excellent job on my math
assignments”; and “I’m certain I can master the skills being taught in my math class.” In the
current study, scores from these items were added to create separate total math self-efficacy
scores for grades 10 and 12.
The ELS 2002 study adapted these items from the self-efficacy PISA 2000 measure
(Burns et al., 2003). However, there were a few differences between the PISA and ELS 2002
math self-efficacy items. For example, the PISA items were not domain specific, whereas the
ELS 2002 items were specific to math. In addition, in the ELS 2002 one PISA self-efficacy item
(“I’m confident I can do an excellent job on my assignments and tests”) was separated into two
items (“I’m confident I can do an excellent job on my math assignments” and “I’m confident I
can do an excellent job on my math tests”) to reflect task specificity (Burns et al., 2003). Another
difference was the labels on the response categories. The PISA response scale indicated students’
agreements with the statements (from disagree to agree), whereas in the ELS 2002 the response
categories reflected frequency of the statements. More specifically, in the ELS 2002 students
answered the question, “How often do these things apply to you?” on a scale from 1 (almost
never) to 4 (almost always). The coefficient alpha in the ELS 2002 study was reported as .93.
Despite differences in the response scale, Fan (2011) reported that math self-efficacy as
measured in ELS 2002 was significantly related to math intrinsic value, utility value, and
educational expectations supporting the measure’s validity.
Generally, the percentage of missing math self-efficacy information was high. One
possibility for this may have been the location of these items, which was towards end of the
student survey (cf. Ingels et al., 2005). Another reason for missing data particularly for grade 10,
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59
was the additional steps taken in the follow-up year to increase the generalizability and sample
size of senior students by freshening the base-year sample and sampling of base-year non-
respondents. Missing data for grade 12 math self-efficacy was the highest (36%). One reason for
this was that students who were no longer enrolled in the base schools in grade 12, such as
dropouts, homeschooled, and early graduates, were not asked about their math self-efficacy in
their modified follow-up surveys. The ELS 2002 study did not impute math self-efficacy scores.
Math achievement (BYTXMIRR and F1TXM1IR). Grade 10 students’ math
performance was measured in the ELS 2002 in two stages (Ingels et al., 2004). In the base-year,
students first completed a multiple-choice routing test (15 math questions), which determined the
difficulty level (low, middle, or high) of the second stage assessment. The second stage test
consisted of 25 to 27 items with a mixture of multiple choice items (90%) and open-ended
responses (10%). The two-stage procedure was used to limit ceiling and floor effects as well as
to increase assessment accuracy within a brief testing period.
Grade 12 math test items were in multiple-choice format and the appropriate difficulty
(low, middle, high) of the test was determined based on students’ grade 10 scores instead of
completing a routing test. For students who were either new to the ELS 2002 study in the follow-
up year or did not have base-year scores, a separate assessment was developed which included
items with broad range difficulty. The grade 12 math tests included 32 items (Ingels et al., 2005).
The math items for grades 10 and 12 that were extracted from prior NELS:88, NAEP,
and PISA assessments were field tested with a sample of 10
th
and 12
th
graders and later modified
accordingly based on the field test results (Burns et al., 2003). The reliability for the grade 10
math test was reported as .92, and for the combined grades 10 and 12 test, it was .92 The math
test items for both grades covered content, such as arithmetic, algebra, geometry,
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60
data/probability, and advance topics, and assessed knowledge/skill, comprehension, and problem
solving. Common items were used in grade 10 and 12, and the base-year math scores were
rescaled to facilitate longitudinal analysis.
For item scoring, Item Response Theory (IRT) based scores were selected to ensure
comparability of ability estimates across tests forms of varying difficulty (Ingels et al., 2004).
Patterns rather than raw number of correct, incorrect, and omitted responses, as well as
characteristics of items, such as their difficulty levels, were used in IRT to estimate ability.
Specifically, the IRT-estimated number right scores, which indicate the number (stated as
probabilities) of correct items students would have received if they were given all 73 (grade 10),
or 85 (grade 12) test items, were used in the current study. The range of scores was from 0 to 85
including for the rescaled grade 10 math achievement scores. About 9% and 17% of scores were
imputed in grades 10 and 12, respectively.
Grade 10 English reading achievement. The reading test was also developed,
administered, and scored similarly as the math test. The reading routing test consisted of 14
reading questions, and the second stage test contained 15 to 17 questions depending on the
difficulty level. The reading assessment consisted of reading passages followed by three to six
multiple choice questions that were either about comprehension, inference/evaluation, or recall
of detail. The passages included literary materials, covered natural and social sciences topics, and
some of them contained graphs. The reported reliability for the reading IRT estimated number
right scores (BYTXRIRR) was .86, and the range of the scores was from 0 to 51. The reading
test was administered only during the base-year and about 10% of the scores were imputed.
Percentage of 10
th
grade students eligible for free or reduced-price lunch (BYA21).
The percentage of sophomores who were from low-income households was assessed through the
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61
base-year school administrator questionnaire. Administrators were asked, “What percentage of
the current 10
th
grade students receives free or reduced-price lunch from your school?”
Percentage of racial/ethnic minority students (CP02PMIN). The percentage of
racial/ethnic minority students in participating schools was not directly measured using school
administrator questionnaires. Instead, external sources that were linked to the ELS 2002, such as
the Common Core of Data and Private School Survey, provided this percentage for the 2001-
2002 academic school year.
Percentage of 10
th
grade ELLs (BYA20). The proportion of ELLs who were also
sophomores in spring 2002 was measured in the base-year school administrator questionnaire.
Administrators responded to the question, “What percentage of the current 10
th
grade students is
Limited English Proficient (LEP) or Non-English Proficient?”
To enable multilevel estimation using the Mplus program (Muthén &Muthén, 2015),
students attending the same school needed to have the same values for school-level student
demographic variables. In other words, if some students from school A had missing values on
percent of ELL sophomores, but others from the same school had valid values for it, the program
would interpret this as an error and it would not estimate multilevel models. Therefore, if there
was school-level student demographic characteristic information available for a school from at
least one student, I recoded missing values for students in the same school to this information.
Aggregated Variables, Sampling Weights, and Imputation
School-level achievement and self-efficacy data. For the initial analyses, school-level
aggregates for the achievement and self-efficacy data were created in the Stata program
(StataCorp LP, 2015) based on student-level achievement or total math self-efficacy scores.
Specifically, the means of student-level data were estimated for students attending the same
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62
schools and these aggregated values were used to estimate descriptive statistics, correlations,
predicting missingness, etc. In multilevel models, instead of using the school-level aggregates as
created above, school cluster (school ID) and stratum information were included in the Mplus
program script for the program to automatically estimate school-level achievement and self-
efficacy data based on the student-level information provided.
Student and school level sampling weights. At the student-level, I selected the
expanded sample panel weight to include all eligible students and use longitudinal data from
grade 10 to 12 (F1XPNLWT). Because this weight dropped students who did not have
longitudinal data, I replaced missing expanded panel weights with expanded cross-sectional
weights that were calculated for the base- (BYEXPWT) and follow-up (F1EXPWT) years. In
addition, I adjusted these weights by multiplying the recoded longitudinal weights by total
sample size divided by the total sum of weights. For the school-level, I used the school level
weight, BYSCHWT, which was estimated only for the base-year.
Using effective sample-size weight provides more accurate parameters for multilevel
SEM rather than using unweighted models or models with weights that sum to the sample size
(Stapleton, 2002). Therefore, at the student-level, I scaled the sampling weight for its sum to be
equal to the effective sample size by selecting the “ecluster” setting for the wtscale option in the
Mplus program (Muthén &Muthén, 1998-2010; Stapleton, 2002). For the school-level, I selected
the “sample” setting for the bwtscale option, which adjusts the school level weights for the
product of weights at both levels to sum to the total sample size, because an effective sample size
adjustment was not available at this level.
Imputation. For missing demographic information, the ELS 2002 study employed
logical imputation (using other information to deduce responses) and weighted sequential hot
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63
deck procedure, and for missing achievement data, it used multiple imputation. To use all
available data and account for student sampling at different stages (e.g., freshened sample and
subsampling), I also imputed missing base-year values with follow-up year values when they
were available.
Data cleaning and most preliminary analyses, including descriptive statistics,
correlations, and detection of outliers and missing data were conducted using the Stata program
(version 12). First, I estimated the descriptive statistics. Second, I checked whether the
multivariate normality assumption was met, which was necessary for the appropriate use of the
Maximum Likelihood method, by examining the distribution of variables, their linear
relationships, and homoscedasticity of residuals. Third, with regards to missing data, I explored
percentages of missing values, patterns of missingness, and predicted missingness using logistic
multiple regressions. Specifically, in predicting missingness, the outcome was missingness in the
endogenous variables, whereas the predictors were all the other variables from the model.
Fourth, I calculated the item loadings of the math self-efficacy measure, internal reliabilities
(Cronbach’s alpha) of this measure, and correlations among variables. Next, confirmatory factor
analysis of the math self-efficacy measure, and multilevel and multi-group analyses were
conducted using the Mplus program (version 6.11). Further details on the specification,
identification, estimation, and evaluation of the multilevel model are provided below.
Multilevel Structural Equation Modeling (SEM)
Multilevel SEM is one statistical approach that can be used to answer this study’s
research questions. SEM refers to a family of statistical techniques that test models that illustrate
relationships among variables based on theory and prior research, one of which is path analysis
(Hoyle, 2012; Kline, 2011). Multilevel SEM enables the estimation of parameters at various
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64
levels to account for clustering (Rabe-Hesketh, Skrondal, & Zheng, 2012). I selected this
statistical approach because in addition to providing information on the within-cluster correlation
(intraclass correlation), and between- and within-cluster variances, it estimates model fit indices
and enables the inclusion of more than one outcome/endogenous variable (Chou, Bentler, &
Pentz, 1998; Kline, 2011; Wendorf, 2002). Multilevel SEM has its limitations, such as requiring
balanced or complete data (Rabe-Hesketh, Skrondal, & Zheng, 2007). However, in light of
investigating interaction effects with clustered data, its advantages suggest that multilevel SEM
may be an appropriate statistical method to examine: (1) the moderating roles of first language
proficiency on the effects of LM status in English reading achievement; and (2) the mediating
roles of English reading and math achievement in the relationship between such interaction and
math self-efficacy. In using multilevel SEM, I checked whether cluster sizes were balanced and
if the dataset had missing data, which are described in later sections.
Specification, identification, estimation, and evaluation of the proposed multilevel
SEM model.
Specification. Based on the literature reviewed I developed a multilevel SEM model that
is displayed in Figures 1 and 2. More specifically the following research findings guided the
development of this model: (1) many LM students have lower reading achievement compared to
native English speakers (e.g., Zarate & Pineda, 2014); (2) LM students’ native language
proficiency increases their L2 achievement including in English reading and math achievement
(e.g., Lindholm-Leary & Block, 2010; Umanski & Reardon, 2014); (3) linguistic skills are
needed in learning and achieving in math (e.g., Grimm, 2008); and (4) prior academic success or
failure experiences influence student self-efficacy (e.g., Usher & Pajares, 2008). The following
findings also formed the base for model development: (1) ELL students’ math achievement and
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65
self-efficacy increase as their English language proficiency improves (e.g., Mosqueda, 2010;
Riconscente, 2014); (2) prior math achievement and self-efficacy predict subsequent
achievement and self-efficacy, respectively (e.g., Caprara et al., 2008; Watts et al., 2014); (3)
achievement and self-efficacy have a reciprocal relationship, including in math (e.g., Caprara et
al., 2011; Williams & Williams, 2010); (4) gender, SES, race/ethnicity relate to students’
achievement and motivation (Berends & Peñaloza, 2010; Kieffer, 2008; Kingston & Lyddy,
2013; Lindberg et al., 2010; Murayama et al., 2013; Vanneman et al., 2009); and (5) school
factors, such as schools’ poverty level, and percentages of racial/ethnic minority and ELL
students all influence math achievement and motivation, particularly those of LM students (e.g.,
Kitsantas et al., 2010; Niehaus & Adelson, 2014).
These separate relationships suggested that English reading and math achievement could
be possible mediators of the relationship between the interaction (LM x L1 proficiency) and
math self-efficacy. Parts of this mediation were explored and supported by Guglielmi (2012), in
particular the relationships between L1 proficiency and English reading, and between English
reading and math achievement. Moreover, based on the recommendation of adjusting for prior
measurements while testing for mediation with two waves of data (Cole & Maxwell, 2003), I
specified the following mediation paths: interaction (LMXL1PROF) -> grade 10 English reading
-> grade 12 math achievement, and grade 10 math achievement -> grade 12 math self-efficacy
while adjusting for grade 10 math achievement and self-efficacy. If stationarity can be
assumed—relationships are the same regardless of when the variables were measured—the path
between grade 10 math achievement and grade 12 math self-efficacy, would be equal to the
unmeasured path between grade 12 math achievement and postsecondary math self-efficacy. In
that case, the product of the coefficients for the mediation paths (read -> gr12 math achievement,
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66
gr10 math achievement -> gr 12 math self-efficacy) would estimate the mediation effect of math
achievement in the relationship between English reading and math self-efficacy. The same
procedure could not be used to test the mediation of reading achievement in the relationship
between interaction term and grade 10 math achievement because reading was measured only in
grade 10, meaning that the path between the interaction and reading could not be adjusted by
prior English reading achievement. Despite the limitations of having only two waves of data to
test for the presence of two mediators, which implied the inability to check for stationarity and
complete mediation, the adjustment for available prior measures, such as math achievement and
self-efficacy, would provide less biased estimates compared to the use of cross-sectional data
(Cole & Maxwell, 2003).
Identification. The school- and student-level sections of the structural multilevel model
are non-recursive because of the correlated disturbances between the grade 12 math achievement
and self-efficacy variables as depicted in Figures 1 and 2. In particular, the residuals of grade 12
math achievement and self-efficacy at the student-level were specified as being correlated
because prior studies (Lewis et al., 2012; Riconscente, 2014) reported that other factors, which
were not included in the current study, such as teacher caring, were related to both math
achievement and self-efficacy for native English speakers and ELLs. Similarly, the disturbances
of math self-efficacy and achievement were correlated at the school-level in order to reflect the
student-level relationship. However, because the model has a bow-free pattern, in which the
endogenous variables with correlated disturbances do not have direct effects on each other, this
non-recursive model can be considered recursive, which means it is identified (Kline, 2011).
That is, sufficient degrees of freedom existed to test the model. By counting estimated
parameters and degrees of freedom, I confirmed that the student- and school-level sections were
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67
identified. More specifically, the school-level model estimates 27 parameters with about 10
degrees of freedom, and the student-level model estimates 51 parameters with about 50 degrees
of freedom.
Estimation. To determine whether multilevel modeling was necessary, I first estimated
the intraclass (within-school) correlation. The intraclass correlation was greater than .10 for the
achievement data. This result indicated that students from the same school had achievement
information that was more similar when compared to students from other schools (Kline, 2011),
which suggested the need for a multilevel analysis.
For estimating the multilevel model, I used the Maximum Likelihood Robust (MLR)
estimator, which adjusts for non-normality and dependence of observations, and provides robust
parameter estimates, standard errors, and Chi-Square test statistic (Muthén &Muthén, 1998-
2010). The sandwich estimator was used to calculate MLR standard errors. Although math self-
efficacy was assessed using a four-point Likert scale, other researchers (Marsh et al., 2006) have
used a very similar measure with the same scale and utilized the ML estimation method.
Consequently, I employed the MLR estimator for model estimation. Moreover, I estimated the
hypothesized multilevel model by sections beginning only with the relationships among
endogenous variables at both levels, followed by subsequent inclusion of main student-level
predictors, student covariates, and school-level predictors. When estimation problems were
encountered while specifying a particular section, the problem was addressed before adding
additional variables to the model.
Evaluation. Model fit indices including Root Mean Square Error of Approximation
(RMSEA), Tucker Lewis Index (TLI), and Comparative Fit Index (CFI) were utilized to assess
model fit. The following suggested cutoff values guided model fit evaluation: CFI and TLI
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68
values greater than or equal to .95, and RMSEA values less than .06 (Hu & Bentler, 1999).
Additionally, I used the Chi-Square test to compare nested models and subsequently adjusted the
Chi-Square values by the scaling correction factor when necessary.
Weights, mediation, and sensitivity analysis. Bias parameters may result from not
including sampling weights to models in which there is unequal sampling probability, as is the
case in the ELS 2002 study (see Asparouhov, 2005; Stapleton, 2002). After estimating a
complete multilevel model, I included school and student sampling weights. Next, I compared
model fit indices and parameter estimates across weighted and unweighted models to determine
whether the inclusion of weights made a difference in them.
In testing the mediating roles of English reading and math achievement, I compared the
model fit indices of models with mediation paths either constrained to zero or freely estimated.
Next, I compared whether these models were significantly different using a Chi-Square
difference test. If the Chi-Square difference test was not significant it would indicate that the
mediation was not supported. In contrast, if the Chi-Square difference test was significant it
would support the mediation.
I conducted sensitivity analysis of the multilevel model results by including centered
variables into the multilevel model, and estimating multi-group models and one-level path
models. More specifically, I estimated the same multilevel model using mean centered variables
(LM, L1 proficiency, and their interaction term). In addition, I estimated a one-level multi-group
model with L1 proficiency as the grouping variable and compared the model fit indices for
models in which the path from LM status to English reading was either freed or constrained to
zero. A multilevel multi-group model could not be estimated because of the Mplus program’s
limitation in estimating this type of model. Lastly, I estimated one-level path models to
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69
determine whether I encountered the same estimation problems as in estimating the multilevel
model.
Power. The accuracy of the statistical tests within the multilevel SEM framework
depends on estimation method, type of standard error, and number of groups (Hox, Maas, &
Brinkhuis, 2010). The estimation method and type of standard error does not influence the
parameter estimates and standard errors at the within-level; however, they lead to differences in
estimates and standard errors at the between-level (Hox et al., 2010). Specifically, using
multivariate normal simulation data Hox and colleagues (2010) reported that the maximum
likelihood estimation with asymptotic standard errors provided unbiased parameters and
confidence intervals when the number of groups was at least 200. Additionally, the authors
caution that maximum likelihood estimation leads to less biased parameters and standard errors
when the assumption of multivariate normality is met and heterogeneity is modeled. Because in
the current study there were over 200 groups (about 740 schools), variables appeared to meet the
normality assumption, and the maximum likelihood robust estimator was used, it was concluded
that there was enough power to estimate unbiased parameters and standard errors.

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Chapter IV: Results
I begin this section by describing the initial analyses. Next, I discuss the results pertaining
to the study’s research questions. With regards to the preliminary analyses, in the section below I
discuss the current study’s descriptive statistics, assumptions of multivariate normality, outliers,
multicollinearity, missing data, psychometric properties of the math self-efficacy measure, and
correlations.
Descriptive Statistics
Table 1 presents the descriptive statistics including means and standard deviations, as
well as kurtosis, skewness, and missing values of demographic, exogenous, and endogenous
variables (weighted and unweighted) for both student and school levels. The mean age of the
participants was 16. Based on the weighted variables, half of sample was female, and more than
half was White, followed by Hispanic, Black/African American, Asian (including Hawaiian and
Pacific Islander), and Other (a combination of multi-race, American Indian, or Alaska Native
students). In terms of language minority status, eighteen percent of the participants were non-
native English speakers (e.g., LM students) among whom 43% were Hispanic, 39% Asian, 10%
White, 4% Black, and 4% Other. In contrast, among native English speakers the majority was
White (66%) followed by Black (15%), Hispanic (9%), Other (6%), and Asian (4%).
Within the whole sample, a smaller percentage of students (2%) completed subsequently
at least grades levels 1 through 5 outside of the U.S., whom I classified as being L1 proficient.
About half of them reported being Hispanic, a quarter Asian, 14% White, 7% Black, and 1%
Other. Among students who did not subsequently complete at least grade levels 1 through 5
outside of the U.S., including native English speakers, 60% were White, 14% Hispanic, 12%
Black, 8% Asian, and 6% Other. Similarly, about 2% of the participants were LM students who
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71
also completed subsequently at least grades 1 through 5 outside of the U.S., and among these
students 57% were Hispanic, 29% Asian, 9% White, 5% Black, and .5% Other. For students who
did not meet this criterion (e.g., reference group), more than half were White, followed by
Hispanic, Black, Asian, and Other. Overall the percentage of Hispanic students was larger
among LM students and those who subsequently completed grades 1 through 5 outside of the
U.S., whereas the percentage of White students was greater among native English speakers and
those who did not subsequently complete grades 1 through 5 outside of the U.S. In contrast to
race, gender was equally distributed across LM status, subsequent completion of at least grades 1
through 5 outside of U.S., and their interaction.
The means for the achievement and self-efficacy data were similar at both student and
school levels although the maximum scores were higher at the student-level. However, there
were more differences in the changes from grade 10 to 12 for math achievement and self-
efficacy. For example, at the student-level math achievement increased, but math self-efficacy
remained the same from grade 10 to 12. In contrast, at the school-level both math achievement
and self-efficacy increased although the increase in self-efficacy was very slight. In addition, at
the school-level, about a quarter of the sophomore students were eligible for FRL, less than 5%
were English language learners, and about 30% of students were from racial/ethnic minority
backgrounds.
Comparison between unweighted versus weighted estimates. Unweighted and
weighted means and SDs are also displayed in Table 1. The mean and standard deviation
estimates tended to be smaller in the weighted model compared to the unweighted model. In
particular, for the exogenous variables at the student-level, most of the means in the unweighted
model were different from the weighted model, in particular for SES, LM status, L1 proficiency,
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72
and percent of Asian students. However, at the school-level, most of the means of exogenous
variables were similar except for percentage of racial/ethnic minority students. With regards to
standard deviations, they were similar across models except for percentage of Asians and L1
proficiency at the student-level, and grade 12 math achievement at the school-level. The means
for the dependent variables at the school-level were similar across weighted and unweighted
models. The differences in estimates may have resulted from the weights, which adjusted for
oversampling of Hispanic and Asian students, and non-response of students and schools.
To obtain accurate parameter estimates and standard errors in structural models, the data
must meet the multivariate normality assumption, which means that variables are normally
distributed, including the joint distributions of paired variables; variables are linearly related; and
residuals are homoscedastic (Kline, 2011). Each of these topics is discussed in detail next.
Running head: NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY
Table 1
Descriptive Statistics
Variable( N ( %( Unweighted(
Mean(
Weighted(
Mean(
Unweighted(
SD(
Weighted(
SD(
Range( Kurtosis( Skewness( No.(
missing(
%(
missing(
Student(Level(
( ( ( ( ( ( ( ( ( ( ( ( ((Age(
15990(
(
16.00(
(
0.58(
(
14.08( 19.08( 6.30( 1.29( 110( 0.70(
((Gender(
16100(
(
0.50( 0.49( 0.50(
(
0.00( 1.00( 1.00( 0.00( 0( 0.00(
((((Male(
8040( 49.92%(
( ( ( ( ( ( ( ( ( ( ((((Female(
8070( 50.08%(
( ( ( ( ( ( ( ( ( ( ((Race(
16100(
( ( ( ( ( ( ( ( ( ( ( ((((White(
8960( 55.64%( 0.56(
(
0.50(
(
0.00( 1.00( 1.05( N0.23(
( ( ((((Black(or(African(((
((((((American( 2160( 13.41%( 0.13( 0.10( 0.34( 0.30( 0.00( 1.00( 5.61( 2.15(
( ( ((((Asian((
1650( 10.23%( 0.10( 0.03( 0.30( 0.16( 0.00( 1.00( 7.89( 2.62(
( ( ((((Hispanic(
2430( 15.09%( 0.15( 0.10( 0.36( 0.30( 0.00( 1.00( 4.80( 1.95(
( ( ((((Other(
910( 5.63%( 0.06( 0.06( 0.23( 0.24( 0.00( 1.00( 15.82( 3.85(
( ( ((Language(Status(
16100(
(
0.18( 0.08( 0.38( 0.27( 0.00( 1.00( 3.85( 1.69( 0( 0.00(
((((Native(English(Speaker (
13250( 82.24%(
( ( ( ( ( ( ( ( ( ( ((((Language(Minority(
2860( 17.76%(
( ( ( ( ( ( ( ( ( ( ((First(Language(((((
((((Proficiency( 12440(
(
0.02( 0.01( 0.14( 0.09( 0.00( 1.00( 51.63( 7.12( 3670( 22.80(
((((((L1(proficient(
230( 1.44%(
( ( ( ( ( ( ( ( ( ( ((((((Non NL1(proficient(
12200( 75.78%(
( ( ( ( ( ( ( ( ( ( ((LMXL1prof(interaction(
12440(
(
0.02(
(
0.12(
(
0.00( 1.00( 63.13( 7.88( 3670( 22.80(
((((((LM(&(L1(proficient(
190( 1.50%(
( ( ( ( ( ( ( ( ( ( ((((((Other( 12250( 98.50%(
( ( ( ( ( ( ( ( ( ( ((SES(
16080(
(
0.04( N0.03( 0.74( 0.70( N2.11( 1.87( 2.35( N0.02( 30( 0.20(
((Math(Achievement((
( ( ( ( ( ( ( ( ( ( ( ( ((((Grade(10(
15800(
(
43.18(
(
14.00( 12.57( 13.74( 82.03( 2.23( 0.12( 300( 1.90(
((((Grade(12(
13580(
(
49.48(
(
15.19( 13.65( 15.20( 82.54( 2.10( N0.08( 2530( 15.70(
((English(Reading((
((((Achievement( 15800(
(
29.90(
(
9.70( 8.86( 10.20( 49.09( 1.97( N0.11( 300( 1.90(
(
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74
Table(1((Continuation)(
Variable( N ( %( Unweighted(
Mean(
Weighted(
Mean(
Unweighted(
SD(
Weighted(
SD(
Range( Kurtosis( Skewness( No.(
missing(
%(
missing(
((Math(SelfNEfficacy(
( ( ( ( ( ( ( ( ( ( ( ( (((( Grade(10(
11650(
(
12.03(
(
4.58( 4.45( 1.00( 20.00( 2.39( 0.03( 4450( 27.60(
(((( Grade(12(
10310(
(
12.96(
(
3.93( 3.88( 1.00( 20.00( 2.40( 0.03( 5790( 36.00(
School(level(
( ( ( ( ( ( ( ( ( ( ( ( ((%(FRL(( 680(
(
23.70( 23.84( 25.26( 24.79( 0.00( 100.00( 3.79( 1.21( 70( 8.70(
((%(Minority(
730(
(
34.03( 24.92( 31.46( 29.75( 0.00( 100.00( 2.31( 0.80( 20( 2.00(
((%(ELLs((
710(
(
3.92( 2.33( 8.03( 6.51( 0.00( 61.00( 15.29( 3.24( 40( 5.00(
((Math(Achievement(
( ( ( ( ( ( ( ( ( ( ( ( (((( Grade(10(
750(
(
42.99( 42.47( 7.40( 3.88( 19.42( 67.00( 3.07( 0.01( 0( 0.00(
(((( Grade(12(
750(
(
48.80( 47.35( 8.18( 4.42( 18.40( 73.50( 3.07( N0.06( 0( 0.00(
((English(Reading((
(((( Achievement( 750(
(
29.94( 29.91( 5.11( 3.04( 16.33( 43.08( 2.70( N0.11( 0( 0.00(
((Math(SelfNEfficacy(
( ( ( ( ( ( ( ( ( ( ( ( (((( Grade(10(
750(
(
11.96( 11.59( 1.69( 0.93( 5.13( 19.00( 4.63( N0.44( 0( 0.00(
(((( Grade(12(
700(
(
12.95( 12.63( 1.38( 1.09( 8.50( 19.50( 4.02( 0.31( 50( 7.10(
Note. Number of observations was rounded to the 10 because of restricted-use data nature. L1 proficient students were operationalized
as those who completed subsequently at least grades 1-5 outside of the U.S. Non-L1 proficiency students were operationalized as
those who did not have this experience. SES = socioeconomic status, FRL = eligible for free or reduced-price lunch; ELLs = English
language learners; minority = racial/ethnic minority students.

Running head: NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY
Multivariate Normality Assumptions
Normal distribution of endogenous variables. Distribution normality was visually
examined by plotting histograms, qqplots, probability plots, and box plots separately for each
endogenous variable at both levels as they are shown in Appendices A and B. I also estimated
their kurtosis and skewness values, which are given in Table 1. Through visual inspection of the
histograms and plots, slight discrepancies from normality were detected for reading achievement
and grades 10 and 12 self-efficacy variables at the student-level. At the school-level, grade 10
math self-efficacy also showed slight discrepancies from normality. However, based on these
variables’ kurtosis and skewness values, which were less than five and close to zero,
respectively, it was concluded that they were approximately normally distributed (Kline, 2011).
In addition, transformations of these variables as shown in Appendices C and D did not look
more normally distributed.
Linearity. The linear relationships among endogenous variables and SES were explored
by plotting scatterplots and residuals plots (residuals vs. predicted, and residuals vs. predictors)
which are provided in Appendices E through H. Based on these plots, at the student-level (see
Appendices E and F), most relationships appeared to be linear, but with slight discrepancies for
the relationships between grade 10 math self-efficacy and grade 12 math achievement, and
between grade 10 and 12 math self-efficacy. At the school-level (see Appendices G and H), the
relationships between grade 10 reading achievement and percent of students eligible for FRL,
and the relationship between grade 12 math achievement and grade 10 math self-efficacy were
slightly less linear. School-level residual plots also indicated slight discrepancies from linearity
for the relationships between residuals (from the model regressing grade 12 math achievement
on grade 10 math achievement, self-efficacy, and reading) and grade 10 math self-efficacy, and
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between residuals (from the model regressing grade 12 math self-efficacy on grade 10 math self-
efficacy and math achievement) and grade 10 math self-efficacy. However, these discrepancies
from linearity did not seem large enough to violate the linearity assumption.
Homoscedasticity of residuals. Based on residuals vs. predicted values scatter plots
displayed in Appendices F and H, at the student-level, residuals appeared slightly
heteroscedastic. In contrast, at the school-level, residuals seemed homoscedastic. The slight
departures from homoscedasticity of the student-level residuals did not seem to violate the
homoscedasticity assumption. Because the slight heteroscedasticity may have resulted from the
presence of outliers, they were examined next.
Outliers, and Leverage and Influential Points.
Outliers were defined as cases having residuals more than three standard deviations from
the mean (Kline, 2011). At the student-level, the percentage of residual outliers ranged from .03
to .6% and at the school-level they ranged from .4% to 1.5%. The small percentage of residual
outliers suggested that their removal was not needed.
Leverage and influential points. The presence of leverage and influential points was
also investigated because they could influence the analyses. High leverage points—observations
with high X values—were defined as having leverage greater than [(2k+2)/n], where k is the
number of predictors and n represents the number of observations (“Stata Web Books, n.d.”).
High influential points, which have both high X and Y values, were defined as observations
having Cook D values greater than 4/n (“Stata Web Books, n.d.”). At the student-level, the
percentage of high leverage points ranged from 9.5% to 31%, and the percentage of influential
points ranged from 5.7% to 7.5%. At the school-level, the percentage of high leverage points
ranged from 7.2% to 12.1%, and the percentage of influential points ranged from 4.8% to 5.9%.
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The percentage of high leverage and influential points at the student-level may have resulted
from the small changes in achievement and self-efficacy data from grades 10 to 12.
Multicollinearity
Inclusion of the interaction term (LMxL1prof) and its components in the multilevel
model could lead to multicollinearity. Therefore, to check for multicollinearity, I ran multiple
regressions in which each student-level endogenous continuous variable was predicted by all
other variables if these predictors were measured before the outcome variable. Based on the R-
Square values, which were lower than .90, and Variance Inflation Factor (VIF) values, which
were lower than 10, there was no indication of multicollinearity among variables at the student-
level (Kline, 2011). In contrast, at the school-level the high VIF values of grades 10 and 12 math
achievement, 15.8 and 12.3, respectively, indicated collinearity among these variables. Although
removing either variable lowered the VIF values of the rest of the variables, I decided to keep
both achievement variables because prior achievement is a good predictor of future achievement
(Hemmings et al., 2010; Park, 2008; Riegle-Crumb & Grodsky, 2010), and their inclusion would
enable the testing of the bi-directionality of the relationship between math achievement and math
self-efficacy in a longitudinal manner at the school-level, which has been investigated scarcely.
Missing Data
The percentage of missing data ranged from 0 to 36% at the student-level, and 0 to 8.7%
at the school-level. As Table 1 shows, the student-level variables that had more than 5% missing
data were the following with increasing missingness: grade 12 math achievement, first language
proficiency, interaction term (LM x L1 proficiency), and grade 10 and 12 math self-efficacy. The
percentage of missing values was lower for the achievement data compared to the self-efficacy
data because I used the imputed achievement data provided by the ELS 2002 study. However,
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78
the ELS 2002 did not impute self-efficacy data. The school-level variables that had more than
5% missing data were percentage of grade 10 students eligible for FRL, and grade 12 math self-
efficacy. Because the percentages of missing values were high I next examined missingness
pattern, data coverage, and prediction of missingness.
Missingness Patterns. Appendix I shows that a total of 96 missing data patterns were
present. It also displays the specific missingness patterns and their frequencies
5
. Specifically,
about 6,260 students were not missing any data. Four prominent missing data patterns had more
than 1,000 students missing data. The number of students who were missing grade 10 math self-
efficacy data was about 1,590, grade 12 math self-efficacy was 1,350 students, first language
proficiency and the interaction term (LM x L1 proficiency) was 1,040, and grade 12 math
achievement and self-efficacy data was 1,010. The proportion of complete data ranged from
50.3% to 100% as shown in Appendix J. The covariance coverage was lowest for the covariance
between grade 10 and 12 math self-efficacy. Most covariances that had lower coverages involved
grade 10 math self-efficacy, grade 12 math self-efficacy, L1 proficiency, or the interaction term.
Predicting Missingness. To test the assumption of completely missing at random, I used
logistic multiple regressions and predicted missingness from the variables in the multilevel
model. Appendix K shows the results from these models. In particular, at the student-level, some
variables (LM x L1proficiency, Black, gender, and Other) perfectly predicted missingness in
grade 10 reading and math achievement while adjusting for LM status, L1 proficiency, SES,
Hispanic, Asian, and grade 12 math self-efficacy and achievement. For example, being an LM
student (with at least grade levels one through five completed subsequently outside of the U.S.),
female, Black, or Other perfectly predicted not having missing values for grade 10 reading and

5
Because the NCES requires rounding to the 10 while reporting results using ELS 2002 restricted-use data,
frequencies lower than 10 were rounded up to 10.
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79
math achievement. In addition, L1 proficiency and grade 12 math achievement predicted
missingness in grade 10 reading and math achievement.
SES, gender, Other, grade 10 math achievement and reading achievement significantly
predicted missingness in grade 12 math achievement, while adjusting for other variables. L1
proficiency, Hispanic, Black, and reading achievement predicted missingness in grade 12 math
self-efficacy, while other variables were also included in the model. Similarly, SES, gender,
Hispanic, Black and reading achievement predicted missingness in grade 10 math self-efficacy
while adjusting for other variables. For example, a one-unit increase in SES was associated with
70% decrease in the odds of missing grade 12 math achievement scores. For a male, the odds of
missing grade 12 math achievement score are 1.14 times larger than the odds for a female.
The results indicated that overall, at the student-level, Hispanic and Black students had
higher odds of missing self-efficacy data compared to the odds of White students. Males had
higher odds of missing achievement data but not self-efficacy data as compared to the odds of
females. Furthermore, students with higher SES had lesser odds of missing math achievement
and self-efficacy data when compared to odds of students from lower SES. Students with higher
reading scores had lesser odds of missing achievement (Odds Ratio (OR) = .98) and self-efficacy
(OR = .99) data than the odds for students with lower reading scores.
At the school-level, grade 10 math achievement and self-efficacy, and grade 12 math
achievement significantly predicted missingness in grade 12 math self-efficacy while adjusting
for reading achievement, and percentages of students eligible for FRL, ELLs, and racial/ethnic
minority students. For example, a one-unit increase in school-level grade 12 math achievement
was associated with a 16% increase in the odds of missing grade 12 math self-efficacy score. To
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80
account for the missing data, its patterns, and missing at random nature, I used all data available
through full information maximum likelihood estimation (Kline, 2011).
Psychometric Properties of Mathematics Self-efficacy Measure
To determine the psychometric properties of the math self-efficacy measure, I estimated
its reliability (Cronbach’s alpha) and tested a one-factor model through Confirmatory Factor
Analysis (CFA) for each grade 10 and 12 data. The Cronbach’s alpha reliability coefficients for
the math self-efficacy measure for grades 10 and 12 were .93 and .91, respectively, indicating
that the measure was reliable.
The model fit indices for the one-factor grade 10 math self-efficacy CFA model were
Chi-Square = 1781, df = 10 (p < .001), CFI = .96, TLI = .92, RMSEA = .175 (Confidence
Interval (CI) = .168 - .181). The standardized loadings (standardized on X and Y values) for the
items were greater than .829 and residual variances ranged from .232-.312 as indicated in Table
2. For grade 12, the model fit indices were slightly higher than for grade 10: Chi-Square = 842,
df = 10 (p < .001), CFI = .98, TLI = .95, RMSEA = .127 (CI = .120 - .135). Standardized
loadings for grade 12 items were greater than .795 and their residual variances ranged from .297-
.368. The adequate model fit indices of the one-factor model for both grade levels, except for the
high RMSEA values, and the high standardized loadings suggested that the math self-efficacy
items loaded into one-factor at each grade level.

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81
Table 2
Math Self-efficacy Items Factor Loadings and Residual Variances for Grades 10 and 12
Item Grade 10 Grade 12
Standardized
Loading
(n=11,650)
Residual
Variance
Standardized
Loading
(n=10,310)
Residual
Variance
I’m confident that I can do an
excellent job on my math tests
.829 .312 .826 .317
I’m certain I can understand the
most difficult material presented in
math texts
.836 .302 .839 .297
I’m confident I can understand the
most complex material presented by
my math teacher
.876 .232 .833 .307
I’m confident I can do an excellent
job on my math assignments
.875 .234 .795 .368
I’m certain I can master the skills
being taught in my math class
.872 .240 .820 .327

Pearson Correlations and Intraclass Correlations
Pearson correlations for both unweighted and weighted variables at student and school
levels are presented in Appendices L and M, and Tables 3 and 4. The correlational coefficients
for the unweighted variables and their corresponding number of observations are presented in
Appendices L and M. Specifically, unweighted correlations among achievement variables were
the highest (above r = .7) and significant (p < .001) for both student and school levels, but they
were higher at the school-level. Although all correlations among endogenous variables and
school-level predictors were significant, some correlations among student-level predictors and
endogenous variables were insignificant. Generally, correlations among predictors and outcomes
were low (lower than r = .4).
With regards to weighted correlations, as shown in Table 3, at the student-level, most
correlations between variables were lower than r = .6, except for the correlations among
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82
achievement variables, which were positive and ranged from r = .693 to .894. Similarly, at the
school-level, which correlational coefficients are displayed in Table 4, the highest correlations
were among achievement variables and these correlations were higher than at the student-level
(range from r = .781 to r = .959). However, most school-level correlations were lower than r =
.7. The significance levels for the weighted correlations were not included because they were not
available from the Mplus output.
Table 3
Weighted Student-Level Correlations

Note. SES = socioeconomic status; RA10 = grade 10 English reading achievement; MA10 =
grade 10 math achievement; MA12 = grade 12 math achievement; MSE12 = grade 12 math self-
efficacy; MSE10 = grade 10 math self-efficacy; LM = language minority status; L1PROF = L1
proficiency; HISP = Hispanic; ASI = Asian; BLK = Black; OTH = Other; WHT = White; G =
gender.

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83
Table 4
Weighted School-Level Correlations
%MIN %ELL %FRL BRA10 BMA10 BMA12 BMSE12 BMSE10
%MIN 1
%ELL 0.405 1
%FRL 0.503 0.249 1
BRA10 -0.284 -0.17 -0.359 1
BMA10 -0.331 -0.105 -0.406 0.792 1
BMA12 -0.295 -0.09 -0.427 0.781 0.959 1
BMSE12 -0.144 -0.079 -0.219 0.515 0.555 0.591 1
BMSE10 0.16 -0.041 -0.104 0.226 0.138 0.108 0.332 1

Note. The letter “B” in front of a variable name represents a school-level variable. % MIN =
percentage of racial/ethnic minority students; % ELL = percentage of ELL sophomores; % FRL
= percentage of sophomores eligible for free or reduced-price lunch; RA10 = grade 10 English
reading achievement; MA10 = grade 10 math achievement; MA12 = grade 12 math
achievement; MSE12 = grade 12 math self-efficacy; MSE10 = grade 10 math self-efficacy.

Based on about 16, 110 students and 740 schools with an average of about 20 students
per school, the intraclass correlation for grade 10 reading achievement and math achievement,
and grade 12 math achievement were higher than .1. They were .23, .228, and .239, respectively.
In contrast, the intraclass correlations for grade 10 and 12 math self-efficacy were .041 and .045,
respectively. The high intraclass correlation for the achievement data suggested that it was
necessary to take into account the clustering nature of the data. Therefore, a multilevel model
was an appropriate technique for the current study.
Multilevel Structural Equation Modeling Analysis
Findings pertaining to research questions. In this section I present results that are
particularly related to the current study’s research questions.
1. Do schools’ poverty level, percentages of racial/ethnic minority students and ELLs relate
to high school students’ math achievement and math self-efficacy?
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84
I had hypothesized that the percentages of grade 10 students eligible for FRL,
racial/ethnic minority students, and grade 10 ELL students would predict math achievement. It
was also expected that percentages of racial/ethnic minority students and ELLs would predict
math self-efficacy.
Figure 3 shows the path diagram of the final weighted multilevel model (mod1, see
Appendix N for Mplus script) for the school-level only with standardized parameter estimates,
which were standardized on X and Y values, and standard errors. The model fit of the weighted
multilevel model was adequate (Chi-Square = 929.6, df = 40, p < .001, RMSEA = .036, CFI =
.934, and TLI = .869, n = 16,080) although not all indices met the a priori established cutoff
scores. In particular the CFI and TLI values were lower than .95. The school-level model
explained the variance of grade 12 math achievement the most (R-Square = .923) and grade 10
math self-efficacy variance the least (R-Square = .012, p > .05).
Based on the school-level results as indicated in Figure 3, the hypotheses were not
supported, with one exception: the significant and negative relationship between percent of
sophomore students eligible for FRL and grade 10 math achievement while adjusting for the
percentages of racial/ethnic minority and ELL students. After adjusting for prior math
achievement, percent of sophomore students eligible for FRL did not predict grade 12 math
achievement. Similarly, the percentages of racial/ethnic minority and ELL students were not
significant predictors of either school math achievement or self-efficacy.

Running head: NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

Figure 3. Final weighted multilevel model displaying school-level relationships among school-level student demographics,
achievement, and self-efficacy and estimated parameters. % MIN = percentage of racial/ethnic minority students; % ELL = percentage
of English language learners in grade 10; % FRL = percentage of sophomores eligible for free or reduced-price lunch; RA10 = grade
10 English reading achievement; MA10 = grade 10 math achievement; MA12 = grade 12 math achievement; MSE12 = grade 12 math
self-efficacy; MSE10 = grade 10 math self-efficacy. The letter “B” in front of a variable name represents a school-level variable.
Black arrows indicate significant paths at p < .05. Gray paths show insignificant paths. Estimates are standardized on x and y values,
except for exogenous variances, which are unstandardized. Standard errors are in parentheses.
614.97
885.28
42.34
-.343(.069)
-.351(.077)
-.096(.066)
.894(.055)
-.042(.05)
.058(.056)
.036(.031)
.025(.018)
-.083(.042)
.105(.09)
-.098(.077)
.252(.177)
.454(.130)
.504(.072)
.248(.093)
.406(.072)
.736(.063)
.078(.114)
.242(.113)
.882(.048)
.833(.057)
.077(.022)
.718(.081)
.988(.016)
BMA10 BMA12
BMSE10 BMSE12
BRA10
% FRL
% MIN
% ELL
Running head: NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY
Additional findings from the school-level model were: (1) the percentage of sophomores
who were eligible for FRL was a negative and significant predictor of English reading
achievement; and (2) grade 10 math achievement was a significant positive predictor of grade 12
math achievement and self-efficacy. The relationship between grade 10 and 12 math
achievement was stronger (double) compared to the relationship between grade 10 and grade 12
math self-efficacy.
2. How are language minority status, first language (non-English) proficiency, English
reading achievement, math achievement, and math self-efficacy related in high school
students while adjusting for race/ethnicity, gender, and SES? More specifically, does first
language (non-English) proficiency relate to English reading achievement?
Based on the reviewed literature, I expected that L1 proficiency, operationalized as
subsequently completing grade levels 1 through 5 outside of the U.S., would be positively and
significantly related to English reading achievement. At the student-level, the weighted
multilevel model (mod1) explained the variance in grade 12 math achievement the most (R-
Square = .806), and grade 10 reading achievement the least (R-Square = .164). Based on the
student-level weighted model results displayed in Figure 4, L1 proficiency was not significantly
related to English reading achievement while adjusting for race/ethnicity, LM status, SES, and
gender.
Results from the mod1 (weighted) model are preferred and emphasized because the
inclusion of sampling weights and all available data adjusted for unequal selection bias and
provided less biased estimates (Asparouhov, 2005; Stapleton, 2002). However, for comparison
purposes I also present results from the unweighted all cases (mod2) and some cases (mod3)
multilevel models in Table 5 (see Appendices O and P for Mplus Scripts for mod2 and mod3,
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87
respectively). One notable difference in findings between mod1 and mod2 was that the
insignificant relationship between L1 proficiency and English reading in mod1 was significant in
mod2, which model fit was adequate as well (Chi-Square = 2087.5, df = 40, p < .001, RMSEA =
.054, CFI = .94, and TLI = .882, n = 16,080).
Running head: NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

Figure 4. Final weighted multilevel model displaying student-level relations among demographic characteristics, achievement, and
math self-efficacy. SES = socioeconomic status; RA10 = grade 10 English reading achievement; MA10 = grade 10 math achievement;
MA12 = grade 12 math achievement; MSE12 = grade 12 math self-efficacy; MSE10 = grade 10 math self-efficacy; LM = language
.074
.008
.091
.025
.088
.056
.495
.25
-.014(.008)
-.049(.009)
-.116(.015)
-.014(.012)
-.167(.014)
-.043(.016)
.302(.016)
-.139(.014)
.014(.012)
-.184(.013)
-.063(.014)
.295(.014)
.753(.011)
.061(.009)
.155(.012)
.045(.008)
.033(.009)
.346(.020)
.246(.016)
.036(.015)
.167(.013)
.614(.01)
.255(.013)
.836(.012)
.834(.011)
.194(.008)
.779(.015)
RA10
MA10 MA12
MSE10 MSE12
LM
L1PROF
LMXL1PROF
SES
G
HISP
ASI
BLK
OTH
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89
minority status; L1PROF = L1 proficiency; LMXL1PROF = interaction between LM and L1PROF; HISP = Hispanic; ASI = Asian;
BLK = Black; OTH = Other; G = gender. Standardized estimates on x and y values are shown except for exogenous variances, which
are unstandardized. Standard errors are in parentheses. Black paths are significant at p < .05. Gray paths represent insignificant
relationships and red arrows indicate the respecified mediation.

Running head: NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY
Table 5
Parameter Estimates for Weighted, Unweighted, and Some Cases Multilevel Models
! Weighted!!
(mod1)!
N=16!080
a!
Unweighted!
(mod2)!
N=16!080
a!
Some!cases!
(mod3)!
N=12!430
a
!!
! Estimate! S.E.! Estimate! S.E.! Estimate! S.E.!
Student!Level! ! ! ! ! ! !
!!RA10!ON! ! ! ! ! ! !
!!!L1PROF! (.014!ns! .008! (.034! .007! G .002!ns! .014!
!!!LM! G .049! .009! G .068! .008! G .053! .009!
!!!LMXl1prof! ! ! ! ! G .038*! .015!
!!!Hispanic! G .116! .015! G .118! .009! G .116! .011!
!!!Asian! (.014!ns! .012! (.024! .01! G .019!ns! .011!
!!!Black! G .167! .014! G .179! .009! G .164! .009!
!!!Other! G .043**! .016! G .041! .008! G .035! .009!
!!!SES! .302! .016! .315! .009! .334! .01!
!MA10!ON! ! ! ! ! ! !
!!Hispanic! G .139! .014! G .137! .01! G .136! .01!
!!Asian! .014!ns! .012! .046! .011! .044! .011!
!!Black! G .184! .013! G .197! .009! G .19! .01!
!!Other! G .063! .014! G .044! .008! G .038! .009!
!!SES! .295! .014! .306! .009! .323! .01!
!MA12!ON! ! ! ! ! ! !
!!MA10! .753! .011! .767! .007! .766! .008!
!!MSE10! .061! .009! .049! .005! .052! .006!
!!RA10! .155! .012! .135! .007! .136! .008!
!!SES! .045! .008! .048! .005! .05! .006!
!!Gender! .033! .009! .025! .004! .028! .005!
!MSE12!ON! ! ! ! ! ! !
!!MSE10! .346! .02! .331! .011! .341! .012!
!!MA10! .246! .016! .234! .011! .229! .012!
!!Gender! .036*! .015! .049! .009! .051! .01!
!MA10!WITH! ! ! ! ! ! !
!!RA10! .614! .01! .623! .006! .61! .007!
!!MSE10! .255! .013! .242! .008! .254! .008!
!MA12!WITH! ! ! ! ! ! !
!!MSE12! .167! .013! .173! .01! .172! .01!
!Mean! ! ! ! ! ! !
!!L1PROF! .091! .006! .139! .006! ! !
!Variance! ! ! ! ! ! !
!!MSE10! 1! 0! 1! 0! 1! 0!
!!L1PROF! 1! 0! 1! 0! ! !
!! ! ! ! ! ! !
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

91
!
Table!5!(Continuation)!
! Weighted!!
(mod1)!
Unweighted!
(mod2)!
Some!cases!
(mod3)!
! Estimate! S.E.! Estimate! S.E.! Estimate! S.E.!
Residual!variance! ! ! ! ! ! !
!!RA10! .836! .012! .813! .007! .799! .008!
!!MA10! .834! .011! .817! .007! .804! .008!
!!MA12! .194! .008! .197! .006! .194! .006!
!!MSE12! .779! .015! .799! .009! .793! .009!
School!Level! ! ! ! ! ! !
!!BRA10!ON! ! ! ! ! ! !
!!!!%!FRL! G.343! .069! G.406! .042! G.404! .045!
!!BMA10!ON! ! ! ! ! ! !
!!!!%!FRL! G.351! .077! G.412! .045! G.378! .05!
!!!!%!Minority! (.096!ns! .066! (.106**! .039! G.119**! .043!
!!BMA12!On! ! ! ! ! ! !
!!!!BMA10! .894! .055! .918! .034! .881! .04!
!!!!BMSE10! G.042!ns! .05! .012!ns! .026! .025!ns! .03!
!!!!BRA10! .058!ns! .056! .023!ns! .036! .049!ns! .045!
!!!!%!Minority! .036!ns! .031! .078! .021! .058!*! .025!
!!!!%!ELL! .025!ns! .018! .033!ns! .018! .047!*! .022!
!!!!%!FRL! (.083!ns! .042! (.111! .024! G.121! .026!
!!BMSE10!ON! ! ! ! ! ! !
!!!!%!Minority! .105!ns! .09! G.065!ns! .078! G.121!ns! .084!
!!!!%!ELL! G.098!ns! .077! G.119!ns! .077! G.106!ns! .085!
!!BMSE12!ON! ! ! ! ! ! !
!!!!BMSE10! .252!ns! .177! .462! .106! .423**! .125!
!!!!BMA10! .454! .13! .218**! .081! .295**! .089!
!!%!FRL!WITH! ! ! ! ! ! !
!!!!%!Minority! .504! .072! .572! .032! .579! .032!
!!!!%!ELL! .248**! .093! .303! .045! .304! .045!
!!%!Minority!WITH! ! ! ! ! ! !
!!!!%!ELL! .406! .072! .433! .041! .434! .041!
!!BMA10!WITH! ! ! ! ! ! !
!!!!BRA10! .736! .063! .801! .028! .792! .03!
!!!!BMSE10! .078!ns! .114! .094!ns! .055! .112!ns! .066!
!!BMA12!WITH! ! ! ! ! ! !
!!!!BMSE12! .242*! .113! .031!ns! .103! .044!ns! .11!
!!Means! ! ! ! ! ! !
!!!%!Minority! .838! .03! 1.086! .027! 1.086! .027!
!!!!%!ELL! .357! .021! .504! .02! .504! .02!
!!!!%!FRL! .963! .047! .95! .026! .949! .026!
!!Intercepts! ! ! ! ! ! !
!!!!BRA10! 10.47! .75! 11.083! .47! 11.607! .544!
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Table!5!(Continuation)!
! Weighted!!
(mod1)!
Unweighted!
(mod2)!
Some!cases!
(mod3)!
! Estimate! S.E.! Estimate! S.E.! Estimate! S.E.!
!!!!BMA10! 11.894! .871! 11.676! .513! 11.703! .542!
!!!!BMA12! .523!ns! .705! .317!ns! .405! .610!ns! .485!
!!!!BMSE12! 3.845!*! 1.905! 8.804! 1.913! 8.73! 2.143!
!!!!BMSE10! 10.748! 1.554! 13.195! .964! 13.313! 1.192!
!!Variances!! ! ! ! ! ! !
!!!!%!Minority!! 1! 0! 1! 0! 1! 0!
!!!!%!ELL! 1! 0! 1! 0! 1! 0!
!!!!%!FRL! 1! 0! 1! 0! 1! 0!
!!Residual!!!
!!!Variances!
! ! ! ! ! !
!!!!BRA10! .882! .048! .835! .034! .836! .036!
!!!!BMA10! .833! .057! .769! .041! .791! .042!
!!!!BMA12! .077! .022! .07! .013! .082! .015!
!!!!BMSE12! .718! .081! .714! .091! .695! .098!
!!!!BMSE10! .988! .016! .975! .02! .963! .027!
!!Model!Fit!Indices! ! ! ! ! ! !
!!!No.!free!!!
!!!!parameters!
65! ! 65! ! 64! !
!!!ChiGSquare!!
!!!(df
a
)!
!!!p!value!
929.6!
(40)
b!
! 2087.5!
(40)
b!
! 1114.1!
(40)
b!
!
!!!RMSEA! .036! ! .054! ! .046! !
!!!CFI! .934! ! .94! ! .963! !
!!!TLI! .869! ! .882! ! .923! !

Note. Standardized estimates on x and y values are shown. Bold!estimates!show!significance!
level!difference!(significant!vs.!nonGsignificant)!between!mod1!and!mod2.!Underlined!
parameters!show!significance!level!difference!between!mod2!and!mod3.!SES =
socioeconomic status; RA10 = grade 10 English reading achievement; MA10 = grade 10 math
achievement; MA12 = grade 12 math achievement; MSE12 = grade 12 math self-efficacy;
MSE10 = grade 10 math self-efficacy; LM = language minority status; L1PROF = L1
proficiency; LMXL1PROF = interaction between LM and L1PROF; ELL = English language
learner; FRL = eligible for free or reduced-price lunch; RMSEA!= Root Mean Square Error of
Approximation;!CFI!=!Comparative!Fit!Index;!TLI!=!Tucker!Lewis!Index;!S.E.!=!standard!
error;!ON!=!regression;!WITH!=!covariance;!df!=!degrees!of!freedom.!!
a
Numbers of observations and df were rounded to the 10 because of NCES’ requirement.
b
The df for mod1 and mod2 were the same and higher than for mod3.
*!p!<.05!,!**!p!!

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2.b. Does first language (non-English) proficiency moderate the relationship between
language minority status and English reading achievement?
L1 proficiency was expected to moderate the relationship between LM status and English
reading achievement. More specifically, it was hypothesized that LM students who had greater
L1 proficiency (subsequently completed at least grade levels 1 through 5 outside of U.S.) would
have higher English reading achievement scores as compared to students without these
experience.
Weighted model mod1, which included all data available, did not support the presence of
this interaction, suggesting that L1 proficiency did not interact with LM status. Table 5 provides
standardized parameter estimates, standard errors, and fit indices for models using either all data
or only observations with information on exogenous variables (mod3). Model fit for mod3 was
good (Chi-Square = 1114.1, df = 40, p < .001, RMSEA = .046, CFI = .963, TLI = .923, n =
12,430). In contrast to mod1, mod3 indicated that the interaction term was a significant predictor
of English reading. Additionally, multi-group models with L1 proficiency as the grouping
variable and only observations with exogenous data supported the presence of the interaction.
Overall, models including all available information such as mod1 and mod2 did not support the
interaction, whereas models including only observations with exogenous variables information,
such as mod3, supported the interaction. Mean centering of LM status, L1 proficiency, and the
interaction term showed similar patterns. Because analyses with only complete observations
would not accurately reflect the whole sample and the percentage of missing data was over 5%
(Graham & Coffman, 2012), I selected the results from mod1, which is the weighted model using
all available information.
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2.c. If such a moderating relationship exists, does the interaction between language status
and first language (non-English) proficiency predict math self-efficacy through English
reading and math achievement?
I hypothesized that the interaction term would be related to grade 12 math self-efficacy
through English reading and math achievement. That is, I expected LM students with L1
proficiency to show higher math self-efficacy compared to other students because of their higher
achievement in English reading and math achievement. Based on model mod1 the hypothesis
was not supported because the interaction term was not significantly related to English reading
achievement.
I further examined whether English reading and math achievement still mediated the
relationship between LM status and grade 12 math self-efficacy. The weighted model with free
estimation of the mediation paths (LM status -> grade 10 English reading -> grade 12 math
achievement, and grade 10 math achievement -> grade 12 math self-efficacy) (mod1) had
adequate fit as described before. The weighted model (mod4) with these mediation paths
constrained to zero had a slightly poorer fit (Chi-Square = 1454.2, df = 50, p < .001, RMSEA =
.044, CFI = .895, TLI = .806, n = 16,080; see Appendix Q for Mplus script). Models mod1 and
mod4 were significantly different (adjusted Chi-Square difference = 472.6, df difference =10, p <
.0001) suggesting that the model with freely estimated mediation was preferred. This result
supported the mediating roles of English reading and math achievement but only in the
relationship between LM status and math self-efficacy. The pattern was the same in models
without sampling weights.
Additional findings. In contrast to the school-level model, most of the hypothesized
relationships were significant at the student-level as shown in Figure 4. The exceptions were that
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95
English reading was not predicted by L1 proficiency and the interaction term, and being Asian
compared to being White was unrelated to grade 10 English reading and math achievement. With
regards to student demographic characteristics, among predictors of English reading and grade
10 math achievement, SES was the strongest and only positive predictor. Compared to White
students, Hispanic, Black and Other students had significantly lower grade 10 reading and math
achievement. Although gender was significantly related to grade 12 math achievement and self-
efficacy, the coefficients were small (b = .033 and b = .036, respectively). Specifically, males
had slightly higher math achievement and felt more self-efficacious in math as compared to
females.
In terms of the relationships between achievement and self-efficacy, grade 10 math and
reading achievement were the greatest positive predictors (b = .753, and b =.155, respectively) of
grade 12 math achievement. Although grade 10 math self-efficacy was also a significant positive
predictor, its coefficient was small (b = .061). For grade 12 math self-efficacy, both grade 10
math achievement and self-efficacy were good predictors (b = .246 and b = .346, respectively).

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Chapter V: Discussion
I begin this section by discussing some significant results from the current study
separately for student and school levels and their implications. Next, I discuss key non-
significant results that are relevant to the research questions, and I provide possible theoretical
and methodological explanations for these unsupported relationships. Lastly, I compare and
contrast relationships among variables across student and school levels.
Supported Relationships
School SES and mathematics achievement. The school-level model shows that school
poverty negatively predicted grade 10 math achievement. This result supports prior findings
(e.g., Mosqueda, 2010; Mosqueda & Maldonado, 2013). School poverty is related to math
achievement probably because student SES is a strong predictor of achievement (e.g., Burnett &
Farkas, 2009; Murayama et al., 2013; Weiss et al., 2010). The relationship between school
poverty and math achievement implies that programs providing resources to schools in low-
income neighborhoods, such as Title I (U.S. Department of Education, 2004), need to be
continuously supported to improve schools’ math achievement. Increasing school-level math
performance is important particularly in light of the school accountability movement (see Figlio
& Loed, 2011).
Reading and mathematics achievement. At the student-level, the model indicates that
grade 10 reading predicted grade 12 math achievement, specifically after adjusting for prior math
achievement, self-efficacy, and other covariates. This result coincides with prior findings (e.g.,
Grimm, 2008; Kyttälä & Björn, 2014; Vukovic & Lesaux, 2013). One explanation for this
finding is that ability in reading and math hails from one common source (Keith, Reynolds,
Patel, & Ridley, 2008). Keith and colleagues (2008) reported that latent factors such as
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97
quantitative reasoning and comprehension knowledge loaded into the higher order g factor. This
finding provides additional support to the relevance of linguistic/literacy skills in learning math.
The relationship between reading and math achievement could have been observed in the present
study because the ELS 2002 math tests likely contained word problems. Research indicated that
linguistic skills were related to math learning particularly within word problems (e.g., Boonen et
al., 2013; Fuchs, et al., 2015). This finding implies that to improve math achievement high
school students need instruction in reading, and particularly in skills of discipline literacy—
specialized content literacy—that are related to learning math content specifically rather than
general reading skills (Shanahan & Shanahan, 2008). In their project, Shanahan and Shanahan
(2008) found that experts in different fields, such as history, math, and chemistry used different
reading strategies while reading texts related to their expertise areas, and consequently the
authors argued that literacy skills relevant to each content area should be taught.
Reciprocal relationship between mathematics achievement and self-efficacy. The
model further indicated that the reciprocal relationship between math achievement and math self-
efficacy was supported as reported by others (Williams & Williams, 2010, in particular with
adolescents’ longitudinal data. However, the influence of math achievement on math self-
efficacy while adjusting for prior math self-efficacy was much stronger than the reverse
relationship while adjusting for prior math achievement. Grade 10 math achievement was likely
related to grade 12 math self-efficacy because math achievement was one form of mastery
experience, which is one key self-efficacy source individuals use to gauge their self-efficacy
(Bandura, 1997). An implication of this finding is that prior math achievement experiences of
linguistically and ethnically diverse high school students significantly influence their subsequent
math self-efficacy. Therefore, educators should focus on scaffolding student’s learning to ensure
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their success in math, and subsequently improve their self-efficacy. For example, educators
could provide timely, frequent, explanatory, and explicit feedback on students’ work based on
how it compares to established goals (Ambrose, Bridges, DiPietro, Lovett, & Norman, 2010;
Krause, Stark, & Mandl, 2009). They could also assess, activate, and build on students’ prior
knowledge by, for example, creating a concept map based on what they know about the topic and
asking them to reflect on how current ideas relate to topics learned previously (Ambrose et al.,
2010; Gutiérrez, 2002; Mayer, 2011).
Unexpectedly, grade 10 self-efficacy was a significant but weak predictor of grade 12
math achievement. This finding contradicts most prior studies’ results (e.g., Alhija & Amasha,
2012; Lee & Stankov, 2010; Liang, 2010). The weak relationship may have resulted from the
current study using longitudinal data and including prior math achievement and self-efficacy into
the same model, which very few studies have done while investigating this relationship (e.g.,
Lewis et al., 2012; Stevens et al., 2004, 2006). For instance, contrary to the current study finding,
Lewis and colleagues (2012) reported a stronger relationship between prior math self-efficacy
and subsequent math achievement as compared to the relationship between prior math
achievement and subsequent math self-efficacy. Because these authors had not included these
relationships within the same model, it is difficult to compare the strengths of these relationships.
Although the study by Lewis and colleagues and the current study were longitudinal and
included prior measures of math achievement and self-efficacy, the inconsistent findings could
have resulted from sampling differences, as Lewis and colleagues only sampled Latino fifth and
six grade students from one school district in California.
The weaker relationship between prior math self-efficacy and subsequent math
achievement observed in the current study suggests that prior knowledge, which is embedded in
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99
the prior achievement construct, is a stronger predictor of future achievement when compared to
math self-efficacy. One possible explanation for this finding is that the stronger predictive power
of math self-efficacy may not have been observed in the current study because of the limited
connection (misalignment) between the math self-efficacy and achievement measures. Bandura
(2006b) argued that the relationship between self-efficacy and the outcome would be stronger if
the self-efficacy measure included items that actually impacted the outcome. More specifically
for the current study, the items from the math self-efficacy measure did not assess how confident
students were about solving the specific tasks or problems given in the math tests. Instead, the
math self-efficacy items measured how confident student were about performing different math
activities given or taught by their math teachers within their math classes. Because the math
assessment was not developed by students’ math teachers, and the tests may not have included
tasks and problems similar to those presented in students’ actual math classes, the relationship
between math self-efficacy and achievement may have been weak. However, the misalignment
between the math self-efficacy and achievement measures seems to play a lesser role on the
influence of prior achievement on subsequent math self-efficacy than vice-versa. The finding
that math self-efficacy is weakly related to math achievement while adjusting for prior
achievement implies that future studies exploring the influence of math self-efficacy on math
achievement should include prior math achievement in longitudinal models to parse out their
influences on math achievement.
Mediating roles of English reading and math achievement. These mediating roles in
the relationship between the interaction—LM status and L1 proficiency—and math self-efficacy
were not supported because of the insignificant relationship between the interaction and English
reading. However, testing these mediating roles in the relationship between LM status and math
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100
self-efficacy showed that these mediations were supported. The mediation may have been
supported because prior studies have supported these relationships (e.g., reading and math
achievement, and math achievement and math self-efficacy) separately (Alhija & Amasha, 2012;
Goforth et al., 2014; Kim & Herman, 2009; Korpershoek et al., 2014; Lesaux & Kieffer, 2010;
Liu, 2009; Panaoura et al., 2010). This finding suggests that high school LM students’ math self-
efficacy could be improved by facilitating their successes not only in math but also in reading.
For instance, educators could focus and scaffold LM students’ reading, and researchers could
develop interventions targeting reading achievement of such students to improve their math self-
efficacy. To improve LM students’ reading skills, their knowledge and use of academic
vocabulary could be increased by exposing them to these words within a context; making
connections between these words and students’ prior knowledge; providing them with other
meanings and uses of these words; and scaffolding their use of these words in new contexts
including in their own writing (Lesaux, Kieffer, Kelley, & Harris, 2014). In addition, LM high
school students’ math self-efficacy could be improved through the following strategies in the
math domain: use of cooperative learning, native language, writing, visuals and manipulatives,
development of linguistic skills, and individualized instruction (Cirillo, Richardson Bruna, &
Herbel-Eisenmann, 2010).
Unsupported Relationships
School proportions of racial/ethnic minority and ELL students, math achievement,
and self-efficacy. The percentage of school racial/ethnic minority student was not related to
math achievement. This finding contradicted results from prior studies (e.g., Benner & Crosnoe,
2011; Benson & Borman, 2010; Brown-Jeffy, 2009; Han & Bridglall, 2009). Because the ELS
2002 did not provide the specific percentage of racial/ethnic minority sophomore students, but
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instead the proportion from the entire student body, the relationship may not have been observed.
The proportion of racial/ethnic minority students of the entire school could have been a measure
too broad and distant to specifically relate to sophomores’ math achievement in grades 10 and
12. That is, the school proportion may not have been similar to the actual proportion of
sophomore students who were from racial/ethnic minority backgrounds. Another explanation
could be that the percentage of racial/ethnic minority students was not high enough in this
sample to relate to students’ math achievement. The mean percentage was 34.03, and the
standard deviation was 31.46. The mean percentages of racial/ethnic minority students for
studies that found support for this relationship ranged from 36% to above 50% (Brown-Jeffy,
2009; Kitsantas et al., 2010), thus much higher than in the study presented here.
Similar to the non-significant relationship between the school proportion of racial/ethnic
minority students and math achievement, the percentage of ELL sophomores was not related to
school-level math achievement. This finding contradicts some studies (e.g., Han & Bridglall,
2009), but also supports others (e.g., Drake, 2014; Mosqueda, 2010; Werblow & Duesbery,
2009). The current study result could have contradicted the finding by Han and Bridglall (2009)
because in their study they investigated this relationship separately for ELLs and native English
speakers. Specifically, they reported that although the proportion of ELLs influenced ELL
students’ mathematics performance, it did not relate to native English speakers’ achievement. In
the current study, this relationship was not examined separately across these subgroups. Another
possible explanation for the discrepancy could be the developmental level of the samples.
Whereas Han and Bridglall sampled elementary school students, the current study focused on
high school students. Moreover, the small percentage of ELL sophomores in the current study
(Mean = 3.92, SD = 2.33) may not have been sufficient to observe this relationship.
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Contrary to the current study’s hypotheses, the proportions of ELL and racial/ethnic
minority students were not related to math self-efficacy. This finding contradicts other findings
that investigated the link between these school factors and self-beliefs, including math self-
efficacy (Kitsantas et al., 2010; Niehaus & Adelson, 2014). The discrepancy could have resulted
from the difference in task specificity among self-beliefs (e.g., Bong & Skaalvik, 2003) and
exclusion of covariates. Broader self-constructs, such as self-concept, appear related to these
school factors, but task-specific constructs, such as self-efficacy, may not be related to them
especially while adjusting for covariates. Based on SCT, it was assumed that larger percentages
of racial/ethnic minority and ELL sophomore students implied a greater number of students of
color and ELL students, who could act as role models (vicarious learning) and in turn influence
the math self-efficacy of students of color and ELLs. The low percentage of sophomore ELLs in
the current sample (Mean = 3.92, SD = 2.33) could have limited the power to detect a significant
relationship between proportion of ELLs and school-level math self-efficacy. Similarly, the low
percentage of racial/ethnic minority students could have influenced the observation of its
relationship to math self-efficacy.
Relationships among LM status, L1 proficiency, and English reading. The finding
that L1 (non-English) proficiency did not moderate the relationship between LM status and
English reading contradicted findings from the cross-linguistic transfer literature (e.g., Greene,
1997; Slavin & Cheung, 2005; Umanski & Reardon, 2014). The lack of support for this
moderation suggests that the influence of L1 proficiency on English reading in LM students,
which has been observed frequently in elementary school students (e.g., Rolstad et al., 2005;
Willig, 1985), is not evident for high school students. One possible explanation of the
significance of this path within elementary school but not in high school is the greater prestige
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the use of English rather than one’s native language receives in higher grade levels. This prestige
could be evident to students by the very limited or lack of support for native language literacy
and instruction in high schools including within ESL courses; teachers’ English only
expectations for assignments; and offering of foreign language courses that target mainly non-
native speakers or native speakers but which is taught mostly in English. In addition, during the
stage when peers are very important and influential in students’ life, high school students may
experience greater social pressure to become proficient in English sooner. Interactions and
communication with native English speakers and language minority students proficient in
English may reflect the school and teacher beliefs that English is a more prestigious language.
Consequently, to fit in, students would put more effort in becoming English proficient at least for
social purposes. These factors may speed language loss in high school students decreasing the
relationship between L1 proficiency and English reading during this developmental period.
However, with a longitudinal sample of middle school English language learners with
limited English proficiency, Guglielmi (2012) reported that students’ self-reported grade 8 L1
proficiency predicted their English reading intercept and slope from grades 8, 10, and 12. In light
of Guglielmi’s finding, the lack of this relationship in the current study, which samples high
school students, may suggest that the relationship between L1 proficiency and English reading
may not be generalizable to all high school LM students, in particular to students with different
English language proficiencies. Another possible explanation for this finding is that cross-
linguistic transfer effects are short-lived within older students. Although some studies (e.g.,
Reese et al., 2000; Sparks, 2009) indicated that the effects were long lasting, the current study’s
result seems to indicate that the effect did not persist across five years. Discrepancies in findings
on the duration of transfer effects could have occurred from sampling different populations
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(elementary school students from one area vs. high school students from a nationally
representative sample) and types of outcomes (foreign language achievement vs. L2 achievement
where L2 is the societal language). Another probable explanation is that general ability masked
the relationship between L1 proficiency and English reading for LM students. Prior studies
argued that including general ability constructs while examining cross-linguistic transfer effects
was crucial because without this covariate it would be difficult both to parse out the influences of
general ability and L1 proficiency, and to examine the possible mediation of general ability in
the relationship between L1 and L2 (August et al., 2006; Carlo, 2009).
In addition, the relationship between L1 proficiency and English reading in LM students
may not have been observed in the current study because of some methodological shortcomings
of the current study. First, the proxy used for L1 proficiency, whether students subsequently
completed at least grade levels 1 through 5 outside of the U.S., may not have been a valid and
specific enough measure of L1 proficiency. More specifically, the proxy could have been too
broad, encompassing other skills than reading, which may not have been transferable to English
reading because some skills transfer only across the same modality (e.g., Spanish reading to
English reading) (e.g., Genesee, Geva, et al., 2006; Melby-Lervåg & Lervåg, 2011). In addition,
this proxy contains limited information on the quality of the literacy instruction in L1. I also
assumed that the instruction received outside of the U.S. was in LM students’ first language
(non-English). The ELS 2002 study did not provide information on the language of instruction
received outside of the U.S. to either confirm or contradict this assumption.
Another challenge with the proxy may have been its dichotomous nature, which reduced
its variability. Although the original variable was ordinal, I created a dummy variable because
treating the ordinal variable as an interval variable did not seem appropriate. More specifically,
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the intervals between higher grade levels completed outside of the U.S. would have a greater
impact on English reading than intervals between lower grade levels. Despite the limitations of
this L1 proxy, Guglielmi (2012) used it as a covariate in his NELS study and indicated that it
predicted the latent L1 proficiency factor, which included reading, writing, speaking, and
understanding. The number of grade levels completed outside of the U.S., which was treated as
an interval variable in the study by Guglielmi, was significantly correlated to L1 reading.
Furthermore, the number of grade levels completed outside of the U.S. negatively predicted the
reading achievement intercept, but positively predicted the math achievement intercept in the
SEM model that included the latent L1 proficiency factor. Therefore, Guglielmi’s finding
supports the current study’s use of number of grade levels completed outside of the U.S. as a
proxy for L1 proficiency. The relationships found in Guglielmi’s study may not have been
observed in the current study however; because the author had specifically sampled middle
school LM students with limited English language proficiency and had treated the L1 proxy as
an interval variable, whereas in the current study LM students were high school students with
varying levels of English language proficiency, and the L1 proxy was a dichotomous variable.
Second, the large imbalance in sample sizes between LM students with L1 proficiency
and the reference group, which included native English speakers who did not have any schooling
abroad, could have resulted in a lack of power to detect the relationship between L1 proficiency
and English reading in LM students. Specifically, in the current study the number of LM students
who subsequently completed at least grades 1 through 5 outside of the U.S. (e.g., L1 proficient)
was 190, compared to 12,440 students in the reference group. Because the ELS dataset is a
clustered data and all schools did not have LM students enrolled in them, I did not sample only
LM students for the current study. Sampling only LM students with this type of data would have
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given results limited in generalizability since schools, which were sampled through stratified
cluster random sampling, without LM student enrollment would have been dropped. In addition,
sampling only LM students would have provided inaccurate standard errors if these were
calculated without taking into consideration the removal of those schools. Moreover, the
sampling weights included in the ELS 2002 study could not be used without their further
modification because in their estimation of the sampling weights, the ELS 2002 study did not
consider the removal of schools without LM students (cf. Johnson & Christensen, 2012).
Consequently, in the current study a reference group was utilized to estimate less biased standard
errors despite the imbalance in sample sizes.
Third, heterogeneity of LM students may have weakened the relationship between L1
proficiency and English reading. Because of the small LM sample size with L1 proficiency, I
aggregated students with different native languages. The effects of cross-linguistic transfer may
not have been observed in the current study because the transfer that actually may have occurred
across similar languages was countered by a lack of transfer across languages with very different
writing systems. Previous studies indicated that transfer occurs more readily across similar
orthographic languages (Jeon & Yamashita, 2014; Melby-Lervåg & Lervåg, 2011). For example,
Guglielmi (2012) reported that L1 proficiency predicted English reading intercept and slope for
Hispanic ELL students, but not for non-Hispanic ELL students including Asian/Pacific Islander
students. Therefore, combining LM students with different native languages may have diluted
the predictive power of L1 proficiency on English reading within LM students.
Fourth, the temporal distance between the measurement of L1 proficiency and English
reading could have been too large. LM students with L1 proficiency—subsequently completed at
least grades 1 though 5 outside of the U.S.—could have been attending schools in the U.S. since
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grade 6. If the cross-linguistic transfer effect was short, the influence of L1 may not have been
observed in the current study because (1) English reading was measured in grade 10, and (2) the
L1 proficiency of students with different lengths of residence in the U.S. were combined.
Genesee et al. (2005) posited that transfer happened more frequently when students start learning
L2. The effect of L1 that might have happened for students beginning their U.S. instruction in
grade 9 may have been overcome by the limited transfer occurring for students who had this
instruction since grade 6 and relied more in the L2 skills.
Comparing Results Across Student and School Levels
With regards to comparing and contrasting relationships among constructs across levels, I
found that most of these links were absent at the school-level, but present at the student-level.
For example, grade 10 math achievement predicted grade 12 math achievement and math self-
efficacy in both student and school levels. Although at the school-level, grade 10 reading and
math self-efficacy did not predict grade 12 math achievement, and grade 10 math self-efficacy
did not relate to grade 12 math self-efficacy these constructs were related at the student-level.
A possible explanation for the lack of relationship between English reading and math
achievement at the school-level could be school differences in math curriculum and instruction.
Some schools may already integrate literacy skill instruction into their math teaching, which
probably lessens the influence of English reading on math achievement at the school-level. For
example, García, Flores, and Chu (2011) observed a small high school in which bilingual-ESL
instructors worked collaboratively with content area instructors and taught lessons jointly to meet
literacy and linguistic objectives within content areas. In this school, depending on students’
English proficiency, students were allowed to give answers in their native language, and they
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were given additional native language support, such as translations and summaries of materials
in Spanish.
The relationship between school-level grade 10 and 12 math self-efficacy was
insignificant after adjusting for prior math achievement probably because the availability of self-
efficacy sources other than mastery experience (math achievement) across schools was limited,
and therefore this diffused the influence of prior self-efficacy. For instance, in some schools
students may be homogenous in terms of race/ethnicity, home language, SES or other
demographic characteristic, which would limit the presence of similar student models for
students from minority backgrounds. Subsequently, this homogeneity would limit a vicarious
source that could impact these students’ self-efficacy. Similarly, teachers and peers from some
schools may not provide (or give very limited) verbal persuasion and feedback on students’
performance for them to either increase or decrease their self-efficacy. Schools could also be
providing environments in which many students do not engage in any particular positive or
negative emotions while learning math. These types of contexts provide students with limited
information based on sources such as vicarious experiences, persuasion, and physiological and
affective states for them to gauge their self-efficacy. Consequently, an absent or weak
relationship between prior and subsequent math self-efficacy at the school-level could be
suppressed by the stronger relationship between school-level prior math achievement and self-
efficacy.
Another possibility is that relationships observed at the student-level may not have been
present at the school-level because most studies and frameworks exploring these relationships
focused exclusively on the individual-level (e.g., Boonen et al. 2013; Caprara et al., 2008;
Riconscente, 2014). One methodological explanation for the insignificant path from school-level
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grade 10 math self-efficacy to grade 12 math achievement from the current study, but which was
supported in another study (Kitsantas et al., 2010), could be the nature of the data used. While
the study by Kitsantas and colleagues (2010) used cross-sectional data, the current study used
longitudinal data and adjusted for prior math achievement, which methodological strategies
provide more accurate estimates (Cole & Maxwell, 2003). The differences in the relationships
among English reading, math achievement, and math self-efficacy at both levels imply that
student-level research does not necessarily reflect the relationships of these constructs at higher
levels. Assuming that relationships observed at student levels would be consistent at school
levels appears to be inappropriate. Therefore, studies should test whether the links found at the
student-level are generalizable at higher levels.
Implications
In the discussion section above, I have briefly discussed implications for several main
findings including: (1) support for programs providing resources to schools with high percentage
of low-income students; (2) increase success opportunities in math to develop the math self-
efficacy of linguistically and ethnically diverse high school students; and (3) raise high school
LM students’ math self-efficacy by improving both their reading and math achievement. Here I
focus on several implications that may have the most impact on theory, research, policy, and
practice with regards to cross-language transfer and math learning. Specifically, one theoretical
implication of the current study is that cross-linguistic transfer models may not be relevant across
all age groups. More specifically, cross-linguistic transfer may not occur in older students,
including high school LM students with differing English language proficiencies. Consequently,
future researchers need to test these models with older students, particularly with LM students
with varying English language proficiency, and revise them to increase their generalizability
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across different ages. With regards to research, one implication is that for LM students, the use
of years of schooling in their home country as a proxy of L1 proficiency is problematic. As
discussed previously, the number of grade levels completed outside of the U.S. was a broad
construct that provided limited information on the L1 literacy instruction and linguistic skills of
LM students. Another research implication is with regards to the construction of questionnaires.
For example, the percentage of missing math self-efficacy data in ELS 2002 was high because
these items were towards the end of the survey and consequently some struggling readers tended
not to answer these items. Researchers need to consider how location and number of items could
influence data missingness, which ultimately relates to the accuracy of studies’ analyses and
conclusions.
The current study provides implications in the areas of policy and practice as well. In
terms of policy, the finding that L1 proficiency is not related to English reading of a nationally
representative sample of high school students questions allocation of resources for native
language instruction at the high school level aimed at improving second language learning,
which has been proposed by some researchers (e.g., Menken & Kleyn, 2010). One practical
implication based on the significant path between English reading and math achievement while
statistically adjusting for prior math achievement is that high school math educators should
become more cognizant of the influence of linguistic and literacy skills on math learning and
achievement. This association is not only relevant for ELL students and within word problem
contexts but all students and any mathematical context. Therefore educators can integrate
teaching of particular literacy skills that are specifically needed to increase access to and
comprehension of math content, particularly for high school students (Ahn et al., 2015; Carter &
Dean, 2006). In particular, math educators could determine which linguistic skills are necessary
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to better comprehend the math topics that are being covered, and explicitly teach these skills to
students. Some instructional examples for teaching these skills are vocabulary instruction,
reading aloud followed by discussions, and developing mathematics-structured note-taking, in
which students include big ideas, explanations, formulas/graphs, and definitions (e.g., Carter &
Dean, 2006; Shanahan & Shanahan, 2008). Educators could also ensure that math assessments
do not include items that confound math with language achievement (see Abedi & Lord, 2001,
Martiniello, 2008). Although the current study provides further knowledge on the relationships
among LM status, L1 proficiency, English reading, math achievement, and math self-efficacy of
linguistically and ethnically diverse high school students, it has some limitations that need to be
considered in interpreting the results.
Limitations
I have already addressed some methodological limitations in the discussion section
including the low school percentage of ELL sophomores, the broad racial/ethnic minority student
proportion measure, and the misaligned measures of math self-efficacy and achievement.
Additional limitations can be identified in: the large time lag between L1 proficiency and English
reading measurements, the dichotomous L1 proficiency variable, the broad L1 proficiency
measure, and the misalignment in literacy skills between L1 proficiency and English reading
measures. These limitations were related to the secondary data analysis nature of the current
study. In the section below, I expand on two limitations that were mentioned earlier and describe
additional limitations with regards to the methods and generalizability of the current study.
One substantial limitation was the lack of a direct measure of LM students’ native
language proficiency. Its proxy, whether students subsequently completed at least grade levels 1
through 5 outside of the U.S., may not necessarily correspond to students’ native language
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proficiency, particularly in literacy, and may not be reliable for students who studied across
different non-U.S. contexts. For example, students who completed the same number of grade
levels abroad may have different native language proficiencies depending on the type of literacy
instruction they received outside of the U.S.
The current study was also limited in that it included only two waves of math
achievement and self-efficacy data, and one time measure of English reading, to test mediation
of two constructs—English reading and math achievement. Ideally to test this type of mediation
one would use at least three data waves and include English reading measures for all waves to
test the assumptions of stationarity and equilibrium for the results to be less biased (Cole &
Maxwell, 2003). However, the number of waves used in the current study was restricted by the
data available in the ELS 2002.
The large amount of missing data for L1 proficiency, and grade 10 and 12 math self-
efficacy, 23%, 28%, and 36%, respectively, was another shortcoming of the ELS 2002 dataset. I
addressed the missing data issue by looking at patterns of missingness, predicting missingness,
and using all data available to decrease biases in the estimates. However, it could be possible that
the results from this study could be different from another study in which the percent of missing
values was much smaller. Another shortcoming was the use of only self-reports to assess
students’ math self-efficacy because this type of measure could be influenced by social
desirability. This type of measure also suffers from issues related to developmental
appropriateness, construct definition, and measurement/reliability (Fulmer & Frijters, 2009).
With regards to generalizability, the current study does not generalize to all sophomore
students despite the participants being sampled from a nationally representative pool of students.
Specifically, the non-response rate for math self-efficacy items was over 15 percent because they
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were located at the end of the survey (Ingels et al., 2004). Poor readers with lower reading and
mathematics achievement were less likely to answer high non-response items, such as math self-
efficacy items (Ingels et al., 2004). Consequently, the current findings may not generalize to this
particular subgroup. In addition, the results would not be generalizable to students with very
limited English proficiency because they had been excluded from the ELS 2002 study if school
staff believed that their limited English proficiency would prevent them from accurately
completing the surveys and assessments (Ingels et al., 2004).
Many of the limitations mentioned above pertain to the secondary data analysis nature of
the current study. The study’s limitations highlight the shortcomings of the ELS 2002 as a
national dataset specifically in exploring relationships among first and second language
proficiencies, L2 achievement, and L2 motivation. First, the ELS 2002 does not provide a direct
L1 language proficiency measure and information on the percentage of sophomore students who
were from racial/ethnic minority backgrounds. Second, it includes limited data waves for math
and reading achievement, and math self-efficacy hindering the investigation of their longitudinal
relationships. Third, its survey design, in particular its length, item location, and language, as
well its administration limited available data for certain student groups. Fourth, the sampling of
participants from high school and on, and the use of varying measures of achievement, did not
allow unbiased investigation of longitudinal relationships including mediations, and examination
of the influence of L1 proficiency on L2 achievement at different developmental levels,
including the transition from middle to high school. Lastly, the procedure used in developing
math self-efficacy and math test items, which did not consider the alignment of these items,
prevented the observation of stronger relationships among these constructs.
Future Research
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The aforementioned limitations point to possible directions for future studies
investigating relationships among LM status, L1 proficiency, English reading, math
achievement, and math self-efficacy of linguistically and ethnically diverse high school students.
In future studies researchers could use native language assessments collected no longer than a
year before collecting achievement data to test the moderation of L1 proficiency in the
relationship between LM status and English reading. They could also include at least three waves
of data with outcomes measured for all waves to test mediation of English reading and math
achievement in the relationship between LM status and/or interaction and math self-efficacy.
Moreover, studies could incorporate observational data or peer/teacher measure of students’
math self-efficacy in addition to students’ self-reports to triangulate the data. Researchers could
use math achievement and self-efficacy measures with more aligned items to increase the
predictability of math self-efficacy.
For researchers interested in school-level analyses, they could purposely sample schools
with high percentage of ELLs and racial/ethnic minority students from the relevant grade level to
examine their relationships to school-level math achievement and self-efficacy. To increase
participation of ELLs and expand the generalizability of results, researchers could provide math
self-efficacy measures and achievement tests in students’ native languages, which are equivalent
to the English versions. To increase the generalizability of these findings to poor readers as well,
the math self-efficacy items could be either located at the beginning of the survey, be scattered
throughout the survey, or survey items could be read to students. Future studies could
additionally incorporate more balanced LM students and native English speakers sample sizes to
ensure that there is enough power to observe the expected relationships. Lastly, if a large sample
size of LM students is available, analyses could be disaggregated by students’ type of native
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language to compare and contrast the influence of L1 proficiency on English reading across
varying native languages.
Contribution of the Study
Despite the limitations discussed above, the current study adds to the existing literature
on cross-linguistic transfer, math achievement and motivation, particularly of linguistically and
ethnically diverse high school students. In terms of theory building, the current study
comprehensibly connected motivation theories, such as SCT, with theories from second language
learning, such as the developmental interdependence hypothesis and common underlying
proficiency model. Previously this connection had received only limited attention particularly for
contexts in which L2 is the societal language (Dixon et al., 2012), despite the importance of
native language instruction and motivation in learning (e.g., Bong et al., 2012; Lindholm-Leary
& Block, 2010; Valentino & Reardon, 2015; Wolters et al., 2014). More specifically, this study
suggested that L1 proficiency did not appear to influence English reading in high school LM
students with varying English language proficiency. Another contribution is the examination of
how school-level demographic factors including school percentages of racial/ethnic minority and
ELL students influence school math achievement and math self-efficacy, and the finding that
they were not related to these outcomes. The study also explored longitudinal relationships
among math achievement and self-efficacy variables at the school and student levels, which has
been done infrequently, and it suggested that their relationships were not the same across levels.
The study contributed to the literature by further supporting the finding that linguistic
skills, and reading in particular, influenced math achievement of all students even after adjusting
for prior math achievement. Other additions to the existing knowledge were the findings that
although the reciprocal and longitudinal relationship between math achievement and self-
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efficacy was supported, math achievement had a greater influence on self-efficacy than vice-
versa. Another contribution was the finding that English reading and math achievement mediated
the relationship between LM status and math self-efficacy. This finding suggested that math self-
efficacy, particularly of LM students, could be improved through their successes in reading and
math.
Conclusion
Despite the growing number of LM students in the U.S. and the math underachievement
of many of these students, the link between native language proficiency and second language
learning, and the relationship between math achievement and motivation, only a limited number
of studies have examined these relationships in one comprehensive model. In particular, these
connections have been explored less with adolescents from diverse linguistic and ethnic
backgrounds. The current study investigated the relationships among LM status, L1 proficiency,
English reading, math achievement, and math self-efficacy from a nationally representative
sample of high school students using the ELS 2002 dataset. In particular, the study explored the
moderating role of L1 proficiency in the relationship between LM status and English reading,
and the mediating roles of English reading and math achievement in the relationship between
such interaction and math self-efficacy. The findings from this study provide further knowledge
educators can use to develop more targeted interventions to support math achievement and self-
efficacy, particularly of linguistically and ethnically diverse adolescents. Providing academic
supports to these students could increase their math motivation and achievement, and ultimately
their participation in STEM fields.

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Running head: NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY
Appendix A
Histograms and Plots for Student-Level Outcomes
Grade 10 Math Achievement
Histogram Qqplot Probability Plot Boxplot

Grade 12 Math Achievement
Histogram Qqplot Probability Plot Boxplot

0 200 400 600 800
Frequency
20 40 60 80
math IRT est no. right by scores w f1 adj
0 50 100
math IRT est no. right by scores w f1 adj
0 50 100
Inverse Normal
0.00 0.25 0.50 0.75 1.00
Normal F[(machv-m)/s]
0.00 0.25 0.50 0.75 1.00
Empirical P[i] = i/(N+1)
20 40 60 80
math IRT est no. right by scores w f1 adj
0 200 400 600
Frequency
20 40 60 80
math IRT estimated number right scores for F1
0 20 40 60 80 100
math IRT estimated number right scores for F1
0 20 40 60 80 100
Inverse Normal
0.00 0.25 0.50 0.75 1.00
Normal F[(machvf1-m)/s]
0.00 0.25 0.50 0.75 1.00
Empirical P[i] = i/(N+1)
20 40 60 80
math IRT estimated number right scores for F1
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

144
Appendix A (Continuation)
Grade 10 Math Self-efficacy
Histogram Qqplot Probability Plot Boxplot

Grade 12 Math Self-efficacy
Histogram Qqplot Probability Plot Boxplot

0 500 1000 1500 2000
Frequency
0 5 10 15 20
by math self-efficacy composite
-10 0 10 20 30
by math self-efficacy composite
-10 0 10 20 30
Inverse Normal
-10 0 10 20 30
by math self-efficacy composite
-10 0 10 20 30
Inverse Normal
0 5 10 15 20
by math self-efficacy composite
0 500 1000 1500
Frequency
0 5 10 15 20
f1 math self-efficacy composite
0 10 20 30
f1 math self-efficacy composite
0 10 20 30
Inverse Normal
0.00 0.25 0.50 0.75 1.00
Normal F[(f1_mse-m)/s]
0.00 0.25 0.50 0.75 1.00
Empirical P[i] = i/(N+1)
0 5 10 15 20
f1 math self-efficacy composite
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

145
Appendix A (Continuation)
Grade 10 Reading Achievement
Histogram Qqplot Probability Plot Boxplot

0 200 400 600
Frequency
10 20 30 40 50
reading achievement IRT estimate number right base year score
0 20 40 60 80
reading achievement IRT estimate number right base year score
0 20 40 60 80
Inverse Normal
0.00 0.25 0.50 0.75 1.00
Normal F[(read_ach-m)/s]
0.00 0.25 0.50 0.75 1.00
Empirical P[i] = i/(N+1) 10 20 30 40 50
reading achievement IRT estimate number right base year score
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

146
Appendix B
Histograms and Plots for School-Level Outcomes
Grade 10 Math Achievement
Histogram Qqplot Probability Plot Boxplot

Grade 12 Math Achievement
Histogram Qqplot Probability Plot Boxplot

0 20 40 60 80 100
Frequency
20 30 40 50 60 70
school level mean by math achievement
20 30 40 50 60 70
school level mean by math achievement
20 30 40 50 60 70
Inverse Normal
0.00 0.25 0.50 0.75 1.00
Normal F[(mmath_by-m)/s]
0.00 0.25 0.50 0.75 1.00
Empirical P[i] = i/(N+1)
20 30 40 50 60 70
school level mean by math achievement
0 20 40 60 80
Frequency
20 40 60 80
school level mean f1 math achievement
20 40 60 80
school level mean f1 math achievement
20 40 60 80
Inverse Normal
0.00 0.25 0.50 0.75 1.00
Normal F[(mmath_f1-m)/s]
0.00 0.25 0.50 0.75 1.00
Empirical P[i] = i/(N+1)
20 40 60 80
school level mean f1 math achievement
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

147
Appendix B (Continuation)
Grade 10 Math Self-efficacy
Histogram Qqplot Probability Plot Boxplot

Grade 12 Math Self-efficacy
Histogram Qqplot Probability Plot Boxplot

0 50 100
Frequency
5 10 15 20
school level mean by math self-efficacy composite
5 10 15 20
school level mean by math self-efficacy composite
5 10 15 20
Inverse Normal
0.00 0.25 0.50 0.75 1.00
Normal F[(mmse_by-m)/s]
0.00 0.25 0.50 0.75 1.00
Empirical P[i] = i/(N+1)
5 10 15 20
school level mean by math self-efficacy composite
0 20 40 60 80 100
Frequency
5 10 15 20
school level mean f1 math self-efficacy composite
5 10 15 20
school level mean f1 math self-efficacy composite
8 10 12 14 16 18
Inverse Normal
0.00 0.25 0.50 0.75 1.00
Normal F[(mmse_f1-m)/s]
0.00 0.25 0.50 0.75 1.00
Empirical P[i] = i/(N+1)
5 10 15 20
school level mean f1 math self-efficacy composite
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

148
Appendix B (Continuation)
Grade 10 Reading Achievement
Histogram Qqplot Probability Plot Boxplot

0 20 40 60
Frequency
10 20 30 40 50
school level mean by reading achievement
10 20 30 40 50
school level mean by reading achievement
10 20 30 40 50
Inverse Normal
0.00 0.25 0.50 0.75 1.00
Normal F[(mreadach-m)/s]
0.00 0.25 0.50 0.75 1.00
Empirical P[i] = i/(N+1)
10 20 30 40 50
school level mean by reading achievement
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

149
Appendix C
Histograms of Possible Transformations of Student-Level Variables
Grade 10 Math Achievement Grade 12 Math Achievement

0
2.0e-06 4.0e-06 6.0e-06 8.0e-06
0 200000 400000 600000
cubic
0
1.0e-04 2.0e-04 3.0e-04 4.0e-04
0 2000 4000 6000 8000
square
0
.01 .02 .03
20 40 60 80
identity
0 .1 .2 .3 .4
4 5 6 7 8 9
sqrt
0 .5 1
1.5
2.5 3 3.5 4 4.5
log
0 5 10 15 20
-.3 -.25 -.2 -.15 -.1
1/sqrt
0 20 40 60
-.08 -.06 -.04 -.02 0
inverse
0
500 1000 1500 2000
-.005 -.004 -.003 -.002-.001 0
1/square
0
1.0e+04 2.0e+04 3.0e+04 4.0e+04 5.0e+04
-.0004 -.0003 -.0002 -.0001 0
1/cubic
Density
math IRT est no. right by scores w f1 adj
Histograms by transformation
0
2.0e-06 4.0e-06 6.0e-06
0 200000 400000 600000
cubic
0
5.0e-05 1.0e-04 1.5e-04 2.0e-04 2.5e-04
0 2000 4000 6000 8000
square
0
.005 .01 .015 .02 .025
20 40 60 80
identity
0 .1 .2 .3 .4
4 5 6 7 8 9
sqrt
0 .5 1
1.5
2.5 3 3.5 4 4.5
log
0 5 10 15 20
-.25 -.2 -.15 -.1
1/sqrt
0 20 40 60 80
-.06 -.04 -.02 0
inverse
0
500 1000 1500 2000
-.004 -.003 -.002 -.001 0
1/square
0
2.0e+04 4.0e+04 6.0e+04 8.0e+04
-.0003 -.0002 -.0001 0
1/cubic
Density
math IRT estimated number right scores for F1
Histograms by transformation
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

150
Appendix C (Continuation)
Grade 10 Math Self-efficacy Grade 12 Math Self-efficacy

0
2.0e-04 4.0e-04 6.0e-04 8.0e-04
0 2000 4000 6000 8000
cubic
0
.005.01.015.02
0 100 200 300 400
square
0 .1 .2 .3 .4
0 5 10 15 20
identity
0 .5 1
1.5
2
1 2 3 4 5
sqrt
0 .5 1
1.5
2
2.5
0 1 2 3
log
0 5 10 15
-1 -.8 -.6 -.4 -.2
1/sqrt
0 5 10 15 20
-1 -.8 -.6 -.4 -.2 0
inverse
0 10 20 30 40
-1 -.8 -.6 -.4 -.2 0
1/square
0 10 20 30 40
-1 -.8 -.6 -.4 -.2 0
1/cubic
Density
by math self-efficacy composite
Histograms by transformation
0
2.0e-04 4.0e-04 6.0e-04
0 2000 4000 6000 8000
cubic
0
.005 .01 .015
0 100 200 300 400
square
0 .1 .2 .3
0 5 10 15 20
identity
0 .5 1
1.5
1 2 3 4 5
sqrt
0 .5 1
1.5
2
0 1 2 3
log
0 2 4 6 8 10
-1 -.8 -.6 -.4 -.2
1/sqrt
0 5 10 15 20
-1 -.8 -.6 -.4 -.2 0
inverse
0 10 20 30 40
-1 -.8 -.6 -.4 -.2 0
1/square
0 10 20 30 40
-1 -.8 -.6 -.4 -.2 0
1/cubic
Density
f1 math self-efficacy composite
Histograms by transformation
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

151
Appendix C (Continuation)
Grade 10 Reading Achievement

0
1.0e-05 2.0e-05 3.0e-05
0 50000 100000 150000
cubic
0
2.0e-04 4.0e-04 6.0e-04 8.0e-04
0 500 1000 1500 2000 2500
square
0
.01 .02 .03 .04
10 20 30 40 50
identity
0 .1 .2 .3 .4
3 4 5 6 7
sqrt
0 .5 1
1.5
2 2.5 3 3.5 4
log
0 5 10 15 20
-.3 -.25 -.2 -.15
1/sqrt
0 20 40 60
-.1 -.08 -.06 -.04 -.02
inverse
0
200 400 600 800
-.01 -.008 -.006 -.004 -.002 0
1/square
0
5000
1.0e+04 1.5e+04 2.0e+04
-.001-.0008 -.0006 -.0004 -.0002 0
1/cubic
Density
reading achievement IRT estimate number right base year score
Histograms by transformation
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

152
Appendix D
Histograms of Possible Transformations of School-Level Variables
Grade 10 Math Self-efficacy Grade 12 Math Self-efficacy

0
2.0e-04 4.0e-04 6.0e-04 8.0e-04
0 2000 4000 6000 8000
cubic
0
.005 .01 .015
0 100 200 300 400
square
0 .1 .2 .3 .4
5 10 15 20
identity
0 .5 1
1.5
2
2 2.5 3 3.5 4 4.5
sqrt
0 1 2 3 4
1.5 2 2.5 3
log
0 5 10 15 20 25
-.45 -.4 -.35 -.3 -.25
1/sqrt
0 10 20 30 40
-.2 -.15 -.1 -.05
inverse
0 50 100 150 200 250
-.04 -.03 -.02 -.01 0
1/square
0
500 1000 1500 2000
-.008 -.006 -.004 -.002 0
1/cubic
Density
school level mean by math self-efficacy composite
Histograms by transformation
0
2.0e-04 4.0e-04 6.0e-04 8.0e-04
0 2000 4000 6000 8000
cubic
0
.005 .01 .015
0 100 200 300 400
square
0 .1 .2 .3 .4
5 10 15 20
identity
0 .5 1
1.5
2
2.5
3 3.5 4 4.5
sqrt
0 1 2 3 4 5
2.2 2.4 2.6 2.8 3
log
0 10 20 30 40
-.35 -.3 -.25 -.2
1/sqrt
0 20 40 60
-.12 -.1 -.08 -.06 -.04
inverse
0
100200300400
-.015 -.01 -.005 0
1/square
0
1000 2000 3000 4000
-.0015 -.001 -.0005 0
1/cubic
Density
school level mean f1 math self-efficacy composite
Histograms by transformation
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153
Appendix E
Bivariate Scatter Plots for Student-Level Variables
Grade 10 Reading Achievement vs. SES Grade 12 Math Achievement vs. Grade 10 Reading Achievement

Grade 12 Math Achievement vs. Grade 10 Math Achievement Grade 12 Math Achievement vs. Grade 10 Math Self-efficacy

10 20 30 40 50
-2 -1 0 1 2
SES
reading achievement IRT estimate number right base year score lowess read_ach by_ses
Fitted values
20 40 60 80
10 20 30 40 50
reading achievement IRT estimate number right base year score
math IRT estimated number right scores for F1 lowess machvf1 read_ach
Fitted values
20 40 60 80 100
20 40 60 80
math IRT est no. right by scores w f1 adj
math IRT estimated number right scores for F1 lowess machvf1 machv
Fitted values
20 40 60 80
0 5 10 15 20
by math self-efficacy composite
math IRT estimated number right scores for F1 lowess machvf1 by_mse
Fitted values
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154
Appendix E (Continuation)
Grade 12 Math Achievement vs. SES Grade 12 Math Self-efficacy vs. Grade 10 Math Self-efficacy

Grade 12 Math Self-efficacy vs. Grade 10 Math Achievement Grade 10 Math Achievement vs. SES

20 40 60 80
-2 -1 0 1 2
SES
math IRT estimated number right scores for F1 lowess machvf1 by_ses
Fitted values
0 5 10 15 20
0 5 10 15 20
by math self-efficacy composite
f1 math self-efficacy composite lowess f1_mse by_mse
Fitted values
0 5 10 15 20
20 40 60 80
math IRT est no. right by scores w f1 adj
f1 math self-efficacy composite lowess f1_mse machv
Fitted values
20 40 60 80
-2 -1 0 1 2
SES
math IRT est no. right by scores w f1 adj lowess machv by_ses
Fitted values
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155
Appendix F
Residual Plots for Student-Level Variables
Grade 10 Reading Achievement Regressed on LM L1 proficiency lmXl1prof SES Hispanic Asian Black Other
Residuals vs. SES Residual vs. Fitted Values

Grade 10 Math Achievement Regressed on SES Hispanic Asian Black Other
Residuals vs. SES Residuals vs. Fitted Values

-40 -20
0 20 40
-2 -1 0 1 2
SES
residuals for read ON by_lm l1prof lmXl1prof by_ses hispanic asian black other lowess resid10 by_ses
-40 -20
0 20 40
10 20 30 40
predicted values for read ON by_lm l1prof lmXl1prof by_ses hispanic asian black
residuals for read ON by_lm l1prof lmXl1prof by_ses hispanic asian black other lowess resid10 pred10
-40 -20
0 20 40
-2 -1 0 1 2
SES
residuals for machv ON by_ses hispanic asian black other lowess resid2 by_ses
-40 -20
0 20 40
20 30 40 50 60
predicted values for machv ON by_ses hispanic asian black other
residuals for machv ON by_ses hispanic asian black otherlowess resid2 pred2
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156
Appendix F (Continuation)
Grade 12 Math Achievement Regressed on Grade 10 Reading Grade 10 Math Achievement Grade 10 self-efficacy SES Gender
Residuals vs. Grade10 Math Achievement Residuals vs. Grade 10 Reading Achievement

Residuals vs. Grade 10 Math Self-efficacy Residual vs. SES

-40 -20
0 20 40
20 40 60 80
math IRT est no. right by scores w f1 adj
residuals for machvf1 ON machv read_ach by_mse by_ses by_sex lowess resid3 machv
-40 -20
0 20 40
10 20 30 40 50
reading achievement IRT estimate number right base year score
residuals for machvf1 ON machv read_ach by_mse by_ses by_sex lowess resid3 read_ach
-40 -20
0 20 40
0 5 10 15 20
by math self-efficacy composite
residuals for machvf1 ON machv read_ach by_mse by_ses by_sex lowess resid3 by_mse
-40 -20
0 20 40
-2 -1 0 1 2
SES
residuals for machvf1 ON machv read_ach by_mse by_ses by_sex lowess resid3 by_ses
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157
Appendix F (Continuation)
Residuals vs. Predicted Values

Grade 12 Math Self-efficacy Regressed on Grade 10 Math Self-efficacy Grade 10 Math Achievement Gender
Residuals vs. Grade 10 Math Self-efficacy Residuals vs. Grade 10 Math Achievement

-40 -20
0 20 40
20 40 60 80 100
predicted values for machvf1 ON machv read_ach by_mse by_ses by_sex
residuals for machvf1 ON machv read_ach by_mse by_ses by_sex lowess resid3 pred3
-15 -10 -5 0 5 10
0 5 10 15 20
by math self-efficacy composite
residuals for f1_mse ON by_mse machv by_sex lowess resid4 by_mse
-15 -10 -5 0 5 10
20 40 60 80
math IRT est no. right by scores w f1 adj
residuals for f1_mse ON by_mse machv by_sex lowess resid4 machv
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158
Appendix F (Continuation)
Residuals vs. Fitted Values

-15 -10 -5 0 5 10
8 10 12 14 16 18
predicted values for f1_mse ON by_mse machv by_sex
residuals for f1_mse ON by_mse machv by_sex lowess resid4 pred4
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159
Appendix G
Bivariate Scatter Plots for School-Level Outcomes
Grade 10 Reading Achievement vs. Percent Free or Reduced-Priced lunch (FRL) Grade 10 Math Achievement vs. Percent FRL

Grade 10 Math Achievement vs. Percent Racial/Ethnic Minority (REM) Grade 10 Math Self-efficacy vs. Percent REM

10 20 30 40 50
0 20 40 60 80 100
recoded pg10frl to replace missing values
school level mean by reading achievement lowess mreadach pg10frl1
Fitted values
20 30 40 50 60 70
0 20 40 60 80 100
recoded pg10frl to replace missing values
school level mean by math achievement lowess mmath_by pg10frl1
Fitted values
20 30 40 50 60 70
0 20 40 60 80 100
recoded pminor w all info from same sch stud
school level mean by math achievement lowess mmath_by pminor1
Fitted values
5 10 15 20
0 20 40 60 80 100
recoded pminor w all info from same sch stud
school level mean by math self-efficacy composite lowess mmse_by pminor1
Fitted values
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160
Appendix G (Continuation)
Grade 10 Math Self-efficacy vs. Percent ELLs Grade 12 Math achievement vs. Grade 10 Reading Achievement

Grade 12 Math Achievement vs. Grade 10 Math Achievement Grade 12 Math Achievement vs. Grade 10 Math Self-efficacy

5 10 15 20
0 20 40 60
recoded pg10ell to replace missing values
school level mean by math self-efficacy composite lowess mmse_by pg10ell1
Fitted values
20 40 60 80
10 20 30 40 50
school level mean by reading achievement
school level mean f1 math achievement lowess mmath_f1 mreadach
Fitted values
20 40 60 80
20 30 40 50 60 70
school level mean by math achievement
school level mean f1 math achievement lowess mmath_f1 mmath_by
Fitted values
20 40 60 80
5 10 15 20
school level mean by math self-efficacy composite
school level mean f1 math achievement lowess mmath_f1 mmse_by
Fitted values
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161
Appendix G (Continuation)
Grade 12 Math Self-efficacy vs. Grade 10 Math Self-efficacy Grade 12 Math Self-efficacy vs. Grade 10 Math Achievement

5 10 15 20
5 10 15 20
school level mean by math self-efficacy composite
school level mean f1 math self-efficacy composite lowess mmse_f1 mmse_by
Fitted values
5 10 15 20
20 30 40 50 60 70
school level mean by math achievement
school level mean f1 math self-efficacy composite lowess mmse_f1 mmath_by
Fitted values
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162
Appendix H
Residual Plots for School-Level Variables
Grade 10 Reading Achievement Regressed on Percent FRL
Residuals vs. Percent FRL Residuals vs. Predicted Values

Grade 10 Math Achievement Regressed on Percent FRL Percent Racial/Ethnic Minority (RM)
Residuals vs. Percent FRL Residuals vs. Percent RM

-15 -10 -5 0 5 10
0 20 40 60 80 100
recoded pg10frl to replace missing values
residuals for mreadach ON pg10frl1 lowess resid5 pg10frl1
-15 -10 -5 0 5 10
20 25 30 35
predicted values for mreadach ON pg10frl1
residuals for mreadach ON pg10frl1 lowess resid5 pred5
-20 -10
0 10 20 30
0 20 40 60 80 100
recoded pg10frl to replace missing values
residuals for mmath_by ON pg10frl1 pminor1 lowess resid6 pg10frl1
-20 -10
0 10 20 30
0 20 40 60 80 100
recoded pminor w all info from same sch stud
residuals for mmath_by ON pg10frl1 pminor1 lowess resid6 pminor1
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163
Appendix H (Continuation)
Residuals vs. Predicted Values

Grade 10 Math Self-efficacy Regressed on Percent Racial/Ethnic Minority (RM) Percent ELLs
Residuals vs. Percent RM Residuals vs. Percent ELLs

-20 -10
0 10 20 30
30 35 40 45 50
predicted values for mmath_by ON pg10frl1 pminor1
residuals for mmath_by ON pg10frl1 pminor1 lowess resid6 pred6
-10 -5 0 5 10
0 20 40 60 80 100
recoded pminor w all info from same sch stud
residuals for mmse_by ON pminor1 pg10ell1 lowess resid7 pminor1
-10 -5 0 5 10
0 20 40 60
recoded pg10ell to replace missing values
residuals for mmse_by ON pminor1 pg10ell1 lowess resid7 pg10ell1
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164
Appendix H (Continuation)
Residuals vs. Predicted Values

Grade 12 Math Achievement Regressed on Grade 10 Math Achievement Grade 10 Math Self-efficacy Grade 10 Reading Achievement
Residuals vs. Grade 10Mmath Achievement Residuals vs. Grade 10 Math Self-efficacy

-10 -5 0 5 10
11.4 11.6 11.8 12 12.2
predicted values for mmse_by ON pminor1 pg10ell1
residuals for mmse_by ON pminor1 pg10ell1 lowess resid7 pred7
-10 -5 0 5 10
20 30 40 50 60 70
school level mean by math achievement
residuals formmath_f1 ON mmath_by mmse_by mreadach lowess resid8 mmath_by
-10 -5 0 5 10
5 10 15 20
school level mean by math self-efficacy composite
residuals formmath_f1 ON mmath_by mmse_by mreadach lowess resid8 mmse_by
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165
Appendix H (Continuation)
Residuals vs. Grade 10 Reading Achievement Residuals vs. Predicted Values

Grade 12 Math Self-efficacy Regressed on Grade 10 Math Self-efficacy Grade 10 Math Achievement
Residuals vs. Grade 10 Math Self-efficacy Residuals vs. Grade 10 Math Achievement

-10 -5 0 5 10
10 20 30 40 50
school level mean by reading achievement
residuals formmath_f1 ON mmath_by mmse_by mreadach lowess resid8 mreadach
-10 -5 0 5 10
20 40 60 80
predicted values for mmath_f1 ON mmath_by mmse_by mreadach
residuals formmath_f1 ON mmath_by mmse_by mreadachlowess resid8 pred8
-5 0 5 10
5 10 15 20
school level mean by math self-efficacy composite
residuals mmse_f1 ON mmse_by mmath_by lowess resid9 mmse_by
-5 0 5 10
20 30 40 50 60 70
school level mean by math achievement
residuals mmse_f1 ON mmse_by mmath_by lowess resid9 mmath_by
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166
Appendix H (Continuation)
Residuals vs. Predicted Values

-5 0 5 10
10 12 14 16
predicted values for mmse_f1 ON mmse_by mmath_by
residuals mmse_f1 ON mmse_by mmath_by lowess resid9 pred9
Running head: NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY
Appendix I
Missing Data Patterns and Frequencies
Pattern
Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
PMINOR1 x x x x x x x x x x x x x x x
PG10ELL1 x x x x x x x x x x x x x x x x x
PG10FRL1 x x x x x x x x x x x x x x x x x x
READ_ACH x x x x x x x x x x x x x x x x x x x
MACHV x x x x x x x x x x x x x x x x x x x
MACHVF1 x x x x x x x x x x x
F1_MSE x x x x x
BY_MSE x x x x x x x x x x x x x
L1PROF x x x x x x x x x x
BY_LM x x x x x x x x x x x x x x x x x x x x
HISPANIC x x x x x x x x x x x x x x x x x x x x
ASIAN x x x x x x x x x x x x x x x x x x x x
BLACK x x x x x x x x x x x x x x x x x x x x
OTHER x x x x x x x x x x x x x x x x x x x x
BY_SEX x x x x x x x x x x x x x x x x x x x x
BY_SES x x x x x x x x x x x x x x x x x x x x
LMXL1PRO x x x x x x x x x x
FREQ 6260 1350 1010 10 20 10 1590 390 170 70 10 10 260 500 1040 560 350 300 10 10

Pattern
Number 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
PMINOR1 x x x x x x x x x x x x x x x x x
PG10ELL1 x x x x x x x x x x x x x x x
PG10FRL1 x x x x x x x x x
READ_ACH x x x x x x x x x x x x x x x x
MACHV x x x x x x x x x x x x x x x x
MACHVF1 x x x x x x x x x x x x x x x
F1_MSE x x x x x x x x x
BY_MSE x x x x x x x x
L1PROF x x x x x x x x x x x x x x
BY_LM x x x x x x x x x x x x x x x x x x x x
HISPANIC x x x x x x x x x x x x x x x x x x x x
ASIAN x x x x x x x x x x x x x x x x x x x x
BLACK x x x x x x x x x x x x x x x x x x x x
OTHER x x x x x x x x x x x x x x x x x x x x
BY_SEX x x x x x x x x x x x x x x x x x x x x
BY_SES x x x x x x x x x x x x x x x x x x x x
LMXL1PRO x x x x x x x x x x x x x x
FREQ 10 10 10 10 310 50 470 50 10 10 70 30 10 40 10 20 10 10 40 70

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168
Appendix I (Continuation)

Pattern
Number 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
PMINOR1 x x x x x x x x x x x x x x x x x
PG10ELL1 x x x x x x x x
PG10FRL1 x x x x
READ_ACH x x x x x x x x x x x x x x x x x
MACHV x x x x x x x x x x x x x x x x x
MACHVF1 x x x x x x x x x x x x x x
F1_MSE x x x x x x x x
BY_MSE x x x x x x x
L1PROF x x x x x x x x x
BY_LM x x x x x x x x x x x x x x x x x x x x
HISPANIC x x x x x x x x x x x x x x x x x x x x
ASIAN x x x x x x x x x x x x x x x x x x x x
BLACK x x x x x x x x x x x x x x x x x x x x
OTHER x x x x x x x x x x x x x x x x x x x x
BY_SEX x x x x x x x x x x x x x x x x x x x x
BY_SES x x x x x x x x x x x x x x x x x x
LMXL1PRO x x x x x x x x x
FREQ 20 10 10 20 200 69 50 20 10 20 50 50 30 80 40 20 40 10 20 30

Pattern
Number 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
PMINOR1 x x x x x x x x
PG10ELL1 x x x x x x x x x x x
PG10FRL1 x x x x x x x x x x
READ_ACH x x x x x x x x x x x x x x x
MACHV x x x x x x x x x x x x x x x
MACHVF1 x x x x x x x x x x x x x
F1_MSE x x x x
BY_MSE x x x x x x
L1PROF x x x x x x x x x x
BY_LM x x x x x x x x x x x x x x x x x x x x
HISPANIC x x x x x x x x x x x x x x x x x x x x
ASIAN x x x x x x x x x x x x x x x x x x x x
BLACK x x x x x x x x x x x x x x x x x x x x
OTHER x x x x x x x x x x x x x x x x x x x x
BY_SEX x x x x x x x x x x x x x x x x x x x x
BY_SES x x x x x x x x x x x x x x x x x x
LMXL1PRO x x x x x x x x x x
FREQ 20 20 10 20 10 10 10 20 10 10 10 10 20 10 20 10 10 10 10 10

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169
Appendix I (Continuation)

Pattern
Number 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
PMINOR1 x x x x x x x x
PG10ELL1 x x x x x
PG10FRL1 x x x x x x x x x
READ_ACH x x x x x x x x
MACHV x x x x x x x x
MACHVF1 x x x x x x x x x
F1_MSE x x x x
BY_MSE x x x
L1PROF x x x x x x x
BY_LM x x x x x x x x x x x x x x x x
HISPANIC x x x x x x x x x x x x x x x x
ASIAN x x x x x x x x x x x x x x x x
BLACK x x x x x x x x x x x x x x x x
OTHER x x x x x x x x x x x x x x x x
BY_SEX x x x x x x x x x x x x x x x x
BY_SES x x x x x x x x x x x x x x
LMXL1PRO x x x x x x x
FREQ 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

Note. Frequencies were rounded to the 10. Frequencies lower than 10 are shown as 10. FREQ =
frequencies, PMINOR1 = % racial/ethnic minority students; PG10ELL1 = % ELL sophomores;
PG10FRL1 = % sophomores eligible for free or reduced-price lunch; READ_ACH = grade 10
reading achievement, MACHV = grade 10 math achievement; MACHVF1 = grade 12 math
achievement; F1_MSE = grade 12 math self-efficacy; BY_MSE = grade 10 math self-efficacy;
L1PROF = L1 proficiency; BY_LM = language minority status; BY_SEX = gender; BY_SES =
socioeconomic status; LMXL1PRO = interaction between LM and L1PROF.

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170
Appendix J
Percentage of Complete Data (Covariance Coverage)
PMINOR1 PG10ELL1 PG10FRL1 READ_ACH MACHV
PMINOR1 98.2
PG10ELL1 93.4 94.9
PG10FRL1 90.1 90.6 91.6
READ_ACH 96.3 93.2 89.9 98.1
MACHV 96.3 93.2 89.9 98.1 98.1
MACHVF1 82.9 80.2 77.4 82.7 82.7
F1_MSE 62.9 61.2 59.1 63.9 63.9
BY_MSE 71.3 69.0 66.4 72.4 72.4
L1PROF 75.8 73.5 71.0 76.8 76.8
BY_LM 98.2 94.9 91.6 98.1 98.1
HISPANIC 98.2 94.9 91.6 98.1 98.1
ASIAN 98.2 94.9 91.6 98.1 98.1
BLACK 98.2 94.9 91.6 98.1 98.1
OTHER 98.2 94.9 91.6 98.1 98.1
BY_SEX 98.2 94.9 91.6 98.1 98.1
BY_SES 98.0 94.8 91.5 98.1 98.1
LMXL1PRO 75.8 73.5 71.0 76.8 76.8

MACHVF1 F1_MSE BY_MSE L1PROF BY_LM
MACHVF1 84.3
F1_MSE 64.0 64.0
BY_MSE 61.7 50.3 72.4
L1PROF 67.2 54.3 59.4 77.2
BY_LM 84.3 64.0 72.4 77.2 100.0
HISPANIC 84.3 64.0 72.4 77.2 100.0
ASIAN 84.3 64.0 72.4 77.2 100.0
BLACK 84.3 64.0 72.4 77.2 100.0
OTHER 84.3 64.0 72.4 77.2 100.0
BY_SEX 84.3 64.0 72.4 77.2 100.0
BY_SES 84.3 64.0 72.4 77.2 99.8
LMXL1PRO 67.2 54.3 59.4 77.2 77.2

HISPANIC ASIAN BLACK OTHER BY_SEX
HISPANIC 100.0
ASIAN 100.0 100.0
BLACK 100.0 100.0 100.0
OTHER 100.0 100.0 100.0 100.0
BY_SEX 100.0 100.0 100.0 100.0 100.0
BY_SES 99.8 99.8 99.8 99.8 99.8
LMXL1PRO 77.2 77.2 77.2 77.2 77.2

BY_SES LMXL1PRO
BY_SES 99.8
LMXL1PRO 77.2 77.2
Note. PMINOR1 = % racial/ethnic minority students; PG10ELL1 = % ELL sophomores;
PG10FRL1 = % sophomores eligible for free or reduced-price lunch; READ_ACH = grade 10
reading achievement, MACHV = grade 10 math achievement; MACHVF1 = grade 12 math
achievement; F1_MSE = grade 12 math self-efficacy; BY_MSE = grade 10 math self-efficacy;
L1PROF = L1 proficiency; BY_LM = language minority status; BY_SEX = gender; BY_SES =
socioeconomic status; LMXL1PRO = interaction between LM and L1PROF.
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171
Appendix K
Results of Logistic Multiple Regression Models Predicting Missingness

No. observations
Pseudo R-square
Odds Ratio S.E.
Student Level

English Reading Achievement
1

3610 0.3302

LM

0.78 1.08
L1PROF

783.38*** 1683.59
SES

1.29 1.18
Hispanic

4.94 6.69
Asian

5.43 9.65
MA12

0.87* 0.05
MSE12

1.14 0.17
Constant

0.02 0.05
Grade 10 Math Achievement
1

3610 0.3302

LM

0.78 1.08
L1PROF

783.38** 1683.59
SES

1.29 1.18
Hispanic

4.94 6.69
Asian

5.43 9.65
MA12

0.87* 0.05
MSE12

1.14 0.17
Constant

0.02 0.05
Grade 12 Math Achievement

9570 0.0678

L1PROF

1.68 0.87
LM

0.93 0.11
LMXL1PROF

1.33 0.76
SES

0.70*** 0.04
Gender

1.14* 0.08
Hispanic

1.04 0.11
Asian

0.93 0.14
Black

0.98 0.10
Other

1.55** 0.20
MA10

0.97*** 0.00
RA10

0.98** 0.00
MSE10

0.99 0.01
Constant

0.76 0.11

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172
Appendix K (Continuation)
No. observations Pseudo-R square Odds Ratio S.E.
Grade 10 Math Self-efficacy

8730 0.0408

LM

1.11 0.11
LMXL1PROF

1.58 0.97
L1PROF

1.04 0.60
SES

0.88** 0.04
Gender

1.46*** 0.08
Hispanic

1.35** 0.12
Asian

1.01 0.12
Black

1.79*** 0.15
Other

1.19 0.15
MA10

1.00 0.00
MA12

0.99 0.00
RA10

0.99** 0.00
MSE12

1.00 0.01
Constant

0.52 0.07
Grade 12 Math Self-efficacy

8430 0.0154

LM

0.95 0.10
LMXL1PROF

0.39 0.20
L1PROF

2.73* 1.16
SES

0.96 0.04
Gender

0.93 0.06
Hispanic

1.40** 0.14
Asian

1.24 0.15
Black

1.25* 0.12
Other

1.22 0.16
MA10

1.00 0.01
MA12

0.99 0.00
RA10

0.99* 0.00
MSE10

1.00 0.01
Constant

0.49 0.07
School Level

Grade 12 Math Self-efficacy

660 0.0409

% FRL

1.00 0.01
% MIN

1.00 0.01
% ELL

1.01 0.02
BMA10

0.82* 0.06
BMA12

1.16* 0.07
BMSE10

0.82* 0.08
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173
Appendix K (Continuation)
No. observations Pseudo-R square Odds Ratio S.E.
MRA10

1.01 0.07
No. observations Pseudo-R square Odds Ratio S.E.
Constant 1.44 2.26

Note. Number of observations was rounded to 10. The$letter$“B” denotes school level variables.
SES = socioeconomic status; RA10 = grade 10 English reading achievement; MA10 = grade 10
math achievement; MA12 = grade 12 math achievement; MSE12 = grade 12 math self-efficacy;
MSE10 = grade 10 math self-efficacy; LM = language minority status; L1PROF = L1
proficiency; LMXL1PROF = interaction between LM and L1PROF; ELL = English language
learner; FRL = eligible for free or reduce-price lunch; MIN = racial/ethnic minority; S.E.$=$
standard$error.$
1
LMXL1PROF, Gender, Black and Other were omitted from regression models predicting
missingness in grade 10 reading and math achievement because they perfectly predicted
missingness while adjusting for other variables.

*** p < .001, **p < .01, *p < .05

NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

174
Appendix L
Unweighted Student-Level Correlations

SES RA10 MA10 MA12 MSE12 MSE10
SES 1

16080

RA10 0.432*** 1

15800 15800

MA10 0.43*** 0.73*** 1

15800 15800 15800

MA12 0.44*** 0.72*** 0.91*** 1

13580 13330 13330 13580

MSE12 0.14*** 0.19*** 0.35*** 0.38*** 1

10310 10290 10290 10310 10310

MSE10 0.15*** 0.2*** 0.36*** 0.35*** 0.42*** 1

11650 11650 11650 9940 8100 11650
L1PROF -0.05*** -0.1*** -0.05*** -0.03*** 0.02* 0.01

12430 12370 12370 10830 8740 9570
LM -0.21*** -0.19*** -0.1*** -0.08*** -0.01 -0.01

16080 15800 15800 13580 10310 11650
LMXL1PROF -0.07*** -0.1*** -0.06*** -0.04*** 0.02 -0.003

12430 12370 12370 10830 8740 9570
HISP -0.23*** -0.19*** -0.2*** -0.19*** -0.04*** -0.04***

16080 15800 15800 13580 10310 11650
ASI -0.01 -0.01 0.1*** 0.11*** 0.02 0.05***

16080 15800 15800 13580 10310 11650
BLK -0.13*** -0.21*** -0.25*** -0.24*** -0.02 -0.03***

16080 15800 15800 13580 10310 11650
OTH -0.003 -0.01 -0.02** -0.02* -0.02 -0.02

16080 15800 15800 13580 10310 11650
WHT 0.27*** 0.29*** 0.26*** 0.23*** 0.04*** 0.03**

16080 15800 15800 13580 10310 11650
G 0.02** -0.07*** 0.06*** 0.07*** 0.1*** 0.11***

16080 15800 15800 13580 10310 11650
Note. Number of observations was rounded to the 10. SES = socioeconomic status; RA10 =
grade 10 English reading achievement; MA10 = grade 10 math achievement; MA12 = grade 12
math achievement; MSE12 = grade 12 math self-efficacy; MSE10 = grade 10 math self-efficacy;
L1PROF = L1 proficiency; LM = language minority status; LMXL1PROF = interaction term
between LM and L1PROF; HISP = Hispanic; ASI = Asian; BLK = Black; OTH = Other; WHT =
White; G = gender.
*** p < .001, ** p < .01,* p < .05

NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

175
Appendix M
Unweighted School-Level Correlations

%MIN %ELL %FRL BRA10 BMA10 BMA12 BMSE10 BMSE12
%MIN 1

15810

%ELL 0.44*** 1

15040 15290

%FRL 0.57*** 0.31*** 1

14520 14590 14760

BRA10 -0.52*** -0.35*** -0.6*** 1

15810 15230 14760 16110

BMA10 -0.51*** -0.26*** -0.62*** 0.89*** 1

15810 15290 14760 16110 16110

BMA12 -0.47*** -0.25*** -0.63*** 0.86*** 0.96*** 1

15810 15290 14760 16110 16110 16110

BMSE10 -0.1*** -0.09*** -0.27*** 0.33*** 0.37*** 0.37*** 1

15810 15290 14760 16110 16110 16110 16110

BMSE12 -0.14*** -0.12*** -0.24*** 0.34*** 0.38*** 0.4*** 0.41*** 1

14750 14290 13800 15040 15040 15040 15040 15040
Note. Number of observations was rounded to the 10. The letter “B” in front of a variable name
represents a school-level variable. % MIN = percentage of racial/ethnic minority students; %
ELL = percentage of ELL sophomores; % FRL = percentage of sophomores eligible for free or
reduce-price lunch; RA10 = grade 10 English reading achievement; MA10 = grade 10 math
achievement; MA12 = grade 12 math achievement; MSE12 = grade 12 math self-efficacy;
MSE10 = grade 10 math self-efficacy.
*** p < .001, ** p < .01,* p < .05

NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

176
Appendix N
Mplus Script for Final Weighted Multilevel Model (mod1)
TITLE:
two level model with L1 (recoded as l1prof,
which shows at least subsequent completion of grades 1-5 outside
US), and by_lm as main predictor.
INCLUDED STUDENT RACE/ETHNICITY COVARIATES: hispanic,
asian, black, other. white as reference. AND GENDER, SES
ADDED school level predictors.
ADDED SAMPLING WEIGHTS AT STUDENT AND SCHOOL LEVELS
used endogenous variables from student level

removed estimation of corr btw lm & l1prof, lm & asian, lm & hispanic,
l1prof & hispanic, l1prof & asian, ses & other, hispanic & ses,
ses & l1prof, black & ses, ses & lm
estimated mean and variance of l1prof b/c 3668 cases missing on x
variables were dropped.

multilev 8v1.0052915

Data:
File is C:\Users\polikoff\Desktop\Elena\ELS\subset8.dta.dat ;
Variable:
Names are
STU_ID STRAT_ID SCH_ID read_ach machv machvf1 bynonusg by_mse f1_mse
mmath_by mmath_f1 mmse_by mmse_f1 by_sex by_lm by_race black white
asian hispanic by_ses age_2002 nonusg byschwt rpanelwt mreadach
lmnonusg
pminor1 pg10ell1 pg10frl1 l1 muchl1 nol1 l1prof somel1 natprof noout
someout muchout other;

usevariables are
read_ach machv machvf1 f1_mse by_mse l1prof by_lm hispanic asian black
other
by_ses by_sex pminor1 pg10ell1 pg10frl1 STRAT_ID rpanelwt byschwt;

cluster is SCH_ID;

!adding within and between sampling weights
stratification = STRAT_ID;
!within level
weight = rpanelwt;
wtscale=ecluster;
!between level
bweight= byschwt;
bwtscale=sample;

between are pminor1 pg10ell1 pg10frl1 ;

within are
l1prof by_lm hispanic asian black other by_sex by_ses;

Missing are all (-9999) ;

NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

177
analysis:
type=twolevel complex;

model:
%within%
read_ach ON l1prof by_lm hispanic asian black other by_ses;
machv ON hispanic asian black other by_ses;
machvf1 ON machv by_mse read_ach by_ses by_sex;
f1_mse ON by_mse machv by_sex ;

!l1prof WITH by_lm; to see if lm variance problem gets solved
!l1prof WITH hispanic; hispanic variance problem
!l1prof WITH asian; asian variance problem

!by_lm WITH hispanic; covar hispanic & lm problem
!by_lm WITH asian; cov asian & lm problem

!by_ses WITH hispanic; problen w this covar
!by_ses WITH black; problem w this covar
!by_ses WITH other; problem w this covar
!by_ses WITH by_lm;
!by_ses WITH l1prof; ses variance problem

machv WITH read_ach;
machv WITH by_mse;
machvf1 WITH f1_mse;

[l1prof]; l1prof; !estimating these to include most cases in analysis

%between%
read_ach ON pg10frl1;
machv ON pg10frl1 pminor1;
machvf1 ON machv by_mse read_ach pminor1 pg10ell1 pg10frl1;
by_mse ON pminor1 pg10ell1;
f1_mse ON by_mse machv;

pg10frl1 WITH pminor1;
pg10frl1 WITH pg10ell1;
pminor1 WITH pg10ell1;

machv WITH read_ach;
machv WITH by_mse;
machvf1 WITH f1_mse;

output:
standardized sampstat res tech1;

NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

178
Appendix O
Mplus Script for Final Unweighted All Cases Multilevel SEM Model (mod2)
TITLE:
two level model with L1 (recoded as l1prof,
which shows at least subsequent completion of grades 1-5 outside
US), and by_lm as main predictor.
INCLUDED STUDENT RACE/ETHNICITY COVARIATES: hispanic,
asian, black, other. white as reference. AND GENDER, SES

ADDED school level predictors.
used endogenous variables from student level

removed estimation of corr btw lm & l1prof, lm & asian, lm & hispanic,
l1prof & hispanic, l1prof & asian, ses & other, hispanic & ses,
ses & l1prof, black & ses, ses & lm
estimated mean and variance of l1prof b/c 3668 cases missing on x
variables were dropped.

multilev 3_v1.29 052515

Data:
File is C:\Users\polikoff\Desktop\Elena\ELS\subset8.dta.dat ;
Variable:
Names are
STU_ID STRAT_ID SCH_ID read_ach machv machvf1 bynonusg by_mse f1_mse
mmath_by mmath_f1 mmse_by mmse_f1 by_sex by_lm by_race black white
asian hispanic by_ses age_2002 nonusg byschwt rpanelwt mreadach
lmnonusg
pminor1 pg10ell1 pg10frl1 l1 muchl1 nol1 l1prof somel1 natprof noout
someout muchout other;

usevariables are
read_ach machv machvf1 f1_mse by_mse l1prof by_lm hispanic asian black
other
by_ses by_sex pminor1 pg10ell1 pg10frl1;

cluster is SCH_ID;

between are pminor1 pg10ell1 pg10frl1 ;

within are
l1prof by_lm hispanic asian black other by_sex by_ses;

Missing are all (-9999) ;

analysis:
type=twolevel;

model:
%within%
read_ach ON l1prof by_lm hispanic asian black other by_ses;
machv ON hispanic asian black other by_ses;
machvf1 ON machv by_mse read_ach by_ses by_sex;
f1_mse ON by_mse machv by_sex ;
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

179

!l1prof WITH by_lm; to see if lm variance problem gets solved
!l1prof WITH hispanic; hispanic variance problem
!l1prof WITH asian; asian variance problem

!by_lm WITH hispanic; covar hispanic & lm problem
!by_lm WITH asian; cov asian & lm problem

!by_ses WITH hispanic; problen w this covar
!by_ses WITH black; problem w this covar
!by_ses WITH other; problem w this covar
!by_ses WITH by_lm;
!by_ses WITH l1prof; ses variance problem

machv WITH read_ach;
machv WITH by_mse;
machvf1 WITH f1_mse;

[l1prof]; l1prof; !estimating these to include most cases in analysis

%between%
read_ach ON pg10frl1;
machv ON pg10frl1 pminor1;
machvf1 ON machv by_mse read_ach pminor1 pg10ell1 pg10frl1;
by_mse ON pminor1 pg10ell1;
f1_mse ON by_mse machv;

pg10frl1 WITH pminor1;
pg10frl1 WITH pg10ell1;
pminor1 WITH pg10ell1;

machv WITH read_ach;
machv WITH by_mse;
machvf1 WITH f1_mse;

output:
standardized sampstat res tech1;

NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

180
Appendix P
Mplus Script for Unweighted Some Cases Multilevel Model (mod3)
TITLE:
two level model with L1 (recoded as l1prof,
which shows at least subsequent completion of grades 1-5 outside
US), by_lm and lmXl1prof as main predictor.
covariates at student level: race/ethnicity variables: hispanic, asian
black and other. white as reference, gender, and ses

ADDED school level predictors.
used endogenous variables from student level

removed estimation of corr btw lmXlprof and l1prof, and l1prof and asian,
and l1prof and hispanic, lm & asian, ses & black, ses & lmxl1prof,
ses & hispanic, lm & hispanic, ses & l1prof, ses & other, ses & lm
there are no exogenous correlations estimated

DID NOT INCLUDE CASES MISSING ON X VARIABLES

multilev 4_v1.12 052515

Data:
File is C:\Users\polikoff\Desktop\Elena\ELS\subset8.dta.dat ;
Variable:
Names are
STU_ID STRAT_ID SCH_ID read_ach machv machvf1 bynonusg by_mse f1_mse
mmath_by mmath_f1 mmse_by mmse_f1 by_sex by_lm by_race black white
asian hispanic by_ses age_2002 nonusg byschwt rpanelwt mreadach
lmnonusg
pminor1 pg10ell1 pg10frl1 l1 muchl1 nol1 l1prof somel1 natprof noout
someout muchout other;

usevariables are
read_ach machv machvf1 f1_mse by_mse l1prof by_lm hispanic asian
black other by_sex by_ses pminor1 pg10ell1 pg10frl1 lmXl1prof;

cluster is SCH_ID;

between are pminor1 pg10ell1 pg10frl1;

within are
l1prof by_lm hispanic asian
black other by_sex by_ses lmXl1prof;

Missing are all (-9999) ;

Define:
lmXl1prof = by_lm*l1prof;

analysis:
type=twolevel;

model:
%within%
NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

181
read_ach ON l1prof by_lm lmXl1prof hispanic asian black other by_ses;
machv ON hispanic asian black other by_ses;
machvf1 ON machv by_mse read_ach by_ses by_sex;
f1_mse ON by_mse machv by_sex;

!l1prof WITH by_lm;
!l1prof WITH lmXl1prof;
!by_lm WITH lmXl1prof;

!l1prof WITH hispanic; bc of hispanic variance problem
!l1prof WITH asian; bc of asian variance problem

!by_lm WITH hispanic; b/c of hispanic variance problem
!by_lm WITH asian; bc asian variance problem

! by_ses WITH hispanic; bc covar hispanic & ses problem
! by_ses WITH black; bc ses & black cov problem
!by_ses WITH other;

!by_ses WITH by_lm; problem w ses variance
!by_ses WITH l1prof; model estimate did not terminate normally
! by_ses WITH lmXl1prof; bc ses var problem

machv WITH read_ach;
machv WITH by_mse;
machvf1 WITH f1_mse;

%between%
read_ach ON pg10frl1;
machv ON pg10frl1 pminor1;
machvf1 ON machv by_mse read_ach pminor1 pg10ell1 pg10frl1;
by_mse ON pminor1 pg10ell1;
f1_mse ON by_mse machv;

pg10frl1 WITH pminor1;
pg10frl1 WITH pg10ell1;
pminor1 WITH pg10ell1;

machv WITH read_ach;
machv WITH by_mse;
machvf1 WITH f1_mse;

output:
standardized sampstat res tech1;

NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

182
Appendix Q
Mplus Script for Final Weighted Model with Mediation Paths Constrained to Zero (mod4)
two level model with L1 (recoded as l1prof,
which shows at least subsequent completion of grades 1-5 outside
US), and by_lm as main predictor.
INCLUDED STUDENT RACE/ETHNICITY COVARIATES: hispanic,
asian, black, other. white as reference. AND GENDER, SES
ADDED school level predictors.
ADDED SAMPLING WEIGHTS AT STUDENT AND SCHOOL LEVELS
used endogenous variables from student level

CONSTRAINED TO ZERO THE FOLLOWING PATHS TO CHECK MEDIATION:
lm->read->12 math, 10 math->12 mse

removed estimation of corr btw lm & l1prof, lm & asian, lm & hispanic,
l1prof & hispanic, l1prof & asian, ses & other, hispanic & ses,
ses & l1prof, black & ses, ses & lm
estimated mean and variance of l1prof b/c 3668 cases missing on x
variables were dropped.

multilev 8v1.1 052915

Data:
File is C:\Users\polikoff\Desktop\Elena\ELS\subset8.dta.dat ;
Variable:
Names are
STU_ID STRAT_ID SCH_ID read_ach machv machvf1 bynonusg by_mse f1_mse
mmath_by mmath_f1 mmse_by mmse_f1 by_sex by_lm by_race black white
asian hispanic by_ses age_2002 nonusg byschwt rpanelwt mreadach
lmnonusg
pminor1 pg10ell1 pg10frl1 l1 muchl1 nol1 l1prof somel1 natprof noout
someout muchout other;

usevariables are
read_ach machv machvf1 f1_mse by_mse l1prof by_lm hispanic asian black
other
by_ses by_sex pminor1 pg10ell1 pg10frl1 STRAT_ID rpanelwt byschwt;

cluster is SCH_ID;

!adding within and between sampling weights
stratification = STRAT_ID;
!within level
weight = rpanelwt;
wtscale=ecluster;
!between level
bweight= byschwt;
bwtscale=sample;

between are pminor1 pg10ell1 pg10frl1 ;

within are
l1prof by_lm hispanic asian black other by_sex by_ses;

NATIVE LANGUAGE, MATH, READING, SELF-EFFICACY

183
Missing are all (-9999) ;

analysis:
type=twolevel complex;

model:
%within%
read_ach ON l1prof hispanic asian black other by_ses;
machv ON hispanic asian black other by_ses;
machvf1 ON machv by_mse by_ses by_sex;
f1_mse ON by_mse by_sex ;

! constraining to zero indirect effects
read_ach ON by_lm@0;
machvf1 ON read_ach@0;
f1_mse ON machv@0;

!l1prof WITH by_lm; to see if lm variance problem gets solved
!l1prof WITH hispanic; hispanic variance problem
!l1prof WITH asian; asian variance problem

!by_lm WITH hispanic; covar hispanic & lm problem
!by_lm WITH asian; cov asian & lm problem

!by_ses WITH hispanic; problen w this covar
!by_ses WITH black; problem w this covar
!by_ses WITH other; problem w this covar
!by_ses WITH by_lm;
!by_ses WITH l1prof; ses variance problem

machv WITH read_ach;
machv WITH by_mse;
machvf1 WITH f1_mse;

[l1prof]; l1prof; !estimating these to include most cases in analysis

%between%
read_ach ON pg10frl1;
machv ON pg10frl1 pminor1;
machvf1 ON machv by_mse read_ach pminor1 pg10ell1 pg10frl1;
by_mse ON pminor1 pg10ell1;
f1_mse ON by_mse machv;

pg10frl1 WITH pminor1;
pg10frl1 WITH pg10ell1;
pminor1 WITH pg10ell1;

machv WITH read_ach;
machv WITH by_mse;
machvf1 WITH f1_mse;

output:
standardized sampstat res tech1;

Abstract (if available)

Abstract The under-preparation in math at the high school and college levels, as well as the low participation of ethnically and linguistically diverse individuals in STEM fields are concerning because their preparation for work in these areas is essential for the U.S. to remain competitive in the innovative knowledge economy. While there is now a substantial body of research on this group of students, there remain unresolved questions around the role of linguistic factors, affective variables, and prior achievement. In light of this concern, the purpose of the study was two-fold. One was to examine the moderating role of first language (L1) proficiency on the effects of language minority (LM) status in English reading. The second was to investigate the mediating roles of English reading and math achievement in the relationship between such interaction and math self-efficacy. The study was a secondary analysis of the Education Longitudinal Study (ELS 2002, n =16,110). Using a multilevel SEM analysis the study did not find support for the moderating role of L1 proficiency. However, English reading and math achievement mediated the relationship between LM status and math self-efficacy. These findings provide further knowledge for the development of targeted interventions that aim at increasing the preparation and participation of linguistically and ethnically diverse students in STEM fields.

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Asset Metadata

Creator Son, Elena (author)

Core Title Math achievement and self-efficacy of linguistically and ethnically diverse high school students: their relationships with English reading and native language proficiency

School Rossier School of Education

Degree Doctor of Philosophy

Degree Program Education

Publication Date 08/25/2015

Defense Date 07/27/2015

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Adolescents,English reading,math achievement,math self-efficacy,Motivation,native language,OAI-PMH Harvest,second language learning,STEM

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Rueda, Robert (committee chair), McArdle, John (committee member), Sinatra, Gale (committee member)

Creator Email eson@usc.edu,kuteme2@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-170967

Unique identifier UC11276264

Identifier etd-SonElena-3845.pdf (filename),usctheses-c40-170967 (legacy record id)

Legacy Identifier etd-SonElena-3845.pdf

Dmrecord 170967

Document Type Dissertation

Format application/pdf (imt)

Rights Son, Elena

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA