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Statistical Models of Identity and Self-Efficacy in Mathematics on A Statistical Models of Identity and Self-Efficacy in Mathematics on a National Sample of Black Adolescents from HSLS:09 Nathan Napoleon Alexander Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy under the Executive Committee of the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2015 © 2015 Nathan Napoleon Alexander All rights reserved ABSTRACT Statistical Models of Identity and Self-Efficacy in Mathematics on a National Sample of Black Adolescents from HSLS:09 Nathan Napoleon Alexander The research reported in this study examined statistical relations in black adolescents’ identity and self-efficacy beliefs in mathematics. Data for this research study were drawn from the High School Longitudinal Study of 2009 (HSLS:09; Ingels, Dalton, Holder, Lauff, & Burns, 2011) and the study’s first follow-up (Ingels & Dalton, 2013); additional measures were taken from the National Center for Education Statistics’ Common Core of Data (CCD). Data were analyzed using quantitative methods on a nationally representative sample of secondary school students (N = 1,362) across 944 schools in the United States. Although there has been an increase in qualitative research on mathematics identity and mathematics identity development, few researchers have utilized quantitative methods to empirically examine the relationships existing between identity and self-efficacy. Fewer researchers have used panel (longitudinal) data in their investigations. Findings from this study confirmed the literature in that mathematics identity development pathways are informed by students’ mathematics self-efficacy beliefs. Sex differences were also noted. Specifically, males and females experienced divergence in their mathematics identity and mathematics self-efficacy beliefs during high school; however, the returns of these beliefs on a measure of Algebraic proficiency for females were significantly greater than they were for males, although females maintained less positive beliefs over the course of the study. School belonging and engagement significantly predicted shifts in students’ mathematics identity development pathways and were moderated by self-efficacy beliefs, supporting theories that measures of perceived differentiation (e.g., belongingness) are key factors in student motivation and subsequent outcomes. Additional findings underscored the ongoing need for empirical research on students’ peer networks and mathematics teacher’s classroom practices. Overall, results of this study indicated that variations in identity development and self-efficacy beliefs among adolescents extend beyond many theoretical considerations in both their complexity and measured effects when accounting for a host of contextual and psychosocial factors. TABLE OF CONTENTS LIST OF FIGURES ............................................................................................................ ii LIST OF TABLES ............................................................................................................. iii LIST OF ABBREVIATIONS AND ACRONYMS .......................................................... iv ACKNOWLEDGEMENTS .................................................................................................v DEDICATION .................................................................................................................. vii PREFACE ........................................................................................................................ viii CHAPTER 1: INTRODUCTION ........................................................................................1 1.1 Need for the Study ...................................................................................................1 1.2 Purpose of the Study ................................................................................................5 1.3 Procedures of the Study ...........................................................................................7 1.4 Organization of the Study ........................................................................................8 1.5 Definition of Key Terms ..........................................................................................9 CHAPTER 2: REVIEW OF THE LITERATURE ............................................................10 2.1 Race and Mathematics Education ..........................................................................10 2.2 Identity and Identity Development in Mathematics ...............................................14 2.3 Self-efficacy and Mathematics Self-efficacy .........................................................25 2.4 Correlates of Identity and Self-efficacy in Mathematics .......................................38 2.5 The Mathematics Identity-Efficacy Connection ....................................................61 CHAPTER 3: METHODS .................................................................................................64 3.1 Data ........................................................................................................................64 3.2 Population and Sample ..........................................................................................66 3.3 Variables and Measures .........................................................................................68 3.4 Analysis..................................................................................................................78 3.5 Delimitations ..........................................................................................................79 CHAPTER 4: RESULTS OF THE STUDY ......................................................................81 4.1 Descriptive Analyses .............................................................................................89 4.2 Structural and Measurement Models .....................................................................93 4.3 Regression Models and Additional Tests ............................................................102 4.4 Longitudinal Latent Variable Models ..................................................................107 4.5 Limitations ...........................................................................................................109 CHAPTER 5: DISCUSSION AND CONCLUSION ......................................................110 5.1 Summary of Findings ...........................................................................................110 5.2 Discussion and Conclusion ..................................................................................116 REFERENCES ................................................................................................................125 APPENDICES .................................................................................................................140 LIST OF FIGURES Figure 1: Model of Self-efficacy and Learning Expectations Figure 2: Framework for the Mathematics Identity-Efficacy Connection Figure 3: Confirmatory Factor Analysis (CFA) Measurement Model for Wave 1 Identity and Self-efficacy Factors Figure 4: Confirmatory Factor Analysis (CFA) Measurement Model for Wave 2 Identity and Self-efficacy Factors Figure 5: Model 1: Reciprocal Effects Model (REM) of the Longitudinal Relation of Mathematics Self-efficacy and Mathematics Identity Figure 6: Model 2: Internal Reciprocal Effects Model (IREM) of the Longitudinal Relation of Mathematics Self-efficacy and Mathematics Identity Figure 7: Model 3: Internal Effect Model (IEM) of the Longitudinal Relation of Mathematics Self-efficacy and Mathematics Identity ii LIST OF TABLES Table 1: Study Variable Summary Statistics Table 2: Algebraic Proficiency Levels—HSLS: 2009, Wave 2 Table 3: Structural Equation Model (SEM) symbols and their meanings Table 4: Year of Algebra 1 Enrollment Table 5: Confirmatory Factor Loadings for Mathematics Identity and Mathematics Self- efficacy Factors Table 6: Factor Correlation Matrix Table 7: Seemingly Unrelated Regression Model Table 8: Parameter Estimates for the Reciprocal Effects Model of the Longitudinal Relation between Mathematics Self-efficacy and Mathematics Identity Table 9: Descriptive Statistics for School Belonging, School Engagement, Learning Environment, and Peer Network Measures Table 10: Evaluation of the Reciprocal Effects Model (REM): Longitudinal Relationship between Mathematics Self-efficacy and Mathematics Identity (Correlations, Means, and Standard Deviations) iii LIST OF ABBREVIATIONS AND ACRONYMS AGFI = Adjusted goodness of fit index AIC = Akaike information criterion BIC = Bayesian information criterion CFA = Confirmatory factor analysis CFI = Comparative fit index HSLS = High school longitudinal study IEM = Internal effects model IREM = Internal reciprocal effects model LVSM = Latent variable structural modeling MSE = Mathematics self-efficacy MID = Mathematics identity REM = Reciprocal effects model RMSEA = Root mean square error of approximation SEM = Structural equation modeling SRMR = Standardized root mean squared residual SUR = Seemingly unrelated regression W1STUDENT = HSLS student analytic weight (wave 1) W2W1STU = HSLS student analytic weight (wave 2) WLSMV = Weighted least squares means and variances WMW = Wilcox-Mann Whitney iv ACKNOWLEDGEMENTS It has been an honor and a privilege to conduct this research. I am beholden to all of the individuals who helped me reach this critical point in my career. I am especially grateful for the opportunity to connect my experiences to research
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