THE EFFECT OF TEACHERS’ SOCIAL NETWORKS ON TEACHING PRACTICES AND CLASS COMPOSITION By CHONG MIN KIM A DISSERTATION Submitted to Michigan State University In partial fulfillment of the requirements For the degree of DOCTOR OF PHILSOPHY Measurement and Quantitative Methods 2011 ABSTRACT THE EFFECT OF TEACHERS’ SOCIAL NETWORKS ON TEACHING PRACTICES AND CLASS COMPOSITION By CHONG MIN KIM Central to this dissertation was an examination of the role teachers’ social networks play in schools as living organizations through three studies. The first study investigated the impact of teachers’ social networks on teaching practices. Recent evidence suggests that teachers’ social networks have a significant effect on teachers’ norms, teachers’ learning in communities of practice, distributed leadership, the implementation of innovations, and students’ attainment including student learning and academic achievement in core subjects (Bryk & Schneider, 2002; Coburn & Russell, 2008; Frank et al., 2011; Spillane, 2006; Supovitz, Sirinides & May, 2010). Relatively little research, however, has been carried out on estimating the effect of teachers’ social networks on teaching practices. The results of the first study indicated that the formal organizational structure of the school and teachers’ social network structure at time 1 affect teachers’ social networks at time 2, which affect teachers’ teaching practices at time 3. In conclusion, the first study shows that teachers’ social networks can improve teaching practice by changing formal (grade) and informal (subgroup) structure. The second study explored the effect of teachers’ social networks on class composition. Previous studies show that class composition and peer effects have an important impact on students’ learning (Burns & Mason, 2002; Dreeben & Barr, 1988; Harris, 2010). Methodologically, Value-Added Models have often been used to estimate the teachers’ effects on student academic achievement with the assumption of random sampling and random assignment. Although there were studies about teachers’ assignment between schools (Lankford, Loeb & Wyckoff, 2002) and students’ assignment within schools (Rothstein, 2008), fewer studies have attempted to explore the effects of teachers’ social networks on class composition. The results of the second study indicated that teachers’ social networks affected class composition through non-random assignment of students to teachers with respect to students’ previous academic achievement as well as economic status. Thus, the second research shows that teachers’ social networks can indirectly affect students’ learning by influencing class composition with respect to previous academic achievement as well as economic status. Finally, in the third study, I quantify the robustness of the statistical inferences models in chapters 2 and 3 for valid causal inference. Generally, observational studies may have weak causal inference due to differences in unobserved preexisting conditions as well as time order of cause. By quantifying the impact threshold of a confounding variable, however, we can evaluate the sensitivity of causal claims to an unobserved confounding factor. Thus, this study evaluates Impact Threshold of a Confounding Variable (ITCV) to invalidate the causal inference in chapters 2 and 3. In spite of limitations including missing values, reliability and validity of network measurement, and a limited sample, these chapters offer significant insight into the role teachers’ social networks play in schools as organizations. Through a systematic analysis of the influence of these networks on key aspects of the student experience, this dissertation highlights the importance of teachers’ social networks for teacher behaviors and in learning contexts. Copyright by CHONG MIN KIM 2011 Dedication My dedication goes to my God, to my parents Soon Nyu Shin, Kwang Chul Kim, to my lovely wife Eun Joo, and to my precious daughter Claire Minjeoung. v Acknowledgements I would like to express my deepest and sincere gratitude to chair of both my guidance and dissertation committees, Dr. Kenneth A. Frank, for his endless help and encouragement throughout the duration of my doctoral studies. Without his insightful direction and expertise of social networks and casual inference, it would not have been possible for me to complete this work. Dr. Frank was continually devoted to providing critical and insightful comments and suggestions. It has been delightful for me to conduct various projects about social networks with him. It has been especially valuable for me to learn from his cutting-edge expertise and skills of statistical analyses. It has been a great honor to have him as my advisor. Of particular help in guiding me through my times as a doctoral student and in the completion of my dissertation has been my dissertation committee: Dr. James P. Spillane, Dr. Peter Youngs, Dr. Mark D. Reckase, and Dr. Barbara Schneider. I am especially indebted to Dr. Spillane for providing me with opportunities to expand my scope and knowledge of distributed leadership through valuable internships and projects as well as data for chapter 3 of my dissertation. I was very fortunate to be able to join the Distributed Leadership Study, which led to developing my interests in leadership theories and methods. I would also like to thank Dr. Youngs, who provided kind and detailed guidelines as well as comfort whenever I asked him about writing my dissertation. I have truly enjoy having discussions with him as a colleague and friend. I must thank Dr. Reckase for his generosity and his offering an interesting Value-Added Model course as well as other valuable measurement courses. Dr. Reckase’s courses provided me with learning and practicing measurement expertise and skills through innovative and vi challengeable assignments. It has been a great comfort for me to visit his office to see advice as though from a father. I am very grateful to Dr. Schneider for her valuable directions throughout the course of this study. I would like to express my deepest thanks to Dr. William Penuel and Min Sun. They are wonderful coworkers on catalyzing network expertise study (chapter 2 of my dissertation), funded by the National Science Foundation. In addition, thanks also go to my precious academic mentors: Dr. Sang-Jin Kang, Dr. Won-sup Jang, Dr. Gue-min Lee, and Dr. Kyung-Seok Min. Dr. Kang introduced me to hierarchical linear models including educational statistics studies at Yonsei University and helped me to start the doctoral program at Michigan State University. Dr. Jang also introduced me to social capital studies at Yonsei University. Dr. Lee and Dr. Min provided invaluable advices like big brothers. I would like to express thanks to my colleagues – Jonghwan Lee, Hyesuk Jang, Eun Hye Ham, Dr. Minh Duong, Ha Min Baik, Seung-Hwan Ham, Dr. Wang Jun Kim, Dr. Tae Seob Shin, Hosun Kang, Sun Hee Baik, Eui Jun Jeong, and Jeoung Jin Kang for their support and help. I owe thanks to colleagues at the Distributed Leadership Study in Northwestern University, especially Dr. Becca Lowenhaupt, Dr. Linda C. Lee, Kaleen Healey, and Allison Kenney. Allison read all the chapters of this dissertation and helped me revise my arguments. Finally, my acknowledgements go to my parents (-in-law) for their unconditional love and support and to my wife Eun Joo for her help and trust in me and to my precious daughter Claire Minjeoung. Also thanks to my entire family in Korea, especially my sister Hyun Joo and my brother Yong Gi for their emotional support. vii TABLE OF CONTENTS LIST OF TABLES ......................................................................................................................... xi LIST OF FIGURES ................................................................................................................... xviii Chapter 1: Theory of Teachers’ Social Networks ........................................................................... 1 Chapter 2: The Effect of Teachers’ Social Networks on Teaching Practices ................................ 12 Introduction ............................................................................................................................... 12 Literature Review ...................................................................................................................... 15 1. School and Teacher Effectiveness Studies: Human Capital or Social Capital................... 15 2. Teachers’ Social Networks ................................................................................................. 17 1) Effects on Human Capital or Social Capital................................................................... 17 2) Modeling: Selection or Influence ................................................................................... 18 3. Selection and Influence: Dynamic Modeling..................................................................... 19 Data and Methods ...................................................................................................................... 22 1. Data .................................................................................................................................... 22 2. Measures............................................................................................................................. 23 1) Dependent Variables ....................................................................................................... 23 2) Teacher Attributes ........................................................................................................... 25 3) Network Variables .......................................................................................................... 26 3. Methods .............................................................................................................................. 29 4. Models ................................................................................................................................ 30 1) Selection Model .............................................................................................................. 30 2) Influence Model .............................................................................................................. 33 3) Actor-Oriented Model ..................................................................................................... 35 Results ....................................................................................................................................... 41 1. Selection Model.................................................................................................................. 41 2. Influence Model ................................................................................................................. 44 1) Basic Statistics ................................................................................................................ 44 2) Regression coefficient for the Multilevel Influence Model ............................................ 46 3. Actor-Oriented Model ........................................................................................................ 48 1) Basic Statistics ................................................................................................................ 48 viii 2) Micro Level: Network Analysis ..................................................................................... 50 3) Macro Level: Combination of Networks ........................................................................ 52 Discussion and Conclusion ........................................................................................................ 56 Chapter 3: The Effect of Teachers’ Social Networks on Class Composition................................ 60 Introduction ............................................................................................................................... 60 Literature Review ...................................................................................................................... 63 1. The Impact of Peers and Class Composition on Students’ Learning ................................. 63 2. Class Composition ............................................................................................................. 65 1) Which Factors Affect Class Composition? ..................................................................... 65 2) How Do Principals Compose Classes? ........................................................................... 66 3. Value-Added Models (VAM) ............................................................................................. 68 Data and Methods ...................................................................................................................... 72 1. Data .................................................................................................................................... 72 2. Measures............................................................................................................................. 74 1) Dependent Variables ....................................................................................................... 74 2) Attributes Variables ......................................................................................................... 75 3) Network Variables .......................................................................................................... 76 3. Methods .............................................................................................................................. 78 4. Models ................................................................................................................................ 78 1) Model 1 ........................................................................................................................... 78 2) Model 2 ........................................................................................................................... 80 3) Model 3 ........................................................................................................................... 81 4) Model 4 ........................................................................................................................... 82 5) Model 5 ........................................................................................................................... 83 Results ....................................................................................................................................... 84 1. Heterogeneous Academic Achievement & Economic Status Within and Between Schools ................................................................................................................................................ 85 2. Association Between Teachers’ Social Networks and Their Students’ Previous Academic Achievement & Economic Status .......................................................................................... 86 3. The Effects of Teachers’ Social Networks and Attributes on Non-Random Assignment .. 88 1) Students’ Previous Academic Achievement ................................................................... 88 2) Students’ Previous Economic Status .............................................................................. 94 Discussion and Conclusion ...................................................................................................... 101 ix Chapter 4: Quantifying the Robustness of Inferences about the Effects of Teachers’ Social Networks on Class Composition................................................................................................. 106 Introduction ............................................................................................................................. 106 Literature Review .................................................................................................................... 108 1. Causal Inference Studies in Education ............................................................................. 108 2. Sensitivity Analysis and Robustness Indices (ITCV) ...................................................... 109 Data and Methods .....................................................................................................................112 Results ......................................................................................................................................113 1. Robustness Indices (ITCV) in p2 Selection Models .........................................................113 2. Robustness Indices (ITCV) in Multilevel Models ............................................................116 3. Robustness Indices (ITCV) in Multiple Regression Models ........................................... 120 Discussion and Conclusion ...................................................................................................... 124 Chapter 5: Policy for Teachers’ Social Networks ....................................................................... 126 Appendices .................................................................................................................................. 133 BIBLIOGRAPHY ....................................................................................................................... 195 x LIST OF TABLES Table 2.1 Characteristics of two school improvement paradigms ................................................ 15 Table 2.2 School demographic information 2007-08 ................................................................... 22 Table 2.3 Descriptive statistics of teacher demographics in 2007-08 ........................................... 23 Table 2.4 Items for mathematics problem solving teaching practices .......................................... 24 Table 2.5 Actor-oriented model components ................................................................................ 36 Table 2.6 Multilevel selection model for ten schools ................................................................... 41 Table 2.7 Variance components of unconditional model .............................................................. 44 Table 2.8 Descriptive statistics of multilevel influence model ..................................................... 45 Table 2.9 Correlation among level-one predictors ........................................................................ 46 Table 2.10 Regression coefficients (standard errors) for multilevel model of mathematics problem solving teaching practices including the influences of colleagues. ................................ 47 Table 2.11 Regression standardized coefficients for multilevel model of mathematics problem solving teaching practices including the subgroup mean of influences of colleagues. ................ 47 Table 2.12 Change in mathematics problem solving teaching practices ...................................... 49 Table 2.13 Change in mathematics teaching practices advice networks ...................................... 49 Table 2.14 Effects estimates (standard errors) in mathematics teaching practices advice networks and mathematics teaching practices .............................................................................................. 51 Table 2.15 the mean and variance of estimates in meta-analysis ................................................. 53 Table 2.16 Results comparison among P2, HLM, and SIENA ..................................................... 54 Table 3.1 Summary of theories and implications.......................................................................... 63 Table 3.2 Strategies principals use to assign students to class and number of principals reporting xi their use ......................................................................................................................................... 67 Table 3.3 School and student characteristics in 30 elementary schools in 2006~2007 ................ 72 Table 3.4 Teacher characteristics in 30 elementary schools in 2006~2007 .................................. 73 Table 3.5 Descriptive statistics of teachers with at least 10 students except the first grade ......... 84 Table 3.6 Two-level (classes nested in schools) unconditional models ........................................ 85 Table 3.7 Correlation matrix ......................................................................................................... 87 Table 3.8 Effects of teachers’ ELA or Math networks on students’ previous ELA achievement . 89 Table 3.9 Effects of teachers’ combined networks on students’ ELA previous achievement ....... 90 Table 3.10 Effects of teachers’ ELA or Math networks on students’ previous Math achievement ....................................................................................................................................................... 92 Table 3.11 Effects of teachers’ combined networks on students’ math previous achievement ..... 93 Table 3.12 Effects of teachers’ ELA or Math networks on students’ previous free/reduced lunch ....................................................................................................................................................... 95 Table 3.13 Effects of teachers’ combined networks on students’ previous free/reduced lunch .... 96 Table 3.14 Effects of teachers’ social networks on class composition in model 1 to model 5 ..... 97 Table 3.15 Adjusted R-square in model 1 to model 5 ................................................................... 98 Table 3.16 Adjusted R-square change in model 1 to model 5 ...................................................... 99 Table 4.1 Regression coefficient (standard error) of selection models ........................................113 Table 4.2 Impact threshold of a confounding variable (ITCV) in selection models ....................115 Table 4.3 Regression standardized coefficients for multilevel model of mathematics problem solving teaching practices including the influences of colleagues. .............................................117 Table 4.4 Impact threshold of a confounding variable (ITCV) in influence models ...................119 Table 4.5 Corrected impact threshold of a confounding variable (ITCV) with other covariates in xii influence models ......................................................................................................................... 120 Table 4.6 Regression coefficients (t-ratio) of teachers’ social networks in model 1, 4, and 5.... 120 Table 4.7 Impact threshold of a confounding variable (ITCV) in multiple regression models .. 122 Table 4.8 Corrected impact threshold of a confounding variable (ITCV) with other covariates 123 Table 5.1 The hypotheses, results, and robustness indices of this dissertation. .......................... 126 Table A.1 Model 1 in effects of teachers' ELA networks on students' previous ELA achievement ..................................................................................................................................................... 134 Table A.2 Model 2 in effects of teachers' ELA networks on students' previous ELA achievement ..................................................................................................................................................... 135 Table A.3 Model 3 in effects of teachers' ELA networks on students' previous ELA achievement ..................................................................................................................................................... 136 Table A.4 Model 4 in effects of teachers' ELA networks on students' previous ELA achievement ..................................................................................................................................................... 137 Table A.5 Model 5 in effects of teachers' ELA networks on students' previous ELA achievement ..................................................................................................................................................... 138 Table A.6 Model 1 in effects of teachers' Math networks on students' previous ELA achievement ..................................................................................................................................................... 139 Table A.7 Model 2 in effects of teachers' Math networks on students' previous ELA achievement ..................................................................................................................................................... 140 Table A.8 Model 3 in effects of teachers' Math networks on students' previous ELA achievement ..................................................................................................................................................... 141 Table A.9 Model 4 in effects of teachers' Math networks on students' previous ELA achievement ..................................................................................................................................................... 142 Table A.10 Model 5 in effects of teachers' Math networks on students' previous ELA achievement ................................................................................................................................ 143 Table A.11 Model 1 in effects of teachers' combined networks on students' previous ELA achievement ................................................................................................................................ 144 xiii Table A.12 Model 2 in effects of teachers' combined networks on students' previous ELA achievement ................................................................................................................................ 145 Table A.13 Model 3 in effects of teachers' combined networks on students' previous ELA achievement ................................................................................................................................ 146 Table A.14 Model 4 in effects of teachers' combined networks on students' previous ELA achievement ................................................................................................................................ 147 Table A.15 Model 5 in effects of teachers' combined networks on students' previous ELA achievement ................................................................................................................................ 148 Table A.16 Model 1 in effects of teachers' ELA networks on students' previous Math achievement ................................................................................................................................ 149 Table A.17 Model 2 in effects of teachers' ELA networks on students' previous Math achievement ................................................................................................................................ 150 Table A.18 Model 3 in effects of teachers' ELA networks on students' previous Math achievement ................................................................................................................................ 151 Table A.19 Model 4 in effects of teachers' ELA networks on students' previous Math achievement ................................................................................................................................ 152 Table A.20 Model 5 in effects of teachers' ELA networks on students' previous Math achievement ................................................................................................................................ 153 Table A.21 Model 1 in effects of teachers' Math networks on students' previous Math achievement ................................................................................................................................ 154 Table A.22 Model 2 in effects of teachers' Math networks on students' previous Math achievement ................................................................................................................................ 155 Table A.23 Model 3 in effects of teachers' Math networks on students' previous Math achievement ................................................................................................................................ 156 Table A.24 Model 4 in effects of teachers' Math networks on students' previous Math achievement ................................................................................................................................ 157 Table A.25 Model 5 in effects of teachers' Math networks on students' previous Math achievement ................................................................................................................................ 158 xiv Table A.26 Model 1 in effects of teachers' combined networks on students' previous Math achievement ................................................................................................................................ 159 Table A.27 Model 2 in effects of teachers' combined networks on students' previous Math achievement ................................................................................................................................ 160 Table A.28 Model 3 in effects of teachers' combined networks on students' previous Math achievement ................................................................................................................................ 161 Table A.29 Model 4 in effects of teachers' combined networks on students' previous Math achievement ................................................................................................................................ 162 Table A.30 Model 5 in effects of teachers' combined networks on students' previous Math achievement ................................................................................................................................ 163 Table A.31 Model 1 in effects of teachers' ELA networks on students' previous free/reduced lunch............................................................................................................................................ 164 Table A.32 Model 2 in effects of teachers' ELA networks on students' previous free/reduced lunch............................................................................................................................................ 165 Table A.33 Model 3 in effects of teachers' ELA networks on students' previous free/reduced lunch............................................................................................................................................ 166 Table A.34 Model 4 in effects of teachers' ELA networks on students' previous free/reduced lunch............................................................................................................................................ 167 Table A.35 Model 5 in effects of teachers' ELA networks on students' previous free/reduced lunch............................................................................................................................................ 168 Table A.36 Model 1 in effects of teachers' Math networks on students' previous free/reduced lunch............................................................................................................................................ 169 Table A.37 Model 2 in effects of teachers' Math networks on students' previous free/reduced lunch............................................................................................................................................ 170 Table A.38 Model 3 in effects of teachers' Math networks on students' previous free/reduced lunch............................................................................................................................................ 171 Table A.39 Model 4 in effects of teachers' Math networks on students' previous free/reduced lunch............................................................................................................................................ 172 Table A.40 Model 5 in effects of teachers' Math networks on students' previous free/reduced xv lunch............................................................................................................................................ 173 Table A.41 Model 1 in effects of teachers' combined networks on students' previous free/reduced lunch............................................................................................................................................ 174 Table A.42 Model 2 in effects of teachers' combined networks on students' previous free/reduced lunch............................................................................................................................................ 175 Table A.43 Model 3 in effects of teachers' combined networks on students' previous free/reduced lunch............................................................................................................................................ 176 Table A.44 Model 4 in effects of teachers' combined networks on students' previous free/reduced lunch............................................................................................................................................ 177 Table A.45 Model 5 in effects of teachers' combined networks on students' previous free/reduced lunch............................................................................................................................................ 178 Table A.46 Model 1 in effects of teachers' attributes on students' previous ELA achievement . 179 Table A.47 Model 2 in effects of teachers' attributes on students' previous ELA achievement . 180 Table A.48 Model 3 in effects of teachers' attributes on students' previous ELA achievement . 181 Table A.49 Model 4 in effects of teachers' attributes on students' previous ELA achievement . 182 Table A.50 Model 5 in effects of teachers' attributes on students' previous ELA achievement . 183 Table A.51 Model 1 in effects of teachers' attributes on students' previous Math achievement. 184 Table A.52 Model 2 in effects of teachers' attributes on students' previous Math achievement . 185 Table A.53 Model 3 in effects of teachers' attributes on students' previous Math achievement . 186 Table A.54 Model 4 in effects of teachers' attributes on students' previous Math achievement. 187 Table A.55 Model 5 in effects of teachers' attributes on students' previous Math achievement . 188 Table A.56 Model 1 in effects of teachers' attributes on students' previous free/reduced lunch 189 Table A.57 Model 2 in effects of teachers' attributes on students' previous free/reduced lunch 190 Table A.58 Model 3 in effects of teachers' attributes on students' previous free/reduced lunch 191 xvi Table A.59 Model 4 in effects of teachers' attributes on students' previous free/reduced lunch 192 Table A.60 Model 5 in effects of teachers' attributes on students' previous free/reduced lunch 193 xvii LIST OF FIGURES Figure 1.1. The relationship among the formal structure of school and teachers’ social networks. 2 Figure 1.2. The relationship among teachers’ network structure and teachers’ social networks. ... 3 Figure 1.3. The effects of teachers’ social networks on teachers’ human capital and students’ formal structure. .............................................................................................................................. 4 Figure 1.4. The relationship among teachers’ human capital, teachers’ social networks, and structure........................................................................................................................................... 5 Figure 1.5. The relationship among students’ human capital, structure, and social networks. ....... 5 Figure 1.6. The relationship among human capital, structure, and social networks. ...................... 6 Figure 1.7. Three models of selection, influence, and actor-oriented in chapter 2. ........................ 8 Figure 1.8. The relationship among teachers’ attributes, teachers’ social networks, and class composition in chapter 3. ................................................................................................................ 9 Figure 1.9. The structure of dissertation: theories, models, causal Inferences, and policy. ...........11 Figure 2.1. The structure of variables in multilevel p2 selection modeling ................................. 30 Figure 2.2. The structure of variables in multilevel influence modeling ...................................... 34 Figure 5.1. Conceptual framework of this study......................................................................... 129 Figure 5.2. The relationship between human capital and social networks at different level ...... 131 xviii Chapter 1: Theory of Teachers’ Social Networks Salancik (1995) pointed out that there was a need for a good network theory of organization and suggested that “A network theory should do either of two things: (1) It should propose how adding or subtracting a particular interaction in an organizational network will change coordination among actors in the network; or (2) It should propose how a network structure enables and disenables the interactions between two parties” (p. 348). After criticism by Salancik (1995), some researchers tried to summarize the relevant network theories in communication (Contractor, Wasserman & Faust, 2006; Monge & Contractor, 2003) as well as organization (Kilduff & Tsai, 2003). With respect to building network theories in organization, Kilduff & Tsai (2003) classified three strategies which previous researchers have used. The first strategy is to import graph theory from mathematics and balance theory from social psychology. The second strategy is to use home-grown concepts such as the strength of weak ties (Granovetter, 1973) or structural role theory (Burt, 1992). The third strategy is to export network concepts into previous organizational theories such as resource dependence (Pfeffer & Salanick, 1978) or transaction cost economics (Williamson, 1981). With respect to building network theories in education, educational researchers have focused on social capital theory (Daly, 2010; Finnigan & Daly, 2010; Frank, Zhao, & Borman, 2004; Moolenaar & Sleegers, 2010; Penuel et. al, 2009, 2011) and have emphasized the importance of teachers’ social capital. A recent study by Häuberer (2011) pointed out the weakness of Bourdieu’s, Coleman’s, Putnam’s, Burt’s and Lin’s approach to social capital and proposed a formalized concept of social capital. This researcher insisted that a social capital concept applies to a hierarchically structured society with the key notion of “a resource embedded in social relationships.” 1 Like the above definition of social capital, teachers’ social capital should include teachers’ social networks. Even if the teachers’ social capital is very important, we can’t improve teachers’ social capital without changing teachers’ social networks because the core element of social capital is social networks. Even though many researchers have used social capital theories to build social networks theories in education, they did not build specific social network theories which are appropriate in elementary school contexts. To propose and test specific social network theories in elementary school, this dissertation proposes hypotheses which investigate the relationship among social networks, structure, hierarchy and time when we examine the effects of teachers’ social networks. Hypothesis 1-1: Previous formal organizational structure at a higher level (level 2) affects the formation of new ties of social networks at a lower level (level 1), as shown in Figure 1.1. For example, the formal organizational structure of the school (e.g., grade level) affects the formation of new ties in teachers’ social networks (See chapter 2). If we can change the formal organizational structure at higher levels, we can change the formation of new ties in social networks at lower levels. Structure Hierarchy &Time Formal Structure of School (e.g., Grade level) School Time 1 Social Networks Teachers’ Social Networks Teacher Time 2 Figure 1.1. The relationship among the formal structure of school and teachers’ social networks. 2 Hypothesis 1-2: Previous network structure at a higher level (level 2) affects the formation of new ties of social networks at a lower level (level 1), as shown in Figure 1.2. For example, teachers’ formal or informal network structure (e.g. formal grade level meeting or informal cohesive subgroup) at time 1 affect the formation of new ties in teachers’ social networks at time 2 (See chapter 2). If we can change the formal network structure of higher levels (e.g. formal cross grade level meeting) in 2010, we can change the formation of new ties in social networks of lower levels in 2011. Social Networks Structure Teachers’ Network Structure (e.g., cohesive subgroups) Teachers’ Social Networks Hierarchy &Time Teacher Time 1 Teacher Time 2 Figure 1.2. The relationship among teachers’ network structure and teachers’ social networks. Hypothesis 2-1: Social networks at level 3 in time 1 affect human capital at level 2 in time 2, as shown in Figure 1.3. In other words, teachers’ social network at level 3 in 2010 can affect teachers’ human capital at level 2 in 2011 (See chapter 2). Hypothesis 2-2: Previous social networks at a higher level (level 2) affect current formal organizational structure at a lower level (level 1), as shown in Figure 1.3. For example, teachers’ social networks at time 1 would influence students’ formal organizational structure (e.g. class composition) at time 2 (See chapter 3). If we can change the social networks at level 2 at time 1, we can change the formal organizational structure at level 1 at time 2. 3 Human Capital Social Networks Structure Hierarchy & Time Teachers’ Social Networks Teacher Time 1 Teachers’ Human Capital (e.g., teaching practices) Teacher Time 2 Students’ Formal Structure (e.g., class composition) Student Time 2 Figure 1.3. The effects of teachers’ social networks on teachers’ human capital and students’ formal structure. Hypothesis 3-1: Through hypothesis 1-1, 1-2, and 2-1, previous formal organizational structure and formal or informal network structure at level 3 in time 1 affect the human capital at level 1 from a coevolution perspective. As shown in Figure 1.4, the formal organizational structure of school at level 3 in 2009 can affect teachers’ social network at level 2 in 2010 (hypothesis 1-1) and teachers’ formal or informal network structure at level 3 in 2009 can affect teachers’ social network at level 2 in 2010 (hypothesis 1-2), which can affect teachers’ human capital at level 1 in 2011 (hypothesis 21). Hypothesis 3-2: Through hypothesis 1-1, 1-2, and 2-2, previous formal organizational structure and formal or informal network structure at level 3 in time 1 affect the formal organizational structure of level 1 from a coevolution perspective. As shown in Figure 1.4, the formal organizational structure of the school at level 3 in 2009 and teachers’ formal or informal network structure at level 3 in 2009 can affect teachers’ social network at level 2 in 2010, which can affect students’ formal organizational structure at level 1 in 2011 (hypothesis 2-2). 4 Social Networks Structure Hierarchy & Time Formal Structure of School & Teachers’ Network Structure Human Capital School Time 1 Teachers’ Social Networks Teacher Time 2 Teachers’ Human Capital (e.g., teaching practices) Teacher Time 3 Students’ Formal Structure Student (e.g., class composition) Time 3 Figure 1.4. The relationship among teachers’ human capital, teachers’ social networks, and structure. Hypothesis 3-1 can be applied to students as shown in Figure 1.5. Instead of testing the hypothesis, this dissertation presented relevant previous results (See chapter 3). Human Capital Social Networks Structure Students’ Formal Structure (e.g., class composition) Students’ Social Networks Students’ Human Capital (e.g., achievement) Hierarchy & Time Student Time 1 Student Time 2 Student Time 3 Figure 1.5. The relationship among students’ human capital, structure, and social networks. Hypothesis 4: Human capital at a higher level (level 3) and time 2 can affect human capital at a lower level (level 1). Instead of testing hypothesis 4, this dissertation presents relevant previous results of school and teacher effectiveness studies (See chapter 2). 5 Taken together, the formal organizational structure of the school and teachers’ network structure can affect teachers’ social networks (Hypothesis 1-1 and 1-2), which can directly affect not only teachers’ human capital (Hypothesis 2-1) but also students’ formal structure (Hypothesis 2-2). Indirectly, teachers’ social network can affect students’ human capital through changing students’ social network (Hypothesis 3-1) as well as changing teachers’ human capital (Hypothesis 4). Social Networks Structure Hierarchy & Time Formal Structure of School & Teachers’ Network Structure Human Capital School Time 1 Teachers’ Social Networks Teacher Time 2 Teachers’ Human Capital (e.g., teaching practices) Teacher Time 3 Students’ Formal Structure (e.g., class composition) Students’ Social Networks Student Time 3 Student Time 4 Students’ Human Capital (e.g., achievement) Student Time 5 Figure 1.6. The relationship among human capital, structure, and social networks. Specifically, in chapter 2, first, I examine whether or not the formal organizational structure of school and teachers’ network structure can affect teachers’ social networks (Hypothesis 1-1 and 1-2) through selection models. Second, I test whether or not teachers’ social 6 networks can directly affect teaching practices (Hypothesis 2-1) through influence and dynamics models. Numerous researchers have found that the quality of teaching has an important impact on students’ learning (Brophy & Good, 1986; Kyriakides et al., 2008; Nye et al., 2004; Teddlie & Reynolds, 2000). In recent years, many studies have reported positive outcomes of teachers’ social networks as well, including teachers’ norms (Bryk & Schneider, 2002); teachers’ learning in communities of practice (Coburn & Russell, 2008); distributed leadership (Spillane, 2006); implementation of innovations (Frank, Zhao & Borman, 2004; Frank et al., 2011; Penuel et al., 2007, 2009); and students’ attainment, students’ learning, and academic achievement in core subjects (Hadfield & Jopling, 2007; Supovitz, Sirinides & May, 2010). Methodologically, however, previous studies have not estimated the dynamic interplay of teachers’ social networks and teaching practices. Thus, the study in chapter 2 takes into account models of selection (the pattern of relations in a social network as a function of attributes of people) and influence (attributes of people as a function of relations in the social networks) in one dynamic model. The main research questions are: How do the mathematics teaching practices advice network and mathematics teaching practices change over two years? What can explain these changes? Three models of selection, influence, and actor-oriented models are shown in Figure 1.7. Previous studies have investigated the effect of teachers’ attributes (e.g., efficacy) on teaching practices without considering teachers’ networks. Newer influence models have studied the effect of teachers’ networks on teaching practices, which suggest that there are significant effects of teachers’ networks on teaching practices after controlling for teachers’ attributes (e.g., Frank et al., 2004). In addition, selection models have examined the effect of teachers’ attributes on teachers’ networks, which have shown which characteristics of actors are related to the formation 7 of teachers’ networks (e.g., Frank, 2009). Teachers’ Attributes Selection models Previous studies Teaching Practices Teachers’ Networks Influence Models Selection + Influence = Actor-oriented Models Figure 1.7. Three models of selection, influence, and actor-oriented in chapter 2. Therefore, teachers’ networks could be a dependent or independent variable depending on whether the model focuses on selection or influence. In order for both teachers’ networks and teaching practices to interchange roles as dependent and independent variables, actor-oriented models have investigated both the change of teachers’ networks and teaching practices. I investigate another aspect of schooling influenced by teachers’ social networks. In chapter 3, I test whether or not teachers’ social networks can affect class composition (Hypothesis 2-2) because classroom composition and peer effects influence students’ learning (Burns & Mason, 2002; Dreeben & Barr, 1988; Harris, 2010). Methodologically, Value-Added Models have been used to estimate teachers’ effects on student academic achievement relying on the assumption of random sampling and random assignment. Although some studies have explored teachers’ assignment between schools (Jackson, 2009; Lankford, Loeb & Wyckoff, 2002; Miller, 2009) and students’ assignment within schools (Koedel & Betts, 2009; Monk, 1987; Rothstein, 2008), little is understood about the mechanisms of assignment of teachers to students that account for teachers’ social networks. In this chapter, I analyze the effect of teachers’ social networks on students’ assignment with respect to students’ previous academic achievement in core subjects (English/language arts 8 and mathematics) as well as students’ previous economic status in elementary schools. The main research question is: Do teachers’ social networks affect class composition through non-random assignment of students to teachers with respect to students’ previous academic achievement and economic status? The relationships among teachers’ attributes, teachers’ social networks, and class composition are shown in Figure 1.8. Few previous studies have examined the effect of teachers’ attributes (e.g., teaching experience) and social networks (e.g., advice networks) on class composition with respect to students’ previous academic achievement and economic status. Thus, first, this study investigated the effect of teachers’ attributes on class composition. Second, the effects of teachers’ social networks on class composition are presented after controlling for teachers’ attributes. Teachers’ Attributes Teachers’ Social Networks Class Composition Figure 1.8. The relationship among teachers’ attributes, teachers’ social networks, and class composition in chapter 3. This study is significant in that it informs whether or not teachers use their social networks to affect class composition. This information will be important when we test whether or not teachers’ social network can indirectly affect students’ human capital through changing students’ social network. In chapter 4, I quantify the robustness of the statistical inferences models in chapters 2 and 3 for valid causal inference. Previous studies present three conditions for valid causal 9 inference which are 1) there is an association between cause and effect, 2) cause preexists before effect, and 3) there is no confounding variable which affect cause and effect. Generally, observational studies may have weak causal inference due to differences in unobserved preexisting conditions. By quantifying the impact threshold of a confounding variable, however, we can evaluate the sensitivity of causal claims to an unobserved confounding factor. Thus, this study evaluates the Impact Threshold of a Confounding Variable (ITCV) to invalidate the causal inference in chapters 2 and 3. This dissertation consists of the three studies described above, as well as a concluding chapter. The chapters were organized similarly as follows. Chapter 2 describes the first study. In this chapter, I first introduce school effectiveness and social network studies. Also, the dynamic of teachers’ social networks and behavior are presented with a focus on empirical studies using actor-oriented models. Second, data and method are presented including sample, dependent variables, independent variables, selection models, influence models and actor-oriented models. Third, the results of selection, influence and actor-oriented models are shared. Finally, the discussion and conclusion are presented. In chapter 3, I discuss the second study. In this chapter, class composition studies and value-added models are introduced. Second, I present my data and methods including sample, dependent variables, and independent variables, including teachers’ attributes and social networks. Third, the results about relationships between teachers’ social networks and class composition are presented. Finally, the discussion and conclusion are shared. In chapter 4, I present the third study. Causal inference and robustness indices studies are offered. In this chapter, the same data and measures as chapter 2 and 3 are used. Second, the 10 results of ITCV in chapter 2 and 3 models are presented. Finally, the discussion and conclusion are included. In the concluding chapter 5, policies for teachers’ social networks are suggested as they relate to teachers and students in instruction and learning contexts; (1) the policy of organizational structure of school such as grade and class formation and (2) the policy of formal network structure such as grade level meetings. In summary, the purposes of this dissertation are to test hypotheses regarding teachers’ social networks through three empirical studies. In addition, the structure of this dissertation is shown as Figure 1.9. Social Networks Theory Chapter 1 Network Structure Hierarchy Time Social Networks Models Chapter 2 & 3 Selection Influence Dynamic Social Networks Causal Inferences Chapter 4 Cause preceded effect Association No confounding variables Social Networks Policy Chapter 5 Policy of formal organizational & network Structure Figure 1.9. The structure of dissertation: theories, models, causal Inferences, and policy. Taken together, these chapters offer significant insight into the role teachers’ social networks play in schools as living organizations. Through a systematic analysis of the influence of these networks on key aspects of the student experience, this dissertation highlights the importance of teachers’ social networks for teacher behaviors and in learning contexts. 11 Chapter 2: The Effect of Teachers’ Social Networks on Teaching Practices Introduction How can we improve student achievement? Over the last two decades, school effectiveness studies have emphasized the classroom effect relative to school effects in elucidating variation on student achievement in affective as well as cognitive outcomes (Teddlie & Reynolds, 2000). Moreover, the most important factor at the classroom level is quality of teaching practices (Brophy & Good, 1986; Nye et al., 2004). In addition, studies about teachers’ cognition have emphasized teachers’ knowledge, beliefs, and decision-making (Calderhead, 1996). Specifically, some studies identified the effect of teachers’ knowledge (e.g., pedagogical or content knowledge) on gains in academic achievement (Hill, Rowan & Ball, 2005; Kennedy, Ahn & Choi, 2008). We can improve teachers’ knowledge to improve teaching practices. Recent studies have emphasized not only the role of professional development (Darling-Hammond et al., 2009; Desimone, 2009; Guskey & Yoon, 2009; Heck et al., 2008; Ingvarson et al., 2005; Wayne et al., 2008), but also the importance of teachers’ social capital (Daly, 2010; Finnigan & Daly, 2010; Frank, Zhao, & Borman, 2004; Moolenaar & Sleegers, 2010; Penuel et. al, 2009, 2011). Social network researchers suggest that teachers’ social networks have a significant effect on teachers’ norms, teachers’ learning in communities of practice, distributed leadership, and the implementation of innovations (Bryk & Schneider, 2002; Coburn & Russell, 2008; Penuel et al., 2007; Spillane, 2006). Specifically, recent studies pointed out peer’s influence on teaching practices through social interactions (Sun & Frank, 2011) and human capital spillovers (Jackson & Bruegmann, 2009). In other words, teachers’ social networks can affect teachers’ human 12 capital (e.g., teaching practices) as well as social capital (e.g., trust). How can we estimate the effect of teachers’ social networks? Social network analysis pertains not only to selection modeling of the pattern of relations in a social network as a function of attributes of people, but also to influence modeling of attributes of people as a function of relations in the social network (Frank, 1998). Selection modeling can be implemented through qualitative studies or statistical models such as p1, p2, or p*. In addition, influence modeling can be implemented through qualitative studies or statistical modeling such as multilevel models. There have been considerable advances in estimating the effects on and of teachers’ social networks through selection and influence modeling, but the previous studies were confined to estimate two models separately (Penuel et al., 2007, 2008). Actor-oriented models, however, can be evaluated for co-evolving social networks and individual behaviors (Bunt & Groenewegen, 2007; Burk et al., 2007; Pearson et al., 2006; Snijders, 2001; Steglich et al., 2006, 2010). These models can be estimated through SIENA (Simulation Investigation for Empirical Network Analysis). Although are were a few studies using dynamic modeling in education (Daly & Finnigan, 2011; Orlina, 2010), little research has been carried out to estimate the effects of teachers’ social networks and teaching behavior simultaneously in education by using longitudinal data. I will do so in this study. Thus, the purpose of this study is to examine the effects of teachers’ social networks on teaching practices through selection, influences and dynamic modeling. This chapter is organized as follows. First, I present school and teacher effectiveness studies briefly and I introduce social network studies especially regarding teachers’ social networks. Also, I present dynamic models of teachers’ social networks and behavior with a focus 13 on empirical studies using actor-oriented models in other fields as theoretical background. Second, data and methods are presented including sample, dependent variables, independent variables, selection model, influence model and actor-oriented models. Third, the results of selection, influence and actor-oriented model are shared. Finally, the discussion and conclusion of the dynamics of teaching practices and advice networks are presented. 14 Literature Review 1. School and Teacher Effectiveness Studies: Human Capital or Social Capital In school effectiveness studies, emphasis has shifted from school effects to classroom effects especially concerning the quality of teaching practices in theoretical models (Kyriakides et al., 2008). Much research has documented that in explaining variation on student achievement in both cognitive and affective outcomes, the classroom effect is more fundamental than the school effect (Teddlie & Reynolds, 2000). Furthermore, quality of teaching practices is the most significant factor at the classroom level (Brophy & Good, 1986). Teddlie and Reynolds (2000) described this paradigm shift in Table 2.1. Table 2.1 Characteristics of two school improvement paradigms Characteristics 1960s Orientation Top down Knowledge Base Elite knowledge Target Organization or curriculum based Outcomes Pupil outcome orientated Goals Outcomes as given Focus School Methods Quantitative Site Outside school Focus Part of school Source: Teddlie and Reynolds, 2000. p. 214 1980s Bottom up Practitioner Knowledge Process based School process orientated Outcomes problematic Teacher Qualitative Within school Whole school In addition, studies concerning teachers’ cognition (e.g., efficacy) have been conducted over three decades emphasizing teachers’ knowledge and beliefs, thinking, and decision-making (Calderhead, 1996). Recently, three dimensions of teacher quality as personal resources, performance and effectiveness were conceptualized (Kennedy, 2007). Among the three dimensions, personal resources focus on teachers’ beliefs as well as teachers’ knowledge. 15 Specifically, some studies pointed out the effect of teachers’ knowledge (e.g., pedagogical or content knowledge) on gains in academic achievement (Hill, Rowan & Ball, 2005; Kennedy, Ahn & Choi, 2008). Through pre-service teacher education and in-service teacher training, we can improve knowledge and teaching practices. With respect to in-service teacher training, recent studies have emphasized the role of professional development activities which included content focus, active learning, coherence, duration, and collective participation because these professional development can increase teachers’ knowledge, which lead to changes in teaching practices and students’ achievement (Desimone, 2009). In addition to professional development, educational research has focused on social capital (Coleman, 1988) and emphasized the importance of teachers’ social capital (Daly, 2010; Finnigan & Daly, 2010; Frank, Zhao, & Borman, 2004; Moolenaar & Sleegers, 2010; Penuel et. al, 2009, 2011). Specifically, recent studies pointed out peer’s influence on teaching practices through social interactions (Sun & Frank, 2011) and human capital spillovers (Jackson & Bruegmann, 2009). In summary, the results of school and teacher effectiveness studies indicate that teaching practices can be improved through not only teachers’ professional development but also teachers’ social interactions and human capital spillovers. Therefore, teachers’ human capital at time 1 can affect students’ human capital at time 2. 16 2. Teachers’ Social Networks 1) Effects on Human Capital or Social Capital In recent years, many studies have reported effects of teachers’ social networks, including teachers’ norms (Bryk & Schneider, 2002), innovative climate (Moolenaar & Sleegers, 2010), teachers’ learning in communities of practice (Coburn & Russell, 2008), distributed leadership (Spillane, 2006; Penuel et al, 2010), implementation of innovations (Frank, Zhao & Borman, 2004; Penuel et al., 2007, 2009) and students’ attainment, students’ learning and academic achievement in core subjects (Hadfield & Jopling, 2007; Supovitz, Sirinides & May, 2010). In other words, teachers’ social networks can affect teachers’ human capital (e.g., teaching practices) as well as social capital (e.g., trust). With respect to the effects on social capital, through multilevel modeling, one study found that the relational dynamics in each school community significantly influenced meaningful school improvement efforts and relational trust, facilitators of social capital, in very disadvantaged urban school communities (Bryk & Schneider, 2002). In addition, the role of social networks through trust in supporting an innovative climate was pointed out through multilevel modeling using Dutch schools data (Moolenaar & Sleegers, 2010). Another study found that prior professional relations and proximity were key factors for trust between teachers through qualitative methods (Coburn & Russell, 2008). In summary, these results indicated that teachers’ social networks can affect trust as a facilitator of teachers’ social capital (Bryk & Schneider, 2002; Coburn & Russell, 2008; Moolenaar & Sleegers, 2010). With respect to the effects on human capital, a recent study found that formal routines and informal interactions can explain how leadership is manifest in everyday practices in schools (Spillane, 2006). In explaining the significance of leadership expertise, this study emphasized 17 that if expertise is distributed, the school leader rather than the individual leader would be the most appropriate unit for considering the improvement of leadership expertise. Other studies have shown how teachers’ social networks facilitated change in school reform practices through distributed leadership (Spillane, 2006) and teachers to teachers influences (Penuel et al., 2010). Furthermore, recent studies pointed out peer’s influence on teaching practices through social interactions (Sun & Frank, 2011) and human capital spillovers (Jackson & Bruegmann, 2009). In other words, these results indicated that teachers’ social networks can affect teaching practices as teachers’ human capital. 2) Modeling: Selection or Influence Due to the improvement of multilevel statistical methods, the school effects model overcame the choice of unit of analysis, misestimated standard errors, heterogeneity of regression, ecological fallacy and poor precision (Hopkins, 1982; Bryk & Raudenbush, 1992; Goldstein, 1986). Finally, multilevel models solved the discordance between theoretical model and methodological model of school effect. Although multilevel models have improved the studies of school and teacher effects, there was also a certain level of limitation because previous models did not consider teachers’ social interaction as peer effects. Methodologically, previous models did not consider interdependence among teachers, which lead to biased estimates of school and teachers’ effects. Social network analysis, however, can consider interdependencies among teachers represented by social network data. Frank (1998) noted that even though multilevel models may integrate distinctiveness ascribed to both students and schools as organizations, they have not been applied to include aspects of the interaction among individuals defining the social contexts 18 in which individuals teach and learn. To solve this limitation, relations among the people in schools have been analyzed by using social network analysis in education (Frank, 1995, 1998; Frank, Zhao and Borman, 2004). Social network analysis pertains not only to selection modeling of the pattern of relations in a social network as a function of attributes of people, but also to influence modeling of attributes of people as a function of relations in the social network (Frank, 1998). While selection models can help us to recognize the creation of social contexts, influence models can help us to understand the effects of those contexts on individuals (Frank, 1998). Selection modeling can be implemented through qualitative studies or statistical modeling such as p1 model (Holland & Leinhard, 1981), p2 model (Van Duijn, Snijders, & Zijlstra, 2004), or p* model (Wasserman & Pattison, 1996). In addition, influence modeling can be implemented through qualitative studies or statistical modeling such as multilevel models (Bryk & Raudenbush, 1992; Goldstein, 1986). In summary, there have been considerable advances in estimating the effects of teachers’ social networks through selection and influence modeling, but previous studies were confined to choosing appropriate methods; thus estimating two models separately (Penuel et al., 2007, 2008). In other words, although both selection and influence processes probably occur among teachers, previous studies have presented the models of selection and influence in isolation. 3. Selection and Influence: Dynamic Modeling Frank et al. (2010) summarized that “A full dynamic conceptualization accounts for actors’ behaviors as outcomes influenced by actors’ attributes or network (influence model), and actors’ networks as outcomes influenced by actors’ attributes or behaviors (selection model)” (p. 19 230). In addition, Frank et al. (2010) argued that the network processes (e.g., knowledge flow) as both predictor and outcome could be modeled by dynamic models, which are helpful when tracking how resources flow through a network. By using simulation, parameters of actororiented modes are estimated from the relations and behaviors at time 1 based on random sequences in order to approximate the network and behaviors at time 2 (Frank et al., 2010). Recent studies using actor-oriented modeling showed that both selection and influence processes played a crucial role regarding behavioral of dynamics (Bunt & Groenewegen, 2007; Burk et al., 2007; Pearson et al., 2006; Steglich et al., 2006). Steglich et al. (2006) investigated the joint dynamics of taste in music, alcohol consumption, and friendship ties among adolescents to assess selection and influence processes by using actor-oriented models and data from teenage friends and lifestyle. This consisted of 129 cohorts of pupils at a school in the west of Scotland starting in 1995 with pupils aged 13 and ending in 1997. They found that a majority of pupils listened to music in techno and rock scales and rock preferences seemed to coincide with higher social status. Also, they found that there was a small exceptional group of mainly girls, listening to music styles in the classical scale, barely drinking alcohol, and being avoided by most of their schoolmates. Steglich et al. (2006) assessed selection and influence processes using actor-oriented models, but there are some theoretical limitations because they focused on friendship among students and students’ behavior without considering organizational effects (e.g., class formation). Burk et al. (2007) examined the co-evolution of friendship networks and delinquent behaviors in a longitudinal sample of Swedish adolescents of 260 students (132 males and 128 females) attending 52 classrooms in 9 schools in a small city in central Sweden for four annual waves. By using actor-oriented network models and longitudinal social network analysis, they 20 found that both selection and influence processes played a substantive role in the observed dynamics of delinquent behaviors, with influence having a relatively stronger role than selection (especially in reciprocated friendships). A methodological strong point was that Burk et al. (2007) examined the co-evolution of friendship networks and delinquent behaviors by using actororiented models, but a theoretical weak point was that they focused on friendship among students and delinquent behaviors without considering organizational effects (e.g., formal structure of school). In addition, Snijders and Baerveldt (2003) proposed a multilevel approach to investigating the evolution of multiple networks. They assumed that the basic evolution process was the same with different parameter values between different networks. By using stochastic actor-oriented models and Markov Chain Monte Carlo methods, this study showed that delinquent behavior similarity had a positive effect on both tie formation and tie dissolution. Although there were a few studies using actor-oriented models in education (Daly & Finnigan, 2011; Orlina, 2010), little research has been carried out to estimate the dynamic of teachers’ social networks and teaching behavior simultaneously in education by using longitudinal data. I will do so in this study. As described in chapter 1, this study will test the following two hypotheses to examine the effects of teachers’ social networks on teaching practices. Hypothesis 1-1 and 1-2: The formal organizational structure of the school and teachers’ social network structure at time 1 would affect the formation of new ties of teachers’ social networks at time 2. Hypothesis 2-1: Teachers’ social networks at time 1 can affect teachers’ teaching practices at time 2. 21 Data and Methods 1. Data The current study is part of a broader project funded by the National Science Foundation to investigate catalyzing network expertise. This sub-study consists of a total of 10 elementary or middle schools. The schools are all located in California, in urban and suburban areas near major cities in Northern and Southern California. Table 2.2 School demographic information 2007-08 ID Student Enrollment % White FTE Teachers 1 441 56.0% 25 3 898 0.7% 43 8 542 14.6% 27 26 538 27.1% 26.8 39 619 37.6% 33.3 45 239 77.8% 14.6 47 580 74.8% 24.8 48 554 70.6% 22.2 53 342 64.6% 19.2 54 288 25.7% 18.6 Notes: All schools met AYP in mathematics in school year of 2007-08 Title I School No Yes No Yes No No No No No Yes The student enrollment size ranged from 288 to 898 as shown in Table 2.2. Five schools had a majority of White student population. The full-time equivalent (FTE) teachers ranged from 14.6 to 43. There were three Title I schools and all schools met AYP in mathematics in the school year of 2007-08. All faculty members were surveyed in the 10 schools in 2007 and 2008. But the sample in the final analysis differs depending on methods for treating missing values among models. The average teaching experience of the sample was 14.5 years, the mean of years working at the 22 current school was 9 years, and 95% of the teachers had full certification (advanced professional, regular/standard/probationary) in their main assignment field shown in Table 2.3. Table 2.3 Descriptive statistics of teacher demographics in 2007-08 Variables Teaching experience (N=198) Mean of number of years teaching Mean of years working at the current school Teacher credential status (N=208) No state certification (no certificate or certificate not from California) Percentage of partial certification (temporary, provisional, or emergency state certificate) Percentage of full certification (advanced professional, regular/standard / probationary) Characteristics 14.5 9.0 3.4% 1.9% 94.7% 2. Measures 1) Dependent Variables Teachers’ mathematics teaching practices advice network: the dependent variable in the selection model is the ties of interaction for each colleague (4-point scales: once or twice a year, monthly, weekly, and daily) in 2008 based on the following question: Which colleagues in this school have helped you in the past twelve months with respect to increasing the STAR mathematics test scores? For the actor-oriented model, dependent variables are the ties of interaction for each colleague in both 2007 and 2008. For analysis, the dependent variable was recoded as 0 =”no tie” and 1 =”tie existence” among teachers within a school. Teachers’ mathematics teaching practices: the dependent variable in the influence model is the extent to which teachers adopted specific teaching practices in mathematics teaching practices in 2008. In the 2008 survey, each teacher was asked to rate how often they had students do a series of activities as part of mathematics instruction during the last month on a five-point scale (1= “almost never,” 2= “1 or 2 times a month,” 3= “1 or 2 times a week,” 4= “almost every 23 day,” and 5= “one or more times a day.”). Seven items were aggregated into a practice variable based on the results of a factor analysis (α=0.87). In the 2007 survey, each teacher was asked to rate how often he or she had students do a series of activities as part of mathematics instruction during the last week on a five-point scale (1= “not at all,” 2= “1 or 2 times,” 3= “3 or 4 time,” 4= “5 or 6 times,” and 5= “more than 6 times.” α=0.88). Because there was a difference in scales between 2007 (during the last week) and 2008 (during the last month), the dependent variables for the actor-oriented model are recoded on a six-point scale (0=”Not at all,” 1=”1 or 2 times a month,” 2=”1 or 2 times a week,” 3=”3 or 4 times a week,” 4=”almost every day,” and 5=”one or more times a day.”). Table 2.4 Items for mathematics problem solving teaching practices Items in 2007 and 2008 2007 Mean (SD) 1. Solve problems that have many possible correct 1.72 (1.48) answers 2. Solve problems in which students have to figure out 2.49 (1.60) what method to use to solve them 3. Describe the procedure or algorithm they used to 2.10 (1.55) solve a problem 4. Explain why a procedure or algorithm they used 1.80 (1.58) worked to solve a problem 5. Prove that a particular method for solving a 1.52 (1.62) problem is valid 6. Analyze similarities or differences among methods 1.23 (1.39) and types of problems 7. Practice answering questions that are in the same 1.71 (1.77) format as the STAR test Recoded Scales 0:Not at all 1:1 or 2 times a month 2:1 or 2 times a week 3:3 or 4 times a week 4:almost every day 5:one or more times a day 2008 Mean (SD) 2.44 (1.45) 3.45 (1.13) 3.27 (1.21) 3.07 (1.34) 3.10 (1.29) 2.77 (1.32) 2.04 (1.68) Valid N (listwise) =142 Cronbach’s α =0.88 (2007) 24 Valid N (listwise) =144 Cronbach’s α =0.87 (2008) 2) Teacher Attributes Prior teaching practices in 2007: For the influence model, it could be that the extent to which the teachers included a mathematics teaching practices in 2008 depended on teaching experience with math in 2007. Therefore, we controlled for prior teaching practices. Mathematics Professional Development in 2008: Mathematics professional development was controlled in the models because learning new expertise through professional development could affect mathematics advice networks and teaching practices. For the selection and influence model, the question was, in the past year, how much professional development have you had in mathematics? The variable scales from 0 to 3 (0= “None at all,” 1= “1-8 hours,” 2= “9-16 hours,” 3= “more than 16 hours,” and 9=”unsure.”). Mathematics teaching efficacy in 2007: Mathematics teaching efficacy may be related to teaching practices because this specific belief measure teachers’ capacities to affect a student’s achievement. For the selection, influence and actor-oriented models, teachers were asked to rate the extent to which they agreed with each of the following statements (1= “Strongly disagree,” 2= “Disagree,” 3= “Agree,” 4= “Strongly Agree,” and 9=”Unsure.”): “I am responsible for students’ high achievement in mathematics”, “ Different mathematics teaching methods can affect a student’s achievement”, and “I change my teaching approach if students are not doing well in mathematics”. The average of these items was included in models (α=0.68). Highest Grade Taught in 2008: California schools have mathematics AYP standard which is 37% of students to be proficient by the 2007-08 school year and 47.5% by the 2008-09 school year for students from the third to eighth grades. For the influence model, teachers’ highest grade taught in 2008 was controlled because we assume that teachers in higher grade 25 rd th levels (3 -8 ) may respond less to new norms for their teaching practices than teachers in lower nd grades (K-2 ). Mathematics program coordinator role in 2008: For the selection model, teachers were asked if they had a mathematics program coordinator role at the school during the 2007-08 school year and were coded as 0 if the teacher was not a coordinator and 1 if the teacher was a coordinator. Subgroup mean teaching practices in 2007: To investigate how teachers respond to subgroup norms about teaching practices, subgroup mean teaching practices in 2007 was included in the influence model. A sociometric item regarding professional closest colleagues was used to construct subgroup boundaries (Frank 1995, 1996). Frank’s (1995, 1996) network clustering algorithm and software KliqueFinder were used for identifying subgroup membership in 2007. Grand mean centering was used to define the subgroup norm by averaging the extent to which the members of a subgroup implemented mathematics problem solving teaching practices in 2007. Subgroup mean mathematics teaching efficacy in 2007: To investigate how teachers respond to subgroup norms about teaching efficacy, subgroup mean mathematics teaching efficacy in 2007 was included in the influence model. Grand mean centering was used to define the subgroup norm by averaging the extent to which the members of a subgroup believed their own capacities to affect a student achievement in 2007. 3) Network Variables Direct Exposure: For the influence model, in order to estimate the effect of network on teaching practices in mathematics, direct exposure variable was: 26 ∑ Where I is the total number of teacher i' that provided help to teacher i. help is the extent to which teacher i reported receiving help with teaching mathematics from teacher i'. provider ′ s expertise indicates teaching practice in mathematics problem solving in 2007 of teacher i'. represents the ability of the help provider i' to deliver help. As teacher i (help receiver) receive more help from teacher i' (help provider), the direct exposure will increase. In addition, as the help provider has more expertise in mathematics teaching practices and more amounts of help provided to others, the direct exposure will increase. Same subgroup network in 2007: For the selection and actor-oriented models, after identifying cohesive subgroup within each school, same subgroup networks in 2007 were made by coding as 0 if a teacher had different subgroup membership from the other teacher within a school and 1 if a teacher had the same subgroup membership as the other teacher within a school. Same grade taught network in 2008: For the selection and actor-oriented models, same grade taught networks in 2008 were created by coding as 0 if teachers i and i' taught different grade levels and 1 if teachers i and i' taught the same grade level. Total of all common meeting types network in 2008: For the selection model, the 27 following question was used to construct the total of all common meeting types: during the 200708 school year, in what ways and how frequently do you review annual STAR test data, either for students you are currently teaching or for students you taught last year?: ① “I participate in discussions of data as part of a formal meeting with my grade-level team”, ② “I participate in d iscu ssions of d ata as part of a form al m eeting w ith a cross-grad e team ” and ③ “I participate in d iscussions of d ata as part of a full staff m eeting” (0= “never,” 8= “1~8 times this year,” 10= “monthly,” 25= “2~3 times a month,” and 40=”weekly.”). If both teachers answered yes to ① and they taught in the same grade level, this ① was counted for common meeting taking the minimum meeting number between actor i and i'. If both teachers answered yes to ② and they taught different grade levels, common meeting was counted by taking the minimum meeting number of ② between actor i and i'. If both teachers answered yes to ③, a common meeting of ③ was added by taking the minimum meeting number between actor i and i'. Finally, all common meetings were computed as “common meeting= minimum (① for i, ① for i') + minimum (② for i, ② for i') + minimum (③ for i, ③ for i')”. nd For example, if 2 nd 2 grade teacher A answered 8 for ①, 10 for ②, 0 for ③ while other grade teacher B answered 10 for ①, 25 for ②, 0 for ③, all common meetings would be 8 as minimum (8=① for i, 10=① for i') + minimum (0=② for i, 0=② for i') + minimum (0=③ for i, 0=③ for i') = 8+ 0 + 0 because we didn’t count ② due to the same grade level taught. If 2 nd rd grade teacher A answered 8 for ①, 10 for ②, 0 for ③ while other 3 grade teacher B answered 10 for ①, 25 for ②, 0 for ③, all common meetings would be 10 as minimum (0=① for i, 0=① for i') + minimum (10=② for i, 25=② for i') + minimum (0=③ for i, 0=③ for i') 28 = 0+ 10 + 0 because we didn’t count ① due to the different grade level taught. Thus, all common meetings could have the five-scale value as 0, 8, 10, 25, and 40. 3. Methods To investigate selection and influence effects, I used multilevel p2 models and two level HLM models while I used actor-oriented models and multilevel (meta-analysis) actor-oriented models to examine the dynamics in mathematics teaching practices advice networks and mathematics teaching practices by using SIENA (Simulation Investigation for Empirical Network Analysis) software. A series of descriptive statistics were employed to assess network and teaching practice in selection, influence and actor-oriented models. For SIENA outputs, there were three steps which were a convergence check, parameter values and standard errors, and a collinearity check. First, a convergence check was given to consider deviations between simulated values of the statistics and their observed values (Snijders et al., 2008). The manual for SIENA reports that “For results that are to be reported, it is advisable to carry out a new estimation when one or more of the t-ratios are larger in absolute value than 0.1” (p.32). Second, the rate parameter indicates the estimated number of changes per actor between observations. Third, the collinearity check was presented to see whether there was collinearity among variables. Missing values of professional development in 2008 had five missing cases out of 209 cases while teaching efficacy in 2007 had 31 missing cases out of 209 cases. All missing cases were recoded as a zero value in model 1 and model 2 of multilevel selection models. For the influence models, mathematics problem solving teaching practices in 2008 had 56 missing cases out of 209 cases. These were deleted in the two-level HLM models because this was a dependent variable and there was little relevant information for multiple imputation of missing values. After deleting the missing values of the dependent variable, there were two 29 missing cases in teaching efficacy in 2007 and one missing case in the highest grade, which were deleted in two-level HLM models. I used P2 4.0 version software for multilevel p2 models, HLM 6.0 version for two level HLM models, SIENA 3.2 version for actor-oriented models, and SIENA 08.exe for metaanalysis actor-oriented models. 4. Models 1) Selection Model Through selection modeling, I will test the following hypothesis: formal organizational structure of school and teachers’ social network structure at time 1 would affect teachers’ social networks at time 2. Social Networks Structure Attributes Hierarchy & Time Formal Structure of School & Teachers’ Informal Network Structure Teachers’ Social Networks Level 3 Time 1 Teachers’ Formal Structure Level 2 Time 2 Network Teachers’ Efficacy Level 1 Time 1 Teachers’ PD Teachers’ leadership Level 1 Time 2 Figure 2.1. The structure of variables in multilevel p2 selection modeling 30 To test this hypothesis through multilevel p2 selection modeling, I consider not only endogenous network variables (density and reciprocity) but also exogenous attributes variables. In other words, teachers’ efficacy, teachers’ professional development, and teachers’ leadership role will be included in multilevel p2 selection modeling as shown in Figure 2.1. The selection model is a logistic model because the dependent variable is dichotomous. ) indicates whether actor i indicated receiving help from actor The dependent variable ( i'. Then is modelled as a function of the tendency of actor i' to provide help regarding ) and the tendency of i to receive help ( mathematics teaching practices ( ). The model at level 1, for the pair of actors i and i', is: Level 1 (pair): ( [ ] [ ] ) To capture different bases of structuring, dummy variables were included indicating whether school actors had a tie in 2007, were members of the same subgroup, whether they taught in the same grade, and participated in regular meetings. And reciprocity was included to control for the extent to which actor i' provided help to i. The final level 1 model was: ( [ ] [ ] ) + δ1 (prior relationship about mathematics) ii’ + δ2 (prior same subgroup) ii’ + δ3 (same grade teaching assignment) ii’ 31 + δ4 (total of all meeting types in common) ii’ + δ5 (reciprocity: help i’i ). The larger the value of δ1, the more we would infer that the network structure as defined by previous ties about mathematics affects help provided at time 2. The larger the value of δ2, the more we would infer that the network structure as defined by same subgroup memberships affects the patterns of advice sharing. Large values of δ3 and δ4 quantify how help is shaped by the formal organization as represented by grade level and meeting structures. The term δ5 indicates the extent to which actors helped others who had helped them. We modelled the tendencies of school actors to be nominated as providing and receiving help at a separate level: Level 2a (i': provider of help) = γ0(α) + uoi’ . Level 2b (i: receiver of help) = γ0(β) + voi . Here, the random effects uoi’ and voi are assumed to be normally distributed and account for dependencies associated with tendencies to provide or receive help that affect all relations in which a given individual engages. In order to estimate what attributes of the provider and receiver of help account for the patterns observed in teachers’ advice networks, mathematics program coordinator role (in school 3, 45, 48 & 54) and mathematics professional development were included in provider effects in level 2a of model 1 while only mathematics professional development was included in sender effects in level 2b of model 1 because we assume that 32 teachers with the mathematics program coordinator role tend to provide advice more than to receive advice. To estimate the effect of the teaching efficacy of provider and receiver of help, prior mathematics teaching efficacy was added to level 2a and level 2b of model 2. In other words, the only difference in model specification between model 1 and model 2 was to add prior mathematics teaching efficacy into the provider and receiver effect. The final level 2 model was: Level 2a (i': provider of help) = γ0(α) + γ1(α) coordinator role i’ + γ2(α) mathematics professional development i’ +γ3(α) prior mathematics teaching efficacyi’ + uoi’ . Level 2b (i: receiver of help) = γ0(β) + γ1(β) mathematics professional developmenti + γ2(β) prior mathematics teaching efficacyi +voi . In summary, the two-level logistic model was used to account for the dependencies among teachers like two-level HLM model (Generalized linear model) in order to explain the effects of dyadic level (level 1) and provider & receiver level (level 2). 2) Influence Model Through influence modelling, I will test the following hypothesis: Teachers’ Social networks at time 1 can affect teachers’ teaching practices at time 2. To test this hypothesis through multilevel influence modeling, I consider not only teachers’ social networks (direct exposure) and organizational structure of school, but also exogenous teachers’ attributes variables. In other words, teachers’ efficacy, teachers’ professional 33 development, and teachers’ leadership role will be included in multilevel influence modeling as shown in Figure 2.2. Human Capital Social Networks Structure Attributes Level 3 Time 2 Grade level Teachers’ Networks Hierarchy & Time Teachers’ Efficacy Teachers’ Efficacy Level 1 Time 1 Teachers’ PD Teaching practices Level 2 Time 1 Level 1 Time 2 Figure 2.2. The structure of variables in multilevel influence modeling The influence model is a two-level multilevel model (Raudenbush & Bryk, 2002) to investigate the influence of those who helped teachers with mathematics teaching practices within and between subgroups on current teacher’s level of teaching practices, controlling for prior teacher’s level of teaching practices, current professional development, prior mathematics teaching efficacy, and current highest grade taught. Level-1 Model (Teacher level: i) 34 Level-2 Model (Subgroup level: j) 𝑏 𝑏 𝛾 𝑐 𝑞 Where π j is the intercept at level one. πpj (p=1, 2, 3, 4, 5) are the level-1 coefficients that indicate the direction and strength of association between each predictor and teacher i’s math in 2008. eij is the level one residual term. β j is the intercept at level two. β j indicates the effect of mean prior teaching practices in subgroup j on π j . β j indicates the effect of mean prior mathematics teaching efficacy in subgroup j on π γj is the level two residual term. πpj (p=1, 2, 3, 4, 5) βqj are fixed at level two. j. All predictors were centered by using grand-mean to estimate the effect of influence from others. 3) Actor-Oriented Model Dynamic models of social networks have been developed from discrete time to continuous time. Previous dynamic models of social networks in discrete time had assumed evolution of one point to the next as one jump while previous dynamic models of social networks in continuous time had assumed the probability of each network change may depend not on earlier states of the tie (continuous-time Markov chain) but on the entire current set of ties (Snijders et al., 2008). And continuous-time models were proposed by Snijders (1996, 2001) and Snijders & Van Duijn (1997). Snijders (2005) explained that actor-oriented models assumed that actor i controls all outgoing ties and changes in the network occur only one tie at a time. The rate function 35 determines “the moment when actor i changes one of his ties” stochastically and the evaluation function determines “the particular change that he makes”, which can depend on the network structure and on attributes represented by observed covariates. Table 2.5 Actor-oriented model components Change occurrence Rule of change network changes network rate function network evaluation function behavioral changes behavioral rate function behavioral evaluation function Therefore, the actor-oriented model consists of rate and evaluation functions. Network changes consist of a network rate function and a network evaluation function while behavioural changes consist of a behavioural rate function and a behavioural evaluation function shown in Table 2.5. (1) Micro Level: Network Analysis Network data from 2007 and 2008 are used as the dependent variables. Based on the findings from a multilevel p2 model in the present study, actor attributes and dyadic covariates are chosen. Thus, a density effect as an out-degree effect is included into actor-oriented model as 𝑆 and a reciprocity effect is also included in the model as 𝑆 . Additionally, actor-oriented models included network structure effects which can, according to the pattern of existing interaction, identify the extent to which an actor’s decisions about whether to terminate a current network or make a new tie depending on structural effects such as transitive triplets. For example, the tendency towards triadic closure is included in the actor-oriented model as 𝑆 . 36 In order to estimate the effect of dyadic covariates on network change, same subgroup 𝑆 effect is included as and same grade level effect is included as 𝑆 . Both are included in the model after grand-mean centering. In order to estimate a homophily effect, an efficacy similarity measure was included as positive estimate ( 6 6 𝑆6 where 𝑆6 is defined as of efficacy similarity indicates that there is more chance to interact with another who has similar the efficacy measures. A multilevel p2 model in this study indicated that high prior mathematics teaching efficacy is related to more probability of providing and receiving help. But, these results cannot tell us whether or not the similarity of efficacy affects the occurrence of a tie between two actors. In order to estimate homophily selection on the similarity of efficacy, the actor-oriented model included the efficacy similarity effect. Finally, the network rate function was defined as a mathematics advice network change 𝜆𝑛 rate from 2007 to 2008 function, 𝑡 𝜌𝑛 𝑡 while a mathematics advice network evaluation function was defined as a weighted sum of effects, 𝑛 𝑡 ∑6𝑘 𝑆 𝑘 𝑘 . 6 𝑛 𝑡 ∑ 𝑘 𝑆 𝑘 𝑘 𝑆 6 𝑆 𝑆 𝑆6 37 𝑆 𝑆 Where 𝑘 is weight as a statistical parameter expressing the importance of effect k. 𝑆 is a function of the network from the point of view of actor i. 𝑘 𝑆 ∑ elp 𝑆 ∑ elp ∑ elp ∑ℎ elp ℎ elpℎ is density effect defined by the out-degree. elp is reciprocity effect defined by the number of reciprocated relations 𝑆 is transitive triplets effect defined by tendency towards triadic closure. ∑ 𝑆 elp ( 𝑏 ̅̅̅̅̅̅̅̅̅̅̅̅̅) 𝑏 is same subgroup (centered) as a dyadic covariate. ∑ 𝑆 ̅̅̅̅̅̅̅̅) elp ( is same grade taught (centered) as a dyadic covariate. ∑ 𝑆6 elp is prior efficacy related similarity effects as actor-dependent covariate defined by tendency to have ties to similar others (homophily selection on prior efficacy). Behavioural data from 2007 and 2008 are used as the dependent variables. Based on the findings from the two-level HLM model in present study, actor attributes were included in the model as 𝑆 . Behavioural rate function was defined as a teaching practices change rate from 2007 to 2008, 𝜆 𝜌 as a weighted sum of effects, while teaching practices evaluation function was defined ∑𝑘 𝑘 𝑆 𝑘 . 𝑆 Where 𝑘 is weight as a statistical parameter expressing the importance of effect k. 𝑆 𝑘 is a function of the behavior from the point of view of actor i. 38 𝑆 ∑ J is the total number of teacher j that provided help to teacher i. is the extent to which teacher i reported receiving help with teaching mathematics from teacher i. ′ indicates teaching practice in mathematics problem solving in 2007. represents the ability of the help provider j to deliver help. The final actor-oriented models were that the behavior model included teaching practices change rate (speed), teaching practices change tendency, and direct exposure effects while the network model included mathematics teaching practices advice network change rate (speed), density, reciprocity, transitive triplets, same subgroup (centered), same grade (centered), and efficacy similarity. (2) Macro Level: Combination of Networks For combining results of several independent (the set of actors are disjoint, and it may be assumed that there are no direct influences from one network to another) networks, Metaanalysis for multilevel network analysis has been developed (Snijders & Baerveldt, 2003). This analysis consists of the micro level as single network evolution study (Snijders, 2001) and the macro level as combination of these network studies. In addition, this analysis combines the 39 estimates in a meta-analysis according to the methods of Snijders and Baerveldt (2003) and a Fisher-type combination of one-sided p-values. Through dynamic modeling, I will test two hypotheses. To test hypothesis 3-1 through dynamics modeling, however, I consider not only endogenous network variables (density, reciprocity, and transitivity), but also an exogenous attribute variable (teachers’ efficacy). In other words, a transitivity variable will be added to test hierarchy in dynamics modeling but two exogenous attributes (teachers’ professional development and teachers’ leadership role) will be excluded in dynamics modeling. In addition, to test hypothesis 3-2 through dynamics modeling, I only include teachers’ social network. In other words, teachers’ efficacy in both subgroup and individual level, teachers’ professional development, and teachers’ leadership role will be excluded in dynamics modeling. 40 Results 1. Selection Model To test whether or not the formal organizational structure of school and teachers’ social network structure at time 1 affect the formation of new ties of teachers’ social networks at time 2, multilevel selection models were analyzed. Table 2.6 Multilevel selection model for ten schools Parameters μ – Pair (level 1) Prior relationship about mathematics, δ1 Mathematics professional development, γ2 0.02 (0.07) 3.51 (0.52) 0.71 (0.37) 1.88* (0.66) 0.36* (0.12) 0.28* (0.14) 1.86 (0.65) 0.17 (0.14) (α) 2.04* (0.48) -0.00 (0.06) 3.33 (0.48) 0.67 (0.36) 1.79* (0.71) Total of all common meeting types, δ4 δ5 – Reciprocity - Provider variance (level 2a) (α) Mathematics program coordinator role, γ1 1.13* (0.42) 2.08* (0.52) Same grade, δ3 Model 2 -6.89 (1.13) 3.58* (0.75) 1.08* (0.37) Prior same subgroup, δ2 Model 1 -5.59 (0.49) 3.52* (0.74) 0.15 (0.13) 1.91 (0.56) -0.03 (0.18) (α) Prior mathematics teaching efficacy, γ3 - Receiver variance (level 2b) (β) Mathematics professional development, γ1 (β) 0.47* (0.16) Prior mathematics teaching efficacy, γ2 0.08 (0.32) -0.09 (0.35) - Provider-receiver covariance 0.36 (0.29) 1.41 (2.15) -Omega for Random Density Effects Note:* means t-ratio more than 2; The sample size was 209 in model 1 & model 2; Burn-in 4000 and sample size 20000 in MCMC estimation The large negative value as -5.59 and -6.89 of μ (density parameter) in model 1 and model 2 indicated that when all random effects and other parameters are equal to zero, the probability of a network is much smaller than 0.5. In other words, there were sparse mathematics 41 advice networks across ten schools. The large positive value as 3.33 and 3.51 of δ5 (reciprocity parameter) in model 1 and model 2 indicated that actors helped others who had helped them. To estimate the network structure effect, four network covariates were included in the density effect in level 1. The large positive values of 3.52 and 3.58 of δ1 for models 1 and 2 indicated that having a previous tie raises the odds of having a current tie by about 33 times1 in model 1 and about 35 times in model 2. In other words, previous ties about mathematics were much strongly related to the patterns of current advice sharing. In addition, based on the positive values of 1.08 and 1.13 of δ2 for models 1 and 2, having a previous same subgroup membership raises the odds of having a current tie by about two times in model 1 and model 2. In other words, we would infer that same subgroup’ memberships affect the patterns of mathematics advice network. Finally, the positive values of 2.08 and 2.04 of δ3 for models 1 and 2 showed that having a previous same grade membership raises the odds of having a current tie by about seven times in model 1 and model 2. In other words, we would infer that advice networks were shaped by the formal organization as represented by grade level. Model 1 and 2 had, statistically nonsignificant, nearly zero estimate of -0.00 and 0.02 of δ4. In summary, the patterns of advice sharing could be, structurally, affected by 1) prior relationship about mathematics, 2) same grade level, and 3) prior same subgroup membership. The results of including what attributes of the advice provider account for the patterns of networks showed that a mathematics program coordinator role (a positional attribute) was 1Note: I get odds to compute the following formula: 42 . -1=33.78-1=32.78=3,278% statistically significant, positively related to providing advice in both model 1 (1.79) and model 2 (1.88). In other words, having a mathematics program coordinator role raises the odds of providing advice by about five times in model 1 and model 2. Mathematics professional development also was statistically significant, positively related to providing advice in both model 1 (0.36) and model 2 (0.28). An one-unit change in mathematics professional development raises the odds of providing advice by about one half times in model 1 and about one third times in model 2. In addition, the results of including what attributes of advice seeker account for the patterns of networks showed that prior mathematics teaching efficacy (a psychological attribute) was statistically significant and positively related to seeking advice in model 2 as 0.47. A oneunit change in mathematics professional development raises the odds of providing advice by 60% in model 2. In summary, the results of the multilevel selection model across ten schools indicated that new ties involving mathematics teaching practices-related help were predicted strongly by having previous mathematics teaching practices help ties (time), teaching in the same grade level (the organizational structure of the school) and teachers being in the same subgroup (informal network structures) in model 1 and 2 shown in table 2.6. These results are consistent with the study of Zahorik (1987). Zahorik pointed out the importance of same grade teachers as “Teachers who teach at the same grade level understand each other’s problems, can offer practical, specific help, and are close at hand” (p.394). At the same time, a mathematics program coordinator role (a social attribute) and mathematics professional development were related to a significant shift in patterns of interaction in model 1 and 2, shown in table 2.6. In addition, prior mathematics teaching efficacy 43 (a psychological attribute) was related to a significant shift in patterns of interaction in model 2 shown in table 2.6. Overall, this pattern of results suggests that formal organizational structure of school (grade level) and teachers’ social network structure (subgroup) at time 1 affect the formation of new ties of teachers’ social networks at time2. 2. Influence Model To test whether or not teachers’ social networks at time 1 affect teachers’ teaching practices at time 2, two-level and three-level multilevel models were analyzed. Table 2.7 Variance components of unconditional model Three-level Math in 2008 Two-level Individual 0.759 Individual Subgroup 0.255 (26%) Subgroup School 0.00013(0.01%) Math in 2008 0.548 0.175 (24%) The final model was a two-level multilevel model because school variances in threelevel unconditional models were almost zero (0.01%) in Table 2.7. 1) Basic Statistics The sample for the influence model included 150 teachers with 41 subgroups in Table 2.8. For descriptive statistics of level 1, the mean of mathematics teaching practices in 2008 was 2.88 with standard deviation (SD) 0.99 and range 0 to 4.57, while the mean of mathematics teaching practices in 2007 was 1.81 (SD, 1.13) with range 0 to 4.57. The increased means of mathematics teaching practices indicated that there was increased implementation in 44 mathematics teaching practices from 2007 (one or two times a week) to 2008 (three or four times a week) across ten schools while the decreased standard deviation of mathematics teaching practices indicated that there were more uniform mathematics teaching practices in 2008 than 2007 across the ten schools. Table 2.8 Descriptive statistics of multilevel influence model Variable Level-1: Individual Teacher (N=150) Teaching practices in 2008 Teaching practices in 2007 Exposure between 2007 and 2008 Professional development in 2008 Mathematics teaching efficacy in 2007 Highest grade in 2007 Level-2: Subgroup (N=41) Subgroup mean of Teaching practices in 2007 Subgroup mean of math teaching efficacy in 2007 M SD Min Max 2.88 1.81 24.17 1.20 3.43 4.11 0.99 1.13 38.18 0.96 0.46 2.19 0 0 0 0 1 1 4.57 4.57 216 3.00 4.00 9.00 1.84 3.40 0.90 0.31 0.00 2.44 4.29 3.89 The mean of direct exposure between 2007 and 2008 was 24.17 (SD, 38.18) with range 0 to 216, which indicated that there was large variation among teachers because the sum of direct exposure was included instead of the mean of direct exposure. The mean of professional development in 2008 was 1.20 (SD, 0.96), which indicated that teachers across ten schools averaged “one to eight hours” of professional development of mathematics in 2008. The mean of mathematics teaching efficacy in 2007 was 3.43 (SD, 0.46), which showed that teachers, on average, had agreement or strong agreement with statements about prior teaching efficacy. For descriptive statistics at level 2, the subgroup mean of mathematics teaching practices in 2007 was 1.84 (SD, 0.90) with range 0 to 4.29 while the subgroup mean of prior teaching efficacy was 3.40 (SD, 0.31) with range 2.44 to 3.89. In addition, correlations among level-one predictors are shown in Table 2.9. 45 Table 2.9 Correlation among level-one predictors Teaching Teaching practices 08 practices 07 Teaching practices 08 Teaching practices 07 0.51** Exposure (07 to 08) 0.29** 0.34** PD in 08 0.17* 0.26** Efficacy in 07 0.22** 0.14 Highest grade in 08 0.12 0.32** Notes: N=150, * p < .05, ** p < .001. Exposure 0.21** 0.09 0.33** PD 08 0.21** 0.09 Efficacy 07 -0.05 The highest significant correlation was 0.51 between mathematics teaching practices in 2007 and mathematics teaching practices in 2008 while the lowest significant correlation was 0.21 between mathematics professional development in 2008 and mathematics teaching efficacy in 2007. In addition, there was statistically non-significant correlation between mathematics teaching practices in 2008 and highest grade in 2008, which indicated that current grade level may not affect current mathematics teaching practices. 2) Regression coefficient for the Multilevel Influence Model To estimate how much teachers’ social networks at time 1 affect teachers’ teaching practices at time 2, regression coefficients (standard error) for a multilevel influence model were shown in Table 2.10. The results of model 1 showed that prior mathematics teaching practices (coefficient of 0.36) and direct influence (coefficient of 0.005) had significant effects on current mathematics teaching practices. In order to estimate the effect of subgroup mean of prior teaching efficacy on current mathematics teaching practices, model 2 was analyzed and the results indicated that there was a significant effect (coefficient of 0.74) of subgroup mean of prior mathematics teaching efficacy. This was the source of differences in model specification and results between model 1 and model 2, which indicated that even though prior individual teaching efficacy didn’t influence 46 current teaching practices, the mean of each member’s prior teaching efficacy within the same subgroup might be key factor to account for current teaching practices. Table 2.10 Regression coefficients (standard errors) for multilevel model of mathematics problem solving teaching practices including the influences of colleagues. Variable Model 1 Model 2 Level-1: Individual Teacher (N=150) Overall mean Teaching practices in 2008 3.01 (0.20) 2.97 (0.20) Teaching practices in 2007 0.36** (0.07) 0.37** (0.07) Exposure between 2007 and 2008 0.005** (0.002) 0.005** (0.002) Mathematics Professional development in 2008 0.03 (0.07) 0.03 (0.08) Mathematics teaching efficacy in 2007 0.17 (0.17) 0.03 (0.18) Highest grade in 2008 -0.04 (0.04) -0.03 (0.04) Level-2: Subgroup (N=41) Subgroup mean of Teaching practices in 2007 0.14 (0.18) 0.05 (0.17) Subgroup mean of math teaching efficacy in 07 N/A 0.74* (0.34) Note: model 2 includes subgroup mean of mathematics teaching efficacy. * p< .05, ** p< .001. To compare the relative impact of these estimates, standardized coefficients of regression models are presented in Table 2.11. Table 2.11 Regression standardized coefficients for multilevel model of mathematics problem solving teaching practices including the subgroup mean of influences of colleagues. Variable Model 2 Model 3 Level-1: Individual Teacher (N=150) Overall mean Teaching practices in 08 -0.05 -0.04 Teaching practices in 07 0.42** 0.41** Exposure between 07 and 08 0.21* 0.28** Mathematics Professional development in 08 0.03 0.02 Mathematics teaching efficacy in 07 0.01 0.01 Highest grade in 2008 -0.07 -0.06 Level-2: Subgroup (N=41) Subgroup mean of teaching practices in 07 0.05 0.15 Subgroup mean of math teaching efficacy in 07 0.23* 0.22+ Subgroup mean of exposure between 07 and 08 N/A -0.19 Note: model 3 includes subgroup mean of exposure between 2007 and 2008. + p=.055, * p < .05, ** p < .001. 47 The effect of exposure between 07 and 08 was the half times as the effect of prior teaching practices. For a two standard deviations increase in of exposure between 07 and 08, there was a two-fifth standard increase. To estimate norm pressure of exposure between 07 and 08 at the subgroup level, model 3 included subgroup mean of exposure. The similar results indicates that mathematics teaching practices in 2007and direct exposure had influence on conducting mathematics problem solving teaching practices in 2008. Overall, this pattern of results suggests that teachers’ social networks at time 1 affect teachers’ teaching practices at time 2. 3. Actor-Oriented Model To examine how mathematics teaching practices advice network and mathematics teaching practices change over two years and what can explain these dynamics, actor-oriented models were analyzed. 1) Basic Statistics There were increases in levels of mathematics teaching practices from 2007 to 2008 in all but School 48. School 54 had the largest change from 1.58 to 3.06 while school 48 had smallest change from 2.49 to 2.34. The highest school in 2007 was school 47 (2.80) and school 8 (3.20) in 2008 while the lowest school was school 1 (1.50) in 2007 and school 8 (3.20) in 2008. The results of a paired ttest in mathematics teaching practices indicate that there were statistically significant differences between 2007 and 2008 within schools except for school 47 and 48. 48 Table 2.12 Change in mathematics problem solving teaching practices Paired Comparison Schools School mean of math teaching practices in 2007 School mean of math teaching practices in 2008 Mean Differences Std. Deviation School 1 (N=21) 1.50 (0.70) 2.80 (0.78) 1.30** 0.71 School 3 (N=29) 1.88 (1.33) 2.82 (1.21) 0.94** 0.95 School 8 (N=19) 1.87 (0.99) 3.20 (0.81) 1.37** 1.14 School 26 (N=14) 2.00 (1.15) 3.19 (0.91) 1.19** 0.54 School 39 (N=23) 1.54 (1.04) 2.84 (0.90) 1.30** 1.03 School 45 (N=7) 1.84 (1.29) 2.71 (0.66) 0.87* 0.84 School 47 (N=9) 2.80 (1.19) 2.81 (1.21) 0.01 1.67 School 48 (N=5) 2.49 (1.30) 2.34 (1.55) -0.15 0.79 School 53 (N=14) 1.60 (1.05) 2.62 (1.07) 1.02** 1.21 School 54 (N=11) 1.58 (1.17) 3.06 (0.87) 1.48** 0.95 Note: Paired comparison test the difference in school mean of math teaching practices between 2007 and 2008. *p <.05, **p <.001. Table 2.13 Change in mathematics teaching practices advice networks Change in math ties (0: no tie, 1: a tie) 2007 2008 Networks Average Average 0 → 1 1 → 0 1→ 1 degree degree (Formation) (Dissolution) (Constant) School 1 (N=24) 0.217 0.217 4 4 1 School 3 (N=36) 0.629 1.714 49 11 11 School 8 (N=23) 0.682 1.000 14 7 8 School 26 (N=21) 1.100 0.400 3 17 5 School 39 (N=28) 0.926 1.074 15 11 14 School 45 (N=13) 0.333 0.500 4 2 2 School 47 (N=19) 1.222 1.056 4 7 15 School 48 (N=18) 0.235 0.294 4 3 1 School 53 (N=14) 0.923 0.769 3 5 7 School 54 (N=13) 1.667 0.583 0 13 7 Note: Average degree means that the total degrees (ties) are divided by the total number of teachers in each school. In addition, Formation means new ties in 2008 which was no ties in 2007, Dissolution means no ties in 2008 which was ties in 2007, and Constant means ties both in 2007 and 2008. 49 For average degree (total tie divided by sample size) in mathematics networks, there were no changes in number of ties in one school (school 1), increases in five schools (school 3, 8, 39, 45 and 48), and decreases in four schools (school 26, 47, 53, and 54) as shown in Table 2.13. There were no mutual ties in the mathematics network in one school (school 1) for two years. School 47 had average degree of more than one in both 2007 and 2008. In addition, there was no formation of a tie in the mathematics teaching practices advice network from 2007 to 2008 in school 54 while there was dissolution and constant ties in mathematics teaching practices advice network from 2007 to 2008 in all schools. 2) Micro Level: Network Analysis For the convergence check, there was poor convergence in school 26, 45, 47, 48, 53 and 54 because at least one t-ratio was not close to zero with an absolute value more than 0.1. For the collinearity check, there were high collinearities among variables in school 45, 48, and 54. With respect to results of the mathematics network selection model, first, the positive network change rate indicated that there was a change in the mathematics network from 2007 to 2008 across all ten schools. Among ten schools, school 3 had the highest, most statistically significant network change rate of 5.47 as we see average degree change from 0.6 to 1.7 shown in table 2.14. Also, school 45 had the lowest, but statistically non-significant network change rate of 1.29 with average degree change from 0.3 to 0.5. Second, a density effect as an out-degree had negative estimates just as the results of multilevel p2 models while a reciprocity effects had positive estimates among eight schools. Third, transitive triplets’ effect as a network structure effect had positive or negative estimates depending on the school. School 47 had a statistically significant positive estimate of 1.35 which indicated that the transitive triple effect was a key factor driving network change over 50 two years in this school after controlling for same grade level and subgroup effects. Table 2.14 Effects estimates (standard errors) in mathematics teaching practices advice networks and mathematics teaching practices Mathematics teaching practices advice network (selection model) school Transitive Subgroup Grade Efficacy Rate Density Reciprocity triplets (Centered) (Centered) similarity 1 1.89 -7.59 3.62* -1.08 5.04 0.87 -0.21 (1.05) (11.04) (1.54) (17.05) (46.7) (46.1) (2.34) 3 5.47* -2.39* 0.98* 0.64* 1.10* 0.70* -0.18 (1.87) (0.23) (0.48) (0.21) (0.31) (0.29) (0.42) 8 1.69* -3.75* 0.81 1.40 -0.53 3.33* 2.37 (0.54) (1.06) (0.99) (0.80) (0.93) (1.17) (1.27) 26 2.39* -6.79 2.37 -0.23 2.60 0.67 -4.50 (0.80) (3.43) (1.85) (7.59) (3.60) 1.32) (6.60) 39 2.83* -3.37* 0.44 0.70* -0.11 2.04* 2.88 (1.03) (0.52) (0.90) (0.29) (0.60) (0.70) (1.52) 45 1.29 -3.26 1.93 -0.84 2.25 -0.29 0.34 (1.19) (2.16) (1.81) (6.28) (2.68) (1.13) (1.83) 47 1.82* -5.66* -1.78 1.35* 2.79 1.47 1.32 (0.74) (1.74) (2.13) (0.63) (1.83) (1.73) (1.96) 48 2.29 -4.96* -15.07 -4.61 0.96 -1.78 2.77 (2.96) (1.30) (350) (7.97) (1.66) (1.74) (2.83) 53 2.21 -12.00 12.93 -0.65 -10.87 11.36 0.33 (1.48) (67.69) (38.40) (1.00) (38.23) (76.31) (2.38) 54 2.03* -8.69 4.88 0.59 3.29 1.35 -2.90 (0.85) (23.91) (17.95) (2.00) (12.97) (7.81) (23.05) Mathematics teaching practices (influence model) School Rate Tendency Exposure 1 1.95* (0.75) 2.12 (1.86) 0.06 (0.71) 3 2.14* (0.86) 0.88 (0.49) -0.01 (0.01) 8 3.81 (2.55) 1.12 (0.73) 0.001 (0.02) 26 1.80* (0.47) 6.30 (31.6) -0.04 (0.29) 39 4.32* (2.06) 0.81 (0.46) 0.02 (0.02) 45 1.49 (0.88) 9.51 (311) -0.27 (9.58) 47 3.84 (3.25) 0.06 (0.36) 0.003 (0.005) 48 0.45 (0.42) -85.4 (9999) 7.60 (9999) 53 8.63 (8.32) 0.40 (0.24) 0.02 (0.02) 54 2.53 (1.41) 3.38 (22.39) 0.14 (1.88) Note:* means t-ratio more than two. Fourth, only school 3 had a statistically significant positive same subgroup effect, which showed that same subgroup membership was a key factor explaining network change over two 51 years although there was also a statistically significant positive network structure effect and same grade level effect. Fifth, school 8 had statistically significant positive same grade level effect, which indicated that same grade level was a key factor explaining network change over two years. Sixth, there was a positive estimate of efficacy similarity in schools 8, 39, 45, 47, 48 and 53, indicating that there is more chance to interact with others who have similar efficacy within schools while a negative estimate of efficacy similarity in school 1, 3 and 54 indicates that there is less chance to interact with other who have similar efficacy in these schools. With respect to the results of mathematics teaching practices in the influence model, there were, statistically significant, positive changes in rates of mathematics teaching practices in schools 3, 8, 26, 39, 47, and 54 which indicated that teaching practices in mathematics problem solving changed over two years in these schools. In addition, in order to estimate the effect of the network on teaching practices in mathematics problem solving, the exposure variable was made and included in the models. There were statistically non-significant, positive estimates in seven schools. These results were different from the results of the two-level HLM in the present study. It could be due to differences in model specification and the unit of analysis. In the actor-oriented models, mathematics professional development, highest grade, and teaching efficacy were not included when specifying mathematics teaching practices dynamics and unit of analysis in actor-oriented models was each school while that of two-level HLM was teachers and subgroup. 3) Macro Level: Combination of Networks To investigate complex dynamic effects in longitudinal network models based on the results of the selection and influence modeling and generalize the results in each school, 52 multilevel longitudinal network models were analyzed using the Meta-analysis method in SIENA 08.exe. The results of model 1 indicate that the change in mathematics teaching practices advice networks was explained by reciprocity, same subgroup and common grade taught while the results of model 2 indicate that the change in mathematics teaching practices advice network was explained by reciprocated dyad network, transitive triplets’ network, and same subgroup. Table 2.15 the mean and variance of estimates in meta-analysis Parameter Mathematics Teaching Practices Advice Network Change Mathematics Teaching practices Change Network Change Speed Density Reciprocity Transitive triplets Subgroup (Centered) Grade (Centered) Efficacy similarity Teaching practices Change Speed Teaching practices Change Tendency Exposure Parameter Mean (S.E.) 1.89** (0.36) Model 1 Sample school 3, 8, 26, 39, 45, 47, 54 -2.35** (0.17) 1.60** (0.28) 3, 8, 39, 45, 47, 48 1, 3, 8, 26, 39, 45, 47 1.04** (0.23) 0.99** (0.26) 0.49 (0.35) 3, 8, 39, 45, 47, 48 3, 8, 26, 39, 45, 47, 48 1, 3, 8, 39, 45, 47, 48 1, 3, 8, 26, 39, 45, 47, 48, 54 1.04** (0.21) 0.55** (0.23) 0.003 (0.005) Mean (S.E.) 2.05** (0.29) 3, 8, 39, 47 3, 8, 39, 47,53 Model 2 Sample school All school Network Change Speed Mathematics Density -2.72** (0.20) 3, 8, 26, 39, 45, 47, 48 Teaching Reciprocity 1.11* (0.54) 1, 3, 8, 26, 39, 45, 47 Practices Advice Transitive triplets 0.70** (0.16) 3, 8. 39, 47, 53, 54 Network Change Subgroup (Centered) 0.81** (0.26) 3, 8, 26, 39, 45, 47, 48 Grade (Centered) 1.04 (0.57) 3, 8, 26, 39, 45, 47, 48 Efficacy similarity 0.32 (0.36) 1, 3, 8, 39, 45, 47, 48, 53 Teaching practices 1.42** (0.25) 1, 3, 8, 39, 45, 47, 48, 54 Mathematics Change Speed Teaching Teaching practices 0.49** (0.17) 1, 3, 8, 39, 47, 53 practices Change Change Tendency Exposure 0.004 (0.004) 1, 3, 8, 26, 39, 47,53, 54 Note: sample school was included if the standard error of each parameter was less than 5. * p< .05, ** p < .001. 53 A disadvantage of this Meta-analysis is that there are inconsistencies in the results obtained for estimates and tests. For example, there were significant grade effects in three schools in the micro level analysis while there was no significant same grade effect when including the transitive triplets’ effect in the Meta-analysis. Thus, there could be collinearlity problems between same grade taught and the transitive triplets’ effect in model specification. Overall, this pattern of results suggests that the formal organizational structure of the school and teachers’ social network structure at time 1 affect change of teachers’ social networks between time 1 and time 2 and teachers’ social networks at time 1 may affect change of teachers’ teaching practices between time 1 and time 2. Table 2.16 Results comparison among P2, HLM, and SIENA Selection Influence Model Model Parameters Model 2 Model 02 Network Change Speed 1.13* (0.42) Prior same subgroup 2.04* (0.48) Same grade Transitive triplets Teaching practices Change Speed Exposure 0.005* (.002) between 07 and 08 Note: * p< .05, ** p < .001. Actor-oriented Model Model 1 Model 2 1.89** (0.36) 1.42** (0.25) 1.04* (0.23) 0.99* (0.26) 0.81* (0.26) 1.04 (0.57) 0.70* (0.16) 2.05** (0.29) 1.42** (0.25) 0.003 (0.005) 0.004 (0.004) Now, we can compare these results (SIENA) with the result of selection (p2) and influence models (HLM). Main similarity among these results was as follows. With respect to 2Model 0 included only teaching practices in 2007 and exposure between 07 and 08 as level 1 predictors with no level 2 predictor in a two-level multilevel model. 54 results of selection models between p2 and SIENA, two results were similar in that prior same subgroup and same grade were significant factors to explain the change of mathematics problem solving teaching practices advice network. With respect to results of influence models between HLM and SIENA, two results were similar in that exposure between 2007 and 2008 was positively related to change of mathematics problem solving teaching practices, though estimates of exposure between 2007 and 2008 were not statistically significant in actor-oriented models. 55 Discussion and Conclusion After investigating teachers’ social networks through selection, influence, and dynamic modeling, research results indicate that the formal organizational structure of the school and teachers’ social network structure at time 1 affect the formation of new ties of teachers’ social networks at time 2 and teachers’ social networks at time 1 can affect teachers’ teaching practices at time 2. Previous studies (Penuel et al., 2009) showed similar results in selection and influence models except mathematics teaching efficacy. The main differences between previous studies and this research are the significant mathematics teaching efficacy effect in the selection model, subgroup mean mathematics teaching efficacy effect in the influence model and the transitive triplets effect in the actor-oriented model. When controlling for prior tie, p2 models like the SIENA model can estimate network change across two time points. In addition, when including covariates like subgroup networks, p2 models can estimate structural effects. However, p2 models have a limitation in estimating change when we use longitudinal data with more than two time points and the assumption of dyads independence. The methodological advantage of actor-oriented models over the selection and influence model is that we can analyze the network and behavior simultaneously while the disadvantage of actor-oriented model is that the model needs a larger sample size for estimation convergence and complex model specification. If we consider teachers’ turnover within schools, there might be much different patterns across the two years. Even though joining and leaving teachers (teachers’ turnover) within schools might be the main cause of change in relation, there were changes in relations among the 56 same teachers across two years after controlling for teachers’ turnover. In addition, change in relation could be the predictor of change in behavior or change in behavior could be the predictor of change in relation. Therefore, the result of the static model of network or behavior might be different from the result of dynamic model of network and behavior especially when there was transitive process, newcomers’ influence, and the effect of teachers’ turnover within schools. The contextual knowledge for specific situations is more important in order to teach students well. Some teachers can seek this contextual and local knowledge from their own previous teaching practices and others may have more suitable knowledge through trial and error as teaching experience increases. But beginning teachers or new joining teachers may not have local knowledge and need more time and effort to coordinate their teaching practices to specific classes. In this situation, school mentor or subject matter (English or Math) coordinators can provide local knowledge through repeated interaction until the knowledge is partially or completely transferred to new teachers. Frank, Zhao & Borman (2004) reported that through interaction with others, elementary teachers could have more knowledge to adapt computer technology in the classroom. For teachers who do not have local knowledge, professional development could be a good source of general knowledge but professional development without interactions with others might be inefficient way for local knowledge (Frank et al., 2011). One limitation of this study is missing values in networks and attributes data due to teacher turnover. Huisman & Steglich (2008) reported that missing actors have large effects on estimates when analyzing longitudinal network data. They showed that a reduced sample size lead to convergence problems with poor fitting evolution model and to biased parameter estimates. However, this study focuses on the same teachers during two years. Change of 57 network and behavior could be due to not only the external condition, which is turnover (attrition, changing composition), but also internal conditions, which are professional development even if composition was the same as before. Another limitation of this study is reliability and validity of network measurement using name generators because the 2007 survey of this study consists of three social network nominations and the 2008 survey of this study consists of five social network data and mathematics teaching practices advice network were presented as last question in both surveys. Pustejovsky & Spillane (2009) reported that multiplex social network data might be vulnerable to question-order effects. To investigate the relationship between network and behavior, this study used three methods that had different statistical assumptions and different model specifications. Even though actor-oriented models could directly identify triadic or higher-order network effects such as closure, actor-oriented models assumed that actor i controls all outgoing ties and changes network only one tie at a time, and that the probability of each tie change may depend on the entire current network, but not on earlier states of the network (continuous-time Markov chain). These are very strong assumptions because actor i may not or cannot control the outgoing tie especially when there are restrictions in environment, law, institution and policy. In addition, actor i may or can change some or most ties at a time especially when there are big life events like marriage, divorce, moving to another school, state or country, participation in international conferences or workshops, or natural disasters like earthquakes or floods. Therefore, we may need to consider event history analysis in longitudinal network analysis and future studies are needed to address this problem. Also, earlier states of the network might or could affect the current network especially when longitudinal data were collected 58 during less than a month or a year in case of friendship network or religious network. Thus, we may need to take this into account for research design and results interpretations. Though there are some limitations, this chapter shows that teachers’ social network can improve teaching practices by changing formal (grade) and informal (subgroup) structure. 59 Chapter 3: The Effect of Teachers’ Social Networks on Class Composition Introduction Recent studies show that students are non-randomly assigned to their teachers between schools (Jackson, 2009, Lankford, Loeb & Wyckoff, 2002; Miller, 2009) and within schools (Monk, 1987; Rothstein, 2008). One study about assignment of teachers to students between schools shows that teachers tended to move to schools that served high achieving students or high socioeconomic schools (Lankford, Loeb & Wyckoff, 2002). In other words, there was an uneven composition of teacher quality across schools. In addition, the studies about assignment of teachers to students within schools reported that principals and teachers were involved in class formation and composition (Monk, 1987; Burns & Mason, 1995; 1998). Specifically, one study showed that about two-thirds of principals included teachers formally or informally when students were assigned to their classes (Burns & Mason, 1995). Another study showed that classes were purposefully created by the majority of principals (Burns & Mason, 1995). In other words, principals and teachers control student assignment at the elementary level, resulting in potentially uneven composition of students in schools. If so, why is this important? First, peer effects studies show that students’ peers have an important impact on their learning (Burns & Mason, 2002; Harris, 2010), which leads to differences in academic achievement. In addition, student composition could affect not only interaction among students but also interaction among their parents, which can affect students’ social capital. 60 Second, Aptitude-Treatment Interactions (ATI) studies showed that there was a remarkable interaction between students’ aptitudes3 and instructional methods (Cronbach & Snow, 1977; Snow, 1989). In other words, class composition can affect overall students’ aptitudes, which can influence teachers’ instructional methods. Specifically, Cronbach (1957) pointed out “persons should be allocated on the basis of those aptitudes which have the greatest interaction with treatment variables” (p. 681). Another study showed that classroom composition constrained teaching practices and student learning (Dreeben & Barr, 1988). In other words, students’ assignment to their teachers could affect the nature of teaching practices for the whole class from the beginning of year, which might lead to different learning outcomes for students at the end of year. Third, when assessing students’ academic achievement, class composition affects model specification and estimates (Cronbach, 1976). Furthermore, recent results indicate that students’ non-random assignment could influence gain scores, which might produce selection bias and misleading conclusions when evaluating teachers’ effects on gain in students’ academic achievement (Koedel & Betts, 2009; Rivkin & Ishii, 2009; Rothstein, 2009). Although previous studies found students were non-randomly assigned to classrooms (Burns & Mason, 1995, 1998; Heck & Marcoulides, 1989; Heck et al., 1989; Jacob & Lefgren, 2007; Monk, 1987), little effort has been made to explain the mechanism of non-random assignment as a function of teachers’ attributes and teachers’ social networks. Thus, the purpose of this study is to explain the mechanism of assignment of students to teachers. This chapter is organized as follows. First, I introduce studies about the impact of class composition on student learning. Class composition studies and value-added models are reported with a focus on Aptitude refers to “any characteristic of the person that affects his response to the treatment” (Cronbach, 1975, p. 116) 3 61 empirical studies. Second, I present my data and methods including sample, dependent variables, and independent variables, including teachers’ attributes and social networks. Third, the estimates of the relationships between teachers’ social networks and class composition are presented. Finally, the discussion and conclusion are shared. 62 Literature Review 1. The Impact of Peers and Class Composition on Students’ Learning One study summarized whether school peers influence educational outcomes and explored three hypotheses that a) advantaged peers were beneficial for disadvantaged students, b) advantaged peers were harmful for disadvantaged students, and c) peers have no influence on disadvantaged students, as shown in Table 3.1 (Harris, 2010). Harris proposed a “group-based contagion theory in which students benefit from advantaged peers mainly when those peers are in the same group” (p. 1190). In addition, Harris pointed out that “peers indirectly influence one another by affecting the school resources to which they have access, especially the qualifications of the teachers who teach them” (p. 1190). Table 3.1 Summary of theories and implications Theory Disciplinary Perspective Advantaged Peers Beneficial Epidemic Cognitive Institutional-resources Institutional-expectations Disruption Advantaged Peers Harmful Relative deprivation Oppositional culture Signaling Focus-boutique Peers have no influence Home Influences Tracking Source of Peer Influence Sociology Psychology Economics and Political Science Economics Beliefs/values Instrumental Instrumental Instrumental Instrumental Sociology Anthropology and Sociology Economics - Beliefs/values Beliefs/values Instrumental Instrumental Sociology - Source: Harris, 2010, p. 1177 63 In addition, previous Aptitude-Treatment Interactions (ATI) studies have shown that there was a remarkable interaction between students’ aptitudes and instructional methods (Cronbach & Snow, 1977; Snow, 1989). Cronbach (1957) claimed that “persons should be allocated on the basis of those aptitudes which have the greatest interaction with treatment variables” (p. 681). Also, Monk (1987) pointed out that “teachers vary in their ability to achieve success with particular types of pupils, and the composition of a classroom is related to how much a particular child learns” (pp. 167-168). Specifically, Dreeben & Barr (1988) pointed out the mechanism of the effect of classroom composition on students’ learning. Dreeben & Barr described the importance of classroom composition on students’ learning in that “because many low-aptitude students have to work independently at their seats while the teacher provides one group with direct attention, there will be more intrusions and time will be used less productively; as a result, there will be less learning in difficult classes” (p. 133). Although they explained the effect of classroom composition, classroom dynamics and student learning, they didn’t explain which factors affect classroom composition. With respect to relationships between classroom composition and achievement, the study by Burns & Mason (1995) reviewed previous studies and summarized that after controlling for individual scores, there are modest but statistically significant relationships between class mean scores and achievement. In addition, they reported that teacher commitment and motivation may be conditioned by classroom composition. Empirically, another study examined the relationship between class composition and student achievement in 22 elementary schools using hierarchical linear modeling to estimate composition effects (Burns & Mason, 2002). Burns & Mason (2002) 64 argued that higher ability and more independent students were assigned to combination classes4 by principals and teachers, which caused variation in student achievement. Though they compared the single classes to combination classes with respect to composition effects, they didn’t consider teachers’ social networks, which could affect class composition. In summary, previous studies have shown that class composition and peer effects have an important impact on students’ learning (Burns & Mason, 2002; Dreeben & Barr, 1988; Harris, 2010), which lead to differences in academic achievement, although few have focused on the factors that might affect assignment of students to teachers. 2. Class Composition 1) Which Factors Affect Class Composition? The study by Heck et al. (1989) examined principals’ roles concerning teacher and student assignment decisions and proposed a model of the factors that influence these decisions. They pointed out five factors which are teacher student matching, organizational concerns, internal political concerns, parent input, and data sources. Based on this model, another study by Heck & Marcoulides (1989) tested whether the principals’ teacher allocation decisions were affected by district and school size using LISREL methodology; 170 Elementary school principals from three categories of California districts and school sized were selected through random interval sampling methods. They found that the proposed model fit well across schools of all sizes but did not fit well in large districts. In other words, school size does not matter in allocation decisions, consistent with the results of Monk (1987, see also Burns & Mason, 1998). 4 includes students from more than one grade level at the elementary school level as selfcontained classrooms 65 In summary, schools and districts factors (organizational concerns, internal political concerns, data sources, and district size) as well as people (teacher student matching, parent input) could affect class composition. The study by Monk (1987) found that principal involvement (high, medium and low) varied between schools. In the case of low principal involvement, teachers met at the end of the year and distributed students to classes, and determined which teacher would teach which classes (Monk, 1987). One principal said that “Well, if you take [name] then I’d like to have [name]” and another principal “recounted an instance where veteran teachers loaded up a first-year teacher with a disproportionate number of difficult students” (Monk, 1987, p. 173). In addition, the length of a principal’s tenure was positively related to the principal’s involvement in assigning students to teachers (Monk, 1987). 2) How Do Principals Compose Classes? Principals used several general strategies for student assignment, including random assignment, homogeneous classes, balanced classrooms, matching characteristics of students to teachers, and assignments by previous year’s teachers (Monk, 1987). Monk summarized that regardless of principals’ involvement, it was common practice to balance classes with respect to gender and race without balancing classes with respect to achievement levels, learning styles, aptitude for learning, and so on. The study by Burns & Mason (1995) described similar class formation procedures across 22 schools as five steps. First, principal provides teachers with the grade-level configuration template; Second, principal provides teachers with guidelines for class formation; 66 Third, at a grade-level meetings, teachers use student placement cards, usually color coded by gender, to create next year’s classes, sorting students cards according to the principal’s template and guidelines outlined in Steps 1 and 2; Fourth, principal reviews cards, checks for potential conflicts or imbalances not noticed by teachers, incorporates parent requests, addresses any teachers’ concerns; Fifth, fall adjustments are made. (pp. 749-750). In addition, the study by Burns & Mason (1995) reported the strategies principals use to assign students to classes and the numbers of principals reporting their use as shown in Table 3.2; 57 principals (64%) used the planned strategies involving teachers formally or informally. Table 3.2 Strategies principals use to assign students to class and number of principals reporting their use Strategy Number of Principals Strategies requiring little or no planning: 31 (34%) Random assignment 15 (17%) Classes roll over with adjustment 12 (13%) Classes roll over 4 (4%) Planned strategies: 59 (66%) Teachers use promotion card* 32 (36%) Teachers informally create classes+ 20 (22%) Principal and teachers decide together^ 5 (6%) Principal uses promotion cards** 2 (2%) Note *Information cards are completed by teachers for each student. Cards reflect behavioral and academic characteristics of students, and teachers attempt to create classes based on card information. Principals can or will review class assignments and make minor adjustments. + Similar to above but without formal promotion cards. Teachers meet and work cooperatively to share knowledge and characteristics of each student and formulate the best assignment for each student. ^ Principals and teachers meet together and cooperatively use promotion cards or share knowledge about students for final student placements. **Principals are given promotion cards by teachers and principals make student assignments. Source: Burns & Mason, 1995, p. 197. 67 In summary, principals used not only random assignment but also the planned strategies involving teachers formally or informally. However, we don’t know how teachers could influence this process informally. 3. Value-Added Models (VAM) When estimating teacher effects on student learning by using value added models, researchers have focused on a) defining and measuring student learning (Linn, 2005), b) education production functions (Hanushek, 1979, 1986), c) test alignment and domain coverage (Porter, et al., 2007; Webb, 2007), d) scaling and growth modeling (Briggs et al., 2008), e) vertical scaling and multidimensionality (Martineau, 2006), f) the Sanders model (Sanders & Horn, 1994; Wright, Horn, & Sanders, 1997), g) models in experimental studies (Dee, 2004; Kane & Staiger, 2008), and h) model specifications (Harris & Sass, 2006; McCaffrey et al., 2004). Specifically, one study showed that a) student and teacher heterogeneity were the most important issues with which value-added models must contend, b) covariates were inadequate replacements for individual student and teacher effects, and c) random effects models yield inconsistent estimates of model parameters due to correlation between the random effects and explanatory variables in the model (Harris & Sass, 2006). Harris & Sass also noted that the biases introduced by covariate and random effects models extend both to the estimates of the unobserved teacher quality and the effects of time-varying teacher characteristics (experience and professional development) on student achievement. The study of Harris & Sass (2006), however, assumed that measuring interactions and coordination among teachers directly was rarely possible even though social network methods 68 can measure interactions and coordination among teachers. In other words, they assumed that characteristics (attribute) can be measured while networks are difficult to measure. If a model includes network measures as well as attribute measures, the results may be changed significantly. Thus, social network measures need to be considered in value-added models specification. A second study designed a random-assignment experiment in the Los Angeles Unified School District (Kane & Staiger, 2008). Kane & Staiger collected students and teachers in grades two through five and relied on an experiment in which 78 pairs of classrooms (156 classrooms and 3194 students) were randomly assigned between teachers in the school years 2003-04 and 2004-05 in the Los Angeles Unified School District. The results of Kane & Staiger indicated that differences in mean student outcomes within each pair could be predicted by several alternative non-experimental specifications. In addition, they evaluated both the bias and predictive accuracy of the value-added estimates by seven model specifications since controls could improve the precision of estimates or reduce bias respectively. The model specifications were a) end-of-year test scores with no controls, b) endof-year test scores with student/peer controls (included prior scores), c) end-of-year test scores with student/peer controls (included prior scores) and school fixed effects, d) end-of-year test scores with student fixed effects, e) gain scores with no controls, f) with student/peer controls, and g) with student/peer controls and school fixed effects for experimental and non-experimental data. The study of Kane & Staiger (2008), however, excluded teachers with less than three years teaching experience in estimating effects. Excluding teachers who have less than three years teaching experience is important because teacher quality depends on teaching experience. 69 Due to this exclusion, we might have only very qualified teachers with little heterogeneity in teacher effects. Furthermore, if teachers’ social networks vary depending on teaching experience and there is relationship between teachers’ social network and class composition, they need to test assumptions about classroom assignment of students to teachers. In summary, random assignment, which can be implemented through experimental design, is one solution to minimize selection bias when conducting value-added models studies. Second, model specification, which can be implemented through statistical modeling, is the other solution to minimize selection bias. Previous value-added models, however, did not account for teachers’ social networks which might influence random assignment as well as model specification when evaluating the teachers’ effect on gain in students’ academic achievement. Using falsification tests for three widely used value-added modeling specifications, Rothstein (2008, 2010) tested assumptions about classroom assignment of students to teachers, based on the idea that future teachers cannot influence students' past achievement just as future teachers cannot have causal effects on past outcomes. Rothstein (2008, 2010) focused on the cohort of students in the fifth grade in 2000-2001, consisting of 60,740 students from 3,040 fifth grade classrooms and 868 schools from a larger population of 99,071. The data were collected by the North Carolina Education Research Data Center. This study examined end-of-grade math and reading tests from grades 3 through 5. To construct the third grade gain score, this study used “pre-tests” given at the beginning of 3rd grade in place of the second grade scores by standardizing the scale scores separately for each subject-grade-year combination. Also, this study used a restricted sample consisting of 23,415 students from 2,116 classrooms and 598 schools. 70 The strength of Rothstein’s research (2008, 2010) is that it challenges assumptions about random assignment of students to teachers and provides information about falsification of three widely used VAMs. At the same time, one weakness is that it does not explain what kind of factors affect non-random assignment of students to teachers. In particular, previous value-added models did not account for teachers’ social networks which might influence random assignment as well as model specification when evaluating the teachers’ effect on gain in students’ academic achievement. Therefore, as described in chapter 1, this study tests Hypothesis 2-2: Previous social networks at a higher level (level 2) affect current formal organizational structure at a lower level (level 1). To do this, first, this study will test the first null hypothesis: formal organizational structure at students’ level are homogeneous with respect to students’ previous academic achievement and economic status. In other words, students are randomly assigned to their teachers within and between schools with respect to previous academic achievement and economic status. Second, this study will test the second null hypothesis: there is no relationship between teachers’ social networks within schools and their students’ previous academic achievement and economic status. Third, this study will test the third null hypothesis: there is no effect of teachers’ specific social networks within schools on their students’ previous academic achievement economic status. Thus, the primary research question will be addressed in this study as follows: After controlling for teachers’ attributes, do teachers’ social networks affect non-random assignment of students to teachers with respect to students’ previous academic achievement and economic status? 71 Data and Methods 1. Data Data for this analysis are drawn from a larger study of school leadership and management in one public school district in the southeastern United States. Table 3.3 School and student characteristics in 30 elementary schools in 2006~2007 African Free/ Student Student White LEP School Title Ⅰ American Reduced Attendance Enrollment students students Student Lunch 77% 97% 522 93% 2% 0% 1 Yes 47% 95% 641 18% 73% 1% 2 No 81% 97% 785 95% 3% 0% 3 Yes 92% 95% 527 99% 1% 0% 4 Yes 38% 96% 508 44% 48% 0% 5 No 92% 96% 583 97% 1% 0% 6 Yes 83% 96% 519 76% 13% 1% 7 Yes 97% 97% 402 99% 0% 0% 8 Yes 35% 96% 622 34% 40% 11% 9 No 62% 95% 507 45% 44% 0% 10 Yes 92% 96% 370 98% 1% 0% 11 Yes 62% 96% 607 71% 20% 0% 12 Yes 35% 96% 434 24% 67% 1% 13 No 84% 95% 381 99% 0% 0% 14 Yes 19% 96% 611 6% 76% 10% 15 No 77% 96% 628 80% 16% 1% 16 Yes 62% 96% 445 60% 30% 3% 17 No 49% 97% 409 61% 33% 1% 18 No 67% 96% 487 74% 13% 0% 19 Yes 80% 96% 468 86% 10% 0% 20 Yes 28% 96% 870 26% 63% 0% 21 No 23% 96% 372 14% 74% 4% 22 No 66% 96% 354 38% 34% 15% 23 Yes 67% 96% 421 67% 18% 0% 24 Yes 54% 96% 761 49% 37% 0% 25 No 97% 95% 533 99% 1% 0% 26 Yes 94% 96% 603 89% 6% 0% 27 Yes 47% 96% 722 38% 46% 1% 28 No 68% 96% 646 66% 20% 2% 29 Yes 53% 96% 476 58% 30% 4% 30 No Note: Only school 26 did not meet AYP; school 30 was excluded from the final sample 72 Table 3.4 Teacher characteristics in 30 elementary schools in 2006~2007 % Full time % Female % White School Total Teacher 48 100% 88% 63% 1 51 94% 92% 80% 2 59 97% 98% 47% 3 56 91% 95% 32% 4 44 95% 89% 93% 5 53 96% 94% 55% 6 45 93% 98% 53% 7 41 95% 80% 37% 8 52 98% 98% 79% 9 44 100% 93% 64% 10 33 97% 91% 52% 11 54 93% 93% 83% 12 35 97% 97% 83% 13 31 97% 84% 58% 14 51 100% 98% 92% 15 52 98% 98% 87% 16 41 98% 93% 88% 17 32 97% 97% 84% 18 47 89% 89% 60% 19 43 95% 93% 72% 20 71 99% 94% 90% 21 32 100% 84% 88% 22 33 97% 94% 70% 23 45 100% 93% 67% 24 56 93% 96% 79% 25 50 90% 86% 62% 26 57 96% 98% 70% 27 51 96% 94% 88% 28 58 97% 93% 90% 29 41 98% 93% 73% 30 Note: school 30 was excluded from the final sample Years Experience 12 14 13 12 11 10 14 11 11 13 10 15 19 11 16 16 14 14 15 12 17 11 13 13 16 9 10 9 16 14 In the 2006-2007 school year, the Cloverville district served 33,156 students, including 16,214 students at its 30 elementary schools. All schools except one met AYP and student attendance was more than 95%. Three schools had more than 10% Limited English Proficient (LEP) students, as shown in Table 3.3. In addition, most schools had full-time teachers with an 73 average 10 of years of teaching experience. The final sample was 309 self-contained teachers across 29 elementary schools in 2007. 2. Measures The dependent variables were class average English/Language Arts (ELA) achievement in 2006, class average Mathematics achievement in 2006 and class average free/reduced lunch in 2006. The attributes variables were gender, race, education, teaching experience, new teachers at the school, self-contained teachers, professional development, formal leader, the number of formal leadership roles, and several leadership roles. In addition to the attributes variables, the network variables were in-degree in ELA, Math, and combined (ELA plus Math) advice networks in 2007. 1) Dependent Variables Class average English/Language Arts (ELA) test score in 2006 is based on a criterionreferenced-test (CRT) with multiple-choice items. The content weights for the ELA CRT in grade 2 consisted of Grammar/Phonics (60%), Sentence Construction (25%), and Research (15%) while the domains for grades 3 through 5 consisted of Grammar/Sentence Construction (60%) and Research/Writing Process (40%). There were three categories of performance standards: below 800, between 800 and 850, and above 850. Class average Mathematics test score in 2006 is based on a criterion-referenced-test (CRT) with multiple-choice items. The content weights for ELA CRT in grade 2 consisted of Number and Operations (55%), Measurement (15%), Geometry (20%), and Data Analysis and Probability (10%) while the domains for grades 3 through 5 consisted of Number and Operations 74 (50%, 43%, and 38%), Measurement (18%, 17%, and 32%), Geometry (12%, 20%, 10%), Algebra (10%), and Data Analysis and Probability (10%). There were three categories of performance standards: below 300, between 300 and 350, and above 350. Class average free/reduced lunch in 2006 ranged from 0.00 to 1.00 with a mean of 0.61 and a standard deviation of 0.28. 2) Attributes Variables Male was coded as 0= “female” and 1= “male”. Race had two dummy variables. One was white (68%) (0= “non-white” and 1= “white”) and the other was African American (26%) (0= “non-African American” and 1= “African American”). Education in 2007: Teachers were asked if they had a graduate degree (e.g., Master’s degree or Ph.D.) and were coded as 0= “No” and 1= “Yes” Teaching Experience in 2007: Teachers were asked how many years they had taught as a teacher. New teachers at this school in 2007: Teachers with less than one year at their current school were coded as 1. Self-contained teachers in 2007: Teachers were asked if they taught self-contained classrooms; if so, they were coded as 1. ELA and Mathematics Professional Development in 2007: The question was: “Please indicate how many professional development sessions you participated in this year.” The variable scales were from 1 to 4 (1= “None,” 2= “1-2 sessions,” 3= “3-7 sessions,” and 4= “more than 8 sessions.”). 75 Formal leader in 2007: The question was: “Are you formally assigned to perform a leadership role at this school such as assistant principal, reform program coach/facilitator, subject area coordinator or chair, master/mentor teacher, or program coordinator (e.g., Title 1 coordinator)?” The number of formal leadership roles in 2007: The total number of formal leadership roles ranged from 0 to 10. Reading, Literacy, or English program coordinator/chair, Math program coordinator/chair, school improvement coordinator, master/mentor teacher and teacher consultant in 2007: the question was: “What percentage of your time is formally assigned to any of the following leadership roles at this school?” The variable scales from 1 to 6 (1= “0%,” 2= “1-25 %,” 3= “26-50%,” 4= “51-75%,” 5= “76-99%,” and 6= “100 %.”). 3) Network Variables This study used advice networks as a proxy indicator instead of networks about class assignment though advice networks might not be directly related to networks about class assignment. In addition, this study assumed that networks in 2007 were similar to networks in 2006 even though there might be some change due to teacher turnover or dynamic factors. Thus, this study used networks in 2007 instead of 2006 in order to control for new teachers at this school in 2007 and due to data limitations although networks in 2006 might be more precisely related to class assignment in 2007. In other words, if we use teachers’ social networks in 2006, we need to exclude the new teachers in 2007 while if we use teachers’ social networks in 2007, we could control for new teachers in 2007 and show whether or not there was a relationship between current (2007) networks and previous (2006) achievement. 76 In-degree in ELA advice network in 2007: The ELA advice network consisted of the ties of interaction for each colleague (5-point scales: yearly, semiannually, monthly, weekly, and daily) in 2007 based on the following question: To whom do you turn in this school for advice or information about reading/language arts or English instruction? In-degree in ELA advice network measures the number of colleagues that were named as an advice-givers as part of their advice networks. In the case of advice networks, teachers with a higher in-degree in ELA advice networks may be considered the experts in ELA within their school because they are sought more frequently for advice in ELA subject. In-degree in Math advice network in 2007: The Math advice network consisted of the ties of interaction for each colleague (5-point scales: yearly, semiannually, monthly, weekly, and daily) in 2007 based on the following question: To whom do you turn in this school for advice or information about mathematics instruction? In-degree in Math advice network measures the number of colleagues that were named as an advice-givers as part of their advice networks. In the case of advice networks, teachers with a higher in-degree in Math advice networks may be considered the experts in Math within their school because they are sought more frequently for advice in Math subject. In-degree in Combined advice network (ELA plus Math advice networks) in 2007: The Combined network was based on the composite networks of ELA and math networks. In-degree in combined advice network measures the number of colleagues that were named as an advicegivers as part of their advice networks. In the case of advice networks, teachers with a higher indegree in combined advice networks may be considered the experts in ELA or Math within their school because they are sought more frequently for advice in ELA or Math subject. 77 3. Methods To test the first null hypothesis: students are randomly assigned to their teachers within and between schools with respect to previous academic achievement and economic status, twolevel unconditional models were performed. In addition, to test the second null hypothesis: there is no relationship between teachers’ social networks within schools and their students’ previous academic achievement and economic status, correlation analyses were performed. Finally, to test the third null hypothesis: there is no relationship between specific teachers’ social networks within schools and their students’ previous academic achievement economic status, I explored which types of teachers’ social networks and attributes affect non-random assignment between and within schools through multiple regression analyses. For this, I analyze the five models for ELA, Math and Combined networks and compare these results. SAS 9.2 software was used to run two-level unconditional models, compute the in-degree in advice networks, analyze correlation, and run five multiple regression models. 4. Models 1) Model 1 To examine the effect of teachers’ social networks on non-random assignment with respect to students’ academic achievement and economic status, model 1 was specified as a multiple regression model. Model 1 controlled the following teachers’ attributes: gender, race, education, teaching experience, professional development, new teachers. 78 r de r de hool r de hool 6 Where β β is the intercept. indicates the effect of Network Variables (Teachers’ ELA, Math, and combined networks) in 2007 on Dependent Variables (Class average ELA academic achievement score, class average Math academic achievement score, and class average free/reduced lunch) in 2006 nd rd th are the effect of 2 , 3 and 4 grade level on Dependent Variables (Class average ELA academic achievement score, class average Math academic achievement score, and class average free/reduced lunch) in 2006. are the effect of each school on Dependent Variables (Class average ELA academic achievement score, class average Math academic achievement score, and class average free/reduced lunch) in 2006. are the effect of each Attributes Variables(Gender, race, education, teaching experience, professional development, new teachers) on Dependent Variables (Class average ELA academic achievement score, class average Math academic achievement score, and class average free/reduced lunch) in 2006. is the residual term. The larger the value of β , the more we would infer that teachers’ social networks affect non-random assignment with respect to students’ academic achievement and economic status. The larger the value of β β , the more we would infer that each grade level affects non-random assignment with respect to students’ academic achievement and economic status. 79 The larger the value of β β , the more we would infer that each school affects non-random assignment with respect to students’ academic achievement and economic status. The larger the value of β β , the more we would infer that each attribute variable affects non-random assignment with respect to students’ academic achievement and economic status. 2) Model 2 Model 2 added up the formal leader variable. r de r de hool r de hool 6 Where is the effect of Attributes Variable(Formal leader) on Dependent Variables (Class average ELA academic achievement score, class average Math academic achievement score, and class average free/reduced lunch) in 2006. 80 3) Model 3 Model 3 replaced the formal leader variable with the total number of leadership roles. r de r de hool r de hool 6 𝑏 Where is the effect of Attributes Variable(the total number of leadership roles) on Dependent Variables (Class average ELA academic achievement score, class average Math academic achievement score, and class average free/reduced lunch) in 2006. 81 4) Model 4 Model 4 replaced the formal leader variable with coordinator (ELA, Math, or School improvement) roles. r de r de hool r de hool 6 Where is the effect of Attributes Variable(Coordinators) on Dependent Variables (Class average ELA academic achievement score, class average Math academic achievement score, and class average free/reduced lunch) in 2006. 82 5) Model 5 Model 5 replaced the formal leader variable with teacher consultant roles. r de r de hool r de hool 6 Where is the effect of Attributes Variable(teacher consultant) on Dependent Variables (Class average ELA academic achievement score, class average Math academic achievement score, and class average free/reduced lunch) in 2006. With respect to students’ economic status, professional development variables were excluded in five models. 83 Results Descriptive statistics are shown in Table 3.5. Table 3.5 Descriptive statistics of teachers with at least 10 students except the first grade N Min Max Mean SD English Language Arts (ELA) in 2006 309 785 862 820 17 Mathematics in 2006 309 284 380 329 20 Free/reduced lunch in 2006 309 0 1 .61 .27 Male teachers 309 0 1 0.09 0.29 How many years have you worked as a teacher? 307 0 47 12 9.5 White teachers 309 0 1 .70 .46 African American teachers 309 0 1 .24 .43 Graduate degree 309 0 1 .76 .43 New teachers at this school 309 0 1 .24 .43 304 0 3 1.28 .82 306 0 3 1.18 .76 309 0 1 .27 .44 The total number of formal leadership roles 309 0 10 0.82 1.51 Reading, Literacy, or English program coordinator/Chair 309 0 6 0.16 0.70 Math program coordinator/Chair 309 0 6 0.18 0.75 School improvement coordinator 309 0 6 0.13 0.59 Master/mentor teacher 309 0 6 0.44 1.14 Teacher consultant 309 0 6 0.21 0.75 In-degree in ELA advice networks 309 0 8 0.84 1.13 In-degree in Math advice networks 309 0 9 0.88 1.24 In-degree in Combined (ELA plus Math) advice networks 309 0 9 1.24 1.49 Valid N (listwise) 309 Reading/Language Arts or English teaching professional development Mathematics teaching professional development Are you formally assigned to perform a leadership role at this school? 84 The ELA academic achievement score in 2006 was an average of 820 with standard deviation 17, and the range was from 785 to 862 while the Math academic achievement score in 2006 was an average of 329 with standard deviation 20, and the range was from 284 to 380. In addition, the percentage of students eligible for free/reduced lunch in 2006 averaged 0.61 with standard deviation 0.27, and the range from 0 to 1. 1. Heterogeneous Academic Achievement & Economic Status Within and Between Schools After testing the first null hypothesis: students are randomly assigned to their teachers within and between schools with respect to previous academic achievement and economic status, the results of two-level unconditional models indicated that there was statistically significant variation in previous academic achievement among teachers (76%) while there was statistically significant variation in economic status between schools (73%), as shown in Table 3.6 (all are statistically significant at p < .01) Table 3.6 Two-level (classes nested in schools) unconditional models ICC Variance Random Effects S.E. (Ratio) Estimates ELA Schools Random Intercept 24% 63 21 Achievement Classes Residual 76% 199 17 Math Schools Random Intercept Achievement Classes Residual 24% 76% 90 283 31 24 Z value p value 2.96 12.01 0.0015 <.0001 2.92 11.99 0.0018 <.0001 Free/reduced Schools Random Intercept 73% 0.054 0.015 3.58 0.0002 Lunch Classes Residual 27% 0.020 0.002 11.94 <.0001 Note: only self-contained teachers were included into models except the first grade level. ICC means intraclass correlation. In other words, these variance estimates suggest that schools vary in students’ average 85 previous academic achievement both in ELA and mathematics and there is more variation among classes (self-contained teachers) within schools. However, there is even more variation between schools in students’ average previous economic status. In summary, these results indicated that students are non-randomly assigned to their teachers within and between schools with respect to previous academic achievement and economic status. 2. Association Between Teachers’ Social Networks and Their Students’ Previous Academic Achievement & Economic Status To test the second null hypothesis: there is no relationship between teachers’ social networks within schools and their students’ previous academic achievement and economic status, correlation analyses were conducted as shown in Table 3.7. If students were randomly assigned to their teachers regardless of their teachers’ social networks, we would expect no relationship between students’ previous academic achievement and their teachers’ social networks. If we found the association between teachers’ social networks and their students’ previous academic achievement, we could infer that students were non-randomly assigned to their teachers depending on their teachers’ social networks within schools with respect to academic achievement. First, the results showed that the correlation between teachers’ ELA networks in 2007 and students’ ELA achievement in 2006 was statistically significant (0.25, p<.01) and the correlation between teachers’ Math networks in 2007 and students’ Math achievement in 2006 was statistically significant (0.18, p<.01), as shown in Table 3.7. 86 Table 3.7 Correlation matrix ELA in Free lunch in 2006 Free lunch 2006 Math in 2006 Math in 2006 in 2006 In-degree in In-degree in ELA in 2007 Math in 2007 .89** -.61*** -.62*** In-degree in ELA in 2007 .25** .22*** -.20*** In-degree in Math in 2007 .19*** .18** -.14* -.41*** In-degree in Combined .20*** .18** -.18** .76*** .83*** Note: * p < .05, ** p < .01, *** p< .001. Second, the correlation between teachers’ Math networks in 2007 and students’ ELA achievement in 2006 was statistically significant (0.19, p<.001) and the correlation between teachers’ ELA networks in 2007 and students’ Math achievement in 2006 was statistically significant (0.22, p<.001), as shown in Table 3.7. Third, the correlation between teachers’ combined networks in 2007 and students’ ELA achievement in 2006 was statistically significant (0.20, p<.001) and the correlation between teachers’ combined networks in 2007 and students’ Math achievement in 2006 was statistically significant (0.18, p<.01), as shown in Table 3.7. Fourth, the correlation between teachers’ ELA networks in 2007 and Free/reduced lunch in 2006 was statistically significant (-0.20, p<.001) and the correlation between teachers’ Math networks in 2007 and Free/reduced lunch in 2006 was statistically significant (-0.14, p<.05). In addition, the correlation between teachers’ combined networks in 2007 and students’ free/reduced lunch in 2006 was statistically significant (-0.18, p<.01), as shown in Table 3.7. In summary, these results of positive correlation indicated that students were nonrandomly assigned to their teachers depending on their teachers’ social networks with respect to previous academic achievement. In other words, the larger social networks a teacher has within 87 her school, the more academically advantaged students the teacher will have. Additionally, these results of negative correlation indicated that students were nonrandomly assigned to their teachers depending on their teachers’ social networks with respect to previous class average free/reduced lunch. In other words, the larger social networks a teacher has within her school, the more economically advantaged students the teacher will have. 3. The Effects of Teachers’ Social Networks and Attributes on Non-Random Assignment Finally, to test the third null hypothesis: there is no effects of teachers’ particular social networks within schools on their students’ previous academic achievement economic status, I explored which types of teachers’ social networks and attributes affect non-random assignment between and within schools through multiple regression analyses. 1) Students’ Previous Academic Achievement Specifically, to examine which types of teachers’ social networks affect non-random assignment, three types of teachers’ social networks (i.e., ELA, Math, and combined networks) were analyzed respectively in five multiple regression models. (1) ELA Achievement First, after controlling for teachers’ attributes with school and grade-fixed effects, the results of five models indicated that teachers’ ELA networks had a positive effect on non-random assignment with respect to students’ previous ELA achievement. African American teachers, new teachers, and ELA professional development had a negative effect on previous ELA achievement while teaching experiences, a Master’s degree, a formal leader, the total numbers and specific 88 Table 3.8 Effects of teachers’ ELA or Math networks on students’ previous ELA achievement ELA Networks Model 1 Model 2 Model 3 Model 4 Model 5 Male 0.01 0.01 -0.01 0.00 0.00 White 0.02 -0.05 -0.01 0.01 0.00 African American -0.06 -0.13 -0.09 -0.07 -0.09 Master’s degree 0.03 0.03 0.03 0.02 0.03 Teaching experience 0.04 0.03 0.02 0.04 0.02 ELA professional development -0.04 -0.07 -0.06 -0.05 -0.05 New teacher -0.14* -0.10+ -0.12* -0.14* -0.12* Formal leader 0.22*** The total number of leadership roles 0.19*** ELA coordinator 0.08 Teacher consultant 0.16*** In-degree in ELA networks 0.13* 0.10+ 0.10+ 0.12* 0.12* R-Square 0.40 0.44 0.43 0.41 0.42 Model 1 Model 2 Model 3 Model 4 Model 5 Male -0.01 0.00 -0.02 -0.02 -0.01 White 0.02 -0.05 -0.01 0.01 0.00 African American -0.06 -0.13 -0.09 -0.07 -0.09 Master’s degree 0.04 0.04 0.03 0.03 0.04 Teaching experience 0.04 0.03 0.02 0.04 0.02 ELA professional development -0.03 -0.05 -0.05 -0.03 -0.04 New teacher -0.14* -0.11+ -0.13* -0.15** -0.13* Math Networks Formal leader 0.22*** The total number of leadership roles 0.19*** ELA coordinator 0.08 Teacher consultant 0.15** In-degree in Math networks 0.12* 0.06 0.07 0.11* 0.10* R-Square 0.40 0.43 0.43 0.41 0.41 Notes: sample size=300, school and grade level fixed effects models. + p < .10, * p < .05, ** p < .01, *** p < .001. 89 types (e.g., teacher consultant) of leadership roles had a positive effect. In summary, these results indicated that ELA networks had a significant positive effect on non-random assignment between and within schools after controlling for teachers’ attributes, as shown in Table 3.8. Second, after controlling for teachers’ attributes with school and grade fixed effects, the results from five models indicate that teachers’ Math networks had a positive effect on nonrandom assignment with respect to students’ previous ELA achievement. African American teachers, new teachers, and ELA professional development had a negative effect on previous ELA achievement while teaching experience, a Master’s degree, a formal leader, the total numbers and specific types (e.g., teacher consultant) of leadership roles had a positive effect. In summary, these results indicate that Math networks had a positive effect on non-random assignment between and within schools after controlling for teachers’ attributes, as shown in Table 3.8. Table 3.9 Effects of teachers’ combined networks on students’ ELA previous achievement Combined Networks Model 1 Model 2 Model 3 Model 4 Model 5 Male 0.00 0.00 -0.02 -0.01 -0.01 White 0.02 -0.04 -0.01 0.01 0.00 African American -0.05 -0.13 -0.09 -0.07 -0.08 Master’s degree 0.03 0.04 0.03 0.03 0.04 Teaching experience 0.04 0.03 0.02 0.04 0.02 ELA professional development -0.04 -0.06 -0.06 -0.04 -0.05 New teacher -0.14* -0.10+ -0.12* -0.14** -0.12* Formal leader 0.22*** The total number of leadership roles 0.19*** ELA coordinator 0.07 Teacher consultant In-degree in Combined networks 0.15** 0.12* 0.07 R-Square 0.07 0.40 0.43 0.43 Notes: sample size=302, school and grade level fixed effects models. + p < .10, * p < .05, ** p < .01, *** p < .001. 90 0.11* 0.11* 0.41 0.42 Third, after controlling for teachers’ attributes with school and grade fixed effects, the results from five models indicate that teachers’ combined networks had a positive effect on nonrandom assignment with respect to students’ previous achievement, as shown in Table 3.9. African American teachers, ELA professional development, and new teachers had a negative effect on previous ELA achievement while teaching experience, a formal leader, the total numbers and specific types of leadership roles had a positive effect. Overall, this pattern of results suggests that three types of teachers’ social network had a significant positive effect on non-random assignment of students to their teachers with respect to previous ELA academic achievement. (2) Math Achievement First, after controlling for teachers’ attributes with school and grade fixed effects, the results from five models indicate that teachers’ ELA networks had a positive effect on nonrandom assignment with respect to students’ previous Math achievement. Male teachers, African American teachers, and new teachers had a negative effect on previous Math achievement while Math professional development, teaching experience, a formal leader, the total numbers and specific types of leadership roles had a positive effect. In summary, this pattern of results suggests that teachers’ ELA networks had a significant positive effect on non-random assignment of students to their teachers between and within schools with respect to previous Math academic achievement after controlling for teachers’ attributes, as shown Table 3.10. Second, after controlling for teachers’ attributes with school and grade fixed effects, the results from five models indicate that teachers’ Math networks had a statistically non-significant positive effect on non-random assignment with respect to students’ previous Math achievement. 91 Table 3.10 Effects of teachers’ ELA or Math networks on students’ previous Math achievement ELA Networks Model 1 Model 2 Model 3 Model 4 Model 5 Male -0.01 -0.01 -0.02 -0.01 -0.01 White 0.00 -0.06 -0.03 -0.01 -0.02 African American -0.07 -0.14 -0.10 -0.09 -0.10 Master’s degree -0.01 0.00 0.00 -0.01 0.00 Teaching experience 0.03 0.02 0.01 0.03 0.02 Math professional development 0.05 0.02 0.02 0.04 0.03 -0.14** -0.09+ -0.11* -0.14** -0.12* New teacher Formal leader 0.20*** The total number of leadership roles 0.20*** Math coordinator 0.10* Teacher consultant 0.14** In-degree in ELA networks 0.11* 0.08 0.08 0.11* 0.10* R-Square 0.47 0.50 0.50 0.48 0.48 Model 1 Model 2 Model 3 Model 4 Model 5 Male -0.02 -0.02 -0.03 -0.02 -0.02 White 0.00 -0.06 -0.04 -0.01 -0.02 African American -0.07 -0.13 -0.10 -0.08 -0.09 Master’s degree -0.00 0.00 0.00 0.00 0.01 Teaching experience 0.04 0.02 0.02 0.04 0.02 Math professional development 0.04 0.01 0.01 0.03 0.02 -0.14** -0.10+ -0.12* -0.14* -0.12* Math Networks New teacher Formal leader 0.20*** The total number of leadership roles 0.19*** Math coordinator 0.09 Teacher consultant 0.13** In-degree in Math networks 0.11* 0.06 0.06 0.09+ 0.10* R-Square 0.47 0.50 0.50 0.48 0.49 Notes: Sample size=302 (ELA networks) or 303 (Math networks), school and grade level fixed effects models, + p < .10, * p < .05, ** p < .01, *** p < .001. 92 Male teachers, African American teachers, and new teachers had a negative effect on previous Math achievement while Math professional development, teaching experience, a Master’s degree, a formal leader, the total numbers and specific types (e.g., teacher consultant) of leadership roles had a positive effect. In summary, these results indicate that Math networks had a positive effect on non-random assignment between and within schools after controlling for teachers’ attributes, as shown Table 3.10. Table 3.11 Effects of teachers’ combined networks on students’ math previous achievement Combined Networks Model 1 Model 2 Model 3 Model 4 Model 5 Male -0.01 -0.01 -0.03 -0.02 -0.02 White 0.00 -0.06 -0.03 -0.01 -0.02 African American -0.07 -0.13 -0.10 -0.08 -0.09 Master’s degree 0.00 0.00 0.00 -0.01 0.00 Teaching experience 0.03 0.02 0.01 0.03 0.02 Math professional development 0.04 0.01 0.01 0.04 0.02 -0.14** -0.10+ -0.12* -0.14* -0.12* New teacher Formal leader 0.20*** The total number of leadership roles 0.20*** Math coordinator 0.09 Teacher consultant 0.14** In-degree in Combined networks 0.10+ 0.05 0.05 0.08 0.09+ R-Square 0.47 0.50 0.50 0.48 0.49 Notes: sample size=303, school and grade level fixed effects models, + p < .10, * p < .05, ** p < .01, *** p < .001. Third, after controlling for teachers’ attributes with school and grade fixed effects, the results from five models indicate that teachers’ combined networks had, borderline statistically significant, a positive effect on non-random assignment with respect to students’ previous Math 93 achievement. Male teachers, African American teachers, and new teachers had a negative effect on previous Math achievement while Math professional development, teaching experience, a Master’s degree, a formal leader, the total numbers and specific types (e.g., teacher consultant) of leadership roles had a positive effect. In summary, these results indicate that combined networks had a positive effect on non-random assignment between and within schools after controlling for teachers’ attributes, as shown Table 3.11. Overall, significant findings indicate that teachers’ social networks might be a key factor in explaining the non-assignment of students to their teachers with respect to Math academic achievement as well as ELA academic achievement. 2) Students’ Previous Economic Status To estimate the effects of teachers’ social networks on non-random assignment with respect to students’ previous economic status, three types of teachers’ social networks were analyzed. First, after controlling for teachers’ attributes with school and grade fixed effects, the results from five models indicate that teachers’ ELA networks had a statistically non-significant negative effect on students’ previous free/reduced lunch. Only new teachers at their schools had a positive effect on previous free/reduced lunch. In other words, new teachers at their schools had more economically disadvantaged students than other teachers. In summary, these results indicated that ELA networks had a negative effect but essential zero on students’ previous free/reduced lunch between and within schools after controlling for teachers’ attributes, as shown Table 3.12. 94 Table 3.12 Effects of teachers’ ELA or Math networks on students’ previous free/reduced lunch ELA Networks Model 1 Model 2 Model 3 Model 4 Model 5 Male -0.06+ -0.05+ -0.04 -0.05 -0.05 White -0.11+ -0.08 -0.09 -0.10+ -0.10 African American -0.04 -0.00 -0.02 -0.01 -0.02 Master’s degree -0.01 -0.01 -0.01 -0.02 -0.02 Teaching experience -0.04 -0.04 -0.03 -0.03 -0.03 New teacher 0.05 0.02 0.03 0.03 0.03 Formal leader -0.11*** The total number of leadership roles -0.13*** School improvement coordinator -0.13*** Teacher consultant -0.12*** In-degree in ELA networks -0.05 -0.03 -0.03 -0.03 -0.04 R-Square 0.77 0.78 0.78 0.78 0.77 Model 1 Model 2 Model 3 Model 4 Model 5 Male -0.05 -0.05 -0.04 -0.04 -0.04 White -0.11+ -0.08 -0.09 -0.10+ -0.10 African American -0.04 -0.01 -0.02 -0.02 -0.02 Master’s degree -0.02 -0.02 -0.01 -0.02 -0.02 Teaching experience -0.04 -0.04 -0.03 -0.04 -0.03 New teacher 0.04 0.02 0.03 0.03 0.03 Math Networks Formal leader -0.04** The total number of leadership roles -0.12*** School improvement coordinator -0.13*** Teacher consultant In-degree in Math networks R-Square -0.12*** -0.07* -0.01 -0.04 -0.05+ -0.06+ 0.77 0.78 0.78 0.78 0.77 Notes: sample size=305, school and grade level fixed effects models. + p < .10, * p < .05, ** p < .01, *** p < .001. Second, after controlling for teachers’ attributes with school and grade fixed effects, the 95 results from three models (except models 2 and 3) indicate that teachers’ Math networks had a statistically significant negative effect on students’ previous free/reduced lunch. Only new teachers at their schools had a positive effect on previous Free/reduced lunch. In other words, new teachers at their schools had more economically disadvantaged students than other teachers. In summary, these results indicated that Math networks had a negative effect on students’ previous free/reduced lunch between and within schools, after controlling for teachers’ attributes, as shown Table 3.12.Third, after controlling for teachers’ attributes with school and grade fixed effects, the results from five models indicate that teachers’ combined networks had a negative effect on students’ previous free/reduced lunch. Only new teachers at their schools had a positive effect on previous Free/reduced lunch. In other words, new teachers at their schools had more economically disadvantaged students than other teachers, as shown Table 3.13. Table 3.13 Effects of teachers’ combined networks on students’ previous free/reduced lunch Combined Networks Model 1 Model 2 Model 3 Model 4 Model 5 Male -0.05 -0.05 -0.04 -0.05 -0.05 White -0.12+ -0.08 -0.09 -0.11+ -0.10 African American -0.04 -0.01 -0.02 -0.02 -0.02 Master’s degree -0.01 -0.02 -0.01 -0.02 -0.02 Teaching experience -0.04 -0.04 -0.03 -0.03 -0.03 New teacher 0.04 0.02 0.03 0.03 0.03 Formal leader -0.10** The total number of leadership roles -0.12*** School improvement coordinator -0.13*** Teacher consultant In-degree in Combined networks R-Square -0.12*** -0.07* -0.04 -0.04 -0.06+ -0.06+ 0.77 0.78 0.78 0.78 0.77 Notes: sample size=305, school and grade level fixed effects models. + p < .10, * p < .05, ** p < .01, *** p < .001. 96 Overall, significant findings indicate that after controlling for teachers’ attributes with school and grade fixed effects, specific teachers’ social networks (i.e., advice network in math) might be a key factor in explaining the non-assignment of students to their teachers with respect to previous students’ free/reduced lunch. But, evidence is borderline. After comparing the standardized coefficients of network effects, the results showed the similar pattern both in ELA and Math achievement, as shown in Table 3.14. However, the results showed smaller network effects on free/reduced lunch than academic achievement. Specifically, for ELA achievement in model 1, we can interpret that an increase of one standard deviation (i.e., about one in-degree) in ELA networks results, on average, in an increase of 0.13 standard deviation (i.e., about two points, 0.13 × 17=2.2) in ELA achievement. Thus, an increase of four standard deviation (i.e., about four in-degree) in ELA networks results in an increase of about half standard deviation (i.e., about nine points) in class average ELA achievement. Table 3.14 Effects of teachers’ social networks on class composition in model 1 to model 5 Model 1 Model 2 Model 3 Model 4 Model 5 ELA achievement 0.13* 0.10+ 0.10+ 0.12* 0.12* In-degree in ELA networks In-degree in Math networks 0.12* 0.06 0.07 0.11* 0.10* In-degree in Combined 0.12* 0.07 0.07 0.11* 0.11* In-degree ELA networks 0.11* 0.08 0.08 0.11* 0.10* In-degree Math networks 0.11* 0.06 0.06 0.09+ 0.10* In-degree in Combined 0.10+ 0.05 0.05 0.08 0.09+ In-degree ELA networks -0.05 -0.03 -0.03 -0.03 -0.04 In-degree Math networks -0.07* -0.01 -0.04 -0.05+ -0.06+ -0.06+ -0.06+ Math achievement Free/reduced lunch -0.07* -0.04 -0.04 In-degree in Combined Notes: school and grade level fixed effects models, + p < .10, * p < .05. 97 In addition, for Math achievement in model 1, we can interpret that an increase of one standard deviation (i.e., about one in-degree) in Math networks results, on average, in an increase of 0.11 standard deviation (i.e., about two points, 0.11 × 20=2.2) in Math achievement. Thus, an increase of four standard deviations (i.e., about five in-degree) in Math networks results in an increase of about half of a standard deviation (i.e., about nine points) in class average math achievement. Finally, for free/reduced lunch in model 1, we can interpret that an increase of two standard deviations (i.e., about three in-degree) in Combined networks results, on average, in a decrease of 0.14 standard deviation (i.e., about four percentage, 0.14 × 0.27=0.0378) in class average free/reduced lunch. Thus, an increase of four standard deviations (i.e., about six indegree) in Combined networks results in a decrease of about one fourth of a standard deviation (i.e., about eight percentage) in class average free/reduced lunch. Table 3.15 Adjusted R-square in model 1 to model 5 Model 1 Model 2 ELA achievement 0.311 0.353 In-degree in ELA networks Model 3 Model 4 Model 5 0.343 0.315 0.333 In-degree in Math networks 0.310 0.348 0.339 0.313 0.330 In-degree in Combined 0.310 0.348 0.339 0.313 0.330 In-degree ELA networks 0.394 0.429 0.428 0.403 0.411 In-degree Math networks 0.396 0.427 0.426 0.401 0.411 In-degree in Combined 0.393 0.425 0.425 0.399 0.408 In-degree ELA networks 0.733 0.743 0.747 0.749 0.747 In-degree Math networks 0.735 0.744 0.747 0.751 0.748 In-degree in Combined 0.735 0.744 0.747 0.751 0.748 Math achievement Free/reduced lunch 98 To examine how much these models explain the variation of academic achievement and economic status, the adjusted R-square is summarized in Table 3.15. Specifically, for ELA achievement, the model with the most explanation power among five models was model 2 which explained about 35% variation of ELA achievement. For Math achievement, the model with the most explanation power among five models also was model 2 which explained about 43% variation of Math achievement. In addition, for free/reduced lunch, the model with the most explanation power among five models was model 4 which explained about 75% variation of Math achievement. To examine how additionally teachers’ social networks explained the variation of academic achievement and economic status after controlling for teachers’ attributes, the models which excluded teachers’ social networks in model 1 to model 5 were analyzed. Then, adjusted R-square change was computed and summarized in Table 3.16. The detailed results of each model were reported in the Appendices in Table A.1 to A.60. Table 3.16 Adjusted R-square change in model 1 to model 5 Model 1 Model 2 Model 3 ELA achievement 0.011 0.006 0.006 In-degree in ELA networks Model 4 Model 5 0.011 0.011 In-degree in Math networks 0.010 0.002 0.002 0.009 0.007 In-degree in Combined 0.010 0.001 0.002 0.009 0.007 In-degree ELA networks 0.007 0.003 0.003 0.008 0.007 In-degree Math networks 0.009 0.001 0.001 0.006 0.007 In-degree in Combined 0.006 0.002 0.000 0.004 0.004 In-degree ELA networks 0.001 0.000 0.000 0.000 0.001 In-degree Math networks 0.002 0.001 0.000 0.002 0.002 In-degree in Combined 0.003 0.001 0.000 0.001 0.002 Math achievement Free/reduced lunch 99 The results showed that teachers’ social networks additionally explained only a small amount of the variation of academic achievement (about 1%) and economic status (about 0.3%) after controlling for teachers’ attributes. In other words, if we have enough information about relationship between teachers’ social networks and teachers’ attributes, we can control for effects of teachers’ social network on class composition through value-added model specification by including relevant teachers’ attributes. In summary, to explain what kind of factors affect non-random assignment of students to teachers, the primary research question was answered as follows. First, the results of two-level unconditional models indicate that students’ academic achievement and economic status were heterogeneous within and between schools in one district. In other words, students are non-randomly assigned to their teachers within and between schools with respect to students’ previous academic achievement and economic status Second, the results of correlation analysis indicate that the significant association between teachers’ social networks (English/Language Arts and Math networks) and students’ previous academic achievement and economic status existed. In other words, the more social networks teachers have within schools, the more academically as well as economically advantaged students teachers have within schools. Third, the results of multiple regression models show that teachers’ social networks and attributes might be a significant factor in explaining the non-assignment of students to their teachers with respect to students’ academic achievement and economic status. 100 Discussion and Conclusion After examining teachers’ social networks and class composition through multilevel models, correlation analyses, and multiple regression models, the results of this study indicate that social networks at a higher level (level 2) affect formal organizational structure at a lower level (level 1). In detail, first, students are non-randomly assigned to their teachers within and between schools with respect to students’ previous academic achievement and economic status. Second, the larger a teacher’ social networks within school, the more academically as well as economically advantaged students the teacher has. Third, teachers’ social networks and attributes are a significant factor in explaining the non-assignment of students to teachers with respect to students’ academic achievement and economic status. This study reported results consistent with Rothstein’s study (2008) that students were not randomly assigned to their teachers. The main difference between this study and previous studies is that this study focuses on the effect of teachers’ social networks on non-random assignment, which has been ignored in previous studies. In addition, this study presented the results of the effect of teachers’ social networks on non-random assignment after controlling for teachers’ attributes; again, this is different from previous research. However, we can doubt the effect of teachers’ social networks on non-random assignment because good teachers might have larger social networks and better quality students with regard to academic achievement. That is, the networks might be confounded with quality of teaching. Even so, we could use teachers’ social networks as an indicator of good teachers. Just as students’ non-random assignment between schools was a big challenge to efforts to reduce the achievement gap between schools, students’ non-random assignment within schools is also an important challenge to efforts to decrease the achievement gap within schools. 101 With respect to teacher quality, if experienced teachers or teachers with more expertise had more high-achieving students than novice teachers, low-performing students would have less chance to improve their academic achievement. In other words, the uneven distribution of effective teachers within schools could interfere with efforts to reduce the achievement gap among students within schools. In addition, Hanushek and Rivkin (2010) pointed out that “In terms of fairness, any failure to account for sorting on unobservable characteristics would potentially penalize teachers given unobservably more difficult classrooms and reward teachers given unobservably less difficult classrooms” (p. 270). What kind of solutions could be implemented to estimate teachers’ effect on students’ learning using VAMs in observational studies when there is non-random assignment of students to teachers? Steiner et al. (2010) pointed out “total bias reduction in an observational study can be achieved when (a) the outcome-related part of the selection process is quite specific… (b) a set of constructs is available that is individually less successful in bias reduction but comes from within the most crucial domains for bias reduction… and (c) a combination of expert judgment, theory, observation, and common sense is used to arrive at the rich set of domains, constructs within these domains, and even items within these constructs that might explain the selection process and be correlated with the outcome.” (pp. 265-266). Just as Steiner et al. (2010), we can consider two steps to minimize observable selection bias. First, we need to examine the process of non-random assignment. If we can identify the factors which are closely related to non-random assignment within schools, we can minimize the selection bias and lead to relevant conclusions. With respect to non-random assignment within schools, previous studies reported that principals engaged teachers formally or informally and classes were purposefully created by the majority of principals with more substantial differences 102 in class ability. In other words, previous studies showed that most elementary schools have complex assignment processes characterized by combination of principals’, teachers’, or parents’ selection and this study show that social networks among teachers are closely related to nonrandom assignment of students to their teachers. Second, we need to include covariates in VAMs specification because Cronbach’s (1976) study showed that model specification was affected by assignment rules. Specifically, Cronbach (1976) pointed out that “the unit of analysis can make a difference in the estimate of a covariateadjusted treatment mean, when persons or classes have not been assigned to treatments at random or when the number of independent assignments to treatment is small” (p. 13). If we can identify the factors that are closely related to non-random assignment within schools, we can minimize selection bias and help produce relevant conclusions about teacher and school performance. The limitations of this research are a) little explanation of the mechanisms of nonrandom assignment of students to teachers in each grade level, b) data limitation, c) no consideration of the effect of principals and parents when assigning students, d) issues related to the reliability and validity of the data on teachers’ social networks, and e) causal inference. With respect to explanation of mechanisms, this study did not investigate the process of how teachers’ attributes and social networks co-evolve (similar to chapter 2). Maybe teachers’ attributes like being a formal leader could affect teachers’ social networks and teachers using social networks could influence other teachers directly or indirectly when assigning students. Although we can conclude that teachers’ social networks affect non-random assignment, we cannot answer whether teachers’ social networks affect non-random assignment directly or indirectly. With respect to data limitations and consideration of the effect of principals and parents, if we have 103 social network data about principal-teacher networks, we can identify the net effect of teacherteacher networks after controlling for principal-teacher networks and parent-teacher networks. But I did not have access to such data for the purposes of this study. With respect to the reliability and validity of the data on teachers’ social networks, this study used proxy indicators instead of data about actual networks related to class assignment due to data limitations. In addition, the missing values of networks and measurement error of networks caused by question order could impede reliability and validity. With respect to interdependency, selection bias, and identification problems, when using value-added models, selection bias caused by non-random assignment could lead to misleading conclusions by affecting the gain score (Rivkin & Ishii, 2009; Rothstein, 2009). In order to control for selection bias, previous studies suggested the solution as model specification. Hanushek & Rivkin (2010) examined generalizations about using Value-Added Measures of teacher quality and summarized the distribution of teacher effectiveness in various studies. They argued that “although the impact of any classroom sorting on unobservables remains an important and unresolved question, the finding of substantial variation in teacher quality appears be robust to such sorting” (p. 269). However, when there was high interdependence and high selection bias, how can I identify the factors and specify the model? In order to estimate net effects of teachers on students’ learning, new VAMs also need to identify these effects as actor-oriented models can separate selection from influence process. If we cannot disentangle these effects, internal validity will be impaired. Future studies are needed to answer this kind of question. Even though there are some limitations, this research shows what kind of factors affect non-random assignment of students to teachers. In addition, this research shows that teachers’ 104 social networks can affect students’ learning by influencing class composition, which is related to non-random assignment of students to their teachers with respect to previous academic achievement. 105 Chapter 4: Quantifying the Robustness of Inferences about the Effects of Teachers’ Social Networks on Class Composition Introduction Experimental studies can provide strong cause and effect relationship while experimental studies may have weak external validity because of volunteer or convenience samples. In addition, experimental designs need random sampling and random assignment to implement successfully for strong internal validity. Observational studies can provide strong external validity when data are representative of populations while observational studies may have weak causal inference due to differences in unobserved preexisting conditions. In order to minimize observables selection bias, we can use statistical controls in observational studies which are described as fixed effects models, instrumental variables, propensity score matching, and regression discontinuity designs (Schneider et al., 2007). However, there are also problems with these methods: the assumption of fixed effects models that omitted variables are time invariant, identifying good instruments, little or no matched cases across treatment conditions in matching propensity scores, and the assumption of regression discontinuity designs that students in the two groups have similar characteristics (Schneider et al., 2007). To respond to these concerns, we can compute how much of the estimate of effect would have to be attributed to other factors to invalidate the causal claims (Frank, 2000). In other words, we can evaluate the sensitivity of causal claims to an unobserved confounding factor by quantifying the Impact Threshold of a Confounding Variable (ITCV). In addition, Seltzer et al. (2007) extended Frank’s (2000) ITCV for multilevel models. Furthermore, Kelcey (2009) extended Frank’s (2000) robust indices for applying to binominal regression models and 106 proposed methods quantifying the Average Impact Threshold of a Confounding Variable (AITCV) with assumption that weights would change dramatically with a confounding variable in the model. Therefore, this chapter is organized as follows. First, I present causal inference studies and the robust indices method. Second, data and methods are presented including sample, dependent variables, independent variables, and models. Third, the results are presented. Finally, the discussion and conclusion are offered with the importance of including prior information in model specification for valid causal inference. 107 Literature Review 1. Causal Inference Studies in Education Shadish, Cook, & Campbell (2002) pointed out the causal relationship that “In a classic th analysis formalized by the 19 -century philosopher John Stuart Mill, a causal relationship exists if (1) the cause preceded the effect (2) the cause was related to the effect (3) we can find no plausible alternative explanation for the effect other than the cause” (p. 6). The most general way to perform causal inference studies is to conduct randomized experiments. Through randomization, we assume that preexisting differences before treatments could be canceled out in each group. When we cannot randomize the subjects, quasiexperiments would be conducted. In education settings, is randomization possible in practice? Murnane & Willett (2011) argued that “actors in the educational system typically care a lot about which experimental units (whether they be students or teachers or schools) are assigned to particular educational treatments, and they take actions to try to influence these assignments” (pp. 34-35). Thus, “Unfortunately, until fairly recently, most educational researchers did not address their causal questions by conducting randomized experiments or by adopting creative approaches to analyzing data from quasi-experiments. Instead, they typically conducted observational studies” (pp. 31-32). Cook (2002) also claimed that “random assignment is most feasible when: treatments are shorter; they require little or no teacher training; patterns of coordination among school staff are not modified; the demand for a particular educational change outstrips its supply; two or more substantive treatments with similar goals are compared as opposed to the situation when 108 comparing a treatment to a no-treatment; the units receiving different treatments cannot communicate with each other; and when students are the unit of assignment rather than classrooms or whole schools” (p. 184). In addition, Schneider et al. (2007) pointed out that “Implementing experiments with randomized assignment can also present problems for researchers, such as breakdowns in randomization, treatment noncompliance, and attrition” (p. 22). One way to infer cause when using observational data or quasi-experimental designs is through statistical tools that a) control for a covariate using the general linear model as in ANCOVA, b) use an instrument variable, and c) use propensity score matching. However, the assumption of fixed effects models that omitted variables are time invariant may not be valid while identifying good instruments is very hard with respect to instrumental variables (Schneider et al., 2007). The problem of propensity score matching can occur when there are little or no matched cases across treatment conditions whereas the assumption of regression discontinuity designs, that students in the two groups have similar characteristics, should be examined (Schneider et al., 2007). Given these limitations, we seek to quantify the robustness of inferences with respect to violations of assumptions. 2. Sensitivity Analysis and Robustness Indices (ITCV) Rosenbaum (2010) introduced Cornfield et al. (1959) study as the first formal sensitivity analysis in an observational study and explained the concept of sensitivity analysis in that “If the association is strong, the hidden bias needed to explain it is large” (p. 106). In addition, he 109 proposed some models of sensitivity analysis. As a kind of sensitivity analysis, we can quantify how much of the estimate of effect would have to be attributed to other factors to invalidate the causal claims (Frank, 2000). To estimate the impact of an unmeasured confounding variable on our causal claims, Frank (2000) developed three steps for ordinary least square (OLS) regression estimates. The first step is to establish correlation between a dependent variable and one predictor, partialling for all covariates. r  t (n  q  1)  t 2 Where: t taken from the result of multiple regression n is the sample size q is the number of parameters estimated The second step is to define a threshold (r#) as the value of r that is just statistically significant for inference. r#  t critical 2 (n  q  1)  t critical Where: n is the sample size q is the number of parameters estimated tcritical is the critical value of the t-distribution for making an inference r# can also be defined in terms of effect sizes The third step is to calculate the threshold for the impact necessary to invalidate the Inference by defining the impact: k =rx∙cv x ry∙cv and assuming rx∙cv = ry∙cv which maximizes the impact of the confounding variable. 110 rx·y|cv  ITCV  rx·y  rx·cv  ry·cv 1 r 2 y·cv 1 r 2 x·cv  rx·y  k 1 k rx·y  r # 1 | r # | In addition, Seltzer et al. (2007) extended Frank’s (2000) formula to evaluate the impact of unobserved confounding variables on coefficients of predictors in multilevel regression models. Furthermore, Kelcey (2009) extended Frank’s (2000) robust indices for applying to binominal regression models because Frank (2000) developed ITCV in the linear regression models without considering nonlinear regression models. Kelcey (2009) assumed that weights could change dramatically with a confounding variable in the model and proposed the Average Impact Threshold of a Confounding Variable (AITCV). I will apply Frank’s analysis to the results in the previous chapters. 111 Data and Methods I used the same data as in chapters 2 and 3. In addition, the same final selection, influence models, and multiple regression models were used for this study. Specifically, in selection models using p2 models, model 1 did not include prior relationship about math while model 2 was the same as model 2 in selection model in chapter 2. In influence models using a two-level multilevel model, model 1 did not include prior teaching practices and subgroup mean of prior teaching practices whereas model 2 did not include prior teaching practices and model 3 was the same as model 2 in the influence model in chapter 2. In multiple regression models using grade and school fixed effects, I used the same as model 1, 4, and 5 in chapter 3. To quantify ITCV in selection and influence models, first, I estimated the effects in multilevel p2 models, two level HLM models, and multiple regression models. Second, three steps are conducted to compute ITCV in these models. Missing values of professional development at time 2 had five missing cases out of 209 cases while teaching efficacy at time 1 had 31 missing cases out of 209 cases. All missing cases were recoded as zero value in model 1 and model 2 of multilevel selection models. In addition, the missing values of dependent and independent values in the influence models were that teaching practices at time 2 had 56 missing cases out of 209 cases, which were deleted in the two-level HLM models because this was a dependent variable and there was little relevant information for multiple imputation of missing values. After deleting the missing values for the dependent variable, there were two missing cases in teaching efficacy at time 1 and one missing case in highest grade, which were deleted in the two-level HLM models. I used P2 4.0 for multilevel p2 models, HLM 6.0 for two level HLM models, SAS 9.2 for multiple regression models, and Excel 2010 for computing ITCV. 112 Results 1. Robustness Indices (ITCV) in p2 Selection Models To estimate how advice regarding mathematics teaching practices is related to characteristics of the actors, multilevel selection models were analyzed in Table 4.1. Table 4.1 Regression coefficient (standard error) of selection models Model 1 Parameters -7.08 (0.64) μ – Pair (level 1) Prior advice network, δ1 1.38* (0.35) Prior same subgroup, δ2 Model 2 -6.89 (1.13) 3.58* (0.75) 2.02* (0.51) 2.04* (0.48) 0.02 (0.07) 3.31 (0.50) 0.75 (0.33) 1.84* (0.75) 0.02 (0.07) 3.51 (0.52) 0.71 (0.37) 1.88* (0.66) 0.32* (0.14) 0.28* (0.14) 0.12 (0.11) 1.70 (0.55) -0.06 (0.16) 0.15 (0.13) 1.91 (0.56) -0.03 (0.18) Same grade, δ3 Total of all common meeting types, δ4 δ5 – Reciprocity - Provider variance (level 2a) (α) Mathematics program coordinator role, γ1 Mathematics professional development, γ2 (α) (α) Prior mathematics teaching efficacy, γ3 - Receiver variance (level 2b) (β) Mathematics professional development, γ1 1.13* (0.42) (β) 0.47* (0.15) 0.47* (0.16) Prior mathematics teaching efficacy, γ2 0.06 (0.34) -0.09 (0.35) - Provider-receiver covariance 0.37 (0.38) 1.41 (2.15) -Omega for Random Density Effects Note:* means t-ratio more than 2; The sample size was 209 in model 1 & model 2; Burn-in 4000 and sample size 20000 in MCMC estimation However, I am not concerned with multilevel models because the focus is a level 1 predictor as prior same subgroup. In addition, I ignored logistic nature at level 1 with the assumption that the weights would not change dramatically with a confounding variable in the model. Thus, in Model 1 shown in Table 4.1, to estimate the impact of an unmeasured confounding variable on the inference that prior same subgroup affected the current mathematics teaching practices advice network, three steps (Frank, 2000) are conducted as follows: The first 113 step is to establish correlation between prior same subgroup and current mathematics advice network, partialling for all covariates. r  t (n  q  1)  t 2  3.94 (209  9)  3.94 2  0.27 Where: t taken from the result of multiple regression, t=1.38/0.35=3.94 n is the sample size q is the number of parameters estimated The second step is to define a threshold (r#) as the value of r that is just statistically significant for inference. r#  t critical (n  q  1)  t 2 critical  1.96 (200)  1.96 2  0.138 Where: n is the sample size q is the number of parameters estimated tcritical is the critical value of the t-distribution for making an inference r# can also be defined in terms of effect sizes The third step is to calculate the threshold for the impact necessary to invalidate the Inference by defining the impact: k =rx∙cv × ry∙cv and assuming rx∙cv = ry∙cv which maximizes the impact of the confounding rx·y  r # 0.272  0.138 ITCV    0.156 # 1 | r | 1  0.138 If the impact of an unmeasured confound is more than 0.16, then inference would be invalid whereas if the impact of an unmeasured confound is less than 0.16, then inference would be valid. 114 In other words, when we assume rx∙cv = ry∙cv, if a correlation between a prior same subgroup and an unmeasured confound variable (rx∙cv) is more than 0.39 and the correlation between a current mathematics teaching practices advice network and an unmeasured confound variable (ry∙cv) is more than 0.39, then inference would be invalid. In Model 2 shown in Table 4.1, to estimate the impact of an unmeasured confounding variable on the inference that prior mathematics advice network affected current mathematics advice network, three steps (Frank, 2000) are conducted. If the impact of an unmeasured confound is more than 0.21, then inference would be invalid whereas if the impact of an unmeasured confound is less than 0.21, then inference would be valid. In other words, when we assume rx∙cv = ry∙cv, if a correlation between a prior mathematics teaching practices advice network and an unmeasured confound variable (rx∙cv) is more than 0.46 and correlation between a current mathematics teaching practices advice network and an unmeasured confound variable (ry∙cv) is more than 0.46, then inference would be invalid. Table 4.2 Impact threshold of a confounding variable (ITCV) in selection models Model 1 Prior mathematics teaching practices advice network Prior same subgroup 0.16 Same grade 0.15 Note: * more robust than model 1. Model 2 0.21 0.06 0.18* Therefore, we can compare the ITCV within and between models. As shown in Table 4.2, the ITCV of the prior relationship about mathematics was 0.21 in model 2 while ITCV of prior same subgroup was 0.06 in model 2. In addition, ITCV of prior same subgroup changed from 115 0.16 to 0.06 when we included prior mathematics teaching practices advice network. Based on ITCV shown as Table 4.2, we can claim that prior mathematics teaching practices advice network affects a current mathematics teaching practices advice network, which may be little sensitive to other unobserved confounding variables when the correlation between an unobserved confounding variable and a current mathematics teaching practices advice network is less than 0.46. At the same time, we can infer that a prior same subgroup network affects a current mathematics teaching practices advice network, which may be sensitive to other unobserved confounding variables when the correlation between an unobserved confounding variable and a current mathematics teaching practices advice network is less than 0.24. In other words, if the correlation between an unobserved confounding variable and a current mathematics teaching practices advice networks exceeded 0.24, the estimate of prior same subgroup membership would be changed from having a significant t-ratio to having a non-significant t-ratio. Thus, if we did not include confounding variables exceeding correlation 0.24 (e.g., a prior mathematics teaching practices advice network as 0.46) into the model specification, we might have invalid causal claims due to the impact of an omitted confounding variable, which affects t-ratio. 2. Robustness Indices (ITCV) in Multilevel Models To examine whether teachers’ mathematics teaching practices advice networks influence their mathematics teaching practices and what factors explain this influence, two-level multilevel models were analyzed. 116 To compare the effect of teachers’ mathematics teaching practices advice networks on their mathematics teaching practices between with prior information and without prior information, regression standardized coefficients for a multilevel influence model were shown in Table 4.3. The results of model 1 showed that Exposure between 2007 and 2008 (standardized coefficient of 0.28) and subgroup mean of math teaching efficacy in 2007 (standardized coefficient of 0.30) had significant effects on current mathematics teaching practices. To estimate the effect of subgroup mean of prior mathematics teaching practices on current mathematics teaching practices in level 2, model 2 was analyzed and the results indicated that there was a significant effect (standardized coefficient of 0.35) of subgroup mean of prior mathematics teaching practices in level 2. Finally, to estimate the effect of prior mathematics teaching practices and subgroup mean of prior mathematics teaching practices on current mathematics teaching practices in level 1and 2, model 3 was analyzed and the results indicated that there was a significant effect (standardized coefficient of 0.42) of prior mathematics teaching practices in level 1. Table 4.3 Regression standardized coefficients for multilevel model of mathematics problem solving teaching practices including the influences of colleagues. Variable Model 1 Model 2 Model 3 Level-1: Individual Teacher (N=150) Overall mean Teaching practices in 2008 -0.05 -0.02 -0.05 Teaching practices in 2007 0.42** Exposure between 2007 and 2008 0.28* 0.21* 0.21* Mathematics professional development in 2008 0.11 0.07 0.03 Mathematics teaching efficacy in 2007 0.04 0.05 0.01 Highest grade in 2008 0.05 -0.04 -0.07 Level-2: Subgroup (N=41) Subgroup mean of Teaching practices in 2007 0.35* 0.05 Subgroup mean of math teaching efficacy in 07 0.30* 0.18 0.23* Note: model 2 includes subgroup mean of mathematics teaching efficacy. + p= .058, * p < .05, ** p < .001. 117 This was the source of differences in model specification and results among four models, which indicated that prior teaching practices in mathematics problem solving was a key factor to account for current teaching practices in mathematics problem solving. In Model 1 shown in Table 4.3, to estimate the impact of an unmeasured confounding variable on the inference that Exposure between 2007 and 2008 affected current mathematics teaching practices, three steps (Frank, 2000) are conducted as follows. The first step is to establish correlation between Exposure between 2007 and 2008 and current mathematics instruction, partialling for all covariates. r  3.984 (150  6)  3.984  0.32 2 The second step is to define a threshold (r#) as the value of r that is just statistically significant for inference. r#  1.96 (144)  1.96 2  .16 The third step is to calculate the threshold for the impact necessary to invalidate the Inference by defining the impact: k =rx∙cv × ry∙cv and assuming rx∙cv = ry∙cv which maximizes the impact of the confounding variable. .32  .16 ITCV   0.19 1  .16 If the impact of an unmeasured confound is more than 0.19, then inference would be 118 invalid whereas if the impact of an unmeasured confound is less than 0.19, then inference would be valid. Therefore, we can compare ITCV within and between models. As shown in Table 4.4, ITCV of prior teaching practices was 0.264 in model 3 while ITCV of exposure between prior and current changed from 0.19 to 0.09 when we included prior teaching practices and subgroup mean of prior teaching practices. Based on the ITCV, we infer that subgroup mean of prior math teaching efficacy affects current teaching practices, which may be sensitive to other unobserved confounding variables when the correlation between an unobserved confounding variable and a current mathematics teaching practices is more than 0.30. In other words, if the correlation between an unobserved confounding variable and a current mathematics teaching practices exceeded 0.30, the estimate of subgroup mean of prior math teaching efficacy would be changed from having significant tratio to having non-significant t-ratio. Thus, if we cannot identify confounding variables exceeding correlation 0.30 (e.g., prior teaching practices as 0.51), we have a valid causal claims. Table 4.4 Impact threshold of a confounding variable (ITCV) in influence models Variable Model 1 Model 2 Level-1: Individual Teacher Teaching practices in 2007 Exposure between 2007 and 2008 0.19 0.09 Model 3 0.26 0.09 We can quantify the corrected ITCV with other covariates, as shown in Table 4.5. The corrected ITCV of prior teaching practices was 0.107 in model 3 while corrected ITCV of exposure between prior and current was 0.09 in model 3. In addition, ITCV of exposure between prior and current changed from 0.098 to 0.094 when we included prior teaching practices and subgroup mean of prior teaching practices. 119 Table 4.5 Corrected impact threshold of a confounding variable (ITCV) with other covariates in influence models Variable Model 1 Model 3 Level-1: Individual Teacher Teaching practices in 2007 0.107 Exposure between 2007 and 2008 0.098 0.094 To compare with impact of an observed variable, I computed the impact of professional development on exposure, which was 0.04. Therefore, this result indicates that the impact threshold for exposure between 2007 and 2008 on mathematics problem solving teaching practices in 2008 is larger than the impact of professional development on exposure. An unmeasured covariate would have to have a stronger impact than the strongest measured covariate to invalidate the inference. 3. Robustness Indices (ITCV) in Multiple Regression Models Regression Coefficients (t-ratio) of Teachers’ Social Networks in Model 1, 4, and 5 were shown in Table 4.6. Table 4.6 Regression coefficients (t-ratio) of teachers’ social networks in model 1, 4, and 5 Model 1 Model 4 Model 5 ELA achievement (N=300) 0.13* (2.30) 0.12* (2.29) 0.12* (2.28) In-degree in ELA networks 0.12* (2.20) 0.11* (2.14) 0.10* (1.97) In-degree in Math networks Math achievement (N=300) In-degree ELA networks 0.11* (2.07) 0.11* (2.10) 0.10*(2.06) In-degree Math networks 0.11* (2.23) 0.09+ (1.92) 0.10* (2.08) -0.05+ (-1.74) -0.06+ (-1.87) Free/reduced lunch (N=305) -0.07* (-2.09) In-degree Math networks Notes: school and grade level fixed effects models. + p < .10, * p < .05, ** p < .01, *** p < .001. 120 To estimate the impact of an unmeasured confounding variable on the inference that teachers’ ELA network in 2007 affected students’ ELA achievement in Table 4.6, three steps (Frank, 2000) are conducted as follows. The first step is to establish correlation between ELA achievement and ELA network indegree, partialling for all covariates. r  2.30 (300  40)  2.30 2  .142 The second step is to define a threshold (r#) as the value of r that is just statistically significant for inference. r#  1.96 (260)  1.96 2  .121 The third step is to calculate the threshold for the impact necessary to invalidate the Inference by defining the impact: k =rx∙cv × ry∙cv and assuming rx∙cv = ry∙cv which maximizes the impact of the confounding variable. ITCV means Impact Threshold of a Confounding Variable. ITCV  .142  .12  .023 1  .12 If the impact of an unmeasured confound is more than .023, then inference would be invalid whereas if the impact of an unmeasured confound is less than .023, then inference would be valid. In other words, if a correlation between in-degree in ELA networks and an unmeasured 121 confound (rx∙cv) is more than 0.15 and a correlation between ELA achievement and an unmeasured confound (ry∙cv) is more than 0.15, then my causal inference would be invalid. Based on my data, two variables (a formal leader and the total number of leadership roles) had correlations more than 0.15. Therefore, when we included a formal leader or the total number of leadership roles (we could regard these variables as an unmeasured confound in model 1) into models 2 and 3, the effects of teachers’ social networks on class composition was insignificant in models 2 and 3. However, if a formal leader is not a confounding variable, it may be an alternative cause or just a different measure. In addition, for valid causal inference, we need to control for prior formal leader or the total number of leadership roles. With respect to the effects of in-degree in Math networks on previous ELA achievement, the ITCV in model 1 indicated that if a correlation between in-degree in Math networks and an unmeasured confounding variable is more than 0.126 and a correlation between previous ELA achievement and an unmeasured confounding variable is more than 0.126, then my causal inference would be invalid. Table 4.7 Impact threshold of a confounding variable (ITCV) in multiple regression models Model 1 Model 4 Model 5 ELA achievement 0.023 0.023 0.022 In-degree in ELA networks 0.016 0.012 0.000 In-degree in Math networks Math achievement In-degree ELA networks 0.008 0.009 0.007 In-degree Math networks 0.018 N/A 0.008 0.009 N/A N/A Free/reduced lunch In-degree Math networks Note: N/A means non-applicable. 122 We can quantify corrected ITCV with other covariates shown in Table 4.8. With respect to ELA achievement, the corrected ITCV of in-degree in ELA networks was 0.016 in model 1 while the corrected ITCV of in-degree in Math networks was 0.012 in model 1. Table 4.8 Corrected impact threshold of a confounding variable (ITCV) with other covariates Model 1 Model 4 Model 5 ELA achievement 0.016 0.016 0.015 In-degree in ELA networks 0.012 0.009 0.000 In-degree in Math networks Math achievement In-degree ELA networks 0.005 0.005 0.006 In-degree Math networks 0.012 N/A 0.005 0.006 N/A N/A Free/reduced lunch In-degree Math networks Note: N/A means non-applicable. In summary, these results indicated that although the ITCV of in-degree in ELA networks was relatively smaller than the ITCV of variables in other studies, in-degree in ELA networks was a key factor in explaining the effects of teachers’ social networks on class composition through non-random assignment with respect to ELA academic achievement because the correlation between other variables and in-degree in teachers’ social networks were relatively lower than this. In addition, the relatively high positive correlation between formal leader variable and indegree in teachers’ social networks indicate that a formal leader can influence in-degree in teachers’ social networks. In spite of this, we cannot say that every formal leader has high indegree in social networks in their schools. Therefore, we need to check the relationship between teachers’ social networks and teachers’ attributes when explaining the effects on teachers’ social networks on class composition. 123 Discussion and Conclusion After quantifying how much of an estimate would have to be attributed to other factors to invalidate the causal claims by using robust indices (Frank, 2000), results suggests that prior same subgroup (previous informal network structure) were closely related to current teaching practices advice networks (current social networks). When examining which factors affect current teaching practices advice networks, not including prior informal network structure into model specification may lead to invalid causal inference. In addition, results indicate that prior teachers’ social networks had relatively significant influence on conducting current mathematics teaching practices. When we investigate which factors affect current teaching practices, we need to include prior teachers’ social networks because prior teachers’ social networks could be a confounding variable to invalidate our claims. Finally, results indicated that in-degree in ELA networks was a key factor in explaining the effects of teachers’ social networks on class composition through non-random assignment with respect to ELA academic achievement. Therefore, when examining which factors affect class composition, not including teachers’ attributes which affect teachers’ social networks into model specification may lead to invalid causal inference. In addition, we need to control for prior teachers’ social networks and attributes for valid inference. One limitation of this study is to ignore the logistic nature in level 1 when we estimate Impact Threshold of a Confounding Variable (ITCV) in selection models. When weights would not change dramatically with a confounding variable in the model, we could use Impact Threshold of a Confounding Variable (ITCV) formula for linear regression. However, if the weights changed dramatically with a confounding variable in the nonlinear regression model, we might need to use Average Impact Threshold of a Confounding Variable (AITCV) formula 124 for nonlinear regression, which Kelcey (2009) proposed. Therefore, future studies are needed to quantify robust indices with considering weights and we need to develop Impact Threshold of a Confounding Variable (ITCV) for actor-oriented models as exponential models. Another limitation of this study is to ignore measurement error in dependent variables, attributes variables, and network variables when we estimate Impact Threshold of a Confounding Variable (ITCV). When there was large measurement error in these variables, robustness indices might be unreliable. Thus, future studies are needed to account for measurement error in calculating Impact Threshold of a Confounding Variable (ITCV), which can be through latent variable modeling. Even though there are two limitations in this study, this research shows that including prior information in model specification is relatively important for valid causal inference when we estimate which factors affect current mathematics advice network and teaching practices. In addition, we need to check the relationship between teachers’ social networks and teachers’ attributes when we estimate which factors affect class composition. 125 Chapter 5: Policy for Teachers’ Social Networks To propose and test specific social network theories in elementary school, this dissertation proposes four hypotheses which investigate the relationship among social networks, structure, hierarchy and time when we examine the effects of teachers’ social networks, as shown as Table 5.1. Table 5.1 The hypotheses, results, and robustness indices of this dissertation. Structure, Hierarchy, Time, & Social networks Hypothesis Results Robustness Indices H 1-1 H 1-2 H 3-1 H 3-2 Yes Yes Yes Yes Robust Robust H 2-1 H 2-2 Yes Yes Robust A little Robust H4 Yes Effects of structure at level 3 On teachers’ social networks at level 2 On class composition at level 1 Effects of teachers’ social networks at level 2 On teaching practices at level 2 On class composition at level 1 Effects of teachers’ teaching practices at level 2 On students’ achievement at level 1 Note: H 1-1: Formal organizational structure at level 3 affects social networks at level 2. H 1-2: Network structure at level 3 affects social networks at level 2. H 2-1: Social networks at level 2 affect human capital at level 2. H 2-2: Social networks at level 2 affect formal organizational structure at level 1. H 3-1: Formal organizational & network structure at level 3 affects human capital at level 1. H 3-2: Formal organizational & network structure at level 3 affect formal organizational structure at level 1. H 4: Human capital at level 2 can affect human capital at level 1. After investigating teachers’ social networks through selection, influence, and dynamic modeling, the results of chapter 2 indicate that formal organizational structure of school and teachers’ social network structure at time 1 affect the formation of new ties of teachers’ social 126 networks at time 2 (H 1-1 & H 1-2) and teachers’ social networks at time 1 can affect teachers’ teaching practices at time 2 (H 2-1). In addition, the results of school and teacher effectiveness studies indicate that in explaining variation on student achievement in both cognitive and affective outcomes, the classroom effect is more fundamental than the school effect (Teddlie & Reynolds, 2000). Furthermore, the most significant factor at the classroom level is the quality of teaching practices (Brophy & Good, 1986), which can be improved through not only teachers’ professional development but also teachers’ social interactions and human capital spillovers. These results suggest that teachers’ human capital at time 1 can affect students’ human capital at time2 (H 4). After examining teachers’ social networks and class composition through multilevel models, correlation analyses, and multiple regression models, the results of chapter 3 indicate that teachers’ social networks affect students’ formal organizational structure through class composition (H 2-2). In addition, the results of class composition and peer effects studies indicate that class composition and peer effects have an important impact on students’ learning (Burns & Mason, 2002; Dreeben & Barr, 1988; Harris, 2010), which lead to differences in academic achievement (H 3-1). Even though I did not test whether the formal organizational & network structure at level 3 affect the formal organizational structure at level 1, we can infer this based on the results of chapters 2 and 3. In other words, we can conclude that the formal organizational structure of the school (i.e., grade level) and teachers’ informal network structure (i.e., cohesive subgroups) affect class composition through non-random assignment with respect to academic achievement and economic status. Specifically, based on the results of chapter 2, we can infer that teaching practice has a 127 dynamic change process which could be due to internal conditions (setting formal structure) as well as external conditions (e.g., teachers’ turnover rate). In other words, even if teachers’ composition remains constant, change in teachers’ attributes such as teaching grade level or formal leadership role might affect changes in teaching practice, which may lead to improve learning and academic achievement. And, we would be wondering what kind of teachers’ social networks affect teaching practices and why specific types of teachers’ social networks affect teaching practices. Therefore, theoretical models which can conceptualize the framework of different kinds of teachers’ social networks remain to be explained by additional research. After identifying the effects of different kinds of teachers’ social networks and comparing the effects size of each type, we could build up newer models of the improvement of teaching practices. Based on the results of chapter 3, we can infer that teachers’ ELA networks have more effect on non-random assignment with respect to previous math achievement as well as ELA achievement. In addition, formal leadership roles have more effect on non-random assignment with respect to students’ previous economic status as well as academic achievement. Thus, teachers’ attributes and social networks could affect on class composition, which lead to different learning. Based on the results of chapter 4, not including prior teachers’ social networks and attributes into model specification may lead to invalid causal inference when examining which factors affect current teaching practices advice networks and current teaching practices. In addition, not including teachers’ attributes which affect teachers’ social networks into a model sand may lead to invalid causal inferences when examining which factors affect class composition; we need to control for prior teachers’ social networks and attributes for valid inference. 128 Teachers’ Attributes Teaching Practices Students’ Learning Teachers’ Networks Academic Achievement Class Composition Figure 5.1. Conceptual framework of this study Though this dissertation offered empirical evidence to the literature concerning social networks in education, there remains a range of issues to be addressed; 1) the conceptual model of teachers’ social networks, teaching practices and class composition, 2) causal inference and controlling for prior and 3) policies for teachers’ social networks. Based on the results of chapters 2 and 3, the conceptual framework of this study was shown in Figure 5.1. First, teachers’ attributes could affect teachers’ networks, teaching practices, and class composition. Second, teachers’ networks could influence teaching practices and class composition. Third, both teaching practices and class composition could affect students’ learning, which may lead to different academic achievement. Thus, future studies should be directed at examining the effect of teachers’ attributes and networks on teaching practices and class composition including students’ learning and academic achievement. Furthermore, future studies are needed to consider other factors which affect teachers’ social networks such as principal leadership, district policy, and information technology when developing comprehensive models. Based on the results of chapter 4, we can understand the importance of including prior information in model specification for more valid inference to satisfy the first Mill’ condition (cause preceded effect). In other words, if we can control for prior information and model emergence (e.x., the formation of new ties) over time, we have more robust causal inferences 129 than without prior in the models. In addition, we can assess the sensitivity of our causal inference using robustness indices, which is closely related to evaluating interval validity. Based on the results of chapters 2, 3, and 4, policies for teachers’ social networks are suggested as they relate to teachers and students in instruction and learning contexts; (1) the policy of organizational structure of school such as class size and composition and (2) the policy of formal network structure such as grade level meetings. A recent study by Dee & West (2011) examined how non-cognitive skills are affected by class size in the middle school and suggested that there was relationship between smaller eighthgrade classes and improvements of school engagement. Like this, previous policymakers and researchers have argued for class size reduction policies. However, if class, even small class, formed only disadvantaged students in non-cognitive as well as cognitive skills, class size reduction policies might not be effective to improve students’ human capital. Therefore, policymakers need to consider not only class size but also class composition when making effective school reforms policies if large class size constrains students’ social networks, as shown in Figure 5.2. In addition, grade size as the number of classes within one grade level might facilitate or constrain the chance to interact among teachers within grade level. Overall, the school size as the number of students and teachers within one school might facilitate or constrain the chance to interact among teachers. Therefore, policymakers need to consider appropriate grade size as well as school size when making effective school reforms policies if large grade and school size constrains teachers’ social networks. Finally, this framework could be applied to district size and district composition. In other words, if district size (the number of schools within one district) is large and the mean of 130 school achievement and teachers’ quality all are low, these conditions may constrain the chance to interact among superintendents, principals, and teachers. Human Capital Social Networks Size & Composition Principals’ Human Capital (e.g., leadership) Principals’ Social Networks (e.g., interschool network) School Size Composition Teachers’ Human Capital (e.g., teaching practices) Teachers’ Social Networks (e.g., advice network) Grade Size Composition Students’ Human Capital (e.g., academic achievement) Parents’ Human Capital (e.g., SES) Students’ Social Networks (e.g., friendship) Parents’ Social Networks (e.g., PTA ) Class Size Composition Figure 5.2. The relationship betw een hum an capital and social netw orks at d ifferent level As the policy of organizational structure of school is important for teachers’ social networks, we can consider the policy of formal network structure such as within and cross grade level meetings. For facilitating interaction among teachers, policymakers and practitioners (e.g., superintendents or principals) can set up formal meetings within and between schools. However, based on the results of chapter 2, these formal meetings might not be a key factor to form of new ties in teachers’ social networks. Rather, too many formal meetings can be harmful for teachers’ motivation and commitment. Therefore, policymakers and practitioners need to set the standard of formal meeting structure, size, time, and number. For example, the minimum number of formal meeting is once but the maximum number of formal meeting is three within one month. And the appropriate size and time of formal 131 meeting is six teachers within 30 minutes. In other words, teachers and school leaders together can set up these standards from the beginning of the academic year and keep revising these standards to the end of the academic year. Through a systematic analysis and empirical evidence, in spite of some limitations, this dissertation highlights the role of teachers’ social networks in education by showing that 1) social networks can improve teaching practice through changing formal and informal structure and 2) social networks can affect non-random assignment of students to their teachers with respect to previous academic achievement. 132 Appendices 133 Table A.1 Model 1 in effects of teachers' ELA networks on students' previous ELA achievement Model 1 B Beta S. E. t value p value Intercept 812.68 0.00 6.41 126.85 <.0001 Grade 2 -0.54 -0.02 2.81 -0.19 0.8474 Grade 3 -0.34 -0.01 2.83 -0.12 0.9042 Grade 4 4.23 0.10 3.01 1.41 0.1603 School 1003 -1.58 -0.01 7.12 -0.22 0.8244 School 1006 -4.87 -0.02 15.22 -0.32 0.7493 School 1007 23.35 0.37 5.76 4.05 <.0001 School 1009 9.01 0.08 7.13 1.26 0.2077 School 1010 -5.14 -0.07 6.16 -0.83 0.4048 School 1011 2.79 0.04 6.00 0.46 0.6425 School 1012 6.89 0.07 7.14 0.97 0.3353 School 1015 5.90 0.06 6.74 0.87 0.3828 School 1017 13.65 0.17 6.22 2.2 0.029 School 1020 5.60 0.07 6.09 0.92 0.3588 School 1021 1.51 0.02 6.20 0.24 0.8079 School 1023 18.16 0.15 7.61 2.39 0.0177 School 1024 2.33 0.02 7.01 0.33 0.7401 School 1025 8.50 0.07 7.55 1.13 0.2614 School 1026 -6.88 -0.09 6.15 -1.12 0.2646 School 1027 2.36 0.03 6.00 0.39 0.6942 School 1028 -7.39 -0.07 6.88 -1.07 0.2836 School 1029 -0.57 -0.01 6.45 -0.09 0.9291 School 1032 10.12 0.10 6.92 1.46 0.1449 School 1034 -2.25 -0.03 6.57 -0.34 0.7321 School 1035 -1.65 -0.02 6.02 -0.27 0.7842 School 1038 15.64 0.17 6.58 2.38 0.0181 School 1040 15.61 0.17 6.49 2.41 0.0168 School 1041 5.94 0.07 6.40 0.93 0.3542 School 1042 15.97 0.18 6.44 2.48 0.0138 School 1044 -3.38 -0.03 7.47 -0.45 0.6513 School 1047 6.73 0.04 9.47 0.71 0.4778 School 1049 11.26 0.11 6.83 1.65 0.1003 Male 0.51 0.01 3.02 0.17 0.8656 White 0.74 0.02 3.72 0.2 0.8422 African American -2.34 -0.06 3.96 -0.59 0.5552 Master’s degree 1.10 0.03 2.01 0.55 0.5859 Teaching experience 0.07 0.04 0.10 0.73 0.4639 ELA professional development -0.89 -0.04 1.09 -0.81 0.418 New teacher -5.53 -0.14 2.22 -2.49 0.0134 In-degree in ELA networks 1.83 0.13 0.80 2.3 0.0224 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 134 Table A.2 Model 2 in effects of teachers' ELA networks on students' previous ELA achievement Model 2 B Beta S. E. t value p value Intercept 812.57 0.00 6.21 130.84 <.0001 Grade 2 0.22 0.01 2.73 0.08 0.9345 Grade 3 0.07 0.00 2.74 0.02 0.9801 Grade 4 4.51 0.11 2.91 1.55 0.1227 School 1003 -3.49 -0.03 6.92 -0.5 0.6143 School 1006 -3.09 -0.01 14.76 -0.21 0.8344 School 1007 24.69 0.39 5.59 4.41 <.0001 School 1009 9.41 0.09 6.91 1.36 0.1745 School 1010 -5.23 -0.07 5.97 -0.88 0.3818 School 1011 2.82 0.04 5.81 0.48 0.6285 School 1012 7.96 0.08 6.93 1.15 0.2514 School 1015 7.35 0.08 6.55 1.12 0.2624 School 1017 12.28 0.16 6.03 2.03 0.0429 School 1020 6.78 0.09 5.91 1.15 0.2522 School 1021 2.12 0.03 6.01 0.35 0.724 School 1023 18.62 0.16 7.38 2.52 0.0122 School 1024 4.15 0.04 6.81 0.61 0.5432 School 1025 8.53 0.07 7.32 1.16 0.2451 School 1026 -6.56 -0.08 5.96 -1.1 0.2726 School 1027 3.29 0.04 5.82 0.57 0.5721 School 1028 -7.12 -0.07 6.67 -1.07 0.2866 School 1029 -1.31 -0.01 6.25 -0.21 0.8346 School 1032 7.49 0.07 6.74 1.11 0.2676 School 1034 -0.39 0.00 6.38 -0.06 0.9512 School 1035 0.13 0.00 5.85 0.02 0.9822 School 1038 16.63 0.18 6.38 2.61 0.0097 School 1040 15.80 0.17 6.29 2.51 0.0126 School 1041 6.31 0.07 6.20 1.02 0.3097 School 1042 16.71 0.19 6.25 2.68 0.0079 School 1044 -2.91 -0.02 7.24 -0.4 0.6881 School 1047 4.68 0.03 9.20 0.51 0.611 School 1049 12.33 0.12 6.62 1.86 0.0637 Male 0.45 0.01 2.93 0.15 0.8792 White -1.69 -0.05 3.66 -0.46 0.6453 African American -5.10 -0.13 3.89 -1.31 0.1915 Master’s degree 1.29 0.03 1.95 0.66 0.5097 Teaching experience 0.05 0.03 0.10 0.48 0.6304 ELA professional development -1.34 -0.07 1.07 -1.26 0.2088 New teacher -3.82 -0.10 2.19 -1.74 0.0829 Formal leader 8.23 0.22 1.95 4.21 <.0001 In-degree in ELA networks 1.45 0.10 0.78 1.85 0.0648 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 135 Table A.3 Model 3 in effects of teachers' ELA networks on students' previous ELA achievement Model 3 B Beta S. E. t value p value Intercept 812.91 0.00 6.26 129.88 <.0001 Grade 2 0.49 0.01 2.76 0.18 0.86 Grade 3 0.48 0.01 2.77 0.17 0.8623 Grade 4 4.53 0.11 2.94 1.54 0.124 School 1003 -5.32 -0.05 7.03 -0.76 0.4499 School 1006 -2.75 -0.01 14.88 -0.18 0.8537 School 1007 23.69 0.38 5.63 4.21 <.0001 School 1009 8.94 0.08 6.97 1.28 0.2005 School 1010 -6.55 -0.09 6.03 -1.09 0.2788 School 1011 2.56 0.03 5.86 0.44 0.6628 School 1012 7.73 0.07 6.98 1.11 0.2689 School 1015 6.02 0.06 6.59 0.91 0.3614 School 1017 13.10 0.17 6.07 2.16 0.0319 School 1020 5.47 0.07 5.95 0.92 0.3585 School 1021 1.92 0.02 6.05 0.32 0.7513 School 1023 18.01 0.15 7.44 2.42 0.0162 School 1024 2.67 0.03 6.85 0.39 0.6968 School 1025 7.09 0.06 7.39 0.96 0.3383 School 1026 -6.77 -0.09 6.01 -1.13 0.2608 School 1027 0.32 0.00 5.89 0.05 0.9568 School 1028 -8.50 -0.08 6.72 -1.26 0.2075 School 1029 -1.48 -0.02 6.30 -0.24 0.8141 School 1032 8.72 0.08 6.77 1.29 0.1991 School 1034 -2.09 -0.02 6.42 -0.32 0.7455 School 1035 -0.81 -0.01 5.88 -0.14 0.8911 School 1038 15.33 0.17 6.43 2.39 0.0178 School 1040 14.98 0.16 6.34 2.36 0.0189 School 1041 5.52 0.06 6.25 0.88 0.3784 School 1042 14.73 0.17 6.30 2.34 0.0201 School 1044 -3.26 -0.03 7.29 -0.45 0.655 School 1047 2.87 0.02 9.31 0.31 0.7583 School 1049 10.29 0.10 6.67 1.54 0.1243 Male -0.42 -0.01 2.96 -0.14 0.8862 White -0.44 -0.01 3.65 -0.12 0.9036 African American -3.56 -0.09 3.88 -0.92 0.3599 Master’s degree 1.09 0.03 1.97 0.56 0.5788 Teaching experience 0.04 0.02 0.10 0.39 0.6981 ELA professional development -1.22 -0.06 1.07 -1.14 0.2552 New teacher -4.72 -0.12 2.18 -2.16 0.0315 The total number of leadership 2.10 0.19 0.57 3.67 0.0003 roles In-degree in ELA networks 1.42 0.10 0.79 1.8 0.0736 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 136 Table A.4 Model 4 in effects of teachers' ELA networks on students' previous ELA achievement Model 4 B Beta S. E. t value p value Intercept 813.27 0.00 6.40 127.05 <.0001 Grade 2 -0.38 -0.01 2.81 -0.14 0.8916 Grade 3 -0.17 0.00 2.82 -0.06 0.9527 Grade 4 4.06 0.10 3.00 1.35 0.1766 School 1003 -2.87 -0.03 7.15 -0.4 0.6884 School 1006 -3.72 -0.01 15.20 -0.24 0.8068 School 1007 23.15 0.37 5.75 4.03 <.0001 School 1009 9.06 0.08 7.11 1.27 0.2042 School 1010 -5.79 -0.08 6.16 -0.94 0.3482 School 1011 2.96 0.04 5.98 0.49 0.6215 School 1012 7.35 0.07 7.13 1.03 0.3036 School 1015 6.18 0.06 6.73 0.92 0.3596 School 1017 13.89 0.18 6.20 2.24 0.026 School 1020 5.53 0.07 6.07 0.91 0.3628 School 1021 1.66 0.02 6.18 0.27 0.7883 School 1023 18.18 0.15 7.59 2.39 0.0173 School 1024 2.54 0.02 6.99 0.36 0.7168 School 1025 8.44 0.07 7.53 1.12 0.2636 School 1026 -6.79 -0.09 6.14 -1.11 0.2694 School 1027 1.97 0.03 5.99 0.33 0.7422 School 1028 -8.72 -0.08 6.91 -1.26 0.2082 School 1029 -0.57 -0.01 6.43 -0.09 0.93 School 1032 9.90 0.10 6.91 1.43 0.1529 School 1034 -2.17 -0.02 6.55 -0.33 0.7413 School 1035 -1.86 -0.03 6.00 -0.31 0.7574 School 1038 15.59 0.17 6.56 2.38 0.0183 School 1040 15.68 0.17 6.47 2.42 0.016 School 1041 5.37 0.06 6.39 0.84 0.4011 School 1042 15.78 0.18 6.42 2.46 0.0147 School 1044 -3.23 -0.03 7.45 -0.43 0.6653 School 1047 7.15 0.04 9.45 0.76 0.4499 School 1049 11.01 0.11 6.81 1.62 0.1072 Male -0.13 0.00 3.04 -0.04 0.9647 White 0.36 0.01 3.72 0.1 0.923 African American -2.89 -0.07 3.97 -0.73 0.4672 Master’s degree 0.79 0.02 2.02 0.39 0.6949 Teaching experience 0.07 0.04 0.10 0.67 0.502 ELA professional development -0.93 -0.05 1.09 -0.85 0.3974 New teacher -5.67 -0.14 2.22 -2.56 0.0111 ELA coordinator 1.85 0.08 1.20 1.54 0.1246 In-degree in ELA networks 1.82 0.12 0.80 2.29 0.0231 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 137 Table A.5 Model 5 in effects of teachers' ELA networks on students' previous ELA achievement Model 5 B Beta S. E. t value p value Intercept 817.82 0.00 6.52 125.51 <.0001 Grade 2 -0.37 -0.01 2.77 -0.13 0.8931 Grade 3 -0.45 -0.01 2.78 -0.16 0.8713 Grade 4 3.68 0.09 2.96 1.24 0.2152 School 1003 -3.02 -0.03 7.02 -0.43 0.6672 School 1006 -4.18 -0.01 14.98 -0.28 0.7805 School 1007 22.56 0.36 5.67 3.98 <.0001 School 1009 8.44 0.08 7.02 1.2 0.2305 School 1010 -5.72 -0.07 6.07 -0.94 0.3468 School 1011 2.21 0.03 5.90 0.37 0.7081 School 1012 7.04 0.07 7.03 1 0.3169 School 1015 5.31 0.05 6.64 0.8 0.4241 School 1017 13.14 0.17 6.12 2.15 0.0327 School 1020 5.07 0.06 5.99 0.85 0.3981 School 1021 1.19 0.02 6.10 0.2 0.8455 School 1023 17.73 0.15 7.49 2.37 0.0187 School 1024 2.15 0.02 6.90 0.31 0.7553 School 1025 7.40 0.06 7.44 1 0.3206 School 1026 -6.90 -0.09 6.05 -1.14 0.255 School 1027 0.89 0.01 5.92 0.15 0.8804 School 1028 -8.28 -0.08 6.77 -1.22 0.2226 School 1029 -2.06 -0.02 6.36 -0.32 0.7463 School 1032 7.95 0.08 6.85 1.16 0.2465 School 1034 -2.48 -0.03 6.46 -0.38 0.7019 School 1035 -0.77 -0.01 5.93 -0.13 0.8964 School 1038 14.20 0.15 6.49 2.19 0.0295 School 1040 14.37 0.15 6.39 2.25 0.0255 School 1041 5.38 0.06 6.30 0.85 0.3936 School 1042 15.95 0.18 6.34 2.52 0.0124 School 1044 -3.16 -0.03 7.35 -0.43 0.6676 School 1047 4.56 0.03 9.35 0.49 0.6257 School 1049 10.53 0.10 6.72 1.57 0.1183 Male 0.03 0.00 2.97 0.01 0.9926 White -0.08 0.00 3.67 -0.02 0.9821 African American -3.52 -0.09 3.92 -0.9 0.3688 Master’s degree 1.33 0.03 1.98 0.67 0.5018 Teaching experience 0.04 0.02 0.10 0.38 0.7051 ELA professional development -1.08 -0.05 1.08 -1 0.3165 New teacher -4.73 -0.12 2.20 -2.15 0.0327 Teacher consultant 0.91 0.16 0.29 3.11 0.0021 In-degree in ELA networks 1.79 0.12 0.79 2.28 0.0234 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 138 Table A.6 Model 1 in effects of teachers' Math networks on students' previous ELA achievement Model 1 B Beta S. E. t value p value Intercept 812.11 0.00 6.44 126.18 <.0001 Grade 2 -0.49 -0.01 2.81 -0.18 0.8608 Grade 3 0.19 0.01 2.83 0.07 0.9458 Grade 4 4.65 0.11 3.01 1.54 0.1237 School 1003 -2.08 -0.02 7.13 -0.29 0.7702 School 1006 -0.65 0.00 15.11 -0.04 0.9658 School 1007 23.61 0.38 5.76 4.1 <.0001 School 1009 8.73 0.08 7.14 1.22 0.2222 School 1010 -5.60 -0.07 6.17 -0.91 0.3656 School 1011 1.79 0.02 6.01 0.3 0.7667 School 1012 6.96 0.07 7.15 0.97 0.3314 School 1015 5.53 0.06 6.75 0.82 0.4133 School 1017 14.06 0.18 6.21 2.27 0.0243 School 1020 5.93 0.08 6.09 0.97 0.3311 School 1021 1.78 0.02 6.19 0.29 0.7745 School 1023 17.89 0.15 7.62 2.35 0.0196 School 1024 1.92 0.02 7.01 0.27 0.7842 School 1025 9.45 0.08 7.52 1.26 0.2104 School 1026 -7.07 -0.09 6.17 -1.15 0.2524 School 1027 1.14 0.02 6.06 0.19 0.8504 School 1028 -7.57 -0.07 6.88 -1.1 0.2724 School 1029 -0.87 -0.01 6.46 -0.13 0.8928 School 1032 10.44 0.10 6.92 1.51 0.1328 School 1034 -1.93 -0.02 6.57 -0.29 0.7689 School 1035 -1.76 -0.02 6.03 -0.29 0.771 School 1038 15.49 0.17 6.59 2.35 0.0195 School 1040 15.81 0.17 6.49 2.44 0.0155 School 1041 5.60 0.07 6.41 0.87 0.3833 School 1042 15.86 0.18 6.45 2.46 0.0147 School 1044 -2.94 -0.02 7.46 -0.39 0.6943 School 1047 11.11 0.07 9.36 1.19 0.2359 School 1049 12.92 0.12 6.79 1.9 0.0582 Male -0.28 -0.01 3.00 -0.09 0.9253 White 0.68 0.02 3.73 0.18 0.8547 African American -2.36 -0.06 3.96 -0.6 0.5515 Master’s degree 1.38 0.04 2.02 0.68 0.4957 Teaching experience 0.08 0.04 0.10 0.79 0.4274 ELA professional development -0.63 -0.03 1.09 -0.58 0.5624 New teacher -5.66 -0.14 2.22 -2.55 0.0113 In-degree in Math networks 1.55 0.12 0.70 2.2 0.0288 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 139 Table A.7 Model 2 in effects of teachers' Math networks on students' previous ELA achievement Model 2 B Beta S. E. t value p value Intercept 812.49 0.00 6.26 129.83 <.0001 Grade 2 0.17 0.00 2.74 0.06 0.9519 Grade 3 0.40 0.01 2.75 0.14 0.8854 Grade 4 4.75 0.11 2.93 1.62 0.1064 School 1003 -3.85 -0.03 6.94 -0.55 0.5797 School 1006 0.27 0.00 14.70 0.02 0.9852 School 1007 25.17 0.40 5.61 4.49 <.0001 School 1009 9.13 0.08 6.94 1.32 0.1893 School 1010 -5.43 -0.07 6.00 -0.9 0.3664 School 1011 2.22 0.03 5.85 0.38 0.7047 School 1012 8.06 0.08 6.95 1.16 0.2473 School 1015 7.00 0.07 6.57 1.07 0.2877 School 1017 12.90 0.16 6.04 2.13 0.0338 School 1020 6.97 0.09 5.93 1.18 0.2409 School 1021 2.65 0.03 6.02 0.44 0.6598 School 1023 18.43 0.15 7.41 2.49 0.0135 School 1024 3.68 0.04 6.83 0.54 0.5906 School 1025 9.54 0.08 7.32 1.3 0.1934 School 1026 -6.41 -0.08 6.00 -1.07 0.2859 School 1027 2.77 0.04 5.90 0.47 0.6396 School 1028 -7.21 -0.07 6.69 -1.08 0.2823 School 1029 -1.29 -0.01 6.28 -0.21 0.8375 School 1032 7.96 0.08 6.76 1.18 0.2399 School 1034 -0.14 0.00 6.41 -0.02 0.9824 School 1035 0.17 0.00 5.88 0.03 0.9776 School 1038 16.77 0.18 6.42 2.61 0.0095 School 1040 15.96 0.17 6.31 2.53 0.0121 School 1041 6.25 0.07 6.23 1 0.3169 School 1042 17.03 0.19 6.28 2.71 0.0072 School 1044 -2.42 -0.02 7.26 -0.33 0.7389 School 1047 8.01 0.05 9.13 0.88 0.3812 School 1049 13.64 0.13 6.60 2.07 0.0399 Male -0.19 0.00 2.92 -0.06 0.9493 White -1.70 -0.05 3.67 -0.46 0.6432 African American -5.15 -0.13 3.92 -1.32 0.1896 Master’s degree 1.43 0.04 1.96 0.73 0.4676 Teaching experience 0.05 0.03 0.10 0.53 0.5965 ELA professional development -1.11 -0.05 1.06 -1.04 0.297 New teacher -4.20 -0.11 2.19 -1.92 0.0559 Formal leader 8.06 0.22 2.01 4.01 <.0001 In-degree in Math networks 0.84 0.06 0.71 1.19 0.2336 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 140 Table A.8 Model 3 in effects of teachers' Math networks on students' previous ELA achievement Model 3 B Beta S. E. t value p value Intercept 812.72 0.00 6.30 128.95 <.0001 Grade 2 0.45 0.01 2.77 0.16 0.8714 Grade 3 0.82 0.02 2.78 0.3 0.7679 Grade 4 4.79 0.11 2.95 1.62 0.1057 School 1003 -5.64 -0.05 7.05 -0.8 0.4243 School 1006 0.53 0.00 14.80 0.04 0.9716 School 1007 24.10 0.39 5.64 4.28 <.0001 School 1009 8.69 0.08 6.98 1.24 0.2146 School 1010 -6.76 -0.09 6.05 -1.12 0.2649 School 1011 1.92 0.03 5.89 0.33 0.7444 School 1012 7.82 0.08 7.00 1.12 0.2651 School 1015 5.71 0.06 6.60 0.87 0.3878 School 1017 13.61 0.17 6.08 2.24 0.0259 School 1020 5.70 0.07 5.96 0.96 0.3403 School 1021 2.35 0.03 6.06 0.39 0.6987 School 1023 17.82 0.15 7.46 2.39 0.0176 School 1024 2.27 0.02 6.86 0.33 0.7405 School 1025 8.03 0.07 7.38 1.09 0.2774 School 1026 -6.72 -0.09 6.04 -1.11 0.2669 School 1027 -0.27 0.00 5.94 -0.05 0.9637 School 1028 -8.58 -0.08 6.74 -1.27 0.2045 School 1029 -1.53 -0.02 6.33 -0.24 0.8087 School 1032 9.11 0.09 6.79 1.34 0.1807 School 1034 -1.82 -0.02 6.43 -0.28 0.7778 School 1035 -0.80 -0.01 5.90 -0.14 0.8926 School 1038 15.41 0.17 6.45 2.39 0.0176 School 1040 15.15 0.16 6.36 2.38 0.0179 School 1041 5.41 0.06 6.28 0.86 0.3894 School 1042 14.96 0.17 6.32 2.37 0.0188 School 1044 -2.82 -0.02 7.31 -0.39 0.6998 School 1047 6.21 0.04 9.26 0.67 0.5029 School 1049 11.61 0.11 6.66 1.74 0.0824 Male -1.02 -0.02 2.95 -0.35 0.7282 White -0.48 -0.01 3.66 -0.13 0.8957 African American -3.62 -0.09 3.90 -0.93 0.3541 Master’s degree 1.26 0.03 1.98 0.64 0.5256 Teaching experience 0.04 0.02 0.10 0.44 0.6611 ELA professional development -1.00 -0.05 1.07 -0.94 0.3484 New teacher -5.00 -0.13 2.18 -2.3 0.0224 The total number of leadership 2.06 0.19 0.58 3.52 0.0005 roles In-degree in Math networks 0.94 0.07 0.71 1.32 0.1883 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 141 Table A.9 Model 4 in effects of teachers' Math networks on students' previous ELA achievement Model 4 B Beta S. E. t value p value Intercept 812.71 0.00 6.43 126.3 <.0001 Grade 2 -0.35 -0.01 2.81 -0.12 0.901 Grade 3 0.35 0.01 2.83 0.12 0.9025 Grade 4 4.48 0.11 3.01 1.49 0.1375 School 1003 -3.31 -0.03 7.16 -0.46 0.6438 School 1006 0.42 0.00 15.10 0.03 0.9777 School 1007 23.45 0.37 5.74 4.08 <.0001 School 1009 8.77 0.08 7.12 1.23 0.2189 School 1010 -6.20 -0.08 6.17 -1 0.3164 School 1011 1.97 0.03 6.00 0.33 0.7427 School 1012 7.40 0.07 7.14 1.04 0.301 School 1015 5.80 0.06 6.73 0.86 0.3902 School 1017 14.31 0.18 6.20 2.31 0.0217 School 1020 5.87 0.07 6.08 0.97 0.3353 School 1021 1.95 0.02 6.18 0.32 0.7526 School 1023 17.91 0.15 7.60 2.36 0.0192 School 1024 2.12 0.02 6.99 0.3 0.7624 School 1025 9.40 0.08 7.51 1.25 0.2115 School 1026 -6.96 -0.09 6.15 -1.13 0.2588 School 1027 0.82 0.01 6.05 0.14 0.8922 School 1028 -8.84 -0.09 6.92 -1.28 0.2028 School 1029 -0.84 -0.01 6.45 -0.13 0.8964 School 1032 10.24 0.10 6.91 1.48 0.1395 School 1034 -1.85 -0.02 6.56 -0.28 0.7781 School 1035 -1.94 -0.03 6.01 -0.32 0.7473 School 1038 15.46 0.17 6.58 2.35 0.0195 School 1040 15.88 0.17 6.48 2.45 0.0149 School 1041 5.08 0.06 6.41 0.79 0.4282 School 1042 15.71 0.18 6.44 2.44 0.0154 School 1044 -2.78 -0.02 7.45 -0.37 0.7092 School 1047 11.47 0.07 9.34 1.23 0.2206 School 1049 12.67 0.12 6.78 1.87 0.0627 Male -0.89 -0.02 3.02 -0.3 0.7676 White 0.32 0.01 3.73 0.08 0.9324 African American -2.90 -0.07 3.97 -0.73 0.4667 Master’s degree 1.08 0.03 2.03 0.53 0.5956 Teaching experience 0.07 0.04 0.10 0.74 0.4629 ELA professional development -0.67 -0.03 1.08 -0.61 0.5394 New teacher -5.81 -0.15 2.21 -2.62 0.0092 ELA coordinator 1.77 0.08 1.20 1.47 0.1426 In-degree in Math networks 1.50 0.11 0.70 2.14 0.0335 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 142 Table A.10 Model 5 in effects of teachers' Math networks on students' previous ELA achievement Model 5 B Beta S. E. t value p value Intercept 817.19 0.00 6.57 124.38 <.0001 Grade 2 -0.36 -0.01 2.77 -0.13 0.896 Grade 3 0.04 0.00 2.79 0.02 0.9876 Grade 4 4.08 0.10 2.98 1.37 0.1715 School 1003 -3.45 -0.03 7.04 -0.49 0.624 School 1006 -0.06 0.00 14.89 0 0.9968 School 1007 22.96 0.37 5.68 4.04 <.0001 School 1009 8.17 0.07 7.03 1.16 0.2465 School 1010 -6.08 -0.08 6.09 -1 0.319 School 1011 1.33 0.02 5.93 0.22 0.8226 School 1012 7.13 0.07 7.04 1.01 0.3125 School 1015 4.97 0.05 6.65 0.75 0.4559 School 1017 13.66 0.17 6.12 2.23 0.0264 School 1020 5.40 0.07 6.01 0.9 0.3692 School 1021 1.59 0.02 6.10 0.26 0.7949 School 1023 17.49 0.15 7.51 2.33 0.0206 School 1024 1.72 0.02 6.91 0.25 0.8034 School 1025 8.47 0.07 7.42 1.14 0.2548 School 1026 -6.98 -0.09 6.08 -1.15 0.2517 School 1027 -0.07 0.00 5.98 -0.01 0.9911 School 1028 -8.40 -0.08 6.79 -1.24 0.2172 School 1029 -2.20 -0.02 6.38 -0.34 0.7306 School 1032 8.42 0.08 6.86 1.23 0.2207 School 1034 -2.14 -0.02 6.48 -0.33 0.7411 School 1035 -0.86 -0.01 5.95 -0.14 0.8857 School 1038 14.22 0.15 6.51 2.18 0.0298 School 1040 14.62 0.16 6.41 2.28 0.0234 School 1041 5.15 0.06 6.32 0.82 0.4155 School 1042 16.00 0.18 6.36 2.52 0.0125 School 1044 -2.68 -0.02 7.36 -0.36 0.7158 School 1047 8.87 0.05 9.25 0.96 0.3386 School 1049 12.19 0.12 6.70 1.82 0.0699 Male -0.73 -0.01 2.96 -0.25 0.8056 White -0.11 0.00 3.68 -0.03 0.9754 African American -3.53 -0.09 3.93 -0.9 0.3693 Master’s degree 1.57 0.04 1.99 0.79 0.4318 Teaching experience 0.04 0.02 0.10 0.45 0.6517 ELA professional development -0.81 -0.04 1.07 -0.76 0.4479 New teacher -4.97 -0.13 2.20 -2.26 0.0244 Teacher consultant 0.87 0.15 0.29 2.96 0.0033 In-degree in Math networks 1.37 0.10 0.70 1.97 0.0501 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 143 Table A.11 Model 1 in effects of teachers' combined networks on students' previous ELA achievement Model 1 B Beta S. E. t value p value Intercept 812.41 0.00 6.43 126.44 <.0001 Grade 2 -0.37 -0.01 2.82 -0.13 0.8969 Grade 3 0.08 0.00 2.83 0.03 0.9781 Grade 4 4.35 0.10 3.01 1.45 0.1496 School 1003 -2.19 -0.02 7.13 -0.31 0.7586 School 1006 -3.38 -0.01 15.18 -0.22 0.8241 School 1007 23.46 0.37 5.77 4.07 <.0001 School 1009 8.82 0.08 7.14 1.24 0.2179 School 1010 -5.55 -0.07 6.18 -0.9 0.3693 School 1011 1.99 0.03 6.01 0.33 0.7408 School 1012 6.82 0.07 7.15 0.95 0.3408 School 1015 5.63 0.06 6.75 0.83 0.4049 School 1017 13.86 0.18 6.22 2.23 0.0267 School 1020 5.61 0.07 6.09 0.92 0.3582 School 1021 1.58 0.02 6.20 0.25 0.7991 School 1023 17.89 0.15 7.62 2.35 0.0196 School 1024 2.07 0.02 7.01 0.29 0.7685 School 1025 8.16 0.07 7.59 1.07 0.2835 School 1026 -7.43 -0.09 6.19 -1.2 0.2307 School 1027 0.97 0.01 6.07 0.16 0.8727 School 1028 -7.84 -0.08 6.89 -1.14 0.256 School 1029 -1.08 -0.01 6.47 -0.17 0.8675 School 1032 10.41 0.10 6.92 1.5 0.1342 School 1034 -2.49 -0.03 6.58 -0.38 0.7058 School 1035 -2.01 -0.03 6.03 -0.33 0.7394 School 1038 14.87 0.16 6.62 2.25 0.0256 School 1040 15.76 0.17 6.49 2.43 0.0159 School 1041 5.81 0.07 6.41 0.91 0.3656 School 1042 15.63 0.18 6.47 2.42 0.0164 School 1044 -3.25 -0.03 7.47 -0.44 0.6638 School 1047 8.11 0.05 9.41 0.86 0.3894 School 1049 11.76 0.11 6.81 1.73 0.0856 Male -0.04 0.00 3.00 -0.01 0.9894 White 0.83 0.02 3.73 0.22 0.8249 African American -2.12 -0.05 3.97 -0.53 0.5939 Master's degree 1.28 0.03 2.02 0.64 0.5251 Teaching experience 0.07 0.04 0.10 0.72 0.4734 ELA professional development -0.88 -0.04 1.10 -0.8 0.422 New teacher -5.54 -0.14 2.23 -2.49 0.0135 In-degree in Combined networks 1.31 0.12 0.60 2.16 0.0315 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 144 Table A.12 Model 2 in effects of teachers' combined networks on students' previous ELA achievement Model 2 B Beta S. E. t value p value Intercept 812.65 0.00 6.25 130.12 <.0001 Grade 2 0.24 0.01 2.74 0.09 0.93 Grade 3 0.34 0.01 2.75 0.12 0.9025 Grade 4 4.58 0.11 2.93 1.57 0.1185 School 1003 -3.91 -0.04 6.94 -0.56 0.5732 School 1006 -1.24 0.00 14.76 -0.08 0.9332 School 1007 25.09 0.40 5.62 4.46 <.0001 School 1009 9.18 0.08 6.94 1.32 0.1869 School 1010 -5.41 -0.07 6.00 -0.9 0.3679 School 1011 2.33 0.03 5.84 0.4 0.691 School 1012 7.99 0.08 6.96 1.15 0.2517 School 1015 7.06 0.07 6.57 1.07 0.2835 School 1017 12.77 0.16 6.05 2.11 0.0358 School 1020 6.79 0.09 5.93 1.15 0.253 School 1021 2.54 0.03 6.03 0.42 0.6747 School 1023 18.43 0.15 7.41 2.49 0.0135 School 1024 3.77 0.04 6.83 0.55 0.5817 School 1025 8.81 0.07 7.38 1.19 0.2333 School 1026 -6.62 -0.08 6.02 -1.1 0.2721 School 1027 2.66 0.04 5.92 0.45 0.6535 School 1028 -7.36 -0.07 6.70 -1.1 0.2726 School 1029 -1.42 -0.02 6.29 -0.23 0.8221 School 1032 7.93 0.08 6.76 1.17 0.2419 School 1034 -0.45 -0.01 6.42 -0.07 0.9448 School 1035 0.02 0.00 5.89 0 0.9967 School 1038 16.42 0.18 6.45 2.55 0.0115 School 1040 15.93 0.17 6.31 2.52 0.0122 School 1041 6.36 0.07 6.23 1.02 0.3082 School 1042 16.89 0.19 6.30 2.68 0.0078 School 1044 -2.60 -0.02 7.27 -0.36 0.7207 School 1047 6.34 0.04 9.16 0.69 0.489 School 1049 13.00 0.13 6.63 1.96 0.051 Male -0.05 0.00 2.92 -0.02 0.9859 White -1.63 -0.04 3.68 -0.44 0.6578 African American -5.02 -0.13 3.93 -1.28 0.2022 Master's degree 1.38 0.04 1.96 0.7 0.4824 Teaching experience 0.05 0.03 0.10 0.49 0.6272 ELA professional development -1.25 -0.06 1.07 -1.17 0.2429 New teacher -4.12 -0.10 2.20 -1.88 0.0617 Formal leader 8.08 0.22 2.00 4.03 <.0001 In-degree in Combined networks 0.72 0.07 0.61 1.2 0.2329 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 145 Table A.13 Model 3 in effects of teachers' combined networks on students' previous ELA achievement Model 3 B Beta S. E. t value p value Intercept 812.89 0.00 6.29 129.26 <.0001 Grade 2 0.54 0.02 2.77 0.19 0.8462 Grade 3 0.76 0.02 2.78 0.27 0.7851 Grade 4 4.61 0.11 2.95 1.56 0.119 School 1003 -5.72 -0.05 7.05 -0.81 0.4173 School 1006 -1.16 0.00 14.87 -0.08 0.9376 School 1007 23.99 0.38 5.64 4.25 <.0001 School 1009 8.75 0.08 6.99 1.25 0.2116 School 1010 -6.75 -0.09 6.05 -1.12 0.2656 School 1011 2.03 0.03 5.88 0.35 0.7296 School 1012 7.73 0.07 7.00 1.1 0.2703 School 1015 5.78 0.06 6.60 0.88 0.3823 School 1017 13.47 0.17 6.09 2.21 0.0278 School 1020 5.50 0.07 5.96 0.92 0.3572 School 1021 2.21 0.03 6.07 0.36 0.7165 School 1023 17.82 0.15 7.46 2.39 0.0176 School 1024 2.37 0.02 6.86 0.35 0.7297 School 1025 7.20 0.06 7.43 0.97 0.3332 School 1026 -6.96 -0.09 6.05 -1.15 0.2516 School 1027 -0.41 -0.01 5.96 -0.07 0.9447 School 1028 -8.75 -0.08 6.74 -1.3 0.1954 School 1029 -1.68 -0.02 6.33 -0.27 0.7908 School 1032 9.07 0.09 6.79 1.34 0.1825 School 1034 -2.16 -0.02 6.44 -0.34 0.7373 School 1035 -0.96 -0.01 5.91 -0.16 0.871 School 1038 15.01 0.16 6.48 2.32 0.0213 School 1040 15.11 0.16 6.36 2.38 0.0181 School 1041 5.52 0.07 6.27 0.88 0.3792 School 1042 14.79 0.17 6.34 2.33 0.0204 School 1044 -3.03 -0.03 7.31 -0.41 0.6794 School 1047 4.34 0.03 9.27 0.47 0.6399 School 1049 10.88 0.11 6.67 1.63 0.1042 Male -0.88 -0.02 2.95 -0.3 0.7662 White -0.40 -0.01 3.67 -0.11 0.9141 African American -3.47 -0.09 3.90 -0.89 0.3756 Master's degree 1.20 0.03 1.97 0.61 0.5429 Teaching experience 0.04 0.02 0.10 0.39 0.6971 ELA professional development -1.16 -0.06 1.08 -1.08 0.2807 New teacher -4.91 -0.12 2.19 -2.24 0.0257 The total number of leadership 2.07 0.19 0.58 3.55 0.0005 roles In-degree in Combined networks 0.81 0.07 0.61 1.33 0.1832 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 146 Table A.14 Model 4 in effects of teachers' combined networks on students' previous ELA achievement Model 4 B Beta S. E. t value p value Intercept 813.00 0.00 6.42 126.55 <.0001 Grade 2 -0.23 -0.01 2.82 -0.08 0.9351 Grade 3 0.23 0.01 2.83 0.08 0.9346 Grade 4 4.19 0.10 3.01 1.39 0.1647 School 1003 -3.41 -0.03 7.16 -0.48 0.6347 School 1006 -2.22 -0.01 15.17 -0.15 0.8837 School 1007 23.32 0.37 5.75 4.05 <.0001 School 1009 8.86 0.08 7.12 1.24 0.2149 School 1010 -6.15 -0.08 6.18 -1 0.3205 School 1011 2.17 0.03 6.00 0.36 0.7175 School 1012 7.27 0.07 7.14 1.02 0.3098 School 1015 5.89 0.06 6.74 0.87 0.3827 School 1017 14.12 0.18 6.21 2.27 0.0238 School 1020 5.55 0.07 6.08 0.91 0.3619 School 1021 1.77 0.02 6.19 0.29 0.7756 School 1023 17.92 0.15 7.60 2.36 0.0192 School 1024 2.25 0.02 7.00 0.32 0.7478 School 1025 8.16 0.07 7.57 1.08 0.282 School 1026 -7.30 -0.09 6.17 -1.18 0.238 School 1027 0.67 0.01 6.06 0.11 0.912 School 1028 -9.09 -0.09 6.93 -1.31 0.1908 School 1029 -1.04 -0.01 6.46 -0.16 0.8726 School 1032 10.21 0.10 6.91 1.48 0.1407 School 1034 -2.39 -0.03 6.57 -0.36 0.7168 School 1035 -2.18 -0.03 6.02 -0.36 0.7178 School 1038 14.87 0.16 6.61 2.25 0.0253 School 1040 15.83 0.17 6.48 2.44 0.0152 School 1041 5.29 0.06 6.40 0.83 0.4091 School 1042 15.51 0.18 6.46 2.4 0.0171 School 1044 -3.08 -0.03 7.46 -0.41 0.6796 School 1047 8.56 0.05 9.39 0.91 0.363 School 1049 11.55 0.11 6.80 1.7 0.0906 Male -0.66 -0.01 3.03 -0.22 0.8289 White 0.46 0.01 3.73 0.12 0.9025 African American -2.66 -0.07 3.98 -0.67 0.5051 Master's degree 0.99 0.03 2.02 0.49 0.6262 Teaching experience 0.07 0.04 0.10 0.66 0.5089 ELA professional development -0.91 -0.04 1.09 -0.83 0.4072 New teacher -5.70 -0.14 2.23 -2.56 0.011 ELA coordinator 1.75 0.07 1.21 1.45 0.1469 In-degree in Combined networks 1.26 0.11 0.60 2.09 0.0377 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 147 Table A.15 Model 5 in effects of teachers' combined networks on students' previous ELA achievement Model 5 B Beta S. E. t value p value Intercept 817.49 0.00 6.55 124.81 <.0001 Grade 2 -0.24 -0.01 2.78 -0.08 0.9326 Grade 3 -0.05 0.00 2.79 -0.02 0.9846 Grade 4 3.81 0.09 2.97 1.28 0.2012 School 1003 -3.57 -0.03 7.04 -0.51 0.6121 School 1006 -2.55 -0.01 14.95 -0.17 0.8649 School 1007 22.79 0.36 5.68 4.01 <.0001 School 1009 8.25 0.07 7.03 1.17 0.242 School 1010 -6.06 -0.08 6.08 -1 0.32 School 1011 1.49 0.02 5.92 0.25 0.8018 School 1012 7.00 0.07 7.04 0.99 0.3211 School 1015 5.06 0.05 6.65 0.76 0.4478 School 1017 13.44 0.17 6.13 2.19 0.0291 School 1020 5.11 0.06 6.00 0.85 0.3959 School 1021 1.37 0.02 6.11 0.22 0.8229 School 1023 17.48 0.15 7.51 2.33 0.0206 School 1024 1.86 0.02 6.91 0.27 0.7876 School 1025 7.25 0.06 7.48 0.97 0.3334 School 1026 -7.34 -0.09 6.09 -1.2 0.2294 School 1027 -0.29 0.00 6.00 -0.05 0.9614 School 1028 -8.66 -0.08 6.79 -1.28 0.2032 School 1029 -2.44 -0.03 6.39 -0.38 0.7033 School 1032 8.34 0.08 6.85 1.22 0.225 School 1034 -2.66 -0.03 6.48 -0.41 0.6825 School 1035 -1.09 -0.02 5.95 -0.18 0.8546 School 1038 13.61 0.15 6.54 2.08 0.0384 School 1040 14.56 0.16 6.41 2.27 0.0239 School 1041 5.31 0.06 6.31 0.84 0.4008 School 1042 15.75 0.18 6.37 2.47 0.0141 School 1044 -2.98 -0.03 7.36 -0.41 0.6857 School 1047 6.12 0.04 9.29 0.66 0.5106 School 1049 11.12 0.11 6.72 1.66 0.099 Male -0.51 -0.01 2.96 -0.17 0.8622 White 0.01 0.00 3.68 0 0.9983 African American -3.31 -0.08 3.93 -0.84 0.3999 Master's degree 1.49 0.04 1.99 0.75 0.4533 Teaching experience 0.04 0.02 0.10 0.38 0.7061 ELA professional development -1.05 -0.05 1.08 -0.97 0.3328 New teacher -4.83 -0.12 2.21 -2.19 0.0296 Teacher consultant 0.89 0.15 0.29 3.01 0.0029 In-degree in Combined networks 1.19 0.11 0.60 2 0.0467 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 148 Table A.16 Model 1 in effects of teachers' ELA networks on students' previous Math achievement Model 1 B Beta S. E. t value p value Intercept 306.16 0.00 6.87 44.59 <.0001 Grade 2 18.74 0.45 3.11 6.03 <.0001 Grade 3 9.02 0.22 3.11 2.9 0.0041 Grade 4 13.83 0.28 3.33 4.15 <.0001 School 1003 1.29 0.01 7.89 0.16 0.8703 School 1006 15.03 0.04 16.84 0.89 0.3728 School 1007 27.72 0.38 6.24 4.44 <.0001 School 1009 17.06 0.13 7.82 2.18 0.03 School 1010 -2.68 -0.03 6.79 -0.39 0.6938 School 1011 8.37 0.10 6.58 1.27 0.2044 School 1012 8.87 0.07 7.88 1.13 0.261 School 1015 15.34 0.13 7.45 2.06 0.0406 School 1017 23.03 0.25 6.75 3.41 0.0007 School 1020 6.39 0.07 6.67 0.96 0.3386 School 1021 5.35 0.06 6.85 0.78 0.4355 School 1023 14.37 0.10 8.32 1.73 0.0852 School 1024 2.91 0.02 7.73 0.38 0.7071 School 1025 17.23 0.11 9.17 1.88 0.0612 School 1026 -1.75 -0.02 6.72 -0.26 0.7944 School 1027 3.98 0.05 6.63 0.6 0.5485 School 1028 -3.67 -0.03 7.57 -0.49 0.6278 School 1029 0.99 0.01 7.13 0.14 0.8894 School 1032 16.12 0.13 7.57 2.13 0.0342 School 1034 -2.28 -0.02 7.19 -0.32 0.7511 School 1035 2.41 0.03 6.66 0.36 0.7177 School 1038 25.58 0.23 7.21 3.55 0.0005 School 1040 23.74 0.22 7.11 3.34 0.001 School 1041 11.62 0.12 7.03 1.65 0.0997 School 1042 20.27 0.20 6.91 2.93 0.0036 School 1044 -2.55 -0.02 8.24 -0.31 0.7568 School 1047 18.44 0.09 10.49 1.76 0.08 School 1049 16.86 0.14 7.52 2.24 0.0257 Male -0.34 -0.01 3.34 -0.1 0.9197 White 0.16 0.00 4.13 0.04 0.969 African American -3.31 -0.07 4.39 -0.76 0.4508 Master’s degree -0.32 -0.01 2.23 -0.14 0.8871 Teaching experience 0.07 0.03 0.11 0.64 0.5199 Math professional development 1.31 0.05 1.33 0.98 0.3278 New teacher -6.32 -0.14 2.45 -2.58 0.0104 In-degree in ELA networks 1.82 0.11 0.88 2.07 0.0395 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 149 Table A.17 Model 2 in effects of teachers' ELA networks on students' previous Math achievement Model 2 B Beta S. E. t value p value Intercept 306.01 0.00 6.67 45.9 <.0001 Grade 2 19.75 0.48 3.03 6.52 <.0001 Grade 3 9.56 0.23 3.02 3.16 0.0018 Grade 4 14.41 0.29 3.24 4.46 <.0001 School 1003 -0.36 0.00 7.67 -0.05 0.9626 School 1006 17.11 0.05 16.35 1.05 0.2964 School 1007 29.71 0.41 6.08 4.89 <.0001 School 1009 18.01 0.14 7.60 2.37 0.0185 School 1010 -2.74 -0.03 6.59 -0.42 0.6778 School 1011 8.95 0.10 6.39 1.4 0.1625 School 1012 10.40 0.08 7.66 1.36 0.1756 School 1015 17.08 0.15 7.25 2.36 0.0192 School 1017 22.27 0.24 6.55 3.4 0.0008 School 1020 7.95 0.09 6.49 1.23 0.2213 School 1021 6.37 0.07 6.66 0.96 0.3392 School 1023 15.03 0.11 8.08 1.86 0.064 School 1024 5.18 0.04 7.53 0.69 0.4915 School 1025 18.46 0.12 8.90 2.07 0.0391 School 1026 -1.29 -0.01 6.52 -0.2 0.8429 School 1027 5.54 0.06 6.45 0.86 0.3906 School 1028 -3.13 -0.03 7.35 -0.43 0.6709 School 1029 0.48 0.00 6.92 0.07 0.9453 School 1032 13.71 0.11 7.38 1.86 0.0642 School 1034 -0.10 0.00 7.01 -0.01 0.9883 School 1035 4.65 0.05 6.49 0.72 0.4744 School 1038 26.92 0.24 7.01 3.84 0.0002 School 1040 24.29 0.22 6.91 3.52 0.0005 School 1041 12.47 0.12 6.83 1.83 0.0691 School 1042 21.56 0.21 6.71 3.21 0.0015 School 1044 -1.94 -0.01 8.00 -0.24 0.8088 School 1047 15.93 0.08 10.20 1.56 0.1197 School 1049 18.31 0.15 7.31 2.51 0.0128 Male -0.39 -0.01 3.24 -0.12 0.9032 White -2.61 -0.06 4.07 -0.64 0.5216 African American -6.30 -0.14 4.32 -1.46 0.1463 Master’s degree 0.02 0.00 2.17 0.01 0.9909 Teaching experience 0.04 0.02 0.11 0.37 0.7139 Math professional development 0.40 0.02 1.31 0.31 0.7586 New teacher -4.39 -0.09 2.42 -1.81 0.0711 Formal leader 8.99 0.20 2.18 4.12 <.0001 In-degree in ELA networks 1.37 0.08 0.86 1.59 0.1124 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 150 Table A.18 Model 3 in effects of teachers' ELA networks on students' previous Math achievement Model 3 B Beta S. E. t value p value Intercept 306.60 0.00 6.68 45.93 <.0001 Grade 2 20.17 0.49 3.04 6.62 <.0001 Grade 3 10.12 0.24 3.04 3.33 0.001 Grade 4 14.39 0.29 3.24 4.44 <.0001 School 1003 -2.91 -0.02 7.74 -0.38 0.7074 School 1006 17.72 0.05 16.38 1.08 0.2803 School 1007 28.54 0.39 6.07 4.7 <.0001 School 1009 17.39 0.13 7.60 2.29 0.023 School 1010 -4.36 -0.05 6.61 -0.66 0.5101 School 1011 8.56 0.10 6.39 1.34 0.1817 School 1012 10.16 0.08 7.66 1.33 0.1859 School 1015 15.59 0.13 7.25 2.15 0.0323 School 1017 22.98 0.25 6.56 3.5 0.0005 School 1020 6.41 0.07 6.48 0.99 0.3234 School 1021 6.16 0.07 6.66 0.92 0.356 School 1023 14.20 0.10 8.09 1.76 0.0803 School 1024 3.54 0.03 7.52 0.47 0.6378 School 1025 16.67 0.11 8.91 1.87 0.0625 School 1026 -1.63 -0.02 6.53 -0.25 0.8031 School 1027 1.96 0.02 6.46 0.3 0.7622 School 1028 -4.87 -0.04 7.36 -0.66 0.5086 School 1029 0.10 0.00 6.93 0.02 0.988 School 1032 14.77 0.12 7.37 2 0.0461 School 1034 -2.07 -0.02 6.99 -0.3 0.7678 School 1035 3.68 0.04 6.48 0.57 0.5711 School 1038 25.37 0.23 7.01 3.62 0.0004 School 1040 23.23 0.21 6.91 3.36 0.0009 School 1041 11.47 0.11 6.84 1.68 0.0946 School 1042 18.79 0.19 6.72 2.79 0.0056 School 1044 -2.39 -0.02 8.01 -0.3 0.7657 School 1047 13.39 0.07 10.27 1.3 0.1935 School 1049 15.89 0.13 7.31 2.17 0.0307 Male -1.47 -0.02 3.26 -0.45 0.6531 White -1.43 -0.03 4.04 -0.35 0.7237 African American -4.75 -0.10 4.28 -1.11 0.2687 Master’s degree -0.20 0.00 2.17 -0.09 0.9271 Teaching experience 0.03 0.01 0.11 0.25 0.8048 Math professional development 0.40 0.02 1.31 0.31 0.7595 New teacher -5.25 -0.11 2.40 -2.19 0.0292 The total number of leadership 2.57 0.20 0.64 4.04 <.0001 roles In-degree in ELA networks 1.30 0.08 0.86 1.5 0.1347 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 151 Table A.19 Model 4 in effects of teachers' ELA networks on students' previous Math achievement Model 4 B Beta S. E. t value p value Intercept 306.51 0.00 6.82 44.95 <.0001 Grade 2 19.17 0.46 3.09 6.19 <.0001 Grade 3 9.63 0.23 3.10 3.11 0.0021 Grade 4 14.42 0.29 3.32 4.35 <.0001 School 1003 -0.16 0.00 7.86 -0.02 0.9836 School 1006 15.29 0.04 16.72 0.91 0.3611 School 1007 27.74 0.38 6.19 4.48 <.0001 School 1009 16.05 0.12 7.78 2.06 0.04 School 1010 -3.22 -0.04 6.74 -0.48 0.6329 School 1011 8.13 0.09 6.53 1.24 0.2143 School 1012 9.18 0.07 7.82 1.17 0.2414 School 1015 15.43 0.13 7.40 2.08 0.0381 School 1017 22.75 0.24 6.70 3.4 0.0008 School 1020 6.51 0.07 6.62 0.98 0.3265 School 1021 4.37 0.05 6.82 0.64 0.5224 School 1023 14.34 0.10 8.26 1.74 0.0837 School 1024 2.74 0.02 7.67 0.36 0.7213 School 1025 17.80 0.12 9.10 1.95 0.0517 School 1026 -1.78 -0.02 6.67 -0.27 0.7897 School 1027 2.92 0.03 6.60 0.44 0.6586 School 1028 -3.62 -0.03 7.51 -0.48 0.6305 School 1029 0.83 0.01 7.08 0.12 0.9069 School 1032 16.21 0.13 7.52 2.16 0.032 School 1034 -2.17 -0.02 7.14 -0.3 0.761 School 1035 2.06 0.02 6.61 0.31 0.7559 School 1038 25.35 0.23 7.16 3.54 0.0005 School 1040 23.86 0.22 7.06 3.38 0.0008 School 1041 10.19 0.10 7.01 1.45 0.1473 School 1042 19.43 0.19 6.87 2.83 0.005 School 1044 -2.51 -0.02 8.18 -0.31 0.7593 School 1047 18.62 0.09 10.41 1.79 0.0749 School 1049 16.88 0.14 7.46 2.26 0.0246 Male -0.49 -0.01 3.32 -0.15 0.882 White -0.38 -0.01 4.11 -0.09 0.9266 African American -3.97 -0.09 4.37 -0.91 0.3645 Master’s degree -0.46 -0.01 2.22 -0.21 0.8363 Teaching experience 0.07 0.03 0.11 0.62 0.5328 Math professional development 1.08 0.04 1.33 0.81 0.418 New teacher -6.48 -0.14 2.43 -2.67 0.0082 Math coordinator 2.80 0.10 1.27 2.2 0.0288 In-degree in ELA networks 1.84 0.11 0.87 2.1 0.0363 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 152 Table A.20 Model 5 in effects of teachers' ELA networks on students' previous Math achievement Model 5 B Beta S. E. t value p value Intercept 311.62 0.00 7.03 44.34 <.0001 Grade 2 19.07 0.46 3.07 6.21 <.0001 Grade 3 9.02 0.22 3.07 2.94 0.0036 Grade 4 13.31 0.27 3.29 4.05 <.0001 School 1003 -0.01 0.00 7.79 0 0.999 School 1006 15.77 0.05 16.60 0.95 0.3432 School 1007 27.11 0.37 6.16 4.4 <.0001 School 1009 16.68 0.13 7.71 2.16 0.0315 School 1010 -3.34 -0.04 6.70 -0.5 0.6184 School 1011 8.03 0.09 6.49 1.24 0.2169 School 1012 9.12 0.07 7.77 1.17 0.2415 School 1015 14.78 0.13 7.35 2.01 0.0455 School 1017 22.81 0.24 6.65 3.43 0.0007 School 1020 5.90 0.06 6.58 0.9 0.3704 School 1021 5.17 0.06 6.76 0.77 0.4449 School 1023 13.85 0.10 8.21 1.69 0.0926 School 1024 2.81 0.02 7.62 0.37 0.7125 School 1025 16.57 0.11 9.04 1.83 0.068 School 1026 -1.83 -0.02 6.62 -0.28 0.7821 School 1027 2.71 0.03 6.55 0.41 0.6791 School 1028 -4.56 -0.04 7.47 -0.61 0.5422 School 1029 -0.44 0.00 7.04 -0.06 0.9499 School 1032 14.02 0.11 7.50 1.87 0.0629 School 1034 -2.58 -0.02 7.10 -0.36 0.7161 School 1035 3.43 0.04 6.58 0.52 0.6025 School 1038 24.13 0.22 7.12 3.39 0.0008 School 1040 22.57 0.21 7.03 3.21 0.0015 School 1041 11.23 0.11 6.94 1.62 0.1068 School 1042 19.69 0.20 6.81 2.89 0.0042 School 1044 -2.38 -0.02 8.13 -0.29 0.7703 School 1047 15.98 0.08 10.38 1.54 0.1249 School 1049 16.19 0.13 7.42 2.18 0.0299 Male -0.78 -0.01 3.30 -0.24 0.8126 White -0.79 -0.02 4.09 -0.19 0.8471 African American -4.50 -0.10 4.35 -1.04 0.3012 Master’s degree -0.02 0.00 2.20 -0.01 0.9923 Teaching experience 0.03 0.02 0.11 0.32 0.7497 Math professional development 0.77 0.03 1.33 0.58 0.562 New teacher -5.42 -0.12 2.43 -2.23 0.0269 Teacher consultant 0.94 0.14 0.32 2.91 0.004 In-degree in ELA networks 1.79 0.10 0.87 2.06 0.0402 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 153 Table A.21 Model 1 in effects of teachers' Math networks on students' previous Math achievement Model 1 B Beta S. E. t value p value Intercept 306.26 0.00 6.84 44.74 <.0001 Grade 2 18.84 0.46 3.11 6.06 <.0001 Grade 3 9.65 0.23 3.11 3.1 0.0021 Grade 4 14.37 0.29 3.33 4.31 <.0001 School 1003 0.75 0.01 7.87 0.1 0.9238 School 1006 19.04 0.06 16.69 1.14 0.2549 School 1007 27.47 0.38 6.24 4.4 <.0001 School 1009 16.54 0.13 7.80 2.12 0.0349 School 1010 -3.29 -0.04 6.79 -0.48 0.6286 School 1011 7.07 0.08 6.57 1.08 0.2829 School 1012 8.69 0.07 7.87 1.1 0.2702 School 1015 14.80 0.13 7.44 1.99 0.0476 School 1017 23.07 0.25 6.73 3.43 0.0007 School 1020 6.48 0.07 6.66 0.97 0.3314 School 1021 5.41 0.06 6.84 0.79 0.4297 School 1023 13.60 0.10 8.30 1.64 0.1026 School 1024 2.31 0.02 7.70 0.3 0.7645 School 1025 19.47 0.13 9.07 2.15 0.0327 School 1026 -2.52 -0.03 6.73 -0.37 0.7089 School 1027 2.46 0.03 6.67 0.37 0.7125 School 1028 -4.15 -0.03 7.56 -0.55 0.5838 School 1029 0.46 0.00 7.13 0.06 0.9483 School 1032 16.12 0.13 7.56 2.13 0.034 School 1034 -2.47 -0.02 7.19 -0.34 0.7318 School 1035 2.15 0.03 6.65 0.32 0.7473 School 1038 24.99 0.23 7.21 3.47 0.0006 School 1040 23.66 0.22 7.10 3.33 0.001 School 1041 10.98 0.11 7.03 1.56 0.1197 School 1042 19.79 0.20 6.91 2.86 0.0046 School 1044 -2.40 -0.02 8.22 -0.29 0.7703 School 1047 22.72 0.11 10.35 2.2 0.029 School 1049 18.34 0.15 7.47 2.45 0.0148 Male -1.21 -0.02 3.31 -0.37 0.7137 White 0.00 0.00 4.13 0 0.9995 African American -3.14 -0.07 4.39 -0.72 0.4752 Master’s degree 0.03 0.00 2.23 0.01 0.9891 Teaching experience 0.08 0.04 0.11 0.74 0.4616 Math professional development 1.00 0.04 1.33 0.75 0.4535 New teacher -6.40 -0.14 2.43 -2.63 0.009 In-degree in Math networks 1.73 0.11 0.77 2.23 0.0264 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 154 Table A.22 Model 2 in effects of teachers' Math networks on students' previous Math achievement Model 2 B Beta S. E. t value p value Intercept 306.41 0.00 6.67 45.96 <.0001 Grade 2 19.72 0.48 3.04 6.49 <.0001 Grade 3 9.95 0.24 3.03 3.28 0.0012 Grade 4 14.71 0.29 3.25 4.53 <.0001 School 1003 -0.74 -0.01 7.68 -0.1 0.9231 School 1006 20.15 0.06 16.26 1.24 0.2164 School 1007 29.74 0.41 6.10 4.87 <.0001 School 1009 17.52 0.13 7.60 2.3 0.022 School 1010 -3.07 -0.03 6.61 -0.46 0.6433 School 1011 8.09 0.09 6.41 1.26 0.2076 School 1012 10.27 0.08 7.67 1.34 0.1819 School 1015 16.58 0.14 7.26 2.28 0.0231 School 1017 22.52 0.24 6.56 3.43 0.0007 School 1020 7.91 0.08 6.50 1.22 0.2243 School 1021 6.66 0.07 6.67 1 0.3187 School 1023 14.43 0.10 8.09 1.78 0.0755 School 1024 4.56 0.04 7.53 0.61 0.5455 School 1025 20.21 0.13 8.83 2.29 0.0229 School 1026 -1.62 -0.02 6.56 -0.25 0.805 School 1027 4.72 0.05 6.53 0.72 0.4698 School 1028 -3.46 -0.03 7.36 -0.47 0.6389 School 1029 0.28 0.00 6.94 0.04 0.9673 School 1032 13.90 0.11 7.39 1.88 0.061 School 1034 -0.27 0.00 7.02 -0.04 0.9691 School 1035 4.51 0.05 6.51 0.69 0.4893 School 1038 26.67 0.24 7.04 3.79 0.0002 School 1040 24.21 0.22 6.92 3.5 0.0005 School 1041 12.13 0.12 6.86 1.77 0.0781 School 1042 21.47 0.21 6.75 3.18 0.0016 School 1044 -1.72 -0.01 8.01 -0.21 0.8303 School 1047 19.10 0.10 10.12 1.89 0.0602 School 1049 19.40 0.16 7.28 2.66 0.0082 Male -1.06 -0.02 3.22 -0.33 0.7421 White -2.66 -0.06 4.08 -0.65 0.5142 African American -6.17 -0.13 4.34 -1.42 0.1567 Master’s degree 0.21 0.00 2.18 0.09 0.9244 Teaching experience 0.05 0.02 0.11 0.45 0.655 Math professional development 0.23 0.01 1.31 0.18 0.8584 New teacher -4.72 -0.10 2.40 -1.96 0.0507 Formal leader 8.74 0.20 2.24 3.9 0.0001 In-degree in Math networks 0.98 0.06 0.78 1.26 0.2097 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 155 Table A.23 Model 3 in effects of teachers' Math networks on students' previous Math achievement Model 3 B Beta S. E. t value p value Intercept 306.89 0.00 6.67 46 <.0001 Grade 2 20.15 0.49 3.05 6.61 <.0001 Grade 3 10.50 0.25 3.04 3.46 0.0006 Grade 4 14.70 0.29 3.25 4.53 <.0001 School 1003 -3.21 -0.02 7.74 -0.42 0.6784 School 1006 20.57 0.06 16.27 1.26 0.2071 School 1007 28.54 0.39 6.08 4.69 <.0001 School 1009 16.96 0.13 7.60 2.23 0.0265 School 1010 -4.67 -0.05 6.62 -0.7 0.4817 School 1011 7.73 0.09 6.41 1.21 0.2287 School 1012 10.04 0.08 7.67 1.31 0.1919 School 1015 15.18 0.13 7.25 2.09 0.0372 School 1017 23.14 0.25 6.56 3.53 0.0005 School 1020 6.43 0.07 6.49 0.99 0.3223 School 1021 6.37 0.07 6.66 0.96 0.34 School 1023 13.65 0.10 8.09 1.69 0.0926 School 1024 3.02 0.02 7.51 0.4 0.6877 School 1025 18.35 0.12 8.84 2.08 0.0389 School 1026 -2.00 -0.02 6.56 -0.3 0.7609 School 1027 1.17 0.01 6.51 0.18 0.8572 School 1028 -5.16 -0.04 7.37 -0.7 0.4846 School 1029 -0.12 0.00 6.95 -0.02 0.9862 School 1032 14.88 0.12 7.38 2.02 0.0447 School 1034 -2.18 -0.02 7.00 -0.31 0.7557 School 1035 3.54 0.04 6.49 0.55 0.5859 School 1038 25.11 0.23 7.03 3.57 0.0004 School 1040 23.18 0.21 6.92 3.35 0.0009 School 1041 11.13 0.11 6.85 1.62 0.1056 School 1042 18.69 0.19 6.74 2.77 0.006 School 1044 -2.20 -0.02 8.01 -0.27 0.7839 School 1047 16.48 0.08 10.21 1.61 0.1077 School 1049 16.98 0.14 7.29 2.33 0.0206 Male -2.07 -0.03 3.23 -0.64 0.5218 White -1.51 -0.04 4.04 -0.37 0.7082 African American -4.65 -0.10 4.29 -1.08 0.28 Master’s degree 0.00 0.00 2.18 0 0.9995 Teaching experience 0.03 0.02 0.11 0.32 0.7459 Math professional development 0.23 0.01 1.31 0.17 0.8615 New teacher -5.48 -0.12 2.38 -2.3 0.022 The total number of leadership 2.51 0.19 0.65 3.88 0.0001 roles In-degree in Math networks 1.01 0.06 0.78 1.3 0.1939 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 156 Table A.24 Model 4 in effects of teachers' Math networks on students' previous Math achievement Model 4 B Beta S. E. t value p value Intercept 306.79 0.00 6.82 44.99 <.0001 Grade 2 19.14 0.46 3.10 6.18 <.0001 Grade 3 10.12 0.24 3.11 3.26 0.0013 Grade 4 14.81 0.30 3.33 4.45 <.0001 School 1003 -0.50 0.00 7.87 -0.06 0.9493 School 1006 19.33 0.06 16.61 1.16 0.2457 School 1007 27.68 0.38 6.21 4.46 <.0001 School 1009 15.63 0.12 7.78 2.01 0.0456 School 1010 -3.66 -0.04 6.76 -0.54 0.5887 School 1011 6.97 0.08 6.54 1.06 0.288 School 1012 8.99 0.07 7.83 1.15 0.2521 School 1015 14.85 0.13 7.40 2.01 0.046 School 1017 22.98 0.25 6.70 3.43 0.0007 School 1020 6.54 0.07 6.63 0.99 0.3251 School 1021 4.76 0.05 6.81 0.7 0.4851 School 1023 13.57 0.10 8.26 1.64 0.1017 School 1024 2.08 0.02 7.67 0.27 0.7861 School 1025 20.03 0.13 9.03 2.22 0.0274 School 1026 -2.35 -0.03 6.70 -0.35 0.7256 School 1027 1.82 0.02 6.65 0.27 0.7849 School 1028 -4.07 -0.03 7.52 -0.54 0.5887 School 1029 0.46 0.00 7.10 0.06 0.9483 School 1032 16.28 0.13 7.53 2.16 0.0315 School 1034 -2.35 -0.02 7.15 -0.33 0.7426 School 1035 1.93 0.02 6.62 0.29 0.7708 School 1038 24.96 0.23 7.18 3.48 0.0006 School 1040 23.76 0.22 7.07 3.36 0.0009 School 1041 9.88 0.10 7.03 1.41 0.1609 School 1042 19.29 0.19 6.89 2.8 0.0055 School 1044 -2.28 -0.02 8.19 -0.28 0.7813 School 1047 22.80 0.11 10.30 2.21 0.0277 School 1049 18.37 0.15 7.44 2.47 0.0142 Male -1.36 -0.02 3.29 -0.41 0.6811 White -0.47 -0.01 4.12 -0.11 0.9099 African American -3.75 -0.08 4.38 -0.86 0.3921 Master’s degree -0.14 0.00 2.23 -0.06 0.9503 Teaching experience 0.08 0.04 0.11 0.72 0.4713 Math professional development 0.83 0.03 1.33 0.62 0.5333 New teacher -6.69 -0.14 2.42 -2.76 0.0062 Math coordinator 2.38 0.09 1.29 1.85 0.066 In-degree in Math networks 1.50 0.09 0.78 1.92 0.0554 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 157 Table A.25 Model 5 in effects of teachers' Math networks on students' previous Math achievement Model 5 B Beta S. E. t value p value Intercept 311.63 0.00 7.03 44.35 <.0001 Grade 2 19.12 0.46 3.07 6.23 <.0001 Grade 3 9.62 0.23 3.07 3.13 0.0019 Grade 4 13.83 0.28 3.30 4.2 <.0001 School 1003 -0.49 0.00 7.79 -0.06 0.9494 School 1006 19.70 0.06 16.48 1.2 0.233 School 1007 26.98 0.37 6.16 4.38 <.0001 School 1009 16.16 0.12 7.70 2.1 0.0369 School 1010 -3.87 -0.04 6.70 -0.58 0.5642 School 1011 6.82 0.08 6.49 1.05 0.2945 School 1012 8.95 0.07 7.77 1.15 0.2503 School 1015 14.26 0.12 7.35 1.94 0.0533 School 1017 22.92 0.25 6.65 3.45 0.0007 School 1020 5.99 0.06 6.58 0.91 0.3635 School 1021 5.32 0.06 6.75 0.79 0.4313 School 1023 13.11 0.09 8.20 1.6 0.111 School 1024 2.19 0.02 7.61 0.29 0.7734 School 1025 18.82 0.12 8.95 2.1 0.0365 School 1026 -2.49 -0.03 6.64 -0.38 0.7077 School 1027 1.39 0.02 6.60 0.21 0.8333 School 1028 -4.98 -0.04 7.47 -0.67 0.5058 School 1029 -0.84 -0.01 7.05 -0.12 0.9048 School 1032 14.13 0.12 7.50 1.88 0.0607 School 1034 -2.74 -0.03 7.10 -0.39 0.6996 School 1035 3.17 0.04 6.58 0.48 0.6303 School 1038 23.69 0.22 7.14 3.32 0.001 School 1040 22.54 0.21 7.02 3.21 0.0015 School 1041 10.67 0.11 6.95 1.54 0.1257 School 1042 19.34 0.19 6.83 2.83 0.005 School 1044 -2.19 -0.02 8.12 -0.27 0.7873 School 1047 20.23 0.10 10.25 1.97 0.0496 School 1049 17.67 0.14 7.38 2.39 0.0174 Male -1.63 -0.02 3.27 -0.5 0.6187 White -0.91 -0.02 4.09 -0.22 0.8235 African American -4.31 -0.09 4.35 -0.99 0.3225 Master’s degree 0.29 0.01 2.21 0.13 0.8972 Teaching experience 0.05 0.02 0.11 0.42 0.6727 Math professional development 0.50 0.02 1.33 0.38 0.7056 New teacher -5.60 -0.12 2.42 -2.32 0.0212 Teacher consultant 0.91 0.13 0.32 2.79 0.0056 In-degree in Math networks 1.59 0.10 0.77 2.08 0.0389 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 158 Table A.26 Model 1 in effects of teachers' combined networks on students' previous Math achievement Model 1 B Beta S. E. t value p value Intercept 306.14 0.00 6.89 44.46 <.0001 Grade 2 18.91 0.46 3.12 6.06 <.0001 Grade 3 9.47 0.23 3.12 3.04 0.0026 Grade 4 13.99 0.28 3.33 4.2 <.0001 School 1003 0.69 0.01 7.89 0.09 0.9307 School 1006 16.58 0.05 16.79 0.99 0.3243 School 1007 27.84 0.38 6.24 4.46 <.0001 School 1009 16.82 0.13 7.82 2.15 0.0325 School 1010 -3.12 -0.04 6.80 -0.46 0.6474 School 1011 7.57 0.09 6.58 1.15 0.2514 School 1012 8.76 0.07 7.89 1.11 0.2678 School 1015 15.00 0.13 7.46 2.01 0.0454 School 1017 23.26 0.25 6.75 3.45 0.0007 School 1020 6.35 0.07 6.68 0.95 0.3427 School 1021 5.45 0.06 6.86 0.79 0.4274 School 1023 13.98 0.10 8.32 1.68 0.0943 School 1024 2.55 0.02 7.73 0.33 0.7415 School 1025 17.80 0.12 9.15 1.94 0.0529 School 1026 -2.36 -0.03 6.75 -0.35 0.727 School 1027 2.67 0.03 6.70 0.4 0.6904 School 1028 -4.17 -0.03 7.58 -0.55 0.5828 School 1029 0.49 0.00 7.15 0.07 0.9452 School 1032 16.39 0.13 7.58 2.16 0.0314 School 1034 -2.64 -0.03 7.21 -0.37 0.7148 School 1035 2.05 0.02 6.68 0.31 0.7591 School 1038 24.79 0.23 7.25 3.42 0.0007 School 1040 23.83 0.22 7.12 3.35 0.0009 School 1041 11.46 0.11 7.04 1.63 0.1048 School 1042 19.97 0.20 6.93 2.88 0.0043 School 1044 -2.46 -0.02 8.25 -0.3 0.7654 School 1047 19.80 0.10 10.43 1.9 0.0587 School 1049 17.37 0.14 7.51 2.31 0.0215 Male -0.92 -0.01 3.32 -0.28 0.7818 White 0.19 0.00 4.14 0.05 0.9633 African American -3.05 -0.07 4.40 -0.69 0.4897 Master's degree -0.12 0.00 2.24 -0.06 0.956 Teaching experience 0.07 0.03 0.11 0.66 0.5111 Math professional development 1.15 0.04 1.33 0.86 0.3895 New teacher -6.38 -0.14 2.46 -2.6 0.0099 In-degree in Combined networks 1.26 0.10 0.66 1.9 0.0585 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 159 Table A.27 Model 2 in effects of teachers' combined networks on students' previous Math achievement Model 2 B Beta S. E. t value p value Intercept 306.49 0.00 6.70 45.75 <.0001 Grade 2 19.74 0.48 3.04 6.49 <.0001 Grade 3 9.84 0.24 3.03 3.25 0.0013 Grade 4 14.50 0.29 3.25 4.47 <.0001 School 1003 -0.81 -0.01 7.69 -0.11 0.9157 School 1006 19.02 0.06 16.35 1.16 0.2458 School 1007 30.08 0.41 6.10 4.93 <.0001 School 1009 17.64 0.14 7.62 2.32 0.0213 School 1010 -2.92 -0.03 6.62 -0.44 0.6593 School 1011 8.41 0.10 6.41 1.31 0.1903 School 1012 10.36 0.08 7.68 1.35 0.1789 School 1015 16.70 0.14 7.27 2.3 0.0224 School 1017 22.68 0.24 6.57 3.45 0.0006 School 1020 7.86 0.08 6.51 1.21 0.2284 School 1021 6.82 0.07 6.68 1.02 0.3085 School 1023 14.64 0.10 8.10 1.81 0.0719 School 1024 4.67 0.04 7.54 0.62 0.5358 School 1025 19.46 0.13 8.91 2.18 0.0299 School 1026 -1.42 -0.02 6.57 -0.22 0.8295 School 1027 5.03 0.06 6.54 0.77 0.4426 School 1028 -3.44 -0.03 7.37 -0.47 0.6414 School 1029 0.38 0.00 6.96 0.05 0.9569 School 1032 14.04 0.11 7.40 1.9 0.0589 School 1034 -0.29 0.00 7.04 -0.04 0.9669 School 1035 4.56 0.05 6.53 0.7 0.485 School 1038 26.70 0.24 7.07 3.78 0.0002 School 1040 24.30 0.22 6.93 3.51 0.0005 School 1041 12.45 0.12 6.86 1.82 0.0705 School 1042 21.72 0.22 6.76 3.21 0.0015 School 1044 -1.68 -0.01 8.03 -0.21 0.8342 School 1047 17.57 0.09 10.16 1.73 0.085 School 1049 18.96 0.15 7.32 2.59 0.0101 Male -0.92 -0.01 3.23 -0.28 0.7765 White -2.64 -0.06 4.09 -0.64 0.5196 African American -6.23 -0.13 4.36 -1.43 0.1542 Master's degree 0.11 0.00 2.18 0.05 0.9608 Teaching experience 0.04 0.02 0.11 0.4 0.6873 Math professional development 0.30 0.01 1.31 0.23 0.8171 New teacher -4.76 -0.10 2.42 -1.97 0.0504 Formal leader 8.93 0.20 2.24 3.98 <.0001 In-degree in Combined networks 0.60 0.05 0.66 0.9 0.3682 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 160 Table A.28 Model 3 in effects of teachers' combined networks on students' previous Math achievement Model 3 B Beta S. E. t value p value Intercept 306.95 0.00 6.70 45.78 <.0001 Grade 2 20.19 0.49 3.05 6.61 <.0001 Grade 3 10.40 0.25 3.04 3.42 0.0007 Grade 4 14.48 0.29 3.25 4.46 <.0001 School 1003 -3.34 -0.03 7.75 -0.43 0.667 School 1006 19.37 0.06 16.36 1.18 0.2375 School 1007 28.84 0.40 6.08 4.74 <.0001 School 1009 17.09 0.13 7.62 2.24 0.0257 School 1010 -4.56 -0.05 6.63 -0.69 0.4922 School 1011 8.05 0.09 6.41 1.26 0.2103 School 1012 10.12 0.08 7.68 1.32 0.1891 School 1015 15.27 0.13 7.26 2.1 0.0364 School 1017 23.31 0.25 6.57 3.55 0.0005 School 1020 6.34 0.07 6.50 0.98 0.3298 School 1021 6.51 0.07 6.68 0.97 0.3311 School 1023 13.85 0.10 8.10 1.71 0.0884 School 1024 3.12 0.03 7.53 0.41 0.6785 School 1025 17.50 0.11 8.91 1.96 0.0506 School 1026 -1.82 -0.02 6.57 -0.28 0.7825 School 1027 1.38 0.02 6.53 0.21 0.8325 School 1028 -5.18 -0.04 7.38 -0.7 0.4836 School 1029 -0.05 0.00 6.96 -0.01 0.9942 School 1032 15.04 0.12 7.38 2.04 0.0427 School 1034 -2.25 -0.02 7.01 -0.32 0.7485 School 1035 3.56 0.04 6.51 0.55 0.5844 School 1038 25.09 0.23 7.05 3.56 0.0004 School 1040 23.25 0.21 6.93 3.35 0.0009 School 1041 11.44 0.11 6.86 1.67 0.0965 School 1042 18.87 0.19 6.75 2.79 0.0056 School 1044 -2.18 -0.02 8.03 -0.27 0.7858 School 1047 14.80 0.07 10.23 1.45 0.1489 School 1049 16.45 0.13 7.31 2.25 0.0253 Male -1.94 -0.03 3.24 -0.6 0.5499 White -1.45 -0.03 4.05 -0.36 0.72 African American -4.66 -0.10 4.30 -1.08 0.2798 Master's degree -0.11 0.00 2.18 -0.05 0.9614 Teaching experience 0.03 0.01 0.11 0.28 0.7833 Math professional development 0.30 0.01 1.32 0.23 0.8186 New teacher -5.53 -0.12 2.40 -2.3 0.0221 The total number of leadership 2.57 0.20 0.65 3.96 <.0001 roles In-degree in Combined networks 0.64 0.05 0.66 0.97 0.3336 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 161 Table A.29 Model 4 in effects of teachers' combined networks on students' previous Math achievement Model 4 B Beta S. E. t value p value Intercept 306.71 0.00 6.86 44.74 <.0001 Grade 2 19.23 0.47 3.11 6.18 <.0001 Grade 3 9.99 0.24 3.11 3.21 0.0015 Grade 4 14.50 0.29 3.33 4.36 <.0001 School 1003 -0.62 0.00 7.88 -0.08 0.9374 School 1006 17.21 0.05 16.71 1.03 0.3039 School 1007 28.01 0.39 6.21 4.51 <.0001 School 1009 15.82 0.12 7.80 2.03 0.0435 School 1010 -3.53 -0.04 6.77 -0.52 0.6024 School 1011 7.39 0.08 6.55 1.13 0.2604 School 1012 9.06 0.07 7.85 1.15 0.2493 School 1015 15.02 0.13 7.42 2.02 0.044 School 1017 23.13 0.25 6.71 3.45 0.0007 School 1020 6.42 0.07 6.64 0.97 0.3344 School 1021 4.77 0.05 6.83 0.7 0.486 School 1023 13.90 0.10 8.28 1.68 0.0944 School 1024 2.28 0.02 7.69 0.3 0.7667 School 1025 18.60 0.12 9.11 2.04 0.0423 School 1026 -2.22 -0.02 6.71 -0.33 0.7416 School 1027 1.96 0.02 6.67 0.29 0.7692 School 1028 -4.09 -0.03 7.54 -0.54 0.5879 School 1029 0.48 0.00 7.11 0.07 0.946 School 1032 16.53 0.13 7.54 2.19 0.0293 School 1034 -2.49 -0.02 7.17 -0.35 0.7282 School 1035 1.84 0.02 6.64 0.28 0.7824 School 1038 24.78 0.23 7.21 3.44 0.0007 School 1040 23.91 0.22 7.09 3.38 0.0008 School 1041 10.24 0.10 7.04 1.46 0.1467 School 1042 19.42 0.19 6.90 2.81 0.0053 School 1044 -2.32 -0.02 8.21 -0.28 0.7772 School 1047 20.28 0.10 10.38 1.95 0.0517 School 1049 17.53 0.14 7.47 2.35 0.0198 Male -1.11 -0.02 3.30 -0.33 0.7379 White -0.33 -0.01 4.13 -0.08 0.9372 African American -3.70 -0.08 4.39 -0.84 0.4 Master's degree -0.28 -0.01 2.23 -0.13 0.9003 Teaching experience 0.07 0.03 0.11 0.65 0.5151 Math professional development 0.95 0.04 1.33 0.71 0.4761 New teacher -6.69 -0.14 2.45 -2.73 0.0067 Math coordinator 2.50 0.09 1.29 1.94 0.0536 In-degree in Combined networks 1.09 0.08 0.66 1.64 0.1021 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 162 Table A.30 Model 5 in effects of teachers' combined networks on students' previous Math achievement Model 5 B Beta S. E. t value p value Intercept 311.58 0.00 7.06 44.11 <.0001 Grade 2 19.20 0.46 3.08 6.23 <.0001 Grade 3 9.45 0.23 3.07 3.07 0.0023 Grade 4 13.47 0.27 3.30 4.09 <.0001 School 1003 -0.57 0.00 7.80 -0.07 0.9416 School 1006 17.45 0.05 16.58 1.05 0.2935 School 1007 27.31 0.38 6.17 4.43 <.0001 School 1009 16.41 0.13 7.72 2.12 0.0346 School 1010 -3.72 -0.04 6.72 -0.55 0.5801 School 1011 7.27 0.08 6.50 1.12 0.2644 School 1012 9.01 0.07 7.79 1.16 0.248 School 1015 14.43 0.12 7.36 1.96 0.0511 School 1017 23.10 0.25 6.66 3.47 0.0006 School 1020 5.86 0.06 6.59 0.89 0.3751 School 1021 5.36 0.06 6.77 0.79 0.4291 School 1023 13.45 0.10 8.22 1.64 0.1028 School 1024 2.41 0.02 7.63 0.32 0.7521 School 1025 17.28 0.11 9.04 1.91 0.057 School 1026 -2.35 -0.03 6.66 -0.35 0.7246 School 1027 1.57 0.02 6.62 0.24 0.8128 School 1028 -5.01 -0.04 7.48 -0.67 0.5041 School 1029 -0.83 -0.01 7.07 -0.12 0.9062 School 1032 14.36 0.12 7.52 1.91 0.0572 School 1034 -2.90 -0.03 7.11 -0.41 0.6838 School 1035 3.10 0.04 6.60 0.47 0.6393 School 1038 23.49 0.21 7.17 3.28 0.0012 School 1040 22.68 0.21 7.04 3.22 0.0014 School 1041 11.11 0.11 6.95 1.6 0.1114 School 1042 19.50 0.19 6.85 2.85 0.0047 School 1044 -2.25 -0.02 8.14 -0.28 0.7829 School 1047 17.52 0.09 10.32 1.7 0.0909 School 1049 16.77 0.14 7.42 2.26 0.0245 Male -1.36 -0.02 3.28 -0.42 0.6779 White -0.75 -0.02 4.10 -0.18 0.8546 African American -4.24 -0.09 4.37 -0.97 0.3319 Master's degree 0.15 0.00 2.21 0.07 0.947 Teaching experience 0.04 0.02 0.11 0.35 0.7291 Math professional development 0.63 0.02 1.33 0.48 0.6344 New teacher -5.57 -0.12 2.44 -2.28 0.0232 Teacher consultant 0.92 0.14 0.33 2.82 0.0051 In-degree in Combined networks 1.15 0.09 0.65 1.76 0.0788 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 163 Table A.31 Model 1 in effects of teachers' ELA networks on students' previous free/reduced lunch Model 1 B Beta S. E. t value p value Intercept 0.93 0.00 0.06 15.11 <.0001 Grade 2 0.01 0.01 0.03 0.3 0.7635 Grade 3 0.04 0.08 0.03 1.54 0.1255 Grade 4 0.02 0.04 0.03 0.81 0.4179 School 1003 -0.11 -0.06 0.07 -1.5 0.1339 School 1006 -0.25 -0.05 0.15 -1.6 0.1109 School 1007 -0.54 -0.54 0.06 -9.45 <.0001 School 1009 -0.24 -0.13 0.07 -3.36 0.0009 School 1010 0.01 0.01 0.06 0.19 0.8473 School 1011 -0.37 -0.31 0.06 -6.19 <.0001 School 1012 -0.08 -0.05 0.07 -1.14 0.2557 School 1015 -0.24 -0.15 0.07 -3.53 0.0005 School 1017 -0.50 -0.39 0.06 -8.18 <.0001 School 1020 -0.55 -0.42 0.06 -9.02 <.0001 School 1021 -0.03 -0.02 0.06 -0.45 0.6535 School 1023 -0.26 -0.13 0.08 -3.43 0.0007 School 1024 -0.09 -0.05 0.07 -1.31 0.1904 School 1025 -0.71 -0.36 0.08 -9.3 <.0001 School 1026 0.02 0.01 0.06 0.25 0.8008 School 1027 -0.08 -0.07 0.06 -1.38 0.1675 School 1028 0.02 0.01 0.07 0.33 0.742 School 1029 -0.21 -0.14 0.07 -3.19 0.0016 School 1032 -0.18 -0.11 0.07 -2.58 0.0105 School 1034 0.00 0.00 0.07 -0.06 0.9516 School 1035 -0.12 -0.10 0.06 -1.96 0.0508 School 1038 -0.69 -0.45 0.07 -10.49 <.0001 School 1040 -0.29 -0.19 0.07 -4.43 <.0001 School 1041 -0.45 -0.33 0.06 -7.07 <.0001 School 1042 -0.41 -0.30 0.06 -6.51 <.0001 School 1044 -0.03 -0.01 0.08 -0.36 0.7225 School 1047 -0.61 -0.22 0.10 -6.37 <.0001 School 1049 -0.45 -0.26 0.07 -6.46 <.0001 Male -0.05 -0.06 0.03 -1.67 0.0968 White -0.07 -0.11 0.04 -1.78 0.0766 African American -0.02 -0.04 0.04 -0.61 0.5452 Master’s degree -0.01 -0.01 0.02 -0.34 0.7327 Teaching experience 0.00 -0.04 0.00 -1.22 0.222 New teacher 0.03 0.05 0.02 1.37 0.1721 In-degree in ELA networks -0.01 -0.05 0.01 -1.36 0.174 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 164 Table A.32 Model 2 in effects of teachers' ELA networks on students' previous free/reduced lunch Model 2 B Beta S. E. t value p value Intercept 0.94 0.00 0.06 15.53 <.0001 Grade 2 0.00 0.01 0.03 0.13 0.8972 Grade 3 0.04 0.07 0.03 1.49 0.1362 Grade 4 0.02 0.03 0.03 0.79 0.4323 School 1003 -0.09 -0.05 0.07 -1.32 0.1888 School 1006 -0.26 -0.06 0.15 -1.74 0.0825 School 1007 -0.56 -0.55 0.06 -9.88 <.0001 School 1009 -0.25 -0.14 0.07 -3.52 0.0005 School 1010 0.01 0.01 0.06 0.19 0.8488 School 1011 -0.38 -0.31 0.06 -6.36 <.0001 School 1012 -0.09 -0.05 0.07 -1.3 0.1933 School 1015 -0.25 -0.16 0.07 -3.78 0.0002 School 1017 -0.50 -0.38 0.06 -8.2 <.0001 School 1020 -0.56 -0.43 0.06 -9.38 <.0001 School 1021 -0.04 -0.03 0.06 -0.58 0.5612 School 1023 -0.27 -0.14 0.07 -3.6 0.0004 School 1024 -0.11 -0.06 0.07 -1.57 0.117 School 1025 -0.71 -0.36 0.07 -9.51 <.0001 School 1026 0.01 0.01 0.06 0.16 0.873 School 1027 -0.09 -0.08 0.06 -1.56 0.1198 School 1028 0.02 0.01 0.07 0.27 0.7906 School 1029 -0.20 -0.13 0.06 -3.17 0.0017 School 1032 -0.16 -0.09 0.07 -2.35 0.0196 School 1034 -0.02 -0.02 0.06 -0.35 0.724 School 1035 -0.13 -0.11 0.06 -2.24 0.0259 School 1038 -0.70 -0.46 0.06 -10.85 <.0001 School 1040 -0.29 -0.19 0.06 -4.58 <.0001 School 1041 -0.45 -0.34 0.06 -7.31 <.0001 School 1042 -0.42 -0.30 0.06 -6.81 <.0001 School 1044 -0.03 -0.02 0.07 -0.44 0.6609 School 1047 -0.60 -0.22 0.09 -6.32 <.0001 School 1049 -0.46 -0.27 0.07 -6.74 <.0001 Male -0.05 -0.05 0.03 -1.69 0.0924 White -0.05 -0.08 0.04 -1.28 0.2029 African American 0.00 0.00 0.04 -0.05 0.963 Master’s degree -0.01 -0.01 0.02 -0.44 0.6572 Teaching experience 0.00 -0.04 0.00 -1.05 0.2937 New teacher 0.02 0.02 0.02 0.7 0.485 Formal leader -0.07 -0.11 0.02 -3.43 0.0007 In-degree in ELA networks -0.01 -0.03 0.01 -0.95 0.3455 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 165 Table A.33 Model 3 in effects of teachers' ELA networks on students' previous free/reduced lunch Model 3 B Beta S. E. t value p value Intercept 0.93 0.00 0.06 15.58 <.0001 Grade 2 0.00 0.00 0.03 -0.05 0.9641 Grade 3 0.04 0.06 0.03 1.31 0.1912 Grade 4 0.02 0.03 0.03 0.78 0.4358 School 1003 -0.07 -0.04 0.07 -0.96 0.3361 School 1006 -0.27 -0.06 0.15 -1.81 0.0715 School 1007 -0.55 -0.54 0.06 -9.84 <.0001 School 1009 -0.24 -0.13 0.07 -3.48 0.0006 School 1010 0.03 0.02 0.06 0.43 0.6711 School 1011 -0.37 -0.31 0.06 -6.36 <.0001 School 1012 -0.09 -0.05 0.07 -1.31 0.1911 School 1015 -0.24 -0.15 0.07 -3.65 0.0003 School 1017 -0.50 -0.39 0.06 -8.35 <.0001 School 1020 -0.55 -0.43 0.06 -9.28 <.0001 School 1021 -0.03 -0.03 0.06 -0.57 0.5709 School 1023 -0.26 -0.13 0.07 -3.55 0.0005 School 1024 -0.10 -0.06 0.07 -1.42 0.1568 School 1025 -0.69 -0.36 0.07 -9.36 <.0001 School 1026 0.01 0.01 0.06 0.19 0.8469 School 1027 -0.06 -0.05 0.06 -1.06 0.2903 School 1028 0.03 0.02 0.07 0.48 0.6295 School 1029 -0.20 -0.13 0.06 -3.13 0.002 School 1032 -0.17 -0.10 0.07 -2.46 0.0144 School 1034 -0.01 -0.01 0.06 -0.14 0.8903 School 1035 -0.13 -0.11 0.06 -2.16 0.0318 School 1038 -0.69 -0.45 0.06 -10.76 <.0001 School 1040 -0.28 -0.19 0.06 -4.48 <.0001 School 1041 -0.44 -0.33 0.06 -7.21 <.0001 School 1042 -0.40 -0.29 0.06 -6.49 <.0001 School 1044 -0.03 -0.02 0.07 -0.4 0.6883 School 1047 -0.57 -0.21 0.09 -6.06 <.0001 School 1049 -0.44 -0.26 0.07 -6.5 <.0001 Male -0.04 -0.04 0.03 -1.38 0.1702 White -0.05 -0.09 0.04 -1.48 0.1387 African American -0.01 -0.02 0.04 -0.29 0.7691 Master’s degree -0.01 -0.01 0.02 -0.36 0.718 Teaching experience 0.00 -0.03 0.00 -0.9 0.3677 New teacher 0.02 0.03 0.02 0.94 0.3477 The total number of leadership -0.02 -0.13 0.01 -3.97 <.0001 roles In-degree in ELA networks -0.01 -0.03 0.01 -0.79 0.4323 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 166 Table A.34 Model 4 in effects of teachers' ELA networks on students' previous free/reduced lunch Model 4 B Beta S. E. t value p value Intercept 0.86 0.00 0.06 13.95 <.0001 Grade 2 0.01 0.02 0.03 0.52 0.6008 Grade 3 0.05 0.09 0.03 1.88 0.0608 Grade 4 0.03 0.05 0.03 1.19 0.2332 School 1003 -0.11 -0.06 0.07 -1.51 0.1323 School 1006 -0.26 -0.05 0.15 -1.73 0.0852 School 1007 -0.53 -0.53 0.06 -9.62 <.0001 School 1009 -0.23 -0.13 0.07 -3.33 0.001 School 1010 0.02 0.02 0.06 0.34 0.7356 School 1011 -0.37 -0.30 0.06 -6.3 <.0001 School 1012 -0.09 -0.05 0.07 -1.23 0.2196 School 1015 -0.24 -0.15 0.07 -3.7 0.0003 School 1017 -0.49 -0.38 0.06 -8.16 <.0001 School 1020 -0.54 -0.42 0.06 -9.2 <.0001 School 1021 -0.02 -0.01 0.06 -0.27 0.7859 School 1023 -0.26 -0.13 0.07 -3.49 0.0006 School 1024 -0.09 -0.05 0.07 -1.3 0.1939 School 1025 -0.70 -0.36 0.07 -9.43 <.0001 School 1026 0.02 0.01 0.06 0.29 0.7757 School 1027 -0.06 -0.05 0.06 -1.09 0.2749 School 1028 0.04 0.02 0.07 0.54 0.5906 School 1029 -0.21 -0.14 0.06 -3.38 0.0008 School 1032 -0.15 -0.09 0.07 -2.28 0.0236 School 1034 -0.01 0.00 0.06 -0.11 0.9158 School 1035 -0.13 -0.11 0.06 -2.22 0.0274 School 1038 -0.69 -0.45 0.06 -10.76 <.0001 School 1040 -0.27 -0.18 0.06 -4.28 <.0001 School 1041 -0.44 -0.33 0.06 -7.2 <.0001 School 1042 -0.41 -0.29 0.06 -6.72 <.0001 School 1044 -0.01 -0.01 0.07 -0.18 0.8584 School 1047 -0.57 -0.21 0.09 -6.12 <.0001 School 1049 -0.44 -0.26 0.07 -6.61 <.0001 Male -0.04 -0.05 0.03 -1.51 0.1322 White -0.06 -0.10 0.04 -1.68 0.0945 African American -0.01 -0.01 0.04 -0.23 0.817 Master’s degree -0.01 -0.02 0.02 -0.53 0.5974 Teaching experience 0.00 -0.03 0.00 -1 0.3171 New teacher 0.02 0.03 0.02 0.92 0.3573 School improvement coordinator -0.01 -0.13 0.00 -4.36 <.0001 In-degree in ELA networks -0.01 -0.03 0.01 -1.05 0.2943 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 167 Table A.35 Model 5 in effects of teachers' ELA networks on students' previous free/reduced lunch Model 5 B Beta S. E. t value p value Intercept 0.87 0.00 0.06 14.06 <.0001 Grade 2 0.01 0.01 0.03 0.26 0.7987 Grade 3 0.05 0.08 0.03 1.65 0.1003 Grade 4 0.03 0.05 0.03 1.15 0.2521 School 1003 -0.09 -0.05 0.07 -1.28 0.2021 School 1006 -0.26 -0.05 0.15 -1.71 0.088 School 1007 -0.53 -0.53 0.06 -9.58 <.0001 School 1009 -0.24 -0.13 0.07 -3.37 0.0009 School 1010 0.02 0.02 0.06 0.32 0.7497 School 1011 -0.37 -0.30 0.06 -6.26 <.0001 School 1012 -0.08 -0.05 0.07 -1.19 0.2336 School 1015 -0.23 -0.15 0.07 -3.52 0.0005 School 1017 -0.50 -0.39 0.06 -8.32 <.0001 School 1020 -0.55 -0.42 0.06 -9.17 <.0001 School 1021 -0.03 -0.02 0.06 -0.42 0.6765 School 1023 -0.26 -0.13 0.07 -3.48 0.0006 School 1024 -0.09 -0.05 0.07 -1.32 0.1877 School 1025 -0.70 -0.36 0.07 -9.36 <.0001 School 1026 0.01 0.01 0.06 0.24 0.8127 School 1027 -0.07 -0.05 0.06 -1.11 0.2683 School 1028 0.03 0.02 0.07 0.49 0.6246 School 1029 -0.19 -0.12 0.06 -2.97 0.0033 School 1032 -0.15 -0.09 0.07 -2.25 0.0252 School 1034 0.00 0.00 0.06 -0.04 0.968 School 1035 -0.13 -0.11 0.06 -2.19 0.0297 School 1038 -0.68 -0.44 0.06 -10.49 <.0001 School 1040 -0.27 -0.18 0.06 -4.32 <.0001 School 1041 -0.44 -0.33 0.06 -7.18 <.0001 School 1042 -0.41 -0.29 0.06 -6.59 <.0001 School 1044 -0.03 -0.02 0.07 -0.41 0.6834 School 1047 -0.58 -0.21 0.09 -6.23 <.0001 School 1049 -0.44 -0.26 0.07 -6.51 <.0001 Male -0.05 -0.05 0.03 -1.53 0.1274 White -0.06 -0.10 0.04 -1.55 0.1232 African American -0.01 -0.02 0.04 -0.25 0.8015 Master’s degree -0.01 -0.02 0.02 -0.5 0.6184 Teaching experience 0.00 -0.03 0.00 -0.85 0.3961 New teacher 0.02 0.03 0.02 0.88 0.3818 Teacher consultant -0.01 -0.12 0.00 -3.98 <.0001 In-degree in ELA networks -0.01 -0.04 0.01 -1.34 0.1817 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 168 Table A.36 Model 1 in effects of teachers' Math networks on students' previous free/reduced lunch Model 1 B Beta S. E. t value p value Intercept 0.93 0.00 0.06 15.26 <.0001 Grade 2 0.01 0.01 0.03 0.28 0.7798 Grade 3 0.04 0.07 0.03 1.42 0.1562 Grade 4 0.02 0.03 0.03 0.7 0.4834 School 1003 -0.11 -0.06 0.07 -1.46 0.1441 School 1006 -0.27 -0.06 0.15 -1.77 0.0775 School 1007 -0.54 -0.53 0.06 -9.41 <.0001 School 1009 -0.24 -0.13 0.07 -3.35 0.0009 School 1010 0.02 0.01 0.06 0.27 0.7883 School 1011 -0.36 -0.30 0.06 -6.05 <.0001 School 1012 -0.08 -0.05 0.07 -1.12 0.2628 School 1015 -0.24 -0.15 0.07 -3.52 0.0005 School 1017 -0.50 -0.39 0.06 -8.17 <.0001 School 1020 -0.55 -0.43 0.06 -9.09 <.0001 School 1021 -0.03 -0.02 0.06 -0.41 0.6825 School 1023 -0.26 -0.13 0.08 -3.4 0.0008 School 1024 -0.09 -0.05 0.07 -1.3 0.1959 School 1025 -0.71 -0.36 0.08 -9.39 <.0001 School 1026 0.02 0.02 0.06 0.37 0.7124 School 1027 -0.07 -0.06 0.06 -1.14 0.2559 School 1028 0.03 0.02 0.07 0.37 0.7109 School 1029 -0.20 -0.13 0.07 -3.11 0.0021 School 1032 -0.18 -0.10 0.07 -2.57 0.0108 School 1034 0.00 0.00 0.07 -0.06 0.9518 School 1035 -0.12 -0.10 0.06 -1.91 0.057 School 1038 -0.69 -0.45 0.07 -10.43 <.0001 School 1040 -0.29 -0.19 0.06 -4.45 <.0001 School 1041 -0.44 -0.33 0.06 -6.96 <.0001 School 1042 -0.41 -0.29 0.06 -6.43 <.0001 School 1044 -0.03 -0.01 0.08 -0.36 0.7225 School 1047 -0.64 -0.23 0.09 -6.78 <.0001 School 1049 -0.45 -0.27 0.07 -6.65 <.0001 Male -0.05 -0.05 0.03 -1.52 0.1295 White -0.07 -0.11 0.04 -1.78 0.076 African American -0.03 -0.04 0.04 -0.65 0.5177 Master’s degree -0.01 -0.02 0.02 -0.48 0.6292 Teaching experience 0.00 -0.04 0.00 -1.28 0.2017 New teacher 0.03 0.04 0.02 1.26 0.2083 In-degree in Math networks -0.01 -0.07 0.01 -2.09 0.0373 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 169 Table A.37 Model 2 in effects of teachers' Math networks on students' previous free/reduced lunch Model 2 B Beta S. E. t value p value Intercept 0.94 0.00 0.06 15.58 <.0001 Grade 2 0.00 0.01 0.03 0.13 0.8962 Grade 3 0.04 0.07 0.03 1.42 0.1569 Grade 4 0.02 0.03 0.03 0.72 0.4737 School 1003 -0.09 -0.05 0.07 -1.3 0.1949 School 1006 -0.28 -0.06 0.15 -1.86 0.0636 School 1007 -0.55 -0.55 0.06 -9.82 <.0001 School 1009 -0.24 -0.14 0.07 -3.49 0.0006 School 1010 0.01 0.01 0.06 0.24 0.8127 School 1011 -0.37 -0.30 0.06 -6.25 <.0001 School 1012 -0.09 -0.05 0.07 -1.28 0.2002 School 1015 -0.25 -0.16 0.07 -3.76 0.0002 School 1017 -0.50 -0.38 0.06 -8.21 <.0001 School 1020 -0.56 -0.43 0.06 -9.4 <.0001 School 1021 -0.03 -0.03 0.06 -0.56 0.576 School 1023 -0.27 -0.14 0.07 -3.57 0.0004 School 1024 -0.11 -0.06 0.07 -1.54 0.1247 School 1025 -0.71 -0.36 0.07 -9.58 <.0001 School 1026 0.01 0.01 0.06 0.23 0.8157 School 1027 -0.08 -0.07 0.06 -1.39 0.1658 School 1028 0.02 0.01 0.07 0.3 0.7666 School 1029 -0.20 -0.13 0.06 -3.12 0.002 School 1032 -0.16 -0.09 0.07 -2.36 0.019 School 1034 -0.02 -0.01 0.06 -0.34 0.7363 School 1035 -0.13 -0.11 0.06 -2.19 0.0293 School 1038 -0.70 -0.46 0.06 -10.78 <.0001 School 1040 -0.29 -0.19 0.06 -4.58 <.0001 School 1041 -0.45 -0.34 0.06 -7.22 <.0001 School 1042 -0.42 -0.30 0.06 -6.73 <.0001 School 1044 -0.03 -0.02 0.07 -0.44 0.6606 School 1047 -0.62 -0.22 0.09 -6.61 <.0001 School 1049 -0.46 -0.27 0.07 -6.87 <.0001 Male -0.05 -0.05 0.03 -1.59 0.1141 White -0.05 -0.08 0.04 -1.3 0.1936 African American 0.00 -0.01 0.04 -0.1 0.9205 Master’s degree -0.01 -0.02 0.02 -0.53 0.5988 Teaching experience 0.00 -0.04 0.00 -1.1 0.2722 New teacher 0.02 0.02 0.02 0.7 0.4869 Formal leader -0.06 -0.10 0.02 -3.16 0.0018 In-degree in Math networks -0.01 -0.04 0.01 -1.28 0.1999 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 170 Table A.38 Model 3 in effects of teachers' Math networks on students' previous free/reduced lunch Model 3 B Beta S. E. t value p value Intercept 0.93 0.00 0.06 15.64 <.0001 Grade 2 0.00 0.00 0.03 -0.04 0.9666 Grade 3 0.03 0.06 0.03 1.25 0.2113 Grade 4 0.02 0.03 0.03 0.72 0.4736 School 1003 -0.07 -0.04 0.07 -0.96 0.3366 School 1006 -0.28 -0.06 0.15 -1.91 0.057 School 1007 -0.55 -0.54 0.06 -9.8 <.0001 School 1009 -0.24 -0.13 0.07 -3.47 0.0006 School 1010 0.03 0.02 0.06 0.46 0.6468 School 1011 -0.37 -0.30 0.06 -6.26 <.0001 School 1012 -0.09 -0.05 0.07 -1.29 0.1969 School 1015 -0.24 -0.15 0.07 -3.64 0.0003 School 1017 -0.50 -0.39 0.06 -8.34 <.0001 School 1020 -0.55 -0.43 0.06 -9.3 <.0001 School 1021 -0.03 -0.03 0.06 -0.54 0.5879 School 1023 -0.26 -0.13 0.07 -3.53 0.0005 School 1024 -0.10 -0.06 0.07 -1.41 0.1612 School 1025 -0.70 -0.36 0.07 -9.42 <.0001 School 1026 0.02 0.01 0.06 0.26 0.7939 School 1027 -0.06 -0.05 0.06 -0.93 0.3525 School 1028 0.03 0.02 0.07 0.5 0.6168 School 1029 -0.20 -0.13 0.06 -3.08 0.0023 School 1032 -0.17 -0.10 0.07 -2.46 0.0144 School 1034 -0.01 -0.01 0.06 -0.13 0.8928 School 1035 -0.13 -0.11 0.06 -2.12 0.0349 School 1038 -0.69 -0.45 0.06 -10.71 <.0001 School 1040 -0.28 -0.19 0.06 -4.49 <.0001 School 1041 -0.44 -0.33 0.06 -7.14 <.0001 School 1042 -0.40 -0.28 0.06 -6.44 <.0001 School 1044 -0.03 -0.02 0.07 -0.4 0.6887 School 1047 -0.59 -0.21 0.09 -6.31 <.0001 School 1049 -0.44 -0.26 0.07 -6.62 <.0001 Male -0.04 -0.04 0.03 -1.3 0.1932 White -0.06 -0.09 0.04 -1.5 0.1353 African American -0.01 -0.02 0.04 -0.33 0.7418 Master’s degree -0.01 -0.01 0.02 -0.44 0.6587 Teaching experience 0.00 -0.03 0.00 -0.95 0.3439 New teacher 0.02 0.03 0.02 0.9 0.3664 The total number of leadership -0.02 -0.12 0.01 -3.73 0.0002 roles In-degree in Math networks -0.01 -0.04 0.01 -1.18 0.24 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 171 Table A.39 Model 4 in effects of teachers' Math networks on students' previous free/reduced lunch Model 4 B Beta S. E. t value p value Intercept 0.86 0.00 0.06 14.09 <.0001 Grade 2 0.01 0.02 0.03 0.5 0.618 Grade 3 0.05 0.08 0.03 1.79 0.0753 Grade 4 0.03 0.05 0.03 1.1 0.2735 School 1003 -0.10 -0.06 0.07 -1.48 0.1398 School 1006 -0.28 -0.06 0.15 -1.86 0.0634 School 1007 -0.53 -0.52 0.06 -9.58 <.0001 School 1009 -0.23 -0.13 0.07 -3.33 0.001 School 1010 0.02 0.02 0.06 0.4 0.6891 School 1011 -0.36 -0.30 0.06 -6.18 <.0001 School 1012 -0.08 -0.05 0.07 -1.21 0.2255 School 1015 -0.24 -0.15 0.07 -3.69 0.0003 School 1017 -0.49 -0.38 0.06 -8.16 <.0001 School 1020 -0.55 -0.42 0.06 -9.26 <.0001 School 1021 -0.01 -0.01 0.06 -0.23 0.8158 School 1023 -0.25 -0.13 0.07 -3.47 0.0006 School 1024 -0.09 -0.05 0.07 -1.3 0.1963 School 1025 -0.70 -0.36 0.07 -9.5 <.0001 School 1026 0.02 0.02 0.06 0.39 0.6999 School 1027 -0.05 -0.04 0.06 -0.89 0.3729 School 1028 0.04 0.02 0.07 0.57 0.5697 School 1029 -0.21 -0.14 0.06 -3.3 0.0011 School 1032 -0.15 -0.09 0.07 -2.27 0.0239 School 1034 -0.01 0.00 0.06 -0.1 0.9169 School 1035 -0.13 -0.11 0.06 -2.17 0.0309 School 1038 -0.68 -0.45 0.06 -10.7 <.0001 School 1040 -0.27 -0.18 0.06 -4.3 <.0001 School 1041 -0.44 -0.32 0.06 -7.11 <.0001 School 1042 -0.41 -0.29 0.06 -6.64 <.0001 School 1044 -0.01 -0.01 0.07 -0.18 0.8594 School 1047 -0.59 -0.22 0.09 -6.45 <.0001 School 1049 -0.45 -0.26 0.07 -6.77 <.0001 Male -0.04 -0.04 0.03 -1.4 0.1618 White -0.06 -0.10 0.04 -1.68 0.0932 African American -0.01 -0.02 0.04 -0.28 0.7831 Master’s degree -0.01 -0.02 0.02 -0.64 0.5199 Teaching experience 0.00 -0.04 0.00 -1.05 0.2949 New teacher 0.02 0.03 0.02 0.82 0.4128 School improvement coordinator -0.01 -0.13 0.00 -4.29 <.0001 In-degree in Math networks -0.01 -0.05 0.01 -1.74 0.0827 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 172 Table A.40 Model 5 in effects of teachers' Math networks on students' previous free/reduced lunch Model 5 B Beta S. E. t value p value Intercept 0.87 0.00 0.06 14.17 <.0001 Grade 2 0.01 0.01 0.03 0.24 0.8088 Grade 3 0.04 0.07 0.03 1.54 0.1252 Grade 4 0.03 0.04 0.03 1.04 0.2993 School 1003 -0.09 -0.05 0.07 -1.24 0.2143 School 1006 -0.28 -0.06 0.15 -1.88 0.0611 School 1007 -0.53 -0.53 0.06 -9.55 <.0001 School 1009 -0.23 -0.13 0.07 -3.35 0.0009 School 1010 0.02 0.02 0.06 0.38 0.7024 School 1011 -0.36 -0.29 0.06 -6.13 <.0001 School 1012 -0.08 -0.05 0.07 -1.18 0.2398 School 1015 -0.23 -0.14 0.07 -3.5 0.0005 School 1017 -0.50 -0.38 0.06 -8.32 <.0001 School 1020 -0.55 -0.42 0.06 -9.22 <.0001 School 1021 -0.02 -0.02 0.06 -0.39 0.6939 School 1023 -0.25 -0.13 0.07 -3.44 0.0007 School 1024 -0.09 -0.05 0.07 -1.3 0.1954 School 1025 -0.70 -0.36 0.07 -9.45 <.0001 School 1026 0.02 0.02 0.06 0.34 0.737 School 1027 -0.05 -0.04 0.06 -0.9 0.3683 School 1028 0.04 0.02 0.07 0.52 0.6002 School 1029 -0.18 -0.12 0.06 -2.91 0.0039 School 1032 -0.15 -0.09 0.07 -2.25 0.025 School 1034 0.00 0.00 0.06 -0.04 0.9672 School 1035 -0.13 -0.11 0.06 -2.14 0.0335 School 1038 -0.67 -0.44 0.06 -10.44 <.0001 School 1040 -0.27 -0.18 0.06 -4.34 <.0001 School 1041 -0.44 -0.33 0.06 -7.08 <.0001 School 1042 -0.40 -0.29 0.06 -6.52 <.0001 School 1044 -0.03 -0.02 0.07 -0.41 0.6803 School 1047 -0.61 -0.22 0.09 -6.63 <.0001 School 1049 -0.45 -0.26 0.07 -6.7 <.0001 Male -0.04 -0.04 0.03 -1.39 0.1664 White -0.06 -0.10 0.04 -1.55 0.1217 African American -0.01 -0.02 0.04 -0.29 0.7698 Master’s degree -0.01 -0.02 0.02 -0.62 0.5353 Teaching experience 0.00 -0.03 0.00 -0.91 0.3625 New teacher 0.02 0.03 0.02 0.82 0.4144 Teacher consultant -0.01 -0.12 0.00 -3.88 0.0001 In-degree in Math networks -0.01 -0.06 0.01 -1.87 0.062 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 173 Table A.41 Model 1 in effects of teachers' combined networks on students' previous free/reduced lunch Model 1 B Beta S. E. t value p value Intercept 0.94 0.00 0.06 15.26 <.0001 Grade 2 0.01 0.01 0.03 0.22 0.8278 Grade 3 0.04 0.07 0.03 1.45 0.148 Grade 4 0.02 0.03 0.03 0.78 0.4332 School 1003 -0.10 -0.06 0.07 -1.46 0.1454 School 1006 -0.25 -0.05 0.15 -1.61 0.1095 School 1007 -0.54 -0.53 0.06 -9.45 <.0001 School 1009 -0.24 -0.13 0.07 -3.39 0.0008 School 1010 0.02 0.01 0.06 0.26 0.7912 School 1011 -0.37 -0.30 0.06 -6.12 <.0001 School 1012 -0.08 -0.05 0.07 -1.12 0.2618 School 1015 -0.24 -0.15 0.07 -3.54 0.0005 School 1017 -0.50 -0.39 0.06 -8.18 <.0001 School 1020 -0.55 -0.43 0.06 -9.07 <.0001 School 1021 -0.02 -0.02 0.06 -0.38 0.7032 School 1023 -0.26 -0.13 0.08 -3.44 0.0007 School 1024 -0.09 -0.06 0.07 -1.34 0.1826 School 1025 -0.70 -0.36 0.08 -9.19 <.0001 School 1026 0.02 0.02 0.06 0.38 0.7027 School 1027 -0.07 -0.06 0.06 -1.13 0.2584 School 1028 0.03 0.02 0.07 0.38 0.7021 School 1029 -0.20 -0.13 0.07 -3.09 0.0022 School 1032 -0.18 -0.11 0.07 -2.59 0.01 School 1034 0.00 0.00 0.07 -0.02 0.9845 School 1035 -0.11 -0.10 0.06 -1.88 0.061 School 1038 -0.68 -0.45 0.07 -10.35 <.0001 School 1040 -0.29 -0.19 0.06 -4.47 <.0001 School 1041 -0.44 -0.33 0.06 -7.02 <.0001 School 1042 -0.41 -0.29 0.06 -6.42 <.0001 School 1044 -0.02 -0.01 0.08 -0.33 0.7401 School 1047 -0.61 -0.22 0.09 -6.45 <.0001 School 1049 -0.44 -0.26 0.07 -6.49 <.0001 Male -0.05 -0.05 0.03 -1.61 0.1085 White -0.07 -0.12 0.04 -1.82 0.0698 African American -0.03 -0.04 0.04 -0.69 0.489 Master's degree -0.01 -0.01 0.02 -0.44 0.6629 Teaching experience 0.00 -0.04 0.00 -1.2 0.2306 New teacher 0.03 0.04 0.02 1.19 0.2369 In-degree in Combined networks -0.01 -0.07 0.01 -2.06 0.0403 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 174 Table A.42 Model 2 in effects of teachers' combined networks on students' previous free/reduced lunch Model 2 B Beta S. E. t value p value Intercept 0.94 0.00 0.06 15.57 <.0001 Grade 2 0.00 0.00 0.03 0.09 0.9272 Grade 3 0.04 0.07 0.03 1.44 0.1516 Grade 4 0.02 0.03 0.03 0.77 0.4416 School 1003 -0.09 -0.05 0.07 -1.3 0.1961 School 1006 -0.26 -0.06 0.15 -1.75 0.0804 School 1007 -0.55 -0.55 0.06 -9.85 <.0001 School 1009 -0.25 -0.14 0.07 -3.52 0.0005 School 1010 0.01 0.01 0.06 0.23 0.8149 School 1011 -0.37 -0.31 0.06 -6.3 <.0001 School 1012 -0.09 -0.05 0.07 -1.29 0.1995 School 1015 -0.25 -0.16 0.07 -3.77 0.0002 School 1017 -0.50 -0.38 0.06 -8.22 <.0001 School 1020 -0.56 -0.43 0.06 -9.39 <.0001 School 1021 -0.03 -0.03 0.06 -0.54 0.5878 School 1023 -0.27 -0.14 0.07 -3.59 0.0004 School 1024 -0.11 -0.06 0.07 -1.57 0.1187 School 1025 -0.71 -0.36 0.07 -9.42 <.0001 School 1026 0.01 0.01 0.06 0.24 0.8101 School 1027 -0.08 -0.07 0.06 -1.39 0.1666 School 1028 0.02 0.01 0.07 0.3 0.7612 School 1029 -0.20 -0.13 0.06 -3.11 0.0021 School 1032 -0.16 -0.10 0.07 -2.38 0.0182 School 1034 -0.02 -0.01 0.06 -0.31 0.7553 School 1035 -0.13 -0.11 0.06 -2.17 0.0307 School 1038 -0.70 -0.46 0.07 -10.71 <.0001 School 1040 -0.29 -0.19 0.06 -4.6 <.0001 School 1041 -0.45 -0.34 0.06 -7.27 <.0001 School 1042 -0.42 -0.30 0.06 -6.72 <.0001 School 1044 -0.03 -0.02 0.07 -0.43 0.671 School 1047 -0.60 -0.22 0.09 -6.4 <.0001 School 1049 -0.46 -0.27 0.07 -6.76 <.0001 Male -0.05 -0.05 0.03 -1.64 0.1019 White -0.05 -0.08 0.04 -1.32 0.1863 African American -0.01 -0.01 0.04 -0.13 0.8995 Master's degree -0.01 -0.02 0.02 -0.5 0.6196 Teaching experience 0.00 -0.04 0.00 -1.05 0.2942 New teacher 0.01 0.02 0.02 0.65 0.5157 Formal leader -0.06 -0.10 0.02 -3.17 0.0017 In-degree in Combined networks -0.01 -0.04 0.01 -1.26 0.2098 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 175 Table A.43 Model 3 in effects of teachers' combined networks on students' previous free/reduced lunch Model 3 B Beta S. E. t value p value Intercept 0.94 0.00 0.06 15.62 <.0001 Grade 2 0.00 0.00 0.03 -0.08 0.9373 Grade 3 0.03 0.06 0.03 1.27 0.2055 Grade 4 0.02 0.03 0.03 0.77 0.4443 School 1003 -0.07 -0.04 0.07 -0.96 0.3385 School 1006 -0.27 -0.06 0.15 -1.81 0.0714 School 1007 -0.55 -0.54 0.06 -9.82 <.0001 School 1009 -0.24 -0.13 0.07 -3.49 0.0006 School 1010 0.03 0.02 0.06 0.46 0.6477 School 1011 -0.37 -0.30 0.06 -6.31 <.0001 School 1012 -0.09 -0.05 0.07 -1.3 0.1963 School 1015 -0.24 -0.15 0.07 -3.66 0.0003 School 1017 -0.50 -0.39 0.06 -8.35 <.0001 School 1020 -0.55 -0.43 0.06 -9.29 <.0001 School 1021 -0.03 -0.03 0.06 -0.53 0.5997 School 1023 -0.26 -0.13 0.07 -3.55 0.0005 School 1024 -0.10 -0.06 0.07 -1.43 0.1544 School 1025 -0.69 -0.35 0.07 -9.28 <.0001 School 1026 0.02 0.01 0.06 0.27 0.7878 School 1027 -0.06 -0.05 0.06 -0.93 0.3551 School 1028 0.03 0.02 0.07 0.51 0.6117 School 1029 -0.20 -0.13 0.06 -3.07 0.0024 School 1032 -0.17 -0.10 0.07 -2.48 0.0138 School 1034 -0.01 0.00 0.06 -0.11 0.9116 School 1035 -0.12 -0.11 0.06 -2.1 0.0365 School 1038 -0.69 -0.45 0.06 -10.65 <.0001 School 1040 -0.29 -0.19 0.06 -4.5 <.0001 School 1041 -0.44 -0.33 0.06 -7.18 <.0001 School 1042 -0.40 -0.28 0.06 -6.43 <.0001 School 1044 -0.03 -0.01 0.07 -0.39 0.6989 School 1047 -0.57 -0.21 0.09 -6.14 <.0001 School 1049 -0.44 -0.26 0.07 -6.53 <.0001 Male -0.04 -0.04 0.03 -1.35 0.1767 White -0.06 -0.09 0.04 -1.52 0.1298 African American -0.01 -0.02 0.04 -0.36 0.7226 Master's degree -0.01 -0.01 0.02 -0.42 0.6782 Teaching experience 0.00 -0.03 0.00 -0.9 0.3676 New teacher 0.02 0.03 0.02 0.86 0.3918 The total number of leadership -0.02 -0.12 0.01 -3.74 0.0002 roles In-degree in Combined networks -0.01 -0.04 0.01 -1.17 0.2446 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 176 Table A.44 Model 4 in effects of teachers' combined networks on students' previous free/reduced lunch Model 4 B Beta S. E. t value p value Intercept 0.87 0.00 0.06 14.08 <.0001 Grade 2 0.01 0.02 0.03 0.45 0.656 Grade 3 0.05 0.09 0.03 1.81 0.0713 Grade 4 0.03 0.05 0.03 1.17 0.2437 School 1003 -0.10 -0.06 0.07 -1.48 0.1408 School 1006 -0.26 -0.05 0.15 -1.72 0.0862 School 1007 -0.53 -0.53 0.06 -9.61 <.0001 School 1009 -0.23 -0.13 0.07 -3.36 0.0009 School 1010 0.02 0.02 0.06 0.4 0.6904 School 1011 -0.36 -0.30 0.06 -6.24 <.0001 School 1012 -0.08 -0.05 0.07 -1.22 0.2248 School 1015 -0.24 -0.15 0.07 -3.72 0.0002 School 1017 -0.49 -0.38 0.06 -8.17 <.0001 School 1020 -0.54 -0.42 0.06 -9.24 <.0001 School 1021 -0.01 -0.01 0.06 -0.21 0.8362 School 1023 -0.26 -0.13 0.07 -3.5 0.0005 School 1024 -0.09 -0.05 0.07 -1.33 0.1847 School 1025 -0.69 -0.35 0.07 -9.32 <.0001 School 1026 0.02 0.02 0.06 0.4 0.6901 School 1027 -0.05 -0.04 0.06 -0.88 0.3777 School 1028 0.04 0.02 0.07 0.58 0.5624 School 1029 -0.21 -0.14 0.06 -3.29 0.0011 School 1032 -0.15 -0.09 0.07 -2.29 0.0227 School 1034 0.00 0.00 0.06 -0.07 0.9445 School 1035 -0.13 -0.11 0.06 -2.14 0.033 School 1038 -0.68 -0.45 0.06 -10.62 <.0001 School 1040 -0.27 -0.18 0.06 -4.31 <.0001 School 1041 -0.44 -0.33 0.06 -7.16 <.0001 School 1042 -0.41 -0.29 0.06 -6.63 <.0001 School 1044 -0.01 -0.01 0.07 -0.16 0.876 School 1047 -0.57 -0.21 0.09 -6.18 <.0001 School 1049 -0.44 -0.26 0.07 -6.63 <.0001 Male -0.04 -0.05 0.03 -1.48 0.1405 White -0.06 -0.11 0.04 -1.72 0.087 African American -0.01 -0.02 0.04 -0.31 0.7538 Master's degree -0.01 -0.02 0.02 -0.61 0.5443 Teaching experience 0.00 -0.03 0.00 -0.98 0.3265 New teacher 0.02 0.03 0.02 0.75 0.4526 School improvement coordinator -0.01 -0.13 0.00 -4.3 <.0001 In-degree in Combined networks -0.01 -0.06 0.01 -1.74 0.083 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 177 Table A.45 Model 5 in effects of teachers' combined networks on students' previous free/reduced lunch Model 5 B Beta S. E. t value p value Intercept 0.88 0.00 0.06 14.17 <.0001 Grade 2 0.00 0.01 0.03 0.18 0.8547 Grade 3 0.04 0.07 0.03 1.56 0.1189 Grade 4 0.03 0.05 0.03 1.12 0.2652 School 1003 -0.09 -0.05 0.07 -1.24 0.2162 School 1006 -0.26 -0.05 0.15 -1.73 0.0853 School 1007 -0.53 -0.53 0.06 -9.58 <.0001 School 1009 -0.24 -0.13 0.07 -3.39 0.0008 School 1010 0.02 0.02 0.06 0.38 0.703 School 1011 -0.36 -0.30 0.06 -6.19 <.0001 School 1012 -0.08 -0.05 0.07 -1.18 0.2391 School 1015 -0.23 -0.15 0.07 -3.52 0.0005 School 1017 -0.50 -0.38 0.06 -8.33 <.0001 School 1020 -0.55 -0.42 0.06 -9.21 <.0001 School 1021 -0.02 -0.02 0.06 -0.36 0.7157 School 1023 -0.26 -0.13 0.07 -3.48 0.0006 School 1024 -0.09 -0.05 0.07 -1.34 0.1827 School 1025 -0.69 -0.35 0.07 -9.26 <.0001 School 1026 0.02 0.02 0.06 0.35 0.7252 School 1027 -0.05 -0.04 0.06 -0.89 0.3748 School 1028 0.04 0.02 0.07 0.54 0.5918 School 1029 -0.18 -0.12 0.06 -2.89 0.0042 School 1032 -0.15 -0.09 0.07 -2.28 0.0237 School 1034 0.00 0.00 0.06 0 0.9974 School 1035 -0.12 -0.11 0.06 -2.11 0.0359 School 1038 -0.67 -0.44 0.06 -10.36 <.0001 School 1040 -0.28 -0.18 0.06 -4.36 <.0001 School 1041 -0.44 -0.33 0.06 -7.14 <.0001 School 1042 -0.40 -0.29 0.06 -6.5 <.0001 School 1044 -0.03 -0.01 0.07 -0.39 0.6969 School 1047 -0.59 -0.21 0.09 -6.33 <.0001 School 1049 -0.44 -0.26 0.07 -6.55 <.0001 Male -0.04 -0.05 0.03 -1.47 0.1429 White -0.06 -0.10 0.04 -1.59 0.1136 African American -0.01 -0.02 0.04 -0.33 0.7383 Master's degree -0.01 -0.02 0.02 -0.58 0.5612 Teaching experience 0.00 -0.03 0.00 -0.84 0.4021 New teacher 0.02 0.03 0.02 0.74 0.46 Teacher consultant -0.01 -0.12 0.00 -3.9 0.0001 In-degree in Combined networks -0.01 -0.06 0.01 -1.89 0.0601 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 178 Table A.46 Model 1 in effects of teachers' attributes on students' previous ELA achievement Model 1 B Beta S. E. t value p value Intercept 813.63 0.00 6.45 126.24 <.0001 Grade 2 -0.82 -0.02 2.83 -0.29 0.7716 Grade 3 -0.15 0.00 2.85 -0.05 0.9574 Grade 4 4.27 0.10 3.03 1.41 0.1598 School 1003 -2.08 -0.02 7.18 -0.29 0.7721 School 1006 -0.43 0.00 15.22 -0.03 0.9773 School 1007 24.83 0.40 5.77 4.3 <.0001 School 1009 8.51 0.08 7.19 1.18 0.2375 School 1010 -4.95 -0.06 6.21 -0.8 0.426 School 1011 2.58 0.03 6.05 0.43 0.6696 School 1012 7.26 0.07 7.20 1.01 0.3144 School 1015 5.39 0.06 6.79 0.79 0.4285 School 1017 15.11 0.19 6.23 2.42 0.016 School 1020 5.72 0.07 6.14 0.93 0.352 School 1021 3.11 0.04 6.21 0.5 0.6163 School 1023 18.01 0.15 7.67 2.35 0.0197 School 1024 1.49 0.01 7.06 0.21 0.8335 School 1025 10.51 0.09 7.56 1.39 0.166 School 1026 -5.84 -0.07 6.18 -0.94 0.346 School 1027 3.00 0.04 6.04 0.5 0.62 School 1028 -7.34 -0.07 6.93 -1.06 0.2905 School 1029 0.09 0.00 6.49 0.01 0.9889 School 1032 11.12 0.11 6.97 1.6 0.1116 School 1034 -1.79 -0.02 6.62 -0.27 0.7867 School 1035 -1.13 -0.02 6.06 -0.19 0.8525 School 1038 16.62 0.18 6.62 2.51 0.0126 School 1040 15.84 0.17 6.54 2.42 0.0161 School 1041 6.47 0.08 6.44 1 0.3164 School 1042 17.55 0.20 6.46 2.72 0.007 School 1044 -2.34 -0.02 7.51 -0.31 0.7556 School 1047 10.35 0.06 9.42 1.1 0.2727 School 1049 12.97 0.13 6.84 1.9 0.059 Male -0.30 -0.01 3.02 -0.1 0.9215 White 0.60 0.02 3.75 0.16 0.8734 African American -2.71 -0.07 3.99 -0.68 0.497 Master's degree 1.06 0.03 2.03 0.52 0.6012 Teaching experience 0.08 0.04 0.10 0.77 0.4432 ELA professional development -0.55 -0.03 1.09 -0.5 0.6152 New teacher -6.67 -0.17 2.18 -3.05 0.0025 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 179 Table A.47 Model 2 in effects of teachers' attributes on students' previous ELA achievement Model 2 B Beta S. E. t value p value Intercept 813.30 0.00 6.23 130.62 <.0001 Grade 2 0.05 0.00 2.74 0.02 0.9866 Grade 3 0.24 0.01 2.75 0.09 0.9318 Grade 4 4.56 0.11 2.93 1.56 0.1207 School 1003 -3.98 -0.04 6.95 -0.57 0.5673 School 1006 0.45 0.00 14.71 0.03 0.9756 School 1007 25.91 0.41 5.58 4.64 <.0001 School 1009 9.05 0.08 6.94 1.3 0.1937 School 1010 -5.09 -0.07 6.00 -0.85 0.3971 School 1011 2.66 0.04 5.84 0.46 0.6494 School 1012 8.30 0.08 6.96 1.19 0.234 School 1015 7.04 0.07 6.57 1.07 0.2856 School 1017 13.35 0.17 6.04 2.21 0.0279 School 1020 6.94 0.09 5.93 1.17 0.2435 School 1021 3.40 0.04 6.00 0.57 0.5707 School 1023 18.52 0.16 7.41 2.5 0.0131 School 1024 3.59 0.03 6.83 0.52 0.6003 School 1025 10.09 0.08 7.31 1.38 0.1686 School 1026 -5.73 -0.07 5.97 -0.96 0.3382 School 1027 3.84 0.05 5.84 0.66 0.5118 School 1028 -7.07 -0.07 6.70 -1.06 0.2923 School 1029 -0.83 -0.01 6.28 -0.13 0.895 School 1032 8.13 0.08 6.76 1.2 0.2306 School 1034 0.06 0.00 6.41 0.01 0.9923 School 1035 0.63 0.01 5.87 0.11 0.9148 School 1038 17.44 0.19 6.40 2.73 0.0068 School 1040 15.98 0.17 6.32 2.53 0.012 School 1041 6.75 0.08 6.23 1.08 0.2796 School 1042 17.98 0.20 6.24 2.88 0.0043 School 1044 -2.08 -0.02 7.26 -0.29 0.7748 School 1047 7.39 0.04 9.12 0.81 0.4187 School 1049 13.72 0.13 6.61 2.08 0.0389 Male -0.19 0.00 2.92 -0.06 0.949 White -1.92 -0.05 3.67 -0.52 0.6009 African American -5.54 -0.14 3.91 -1.42 0.1576 Master's degree 1.27 0.03 1.96 0.65 0.5176 Teaching experience 0.05 0.03 0.10 0.5 0.6192 ELA professional development -1.11 -0.05 1.06 -1.04 0.2997 New teacher -4.61 -0.12 2.16 -2.13 0.0339 Formal leader 8.65 0.23 1.95 4.44 <.0001 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 180 Table A.48 Model 3 in effects of teachers' attributes on students' previous ELA achievement Model 3 B Beta S. E. t value p value Intercept 813.65 0.00 6.27 129.72 <.0001 Grade 2 0.35 0.01 2.77 0.13 0.9003 Grade 3 0.68 0.02 2.78 0.25 0.8066 Grade 4 4.58 0.11 2.95 1.55 0.1214 School 1003 -5.96 -0.05 7.05 -0.85 0.3987 School 1006 0.76 0.00 14.82 0.05 0.9593 School 1007 24.84 0.40 5.62 4.42 <.0001 School 1009 8.56 0.08 6.99 1.22 0.2221 School 1010 -6.50 -0.09 6.06 -1.07 0.2841 School 1011 2.39 0.03 5.88 0.41 0.6853 School 1012 8.07 0.08 7.01 1.15 0.2507 School 1015 5.65 0.06 6.61 0.85 0.3937 School 1017 14.17 0.18 6.07 2.33 0.0204 School 1020 5.56 0.07 5.97 0.93 0.353 School 1021 3.16 0.04 6.04 0.52 0.6008 School 1023 17.87 0.15 7.47 2.39 0.0174 School 1024 2.06 0.02 6.87 0.3 0.7646 School 1025 8.50 0.07 7.38 1.15 0.2502 School 1026 -5.98 -0.08 6.02 -0.99 0.3214 School 1027 0.66 0.01 5.91 0.11 0.9116 School 1028 -8.54 -0.08 6.75 -1.26 0.2071 School 1029 -1.05 -0.01 6.32 -0.17 0.8688 School 1032 9.38 0.09 6.79 1.38 0.1687 School 1034 -1.73 -0.02 6.44 -0.27 0.7889 School 1035 -0.35 0.00 5.90 -0.06 0.9524 School 1038 16.05 0.17 6.44 2.49 0.0134 School 1040 15.10 0.16 6.37 2.37 0.0184 School 1041 5.89 0.07 6.27 0.94 0.3488 School 1042 15.84 0.18 6.30 2.52 0.0125 School 1044 -2.47 -0.02 7.31 -0.34 0.7358 School 1047 5.33 0.03 9.25 0.58 0.565 School 1049 11.52 0.11 6.67 1.73 0.0852 Male -1.10 -0.02 2.95 -0.37 0.7089 White -0.64 -0.02 3.67 -0.17 0.8626 African American -3.93 -0.10 3.89 -1.01 0.3137 Master's degree 1.07 0.03 1.97 0.54 0.5899 Teaching experience 0.04 0.02 0.10 0.39 0.6951 ELA professional development -0.99 -0.05 1.07 -0.93 0.3545 New teacher -5.52 -0.14 2.15 -2.57 0.0107 The total number of leadership 2.24 0.21 0.57 3.95 <.0001 roles Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 181 Table A.49 Model 4 in effects of teachers' attributes on students' previous ELA achievement Model 4 B Beta S. E. t value p value Intercept 814.22 0.00 6.44 126.45 <.0001 Grade 2 -0.66 -0.02 2.83 -0.23 0.8157 Grade 3 0.02 0.00 2.84 0.01 0.9939 Grade 4 4.10 0.10 3.02 1.36 0.1762 School 1003 -3.39 -0.03 7.21 -0.47 0.6387 School 1006 0.70 0.00 15.20 0.05 0.9634 School 1007 24.62 0.39 5.76 4.28 <.0001 School 1009 8.56 0.08 7.17 1.19 0.2334 School 1010 -5.61 -0.07 6.21 -0.9 0.367 School 1011 2.76 0.04 6.03 0.46 0.648 School 1012 7.71 0.07 7.18 1.07 0.2839 School 1015 5.68 0.06 6.78 0.84 0.4031 School 1017 15.34 0.19 6.22 2.47 0.0143 School 1020 5.66 0.07 6.12 0.92 0.356 School 1021 3.26 0.04 6.19 0.53 0.5991 School 1023 18.02 0.15 7.65 2.36 0.0193 School 1024 1.71 0.02 7.04 0.24 0.8086 School 1025 10.43 0.09 7.54 1.38 0.1681 School 1026 -5.76 -0.07 6.17 -0.93 0.3514 School 1027 2.60 0.04 6.03 0.43 0.6669 School 1028 -8.70 -0.08 6.97 -1.25 0.2131 School 1029 0.09 0.00 6.47 0.01 0.9884 School 1032 10.89 0.11 6.95 1.57 0.1183 School 1034 -1.71 -0.02 6.60 -0.26 0.7959 School 1035 -1.34 -0.02 6.05 -0.22 0.8244 School 1038 16.55 0.18 6.60 2.51 0.0128 School 1040 15.90 0.17 6.52 2.44 0.0154 School 1041 5.89 0.07 6.44 0.92 0.3608 School 1042 17.34 0.20 6.44 2.69 0.0076 School 1044 -2.19 -0.02 7.49 -0.29 0.77 School 1047 10.75 0.06 9.40 1.14 0.2537 School 1049 12.70 0.12 6.82 1.86 0.0637 Male -0.95 -0.02 3.04 -0.31 0.7555 White 0.21 0.01 3.75 0.06 0.9551 African American -3.27 -0.08 3.99 -0.82 0.4141 Master's degree 0.75 0.02 2.03 0.37 0.7121 Teaching experience 0.07 0.04 0.10 0.71 0.4809 ELA professional development -0.59 -0.03 1.09 -0.54 0.5883 New teacher -6.80 -0.17 2.18 -3.12 0.002 ELA coordinator 1.88 0.08 1.21 1.55 0.1214 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 182 Table A.50 Model 5 in effects of teachers' attributes on students' previous ELA achievement Model 5 B Beta S. E. t value p value Intercept 818.82 0.00 6.55 124.95 <.0001 Grade 2 -0.64 -0.02 2.79 -0.23 0.8172 Grade 3 -0.27 -0.01 2.80 -0.1 0.9237 Grade 4 3.71 0.09 2.99 1.24 0.215 School 1003 -3.53 -0.03 7.08 -0.5 0.6184 School 1006 0.16 0.00 14.98 0.01 0.9914 School 1007 24.00 0.38 5.68 4.22 <.0001 School 1009 7.94 0.07 7.07 1.12 0.2624 School 1010 -5.54 -0.07 6.11 -0.91 0.3659 School 1011 2.00 0.03 5.95 0.34 0.7364 School 1012 7.40 0.07 7.08 1.05 0.2969 School 1015 4.81 0.05 6.69 0.72 0.4724 School 1017 14.56 0.18 6.13 2.37 0.0183 School 1020 5.19 0.07 6.04 0.86 0.3912 School 1021 2.75 0.03 6.11 0.45 0.6524 School 1023 17.57 0.15 7.55 2.33 0.0208 School 1024 1.33 0.01 6.94 0.19 0.8486 School 1025 9.35 0.08 7.45 1.25 0.2108 School 1026 -5.89 -0.07 6.08 -0.97 0.3339 School 1027 1.50 0.02 5.96 0.25 0.8021 School 1028 -8.24 -0.08 6.83 -1.21 0.2282 School 1029 -1.43 -0.02 6.40 -0.22 0.8235 School 1032 8.90 0.09 6.89 1.29 0.1975 School 1034 -2.03 -0.02 6.51 -0.31 0.7553 School 1035 -0.25 0.00 5.97 -0.04 0.9663 School 1038 15.14 0.16 6.53 2.32 0.0212 School 1040 14.57 0.16 6.45 2.26 0.0246 School 1041 5.89 0.07 6.34 0.93 0.3536 School 1042 17.49 0.20 6.35 2.76 0.0063 School 1044 -2.14 -0.02 7.39 -0.29 0.7722 School 1047 8.07 0.05 9.29 0.87 0.386 School 1049 12.19 0.12 6.73 1.81 0.0712 Male -0.77 -0.01 2.98 -0.26 0.7963 White -0.23 -0.01 3.70 -0.06 0.9499 African American -3.91 -0.10 3.94 -0.99 0.3229 Master's degree 1.30 0.03 2.00 0.65 0.5155 Teaching experience 0.04 0.02 0.10 0.41 0.6815 ELA professional development -0.76 -0.04 1.08 -0.7 0.484 New teacher -5.82 -0.15 2.17 -2.69 0.0076 Teacher consultant 0.92 0.16 0.30 3.13 0.002 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 183 Table A.51 Model 1 in effects of teachers' attributes on students' previous Math achievement Model 1 B Beta S. E. t value p value Intercept 307.90 0.00 6.86 44.9 <.0001 Grade 2 18.45 0.45 3.13 5.9 <.0001 Grade 3 9.28 0.22 3.13 2.97 0.0033 Grade 4 13.86 0.28 3.35 4.14 <.0001 School 1003 0.65 0.00 7.93 0.08 0.935 School 1006 19.26 0.06 16.82 1.15 0.253 School 1007 28.87 0.40 6.25 4.62 <.0001 School 1009 16.19 0.12 7.86 2.06 0.0404 School 1010 -2.57 -0.03 6.83 -0.38 0.7072 School 1011 7.85 0.09 6.61 1.19 0.2364 School 1012 8.95 0.07 7.93 1.13 0.2599 School 1015 14.60 0.13 7.49 1.95 0.0523 School 1017 24.09 0.26 6.77 3.56 0.0004 School 1020 6.17 0.07 6.71 0.92 0.3586 School 1021 6.80 0.07 6.86 0.99 0.3221 School 1023 13.67 0.10 8.36 1.63 0.1034 School 1024 1.74 0.01 7.76 0.22 0.8225 School 1025 19.87 0.13 9.13 2.18 0.0304 School 1026 -1.14 -0.01 6.75 -0.17 0.8657 School 1027 4.42 0.05 6.67 0.66 0.5075 School 1028 -3.93 -0.03 7.61 -0.52 0.6059 School 1029 1.48 0.01 7.17 0.21 0.8366 School 1032 16.76 0.14 7.61 2.2 0.0286 School 1034 -2.31 -0.02 7.24 -0.32 0.7503 School 1035 2.78 0.03 6.70 0.42 0.6784 School 1038 26.18 0.24 7.25 3.61 0.0004 School 1040 23.62 0.21 7.16 3.3 0.0011 School 1041 11.84 0.12 7.08 1.67 0.0954 School 1042 21.39 0.21 6.93 3.09 0.0022 School 1044 -1.77 -0.01 8.28 -0.21 0.8306 School 1047 21.92 0.11 10.42 2.1 0.0363 School 1049 18.36 0.15 7.53 2.44 0.0154 Male -1.22 -0.02 3.33 -0.37 0.7152 White -0.05 0.00 4.16 -0.01 0.9901 African American -3.59 -0.08 4.41 -0.81 0.4173 Master's degree -0.34 -0.01 2.24 -0.15 0.8783 Teaching experience 0.08 0.04 0.11 0.74 0.4609 Math professional development 1.20 0.05 1.34 0.89 0.373 New teacher -7.51 -0.16 2.40 -3.13 0.0019 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 184 Table A.52 Model 2 in effects of teachers' attributes on students' previous Math achievement Model 2 B Beta S. E. t value p value Intercept 307.29 0.00 6.64 46.3 <.0001 Grade 2 19.58 0.47 3.04 6.45 <.0001 Grade 3 9.78 0.23 3.03 3.23 0.0014 Grade 4 14.47 0.29 3.24 4.46 <.0001 School 1003 -0.92 -0.01 7.69 -0.12 0.9051 School 1006 20.35 0.06 16.28 1.25 0.2123 School 1007 30.66 0.42 6.06 5.06 <.0001 School 1009 17.41 0.13 7.61 2.29 0.023 School 1010 -2.67 -0.03 6.61 -0.4 0.6871 School 1011 8.59 0.10 6.40 1.34 0.181 School 1012 10.53 0.09 7.68 1.37 0.1714 School 1015 16.62 0.14 7.27 2.29 0.023 School 1017 23.01 0.25 6.55 3.51 0.0005 School 1020 7.86 0.08 6.50 1.21 0.2278 School 1021 7.50 0.08 6.64 1.13 0.2596 School 1023 14.54 0.10 8.10 1.8 0.0737 School 1024 4.43 0.04 7.53 0.59 0.5568 School 1025 20.48 0.13 8.84 2.32 0.0213 School 1026 -0.82 -0.01 6.53 -0.13 0.9003 School 1027 5.95 0.07 6.46 0.92 0.3581 School 1028 -3.29 -0.03 7.37 -0.45 0.6555 School 1029 0.81 0.01 6.94 0.12 0.907 School 1032 14.06 0.11 7.40 1.9 0.0583 School 1034 -0.01 0.00 7.03 0 0.9986 School 1035 5.03 0.06 6.50 0.77 0.4398 School 1038 27.43 0.25 7.02 3.91 0.0001 School 1040 24.23 0.22 6.93 3.5 0.0005 School 1041 12.68 0.13 6.85 1.85 0.0653 School 1042 22.45 0.22 6.71 3.35 0.0009 School 1044 -1.33 -0.01 8.02 -0.17 0.8684 School 1047 18.39 0.09 10.11 1.82 0.0702 School 1049 19.49 0.16 7.29 2.67 0.008 Male -1.05 -0.02 3.23 -0.33 0.745 White -2.91 -0.07 4.08 -0.71 0.4769 African American -6.65 -0.14 4.33 -1.54 0.126 Master's degree 0.02 0.00 2.17 0.01 0.9921 Teaching experience 0.05 0.02 0.11 0.43 0.6704 Math professional development 0.28 0.01 1.31 0.21 0.833 New teacher -5.18 -0.11 2.38 -2.18 0.0305 Formal leader 9.43 0.21 2.17 4.34 <.0001 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 185 Table A.53 Model 3 in effects of teachers' attributes on students' previous Math achievement Model 3 B Beta S. E. t value p value Intercept 307.84 0.00 6.64 46.36 <.0001 Grade 2 20.04 0.48 3.05 6.57 <.0001 Grade 3 10.37 0.25 3.04 3.41 0.0008 Grade 4 14.45 0.29 3.25 4.45 <.0001 School 1003 -3.59 -0.03 7.74 -0.46 0.6435 School 1006 20.82 0.06 16.29 1.28 0.2023 School 1007 29.39 0.40 6.06 4.85 <.0001 School 1009 16.80 0.13 7.61 2.21 0.0282 School 1010 -4.38 -0.05 6.63 -0.66 0.5093 School 1011 8.21 0.09 6.40 1.28 0.201 School 1012 10.29 0.08 7.68 1.34 0.1816 School 1015 15.10 0.13 7.26 2.08 0.0384 School 1017 23.71 0.25 6.55 3.62 0.0004 School 1020 6.26 0.07 6.50 0.96 0.3362 School 1021 7.22 0.08 6.64 1.09 0.2781 School 1023 13.70 0.10 8.10 1.69 0.092 School 1024 2.76 0.02 7.52 0.37 0.7133 School 1025 18.48 0.12 8.85 2.09 0.0377 School 1026 -1.20 -0.01 6.54 -0.18 0.8547 School 1027 2.15 0.02 6.48 0.33 0.7398 School 1028 -5.12 -0.04 7.38 -0.69 0.4882 School 1029 0.39 0.00 6.94 0.06 0.9547 School 1032 15.13 0.12 7.38 2.05 0.0414 School 1034 -2.07 -0.02 7.01 -0.3 0.768 School 1035 4.00 0.05 6.49 0.62 0.538 School 1038 25.78 0.23 7.02 3.67 0.0003 School 1040 23.11 0.21 6.93 3.33 0.001 School 1041 11.62 0.12 6.85 1.7 0.0912 School 1042 19.49 0.19 6.72 2.9 0.0041 School 1044 -1.84 -0.01 8.02 -0.23 0.8192 School 1047 15.54 0.08 10.20 1.52 0.1287 School 1049 16.88 0.14 7.30 2.31 0.0216 Male -2.14 -0.03 3.23 -0.66 0.5083 White -1.67 -0.04 4.04 -0.41 0.6809 African American -5.01 -0.11 4.29 -1.17 0.2433 Master's degree -0.21 0.00 2.17 -0.1 0.9229 Teaching experience 0.03 0.01 0.11 0.29 0.7692 Math professional development 0.28 0.01 1.31 0.21 0.8342 New teacher -6.02 -0.13 2.35 -2.57 0.0108 The total number of leadership 2.71 0.21 0.63 4.31 <.0001 roles Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 186 Table A.54 Model 4 in effects of teachers' attributes on students' previous Math achievement Model 4 B Beta S. E. t value p value Intercept 308.26 0.00 6.81 45.26 <.0001 Grade 2 18.87 0.46 3.11 6.06 <.0001 Grade 3 9.89 0.24 3.12 3.17 0.0017 Grade 4 14.45 0.29 3.34 4.33 <.0001 School 1003 -0.80 -0.01 7.91 -0.1 0.9197 School 1006 19.56 0.06 16.70 1.17 0.2425 School 1007 28.90 0.40 6.21 4.65 <.0001 School 1009 15.17 0.12 7.82 1.94 0.0533 School 1010 -3.11 -0.04 6.79 -0.46 0.647 School 1011 7.61 0.09 6.57 1.16 0.248 School 1012 9.26 0.08 7.87 1.18 0.2407 School 1015 14.68 0.13 7.44 1.97 0.0495 School 1017 23.82 0.25 6.72 3.54 0.0005 School 1020 6.28 0.07 6.66 0.94 0.3466 School 1021 5.84 0.06 6.82 0.86 0.3929 School 1023 13.63 0.10 8.31 1.64 0.102 School 1024 1.56 0.01 7.70 0.2 0.8392 School 1025 20.46 0.13 9.07 2.25 0.025 School 1026 -1.16 -0.01 6.70 -0.17 0.8623 School 1027 3.37 0.04 6.64 0.51 0.6117 School 1028 -3.88 -0.03 7.56 -0.51 0.6082 School 1029 1.32 0.01 7.12 0.19 0.8528 School 1032 16.85 0.14 7.56 2.23 0.0267 School 1034 -2.20 -0.02 7.19 -0.31 0.7601 School 1035 2.43 0.03 6.65 0.37 0.7148 School 1038 25.96 0.24 7.20 3.61 0.0004 School 1040 23.74 0.22 7.11 3.34 0.001 School 1041 10.43 0.10 7.06 1.48 0.1407 School 1042 20.57 0.20 6.89 2.99 0.0031 School 1044 -1.72 -0.01 8.22 -0.21 0.8344 School 1047 22.14 0.11 10.35 2.14 0.0333 School 1049 18.38 0.15 7.48 2.46 0.0146 Male -1.38 -0.02 3.31 -0.42 0.6771 White -0.59 -0.01 4.14 -0.14 0.887 African American -4.24 -0.09 4.39 -0.96 0.3359 Master's degree -0.48 -0.01 2.23 -0.22 0.8283 Teaching experience 0.08 0.04 0.11 0.72 0.4719 Math professional development 0.97 0.04 1.33 0.72 0.4695 New teacher -7.68 -0.17 2.38 -3.23 0.0014 Math coordinator 2.77 0.10 1.28 2.16 0.0313 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 187 Table A.55 Model 5 in effects of teachers' attributes on students' previous Math achievement Model 5 B Beta S. E. t value p value Intercept 313.38 0.00 7.02 44.65 <.0001 Grade 2 18.79 0.45 3.09 6.09 <.0001 Grade 3 9.28 0.22 3.09 3.01 0.0029 Grade 4 13.34 0.27 3.31 4.03 <.0001 School 1003 -0.65 0.00 7.84 -0.08 0.9337 School 1006 19.93 0.06 16.58 1.2 0.2305 School 1007 28.23 0.39 6.17 4.58 <.0001 School 1009 15.81 0.12 7.75 2.04 0.0423 School 1010 -3.24 -0.04 6.74 -0.48 0.6309 School 1011 7.52 0.09 6.52 1.15 0.2503 School 1012 9.20 0.08 7.82 1.18 0.2404 School 1015 14.05 0.12 7.39 1.9 0.0584 School 1017 23.85 0.26 6.67 3.57 0.0004 School 1020 5.68 0.06 6.62 0.86 0.3916 School 1021 6.59 0.07 6.76 0.98 0.3304 School 1023 13.15 0.09 8.25 1.59 0.1121 School 1024 1.67 0.01 7.65 0.22 0.8278 School 1025 19.16 0.12 9.01 2.13 0.0344 School 1026 -1.24 -0.01 6.66 -0.19 0.8529 School 1027 3.14 0.04 6.59 0.48 0.6345 School 1028 -4.82 -0.04 7.51 -0.64 0.5217 School 1029 0.02 0.00 7.08 0 0.9973 School 1032 14.62 0.12 7.54 1.94 0.0537 School 1034 -2.61 -0.02 7.14 -0.37 0.7152 School 1035 3.80 0.04 6.61 0.57 0.566 School 1038 24.72 0.22 7.16 3.45 0.0007 School 1040 22.44 0.20 7.07 3.17 0.0017 School 1041 11.44 0.11 6.98 1.64 0.1023 School 1042 20.78 0.21 6.83 3.04 0.0026 School 1044 -1.61 -0.01 8.17 -0.2 0.8442 School 1047 19.38 0.10 10.31 1.88 0.0613 School 1049 17.65 0.14 7.43 2.38 0.0182 Male -1.65 -0.02 3.29 -0.5 0.6161 White -1.01 -0.02 4.11 -0.24 0.8069 African American -4.78 -0.10 4.37 -1.09 0.2752 Master's degree -0.05 0.00 2.22 -0.02 0.9838 Teaching experience 0.04 0.02 0.11 0.41 0.6812 Math professional development 0.66 0.03 1.33 0.49 0.6228 New teacher -6.58 -0.14 2.38 -2.76 0.0062 Teacher consultant 0.95 0.14 0.33 2.91 0.0039 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 188 Table A.56 Model 1 in effects of teachers' attributes on students' previous free/reduced lunch Model 1 B Beta S. E. t value p value Intercept 0.92 0.00 0.06 15.03 <.0001 Grade 2 0.01 0.02 0.03 0.36 0.7192 Grade 3 0.04 0.07 0.03 1.48 0.1408 Grade 4 0.02 0.03 0.03 0.81 0.4212 School 1003 -0.10 -0.06 0.07 -1.45 0.1481 School 1006 -0.27 -0.06 0.15 -1.77 0.0773 School 1007 -0.55 -0.54 0.06 -9.61 <.0001 School 1009 -0.24 -0.13 0.07 -3.29 0.0011 School 1010 0.01 0.01 0.06 0.17 0.8638 School 1011 -0.37 -0.30 0.06 -6.14 <.0001 School 1012 -0.08 -0.05 0.07 -1.15 0.2514 School 1015 -0.24 -0.15 0.07 -3.47 0.0006 School 1017 -0.51 -0.39 0.06 -8.29 <.0001 School 1020 -0.55 -0.42 0.06 -8.99 <.0001 School 1021 -0.04 -0.03 0.06 -0.59 0.5582 School 1023 -0.26 -0.13 0.08 -3.38 0.0008 School 1024 -0.09 -0.05 0.07 -1.22 0.2232 School 1025 -0.72 -0.37 0.08 -9.47 <.0001 School 1026 0.01 0.01 0.06 0.19 0.8533 School 1027 -0.09 -0.07 0.06 -1.43 0.1546 School 1028 0.02 0.01 0.07 0.35 0.7288 School 1029 -0.21 -0.14 0.07 -3.24 0.0014 School 1032 -0.18 -0.11 0.07 -2.63 0.009 School 1034 0.00 0.00 0.07 -0.07 0.9444 School 1035 -0.12 -0.10 0.06 -2 0.0463 School 1038 -0.70 -0.46 0.07 -10.54 <.0001 School 1040 -0.29 -0.19 0.07 -4.41 <.0001 School 1041 -0.45 -0.34 0.06 -7.09 <.0001 School 1042 -0.42 -0.30 0.06 -6.64 <.0001 School 1044 -0.03 -0.02 0.08 -0.42 0.6741 School 1047 -0.63 -0.23 0.09 -6.67 <.0001 School 1049 -0.45 -0.27 0.07 -6.61 <.0001 Male -0.05 -0.05 0.03 -1.51 0.1332 White -0.07 -0.11 0.04 -1.75 0.0813 African American -0.02 -0.03 0.04 -0.55 0.5816 Master's degree -0.01 -0.01 0.02 -0.33 0.739 Teaching experience 0.00 -0.04 0.00 -1.27 0.2055 New teacher 0.04 0.06 0.02 1.72 0.0858 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 189 Table A.57 Model 2 in effects of teachers' attributes on students' previous free/reduced lunch Model 2 B Beta S. E. t value p value Intercept 0.93 0.00 0.06 15.52 <.0001 Grade 2 0.00 0.01 0.03 0.16 0.87 Grade 3 0.04 0.07 0.03 1.45 0.1472 Grade 4 0.02 0.03 0.03 0.78 0.4348 School 1003 -0.09 -0.05 0.07 -1.28 0.2031 School 1006 -0.28 -0.06 0.15 -1.87 0.062 School 1007 -0.56 -0.56 0.06 -10.03 <.0001 School 1009 -0.24 -0.13 0.07 -3.47 0.0006 School 1010 0.01 0.01 0.06 0.18 0.8602 School 1011 -0.37 -0.31 0.06 -6.33 <.0001 School 1012 -0.09 -0.05 0.07 -1.32 0.1887 School 1015 -0.25 -0.16 0.07 -3.75 0.0002 School 1017 -0.50 -0.39 0.06 -8.29 <.0001 School 1020 -0.56 -0.43 0.06 -9.38 <.0001 School 1021 -0.04 -0.03 0.06 -0.68 0.4956 School 1023 -0.27 -0.14 0.07 -3.58 0.0004 School 1024 -0.11 -0.06 0.07 -1.52 0.1297 School 1025 -0.72 -0.37 0.07 -9.66 <.0001 School 1026 0.01 0.01 0.06 0.11 0.9122 School 1027 -0.09 -0.08 0.06 -1.6 0.1111 School 1028 0.02 0.01 0.07 0.28 0.7829 School 1029 -0.20 -0.13 0.06 -3.2 0.0015 School 1032 -0.16 -0.10 0.07 -2.38 0.0181 School 1034 -0.02 -0.02 0.06 -0.37 0.7116 School 1035 -0.14 -0.12 0.06 -2.28 0.0235 School 1038 -0.71 -0.46 0.06 -10.92 <.0001 School 1040 -0.29 -0.19 0.06 -4.58 <.0001 School 1041 -0.46 -0.34 0.06 -7.35 <.0001 School 1042 -0.43 -0.31 0.06 -6.92 <.0001 School 1044 -0.04 -0.02 0.07 -0.49 0.6262 School 1047 -0.61 -0.22 0.09 -6.54 <.0001 School 1049 -0.46 -0.27 0.07 -6.88 <.0001 Male -0.05 -0.05 0.03 -1.58 0.1143 White -0.05 -0.08 0.04 -1.24 0.2153 African American 0.00 0.00 0.04 0.01 0.9923 Master's degree -0.01 -0.01 0.02 -0.44 0.6587 Teaching experience 0.00 -0.04 0.00 -1.08 0.2821 New teacher 0.02 0.03 0.02 0.9 0.3665 Formal leader -0.07 -0.11 0.02 -3.58 0.0004 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 190 Table A.58 Model 3 in effects of teachers' attributes on students' previous free/reduced lunch Model 3 B Beta S. E. t value p value Intercept 0.93 0.00 0.06 15.6 <.0001 Grade 2 0.00 0.00 0.03 -0.02 0.9823 Grade 3 0.03 0.06 0.03 1.27 0.2046 Grade 4 0.02 0.03 0.03 0.78 0.4379 School 1003 -0.07 -0.04 0.07 -0.92 0.3586 School 1006 -0.29 -0.06 0.15 -1.92 0.0554 School 1007 -0.55 -0.55 0.06 -9.97 <.0001 School 1009 -0.24 -0.13 0.07 -3.45 0.0007 School 1010 0.03 0.02 0.06 0.42 0.6746 School 1011 -0.37 -0.31 0.06 -6.34 <.0001 School 1012 -0.09 -0.05 0.07 -1.32 0.1869 School 1015 -0.24 -0.15 0.07 -3.63 0.0003 School 1017 -0.51 -0.39 0.06 -8.44 <.0001 School 1020 -0.55 -0.43 0.06 -9.27 <.0001 School 1021 -0.04 -0.03 0.06 -0.65 0.5154 School 1023 -0.26 -0.13 0.07 -3.53 0.0005 School 1024 -0.09 -0.06 0.07 -1.37 0.1709 School 1025 -0.70 -0.36 0.07 -9.48 <.0001 School 1026 0.01 0.01 0.06 0.15 0.8784 School 1027 -0.06 -0.05 0.06 -1.08 0.2831 School 1028 0.03 0.02 0.07 0.5 0.6189 School 1029 -0.20 -0.13 0.06 -3.15 0.0018 School 1032 -0.17 -0.10 0.07 -2.49 0.0133 School 1034 -0.01 -0.01 0.06 -0.15 0.8843 School 1035 -0.13 -0.11 0.06 -2.19 0.0294 School 1038 -0.69 -0.46 0.06 -10.81 <.0001 School 1040 -0.28 -0.19 0.06 -4.48 <.0001 School 1041 -0.45 -0.33 0.06 -7.24 <.0001 School 1042 -0.40 -0.29 0.06 -6.57 <.0001 School 1044 -0.03 -0.02 0.07 -0.44 0.6596 School 1047 -0.58 -0.21 0.09 -6.24 <.0001 School 1049 -0.44 -0.26 0.07 -6.61 <.0001 Male -0.04 -0.04 0.03 -1.28 0.201 White -0.05 -0.09 0.04 -1.46 0.1449 African American -0.01 -0.02 0.04 -0.25 0.7996 Master's degree -0.01 -0.01 0.02 -0.36 0.7209 Teaching experience 0.00 -0.03 0.00 -0.92 0.3588 New teacher 0.02 0.04 0.02 1.13 0.2597 The total number of leadership -0.02 -0.13 0.01 -4.13 <.0001 roles Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 191 Table A.59 Model 4 in effects of teachers' attributes on students' previous free/reduced lunch Model 4 B Beta S. E. t value p value Intercept 0.85 0.00 0.06 13.93 <.0001 Grade 2 0.02 0.03 0.03 0.57 0.5666 Grade 3 0.05 0.09 0.03 1.85 0.0661 Grade 4 0.03 0.05 0.03 1.2 0.2319 School 1003 -0.10 -0.06 0.07 -1.47 0.1428 School 1006 -0.28 -0.06 0.15 -1.87 0.0627 School 1007 -0.54 -0.53 0.06 -9.75 <.0001 School 1009 -0.23 -0.13 0.07 -3.28 0.0012 School 1010 0.02 0.02 0.06 0.32 0.7457 School 1011 -0.36 -0.30 0.06 -6.26 <.0001 School 1012 -0.09 -0.05 0.07 -1.24 0.2157 School 1015 -0.24 -0.15 0.07 -3.66 0.0003 School 1017 -0.49 -0.38 0.06 -8.26 <.0001 School 1020 -0.54 -0.42 0.06 -9.18 <.0001 School 1021 -0.02 -0.02 0.06 -0.37 0.7093 School 1023 -0.25 -0.13 0.07 -3.45 0.0007 School 1024 -0.08 -0.05 0.07 -1.23 0.2187 School 1025 -0.70 -0.36 0.07 -9.58 <.0001 School 1026 0.01 0.01 0.06 0.23 0.8151 School 1027 -0.07 -0.05 0.06 -1.12 0.2627 School 1028 0.04 0.02 0.07 0.56 0.5785 School 1029 -0.22 -0.14 0.06 -3.42 0.0007 School 1032 -0.16 -0.09 0.07 -2.31 0.0215 School 1034 -0.01 0.00 0.06 -0.11 0.9096 School 1035 -0.13 -0.11 0.06 -2.25 0.025 School 1038 -0.69 -0.45 0.06 -10.81 <.0001 School 1040 -0.27 -0.18 0.06 -4.26 <.0001 School 1041 -0.44 -0.33 0.06 -7.23 <.0001 School 1042 -0.42 -0.30 0.06 -6.83 <.0001 School 1044 -0.02 -0.01 0.07 -0.23 0.8218 School 1047 -0.59 -0.21 0.09 -6.36 <.0001 School 1049 -0.45 -0.26 0.07 -6.74 <.0001 Male -0.04 -0.04 0.03 -1.39 0.1666 White -0.06 -0.10 0.04 -1.66 0.0991 African American -0.01 -0.01 0.04 -0.18 0.8549 Master's degree -0.01 -0.02 0.02 -0.53 0.5993 Teaching experience 0.00 -0.03 0.00 -1.03 0.3025 New teacher 0.03 0.04 0.02 1.18 0.2393 School improvement coordinator -0.01 -0.14 0.00 -4.46 <.0001 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 192 Table A.60 Model 5 in effects of teachers' attributes on students' previous free/reduced lunch Model 5 B Beta S. E. t value p value Intercept 0.86 0.00 0.06 13.98 <.0001 Grade 2 0.01 0.01 0.03 0.31 0.7547 Grade 3 0.04 0.08 0.03 1.59 0.1128 Grade 4 0.03 0.05 0.03 1.14 0.254 School 1003 -0.09 -0.05 0.07 -1.23 0.2214 School 1006 -0.28 -0.06 0.15 -1.88 0.0606 School 1007 -0.54 -0.54 0.06 -9.73 <.0001 School 1009 -0.23 -0.13 0.07 -3.3 0.0011 School 1010 0.02 0.01 0.06 0.3 0.7651 School 1011 -0.36 -0.30 0.06 -6.21 <.0001 School 1012 -0.08 -0.05 0.07 -1.2 0.2296 School 1015 -0.23 -0.14 0.07 -3.46 0.0006 School 1017 -0.51 -0.39 0.06 -8.43 <.0001 School 1020 -0.54 -0.42 0.06 -9.14 <.0001 School 1021 -0.03 -0.03 0.06 -0.55 0.5816 School 1023 -0.25 -0.13 0.07 -3.42 0.0007 School 1024 -0.08 -0.05 0.07 -1.23 0.2196 School 1025 -0.70 -0.36 0.07 -9.53 <.0001 School 1026 0.01 0.01 0.06 0.17 0.8646 School 1027 -0.07 -0.06 0.06 -1.15 0.2509 School 1028 0.03 0.02 0.07 0.51 0.6122 School 1029 -0.19 -0.13 0.06 -3.01 0.0028 School 1032 -0.16 -0.09 0.07 -2.3 0.0221 School 1034 0.00 0.00 0.06 -0.05 0.961 School 1035 -0.13 -0.11 0.06 -2.23 0.0269 School 1038 -0.68 -0.45 0.06 -10.54 <.0001 School 1040 -0.27 -0.18 0.06 -4.3 <.0001 School 1041 -0.45 -0.33 0.06 -7.21 <.0001 School 1042 -0.41 -0.30 0.06 -6.71 <.0001 School 1044 -0.03 -0.02 0.07 -0.47 0.6367 School 1047 -0.60 -0.22 0.09 -6.52 <.0001 School 1049 -0.45 -0.26 0.07 -6.66 <.0001 Male -0.04 -0.04 0.03 -1.37 0.1715 White -0.06 -0.09 0.04 -1.52 0.1302 African American -0.01 -0.01 0.04 -0.2 0.8439 Master's degree -0.01 -0.02 0.02 -0.49 0.6238 Teaching experience 0.00 -0.03 0.00 -0.89 0.3729 New teacher 0.03 0.04 0.02 1.21 0.2285 Teacher consultant -0.01 -0.12 0.00 -4 <.0001 Note: B=unstandardized coefficients, Beta=standardized coefficients, and S.E.=Standard Errors. 193 BIBLIOGRAPHY 194 BIBLIOGRAPHY Briggs, D., Weeks, J., Wiley, E. 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